Finite Group Databasebeta

← #29730371

Subgroups — #29730371

Order 15616.

832 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840070C211yesmaximal, characteristic, complemented
2#21450985C211yesmaximal, characteristic
3#21450741C211yesmaximal, characteristic
4#21447252C211yesmaximal, characteristic, complemented
5#21447247C211yesmaximal, characteristic, complemented
6#21447016C211yesmaximal, characteristic, complemented
7#21447016C211yesmaximal, characteristic, complemented
8#21446788C211yesmaximal, characteristic, complemented
9#21446583C211yesmaximal, characteristic, complemented
10#21446252C211yesmaximal, characteristic, complemented
11#21446242C211yesmaximal, characteristic, complemented
12#21445852C211yesmaximal, characteristic, complemented
13#21445844C211yesmaximal, characteristic, complemented
14#21445650C211yesmaximal, characteristic, complemented
15C2×C61×#435C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519756C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16840070—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29#16839700—21no—
30#16839685—21no—
31#16839685—21no—
32#16839676—21no—
33#16839665—21no—
34#16839658—21no—
35#16839656—21no—
36#16839656—21no—
37#16839632C2211yescharacteristic
38#16839622C2211yescharacteristic
39#16839622C2211yescharacteristic
40#16839622C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839622C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C23×C8×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C61×#435—24no—
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C2×#97×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62#14519769—22no—
63#14519769—22no—
64#14519756—22no—
65#14519756—22no—
66#14519756—41no—
67#14519756—41no—
68C4×Dic122—22no—
69C4×Dic122—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94#14519711—22no—
95#14519709—22no—
96#14519709—41no—
97#14519709—41no—
98#14519702—22no—
99#14519702—22no—
100#14519702—22no—
101#14519702—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109C22×D4×C61D411yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C23×C4×C61C2311yescharacteristic
112C2×M16×C61—22no—
113C2×M16×C61—22no—
114C2×M16×C61—41no—
115C22×C8×C61—22no—
116C22×C8×C61—22no—
117C22×C8×C61—41no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C4×D4×C61—22no—
123C4×D4×C61—22no—
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C4⋊C8×C61D414yes—
129C22⋊C8×C61—22no—
130C22⋊C8×C61—22no—
131#97×C61—22no—
132C24×C61C2×D411yescharacteristic
133C24×C61C2×D411yescharacteristic
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—41no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122#4112yes—
168M16×C61—44no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—44no—
179C2×C8×C61C2×D412yes—
180C2×C8×C61C2×D412yes—
181C2×C8×C61#4112yes—
182C4⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C42×C61C2×D412yes—
186C42×C61#4112yes—
187C22⋊Dic61—42no—
188C22⋊Dic61—42no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191C22⋊Dic61—81no—
192C22⋊Dic61—81no—
193C22⋊Dic61—81no—
194C22⋊Dic61—81no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—42no—
210C4⋊Dic61—81no—
211#11758196—42no—
212#11758196—42no—
213#11758196—42no—
214#11758196—42no—
215#11758196—42no—
216C4×Dic61—42no—
217C4×Dic61—81no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61#13712yes—
232C23×C61C22×D411yescharacteristic
233D4×C61—44no—
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—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61#12112yes—
253C2×C4×C61#12112yes—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C612−1+411yescharacteristic
260C2×Dic61—22no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288C8×C61—22no—
289C8×C61—22no—
290C8×C61—22no—
291C8×C61—41no—
292#25357—611nomaximal, Sylow
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61C2×#12111yescharacteristic
308C22×C61#54611yescharacteristic
309C22×C61#54711yescharacteristic
310C22×C61C2×#13711yescharacteristic
311C22×C61#57511yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57711yescharacteristic
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61#52911yescharacteristic
319C4×C61#54211yescharacteristic
320C4×C61#54611yescharacteristic
321C4×C61#54811yescharacteristic
322Dic61—42no—
323Dic61—81no—
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330C2×#479—611no—
331C2×#435D6111yescharacteristic
332#2562—611no—
333#2538—611no—
334#2077—611no—
335#2076—611no—
336#2035—611no—
337#2035—611no—
338#1992—611no—
339#1948—611no—
340#1894—611no—
341#1892—611no—
342#1809—611no—
343#1807—611no—
344#1763—611no—
345C2×C61—42no—
346C2×C61—81no—
347C2×C61#242711yescharacteristic
348C2×C61#305611yescharacteristic
349C2×C61#306011yescharacteristic
350C2×C61#331711yescharacteristic
351C2×C61#332011yescharacteristic
352C2×C61#334411yescharacteristic
353C2×C61#335011yescharacteristic
354C22×M16C2×D6111yescharacteristic
355C23×C8C2×D6111yescharacteristic
356C2×#125—611no—
357C2×C22⋊Q8—611no—
358C2×C22⋊Q8—611no—
359C2×C4×Q8—611no—
360C2×C4×D4C2×D6111yescharacteristic
361#479—1224no—
362#435—24no—
363C2×#107—611no—
364C2×#106—611no—
365C2×#106—611no—
366C2×#106—611no—
367C2×C4⋊C8C2×D6111yescharacteristic
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×C22⋊C8C2×D6111yescharacteristic
373C2×C22⋊C8C2×D6111yescharacteristic
374C2×#97C2×D6111yescharacteristic
375#400—1221no—
376#396—1221no—
377#396—1221no—
378#394—1221no—
379#390—1221no—
380#389—1221no—
381#388—1221no—
382#388—1221no—
383#384—611no—
384#380—611no—
385#380—611no—
386#380—611no—
387#380—611no—
388#380—611no—
389#380—611no—
390C4×C4⋊C4—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399C61#2535711yescharacteristic, complemented, Sylow
400C22×Q8—611no—
401C22×D4D24411yescharacteristic
402C23×C4C22×D6111yescharacteristic
403C23×C4C22×D6111yescharacteristic
404C2×M16—22no—
405C2×M16—22no—
406C2×M16—41no—
407C22×C8—22no—
408C22×C8—22no—
409C22×C8—41no—
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414#125—1222no—
415#125—1222no—
416C22⋊Q8—1222no—
417C22⋊Q8—1222no—
418C22⋊Q8—2441no—
419C22⋊Q8—2441no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422C4×D4—22no—
423C4×D4—22no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4D24411yescharacteristic
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4D24411yescharacteristic
448C2×C22⋊C4D24411yescharacteristic
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42C22×D6111yescharacteristic
452#107—1222no—
453#106—1222no—
454#106—2441no—
455#106—2441no—
456C4⋊C8C22⋊D6114yes—
457#103—1222no—
458#103—1222no—
459#103—1222no—
460#103—1222no—
461C22⋊C8—22no—
462C22⋊C8—22no—
463#97—22no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471C24C2×D24411yescharacteristic
472C24C2×D24411yescharacteristic
473C2×Q8—612no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—41no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4C2×D24411yescharacteristic
503C22×C4C2×D24411yescharacteristic
504C22×C4C2×D24411yescharacteristic
505C22×C4C2×D24411yescharacteristic
506C22×C4C23×D6111yescharacteristic
507M16—44no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—44no—
518C2×C8#1175822412yes—
519C2×C8C2×C22⋊D6112yes—
520C2×C8C2×C22⋊D6112yes—
521C4⋊C4—22no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—4881no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C4⋊C4—2442no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—4881no—
546C22⋊C4—4881no—
547C22⋊C4—4881no—
548C22⋊C4—4881no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C42—4881no—
554C42—2442no—
555C42#1175822212yes—
556C42C2×C22⋊D6112yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#1451981012yes—
571C23C22×D24411yescharacteristic
572Q8—1222no—
573Q8—1222no—
574Q8—2441no—
575Q8—2441no—
576D4—44no—
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—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—1222no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4#1451975112yes—
620C2×C4#1451975112yes—
621C2×C4C22×C22⋊D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4C2×#1175822411yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C2×C4#1451982211yescharacteristic
626C2×C4#1451982411yescharacteristic
627C8—22no—
628C8—22no—
629C8—22no—
630C8—41no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22C2×#1451975111yescharacteristic
646C22#1684040711yescharacteristic
647C22#1684041611yescharacteristic
648C22C2×#1451981011yescharacteristic
649C22#1684056211yescharacteristic
650C22#1684057411yescharacteristic
651C22#1684057411yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—4881no—
657C4—4881no—
658C4—4881no—
659C4—4881no—
660C4—4881no—
661C4—4881no—
662C4—4881no—
663C4—2442no—
664C4#1684029411yescharacteristic
665C4#1684037711yescharacteristic
666C4#1684040611yescharacteristic
667C4#1684041911yescharacteristic
668C2—42no—
669C2—81no—
670C2#2144939011yescharacteristic
671C2#2145544911yescharacteristic
672C2#2145548111yescharacteristic
673C2#2145794711yescharacteristic
674C2#2145799211yescharacteristic
675C2#2145827611yescharacteristic
676C2#2145836611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#435C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenterC2×#435D61
hypocenterC61#25357
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
1. C2×#16840070C2semidirect
4. #21447252C2semidirect
5. #21447247C2semidirect
6. #21447016C2semidirect
7. #21447016C2semidirect
8. #21446788C2semidirect
9. #21446583C2semidirect
10. #21446252C2semidirect
11. #21446242C2semidirect
12. #21445852C2semidirect
13. #21445844C2semidirect
14. #21445650C2semidirect
399. C61#25357semidirect

Maximal subgroups

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

subgroupindex
C2×#168400702
#214509852
#214507412
#214472522
#214472472
#214470162
#214467882
#214465832
#214462522
#214462422
#214458522
#214458442
#214456502
C2×C61×#4352
#2535761