Finite Group Databasebeta

← #30621821

Subgroups — #30621821

Order 15616.

860 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840789C211yesmaximal, characteristic
2C2×#16840786C211yesmaximal, characteristic, complemented
3#21461917C211yesmaximal, characteristic, complemented
4#21461894C211yesmaximal, characteristic, complemented
5#21461894C211yesmaximal, characteristic, complemented
6#21461874C211yesmaximal, characteristic, complemented
7#21461872C211yesmaximal, characteristic, complemented
8#21461798C211yesmaximal, characteristic
9#21461764C211yesmaximal, characteristic, complemented
10#21461762C211yesmaximal, characteristic, complemented
11#21461618C211yesmaximal, characteristic, complemented
12#21461561C211yesmaximal, characteristic, complemented
13#21461553C211yesmaximal, characteristic, complemented
14#21461485C211yesmaximal, characteristic, complemented
15#21449077C211yesmaximal, characteristic
16C22×#11758217C2211yescharacteristic
17C22×D61⋊C8C2211yescharacteristic
18C23×#92420C2211yescharacteristic
19#16840789—24no—
20#16840786—24no—
21C2×#14519844C2211yescharacteristic
22C2×#14519844C2211yescharacteristic
23C2×C4⋊#92420C2211yescharacteristic
24C2×C4⋊#92420C2211yescharacteristic
25C2×C4⋊#92420C2211yescharacteristic
26C2×#14519834C2211yescharacteristic
27C2×#14519834C2211yescharacteristic
28C2×#14519834C2211yescharacteristic
29C2×C22⋊#92420C2211yescharacteristic
30C2×C22⋊#92420C2211yescharacteristic
31C2×#14519832C2211yescharacteristic
32C2×#14519832C2211yescharacteristic
33C2×C4×#92420C2211yescharacteristic
34C2×C4×#92420C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840628C2211yescharacteristic
37#16840628C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44#16840628C2211yescharacteristic
45#16840628C2211yescharacteristic
46C2×#14519737C2211yescharacteristic
47C2×#14519733C2211yescharacteristic
48#16839703C411yescharacteristic
49#16839697C411yescharacteristic
50#16839696C411yescharacteristic
51#16839691C411yescharacteristic
52#16839690C411yescharacteristic
53#16839678C411yescharacteristic
54#16839677C411yescharacteristic
55C4⋊C4×Dic61C2211yescharacteristic
56#16839613C2211yescharacteristic
57C22⋊C4×Dic61C2211yescharacteristic
58#16839609C2211yescharacteristic
59#16839609C2211yescharacteristic
60C61×#397C411yescharacteristic
61C2×#11758217—22no—
62C2×#11758217—22no—
63C2×#11758217—41no—
64C2×D61⋊C8—22no—
65C2×D61⋊C8—22no—
66C2×D61⋊C8—41no—
67C22×#92420—22no—
68C22×#92420—22no—
69C22×#92420—41no—
70C22×#92420C2311yescharacteristic
71C22×#92420C2311yescharacteristic
72C22×#92420C2311yescharacteristic
73C22×#92420C2311yescharacteristic
74C22×#92420C2311yescharacteristic
75C22×#92420C2311yescharacteristic
76C22×#92420C2311yescharacteristic
77C22×#92420C2311yescharacteristic
78#14519844—22no—
79#14519844—22no—
80C4⋊#92420—22no—
81C4⋊#92420—22no—
82C4⋊#92420—41no—
83#14519834—22no—
84#14519834—22no—
85#14519834—41no—
86C22⋊#92420—22no—
87C22⋊#92420D414yes—
88#14519832—22no—
89#14519832—22no—
90C4×#92420—22no—
91C4×#92420—22no—
92C4×#92420—41no—
93C22×C22⋊D61C2×C411yescharacteristic
94C23×Dic61C2311yescharacteristic
95C22×C4×D61C2311yescharacteristic
96#14519737—22no—
97#14519737—22no—
98#14519733—22no—
99#14519733—22no—
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×#11758196C2×C411yescharacteristic
102C2×#11758196C2×C411yescharacteristic
103C2×#11758196C2×C411yescharacteristic
104C2×#11758196C2×C411yescharacteristic
105C2×#11758196C2×C411yescharacteristic
106C2×C22⋊Dic61C2×C411yescharacteristic
107C2×C22⋊Dic61C2×C411yescharacteristic
108C2×#11758198C2×C411yescharacteristic
109C2×#11758198C2×C411yescharacteristic
110C2×#11758198C2×C411yescharacteristic
111C2×#11758198C2×C411yescharacteristic
112C2×#11758198C2×C411yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C2×C4×Dic61C2311yescharacteristic
116C2×C4×Dic61C2311yescharacteristic
117C2×C4×Dic61C2311yescharacteristic
118#14519684C2×C411yescharacteristic
119#14519684C2×C411yescharacteristic
120#14519684C2×C411yescharacteristic
121#14519684C2×C411yescharacteristic
122#14519684C2×C411yescharacteristic
123#14519684C2×C411yescharacteristic
124#14519684C2×C411yescharacteristic
125C2×C4⋊C4×C61C2×C411yescharacteristic
126C2×C22⋊C4×C61C2×C411yescharacteristic
127C2×C22⋊C4×C61C2×C411yescharacteristic
128C2×C22⋊C4×C61C2×C411yescharacteristic
129#95×C61C2×C411yescharacteristic
130#95×C61C2×C411yescharacteristic
131#95×C61C2×C411yescharacteristic
132C24×C61C22×C411yescharacteristic
133C23×D61C22×C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C2×C22⋊D61—22no—
141C2×C22⋊D61—22no—
142C2×C22⋊D61—22no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C2×D412yes—
154C22×Dic61#4112yes—
155C22×Dic61C2411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162#11758217—44no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420—21no—
180C2×#92420—21no—
181C2×#92420—21no—
182C2×#92420—21no—
183C2×#92420—44no—
184C2×#92420C2×D412yes—
185C2×#92420C2×D412yes—
186C2×#92420#4112yes—
187C2×#92420#4112yes—
188D61⋊C8—44no—
189C4⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—41no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—41no—
200C4⋊Dic61—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—41no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61C4×D412yes—
223C23×C61C23×C411yescharacteristic
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61#13112yes—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61#13112yes—
249C22⋊D61—28no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61C4×D412yes—
271C2×Dic61#13112yes—
272C2×Dic61C22×D411yescharacteristic
273C2×Dic61C2×#4111yescharacteristic
274C2×Dic61C2×#4111yescharacteristic
275C2×Dic61C2×#4111yescharacteristic
276C2×Dic61H(5,𝔽2)11yescharacteristic
277C2×Dic612−1+411yescharacteristic
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282#92420—22no—
283#92420—22no—
284#92420—22no—
285#92420—22no—
286#92420—41no—
287#92420—41no—
288#92420—41no—
289#92420—41no—
290#22901—611nomaximal, Sylow
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61C2×C4×D411yescharacteristic
299C22×C61C4×#4111yescharacteristic
300C22×C61#52011yescharacteristic
301C22×C61C2×#13111yescharacteristic
302C22×C61C2×#13111yescharacteristic
303C22×C61C2×#13111yescharacteristic
304C22×C61#56811yescharacteristic
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C4×C61—22no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61—41no—
317C4×C61—41no—
318Dic61—22no—
319Dic61—22no—
320Dic61—41no—
321Dic61—41no—
322Dic61—41no—
323Dic61—41no—
324Dic61#54611yescharacteristic
325Dic61#54711yescharacteristic
326Dic61#55111yescharacteristic
327Dic61#55211yescharacteristic
328C2×C8×D4—611no—
329C2×#433—611no—
330#2395—611no—
331#1939—611no—
332#1924—611no—
333#1924—611no—
334#1916—611no—
335#1916—611no—
336#1870—611no—
337#1850—611no—
338#1850—611no—
339#1771—611no—
340#1753—611no—
341C8×C22⋊C4—611no—
342#1729—611no—
343C2×C61—42no—
344C2×C61#235911yescharacteristic
345C2×C61#292911yescharacteristic
346C2×C61#293511yescharacteristic
347C2×C61#296611yescharacteristic
348C2×C61#296911yescharacteristic
349C2×C61#299211yescharacteristic
350C2×C61#299411yescharacteristic
351D61—42no—
352C22×M16—611no—
353C23×C8—611no—
354C23×C8—611no—
355C2×C4×D4—611no—
356C2×#117—611no—
357C8×D4—1224no—
358#433—1224no—
359C2×C4⋊C8—611no—
360C2×C4⋊C8—611no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8—611no—
364C2×C22⋊C8—611no—
365C2×C22⋊C8—611no—
366C2×C22⋊C8—611no—
367C2×C22⋊C8—611no—
368C2×C22⋊C8—611no—
369C2×#97—611no—
370C2×#97—611no—
371C2×C4×C8—611no—
372C2×C4×C8—611no—
373C2×C4×C8—611no—
374#400—611no—
375#399—611no—
376#398—611no—
377#397—611no—
378#397—611no—
379#397#692311yescharacteristic
380#394—611no—
381#394—611no—
382C4×C4⋊C4—611no—
383C4×C4⋊C4—611no—
384C4×C22⋊C4—611no—
385C4×C22⋊C4—611no—
386C4×C22⋊C4—611no—
387#336—611no—
388#336—611no—
389#336—611no—
390#336—611no—
391#336—611no—
392#336—611no—
393#336—611no—
394#336—611no—
395#336—611no—
396#336—611no—
397C61#2290111yescharacteristic, complemented, Sylow
398C22×D4—611no—
399C23×C4—611no—
400C23×C4—611no—
401C2×M16—1222no—
402C2×M16—1222no—
403C2×M16—2441no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8—611no—
408C22×C8—611no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8—1222no—
413C22×C8—1222no—
414C22×C8—1222no—
415C22×C8—1222no—
416C22×C8—2441no—
417C22×C8—2441no—
418C4×D4—1222no—
419C4×D4—1222no—
420#117—1222no—
421#117—1222no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4C2×#692311yescharacteristic
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—611no—
432C2×C22⋊C4—611no—
433C2×C22⋊C4—611no—
434C2×C22⋊C4—611no—
435C2×C22⋊C4—611no—
436C2×C22⋊C4C2×#692311yescharacteristic
437C2×C22⋊C4C2×#692311yescharacteristic
438C2×C22⋊C4C2×#692311yescharacteristic
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C4⋊C8—1222no—
445C4⋊C8—1222no—
446C4⋊C8—1222no—
447C4⋊C8—1222no—
448C4⋊C8—2441no—
449C22⋊C8—614no—
450C22⋊C8—1222no—
451C22⋊C8—1222no—
452C22⋊C8—1222no—
453C22⋊C8—2441no—
454#97—1222no—
455#97—1222no—
456C4×C8—1222no—
457C4×C8—1222no—
458C4×C8—2441no—
459#95—611no—
460#95—611no—
461#95—611no—
462#95—611no—
463#95—611no—
464#95—611no—
465#95—611no—
466#95C2×#692311yescharacteristic
467#95C2×#692311yescharacteristic
468#95C2×#692311yescharacteristic
469C24—611no—
470C24C22×#692311yescharacteristic
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—1222no—
474C22×C4—611no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4C22×#692311yescharacteristic
494C22×C4C22×#692311yescharacteristic
495C22×C4C22×#692311yescharacteristic
496C22×C4C22×#692311yescharacteristic
497C22×C4C22×#692311yescharacteristic
498C22×C4C22×#692311yescharacteristic
499M16—2444no—
500C2×C8—612no—
501C2×C8—612no—
502C2×C8—612no—
503C2×C8—612no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—2444no—
525C2×C8—2444no—
526C4⋊C4—22no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—2441no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—2441no—
542C22⋊C4—2441no—
543C42—1221no—
544C42—1221no—
545C42—1221no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42—1221no—
552C42—1221no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—612no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23D4×#692312yes—
567C23C23×#692311yescharacteristic
568D4—1228no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—611no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4#1451986412yes—
619C8—1222no—
620C8—1222no—
621C8—1222no—
622C8—1222no—
623C8—2441no—
624C8—2441no—
625C8—2441no—
626C8—2441no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—1222no—
635C22—1222no—
636C22—1222no—
637C22—2441no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22C2×D4×#692311yescharacteristic
642C22#41×#692311yescharacteristic
643C22#1684083111yescharacteristic
644C22C2×#1451986411yescharacteristic
645C22D61⋊#13111yescharacteristic
646C22D61⋊#13111yescharacteristic
647C22#1684084111yescharacteristic
648C4—22no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—611no—
655C4—611no—
656C4—611no—
657C4—611no—
658C4—1222no—
659C4—1222no—
660C4—2441no—
661C4—2441no—
662C4—2441no—
663C4—2441no—
664C2—42no—
665C2—2442no—
666C2#2146234211yescharacteristic
667C2#2146253611yescharacteristic
668C2#2146256111yescharacteristic
669C2#2146263311yescharacteristic
670C2#2146265411yescharacteristic
671C2#2146275711yescharacteristic
672C2#2146276711yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#397C4
prime radicalC22×C22⋊D61C2×C4
cosocleC22×Dic61C24
hypercenter#397#6923
hypocenterC61#22901
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #397

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16840786C2semidirect
3. #21461917C2semidirect
4. #21461894C2semidirect
5. #21461894C2semidirect
6. #21461874C2semidirect
7. #21461872C2semidirect
9. #21461764C2semidirect
10. #21461762C2semidirect
11. #21461618C2semidirect
12. #21461561C2semidirect
13. #21461553C2semidirect
14. #21461485C2semidirect
397. C61#22901semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168407892
C2×#168407862
#214619172
#214618942
#214618742
#214618722
#214617982
#214617642
#214617622
#214616182
#214615612
#214615532
#214614852
#214490772
#2290161