Finite Group Databasebeta

← #30616134

Subgroups — #30616134

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840768C211yesmaximal, characteristic
2C2×#16840768C211yesmaximal, characteristic
3C2×#16840765C211yesmaximal, characteristic
4C2×#16840726C211yesmaximal, characteristic
5#21461921C211yesmaximal, characteristic, complemented
6#21461921C211yesmaximal, characteristic, complemented
7#21461920C211yesmaximal, characteristic, complemented
8#21461919C211yesmaximal, characteristic, complemented
9#21461918C211yesmaximal, characteristic, complemented
10#21461918C211yesmaximal, characteristic, complemented
11#21461917C211yesmaximal, characteristic, complemented
12#21461916C211yesmaximal, characteristic, complemented
13#21461485C211yesmaximal, characteristic
14C2×#16840628C211yesmaximal, characteristic
15#21448298C211yesmaximal, characteristic
16C22×#11758217C2211yescharacteristic
17C22×#11758217C2211yescharacteristic
18C23×#92420C2211yescharacteristic
19C23×#92420C2211yescharacteristic
20#16840768—24no—
21#16840768—24no—
22#16840765—24no—
23C2×#14519844C2211yescharacteristic
24C2×#14519844C2211yescharacteristic
25C2×#14519844C2211yescharacteristic
26C2×#14519844C2211yescharacteristic
27C2×#14519844C2211yescharacteristic
28C2×#14519844C2211yescharacteristic
29C2×C4⋊#92420C2211yescharacteristic
30C2×C4⋊#92420C2211yescharacteristic
31C2×C4⋊#92420C2211yescharacteristic
32C2×C4⋊#92420C2211yescharacteristic
33C2×C4⋊#92420C2211yescharacteristic
34C2×C4⋊#92420C2211yescharacteristic
35#16840726—24no—
36C2×#14519832C2211yescharacteristic
37C2×#14519832C2211yescharacteristic
38C2×C4×#92420C2211yescharacteristic
39C2×C4×#92420C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44#16840628C2211yescharacteristic
45#16840628C2211yescharacteristic
46#16840628C2211yescharacteristic
47#16840628C2211yescharacteristic
48C2×#14519733C2211yescharacteristic
49C2×#14519733C2211yescharacteristic
50C22×C4×Dic61C2211yescharacteristic
51#16839659C411yescharacteristic
52#16839659C411yescharacteristic
53#16839657C411yescharacteristic
54#16839657C411yescharacteristic
55#16839637C411yescharacteristic
56#16839635C411yescharacteristic
57#16839634C411yescharacteristic
58C4⋊C4×Dic61C2211yescharacteristic
59C4⋊C4×Dic61C2211yescharacteristic
60#16839613C2211yescharacteristic
61#16839613C2211yescharacteristic
62C61×#385C411yescharacteristic
63C2×#11758217—22no—
64C2×#11758217—22no—
65C2×#11758217—22no—
66C2×#11758217—22no—
67C2×#11758217—22no—
68C2×#11758217D414yes—
69C22×#92420—22no—
70C22×#92420—22no—
71C22×#92420—22no—
72C22×#92420—22no—
73C22×#92420—22no—
74C22×#92420Q814yes—
75C22×#92420C2311yescharacteristic
76C22×#92420C2311yescharacteristic
77C22×#92420C2311yescharacteristic
78C22×#92420C2311yescharacteristic
79C22×#92420C2311yescharacteristic
80C22×#92420C2311yescharacteristic
81C22×#92420C2311yescharacteristic
82C22×#92420C2311yescharacteristic
83#14519844—22no—
84#14519844—22no—
85#14519844—22no—
86#14519844—22no—
87#14519844—22no—
88#14519844—22no—
89C4⋊#92420—22no—
90C4⋊#92420—22no—
91C4⋊#92420—22no—
92C4⋊#92420—22no—
93C4⋊#92420—22no—
94C4⋊#92420—22no—
95#14519832—22no—
96#14519832—22no—
97C4×#92420—22no—
98C4×#92420—22no—
99C23×Dic61C2×C411yescharacteristic
100C23×Dic61C2311yescharacteristic
101#14519733—22no—
102#14519733—22no—
103#14519733D414yes—
104#14519733Q814yes—
105C2×C4⋊Dic61C2×C411yescharacteristic
106C2×C4⋊Dic61C2×C411yescharacteristic
107C2×#11758196C2×C411yescharacteristic
108C2×#11758196C2×C411yescharacteristic
109C2×#11758196C2×C411yescharacteristic
110C2×#11758196C2×C411yescharacteristic
111C2×#11758196C2×C411yescharacteristic
112C2×#11758196C2×C411yescharacteristic
113C2×#11758196C2×C411yescharacteristic
114C2×#11758196C2×C411yescharacteristic
115C2×C22⋊Dic61C2×C411yescharacteristic
116C2×C22⋊Dic61C2×C411yescharacteristic
117C2×C4×Dic61—22no—
118C2×C4×Dic61—22no—
119C2×C4×Dic61C2×C411yescharacteristic
120C2×C4×Dic61C2×C411yescharacteristic
121C2×C4×Dic61C2311yescharacteristic
122C2×C4×Dic61C2311yescharacteristic
123C2×C4×Dic61C2311yescharacteristic
124C2×C4×Dic61C2311yescharacteristic
125C2×C4×Dic61C2311yescharacteristic
126C2×C4×Dic61C2311yescharacteristic
127#14519684C2×C411yescharacteristic
128#14519684C2×C411yescharacteristic
129#14519684C2×C411yescharacteristic
130#14519684C2×C411yescharacteristic
131#14519684C2×C411yescharacteristic
132#14519684C2×C411yescharacteristic
133C23×C4×C61C2×C411yescharacteristic
134C2×C4⋊C4×C61C2×C411yescharacteristic
135C2×C4⋊C4×C61C2×C411yescharacteristic
136C2×C22⋊C4×C61C2×C411yescharacteristic
137C2×C22⋊C4×C61C2×C411yescharacteristic
138#95×C61C2×C411yescharacteristic
139#95×C61C2×C411yescharacteristic
140C24×C61C22×C411yescharacteristic
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—41no—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C22×C411yescharacteristic
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—41no—
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C22×C411yescharacteristic
156C22×Dic61C22×C411yescharacteristic
157C22×Dic61C22×C411yescharacteristic
158C22×Dic61C22×C411yescharacteristic
159C22×Dic61C22×C411yescharacteristic
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C22×Dic61C2×Q812yes—
163C22×Dic61C2×Q812yes—
164C22×Dic61#4112yes—
165C22×Dic61#4112yes—
166C22×Dic61C2411yescharacteristic
167#11758217—28no—
168#11758217—44no—
169C2×#92420—28no—
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—21no—
184C2×#92420—21no—
185C2×#92420—21no—
186C2×#92420—21no—
187C2×#92420—21no—
188C2×#92420—21no—
189C2×#92420—21no—
190C2×#92420—44no—
191C2×#92420C2×D412yes—
192C2×#92420C2×D412yes—
193C2×#92420C2×Q812yes—
194C2×#92420C2×Q812yes—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—41no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61C4⋊C414yes—
199C22⋊Dic61—22no—
200C22⋊Dic61C4⋊C414yes—
201C4⋊Dic61—22no—
202C4⋊Dic61—41no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—41no—
208#11758196—41no—
209#11758196C4⋊C414yes—
210#11758196C4⋊C414yes—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61C2×D412yes—
222C4×Dic61C2×D412yes—
223C4×Dic61C2×Q812yes—
224C4×Dic61C2×Q812yes—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61C2×C4⋊C412yes—
230C23×C61C4×D412yes—
231C23×C61C4×Q812yes—
232C23×C61C23×C411yescharacteristic
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—44no—
249C2×C4×C61C2×C4⋊C412yes—
250C2×C4×C61C2×C4⋊C412yes—
251C2×C4×C61#13112yes—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
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—44no—
268C2×Dic61C2×C4⋊C412yes—
269C2×Dic61C2×C4⋊C412yes—
270C2×Dic61C2×C4⋊C412yes—
271C2×Dic61C4×D412yes—
272C2×Dic61C4×Q812yes—
273C2×Dic61#12114yes—
274C2×Dic61C22⋊Q814yes—
275C2×Dic61#12514yes—
276C2×Dic61C4⋊Q814yes—
277C2×Dic61#13112yes—
278C2×Dic61C22×D411yescharacteristic
279C2×Dic61C22×D411yescharacteristic
280C2×Dic61C22×Q811yescharacteristic
281C2×Dic61C22×Q811yescharacteristic
282C2×Dic61C2×#4111yescharacteristic
283C2×Dic61C2×#4111yescharacteristic
284#92420—22no—
285#92420—22no—
286#92420—22no—
287#92420—22no—
288#92420—41no—
289#92420—41no—
290#92420—41no—
291#92420—41no—
292#21148—611nomaximal, Sylow
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61#38914yes—
300C22×C61C22×C4⋊C411yescharacteristic
301C22×C61C2×C4×D411yescharacteristic
302C22×C61C2×C4×Q811yescharacteristic
303C22×C61C2×#13111yescharacteristic
304C22×C61C2×#13111yescharacteristic
305C22×C61#56811yescharacteristic
306C22×C61#56811yescharacteristic
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—41no—
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
319Dic61#38914yes—
320Dic61C2×#12111yescharacteristic
321Dic61C2×C22⋊Q811yescharacteristic
322Dic61C2×#12511yescharacteristic
323Dic61C2×C4⋊Q811yescharacteristic
324C2×#424—611no—
325C2×#424—611no—
326C2×#423—611no—
327C2×#405—611no—
328#2296—611no—
329#1940—611no—
330#1940—611no—
331#1940—611no—
332#1940—611no—
333#1939—611no—
334#1939—611no—
335#1939—611no—
336#1939—611no—
337#1729—611no—
338C2×#336—611no—
339C2×C61—24no—
340C2×C61C2×#38911yescharacteristic
341C2×C61#290611yescharacteristic
342C2×C61#290611yescharacteristic
343C2×C61#293311yescharacteristic
344C2×C61#293311yescharacteristic
345C2×C61#296011yescharacteristic
346C2×C61#296111yescharacteristic
347C22×M16—611no—
348C22×M16—611no—
349C23×C8—611no—
350C23×C8—611no—
351C2×#117—611no—
352C2×#117—611no—
353C22×C42—611no—
354#424—1224no—
355#424—1224no—
356#423—1224no—
357C2×C4⋊C8—611no—
358C2×C4⋊C8—611no—
359C2×C4⋊C8—611no—
360C2×C4⋊C8—611no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8—611no—
368C2×C4⋊C8—611no—
369#405—1224no—
370C2×#97—611no—
371C2×#97—611no—
372C2×C4×C8—611no—
373C2×C4×C8—611no—
374#389—611no—
375#389—611no—
376#389—611no—
377#389—611no—
378#385—611no—
379#385—611no—
380#385—611no—
381#385#692311yescharacteristic
382C4×C4⋊C4—611no—
383C4×C4⋊C4—611no—
384C4×C4⋊C4—611no—
385C4×C4⋊C4—611no—
386#336—611no—
387#336—611no—
388#336—611no—
389#336—611no—
390#336—611no—
391#336—611no—
392#336—611no—
393#336—611no—
394C61#2114811yescharacteristic, complemented, Sylow
395C23×C4—611no—
396C23×C4—611no—
397C23×C4C2×#692311yescharacteristic
398C2×M16—614no—
399C2×M16—1222no—
400C2×M16—1222no—
401C2×M16—1222no—
402C2×M16—1222no—
403C2×M16—1222no—
404C22×C8—614no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8—611no—
408C22×C8—611no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8—611no—
413C22×C8—1222no—
414C22×C8—1222no—
415C22×C8—1222no—
416C22×C8—1222no—
417C22×C8—1222no—
418#117—614no—
419#117—614no—
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⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4C2×#692311yescharacteristic
433C2×C4⋊C4C2×#692311yescharacteristic
434C2×C22⋊C4—611no—
435C2×C22⋊C4—611no—
436C2×C22⋊C4C2×#692311yescharacteristic
437C2×C22⋊C4C2×#692311yescharacteristic
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—1222no—
447C2×C42—1222no—
448C4⋊C8—1222no—
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452C4⋊C8—1222no—
453C4⋊C8—1222no—
454C4⋊C8—1222no—
455C4⋊C8—1222no—
456C4⋊C8—1222no—
457C4⋊C8—1222no—
458C4⋊C8—1222no—
459C4⋊C8—1222no—
460#97—1222no—
461#97—1222no—
462C4×C8—1222no—
463C4×C8—1222no—
464#95—611no—
465#95—611no—
466#95—611no—
467#95—611no—
468#95—611no—
469#95—611no—
470#95C2×#692311yescharacteristic
471#95C2×#692311yescharacteristic
472C24C22×#692311yescharacteristic
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—2441no—
493C22×C4C22×#692311yescharacteristic
494C22×C4C22×#692311yescharacteristic
495C22×C4C22×#692311yescharacteristic
496C22×C4C22×#692311yescharacteristic
497C22×C4C22×#692311yescharacteristic
498C22×C4C22×#692311yescharacteristic
499M16—1228no—
500M16—2444no—
501C2×C8—612no—
502C2×C8—612no—
503C2×C8—612no—
504C2×C8—612no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1228no—
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—1221no—
525C2×C8—1221no—
526C2×C8—2444no—
527C4⋊C4—22no—
528C4⋊C4—41no—
529C4⋊C4—614no—
530C4⋊C4—614no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—2441no—
537C4⋊C4—2441no—
538C4⋊C4—2441no—
539C22⋊C4—22no—
540C22⋊C4—614no—
541C22⋊C4—1222no—
542C22⋊C4C4⋊#692314yes—
543C42—612no—
544C42—612no—
545C42—612no—
546C42—612no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42—1222no—
552C42—1221no—
553C42—1222no—
554C42—1221no—
555C42—1221no—
556C42—1221no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23C2×C4⋊#692312yes—
562C23D4×#692312yes—
563C23Q8×#692312yes—
564C23C23×#692311yescharacteristic
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
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—44no—
581C2×C4—614no—
582C2×C4—614no—
583C2×C4—614no—
584C2×C4—614no—
585C2×C4—612no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—612no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—612no—
592C2×C4—611no—
593C2×C4—611no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—612no—
597C2×C4—1221no—
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—2444no—
613C2×C4C2×C4⋊#692312yes—
614C2×C4C2×C4⋊#692312yes—
615C2×C4#1451986412yes—
616C8—1222no—
617C8—1222no—
618C8—1222no—
619C8—1222no—
620C8—2441no—
621C8—2441no—
622C8—2441no—
623C8—2441no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22#1684071314yes—
631C22C22×C4⋊#692311yescharacteristic
632C22C2×D4×#692311yescharacteristic
633C22C2×Q8×#692311yescharacteristic
634C22C2×#1451986411yescharacteristic
635C22D61⋊#13111yescharacteristic
636C22#1684084011yescharacteristic
637C22#1684084111yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—611no—
645C4—611no—
646C4—611no—
647C4—611no—
648C4—614no—
649C4—1222no—
650C4—1222no—
651C4—1222no—
652C4—2441no—
653C4—2441no—
654C4—2441no—
655C2—24no—
656C2C2×#1684071311yescharacteristic
657C2#2146246411yescharacteristic
658C2#2146246611yescharacteristic
659C2#2146255111yescharacteristic
660C2#2146255311yescharacteristic
661C2#2146260711yescharacteristic
662C2#2146261011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#385C4
prime radicalC24×C61C22×C4
cosocleC22×Dic61C24
hypercenter#385#6923
hypocenterC61#21148
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
5. #21461921C2semidirect
6. #21461921C2semidirect
7. #21461920C2semidirect
8. #21461919C2semidirect
9. #21461918C2semidirect
10. #21461918C2semidirect
11. #21461917C2semidirect
12. #21461916C2semidirect
394. C61#21148semidirect

Maximal subgroups

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

subgroupindex
C2×#168407682
C2×#168407652
C2×#168407262
#214619212
#214619202
#214619192
#214619182
#214619172
#214619162
#214614852
C2×#168406282
#214482982
#2114861