Finite Group Databasebeta

← #30609410

Subgroups — #30609410

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461008C212yesmaximal
2#21461008C212yesmaximal
3#21461005C212yesmaximal
4#21460999C212yesmaximal
5#21460998C212yesmaximal
6#21460998C212yesmaximal
7#21460997C211yesmaximal, characteristic
8#21460995C211yesmaximal, characteristic
9#389×D61C211yesmaximal, characteristic
10C22×C4⋊#6923C2212yes—
11C22×C4×#6923C2212yes—
12C2×(C2×#6923)⋊C4C2212yes—
13C2×(C2×#6923)⋊C4C2211yescharacteristic
14C2×(C2×#6923)⋊C4C2212yes—
15C2×(C2×#6923)⋊C4C2212yes—
16C2×(C2×#6923)⋊C4C2212yes—
17C2×(C2×#6923)⋊C4C2212yes—
18C2×(C2×#6923)⋊C4C2211yescharacteristic
19C2×(C2×#6923)⋊C4C2211yescharacteristic
20C2×(C2×#6923)⋊C4C2212yes—
21C2×(C2×#6923)⋊C4C2212yes—
22C2×(C2×#6923)⋊C4C2212yes—
23C2×(C2×#6923)⋊C4C2212yes—
24C2×(C2×#6923)⋊C4C2212yes—
25C2×(C2×#6923)⋊C4C2211yescharacteristic
26C2×C4⋊C4×D61C2211yescharacteristic
27C2×C4⋊C4×D61C2211yescharacteristic
28C2×C4⋊C4×D61C2212yes—
29C2×C42×D61C2211yescharacteristic
30#16839661C411yescharacteristic, complemented
31#16839660C411yescharacteristic, complemented
32#16839659C412yescomplemented
33#16839658C411yescharacteristic, complemented
34#16839657C412yescomplemented
35#95×D61C2212yes—
36C61×#389C411yescharacteristic, complemented
37C23×#6923C2312yes—
38C23×#6923C2312yes—
39C23×#6923C2312yes—
40C23×#6923C2312yes—
41C2×C4⋊#6923—24no—
42C2×C4⋊#6923—24no—
43C2×C4⋊#6923—42no—
44C2×C4×#6923—42no—
45C2×C4×#6923—42no—
46C2×C4×#6923—42no—
47(C2×#6923)⋊C4—24no—
48(C2×#6923)⋊C4—24no—
49(C2×#6923)⋊C4—24no—
50(C2×#6923)⋊C4—24no—
51(C2×#6923)⋊C4—22no—
52(C2×#6923)⋊C4—41no—
53(C2×#6923)⋊C4—41no—
54(C2×#6923)⋊C4—42no—
55(C2×#6923)⋊C4—42no—
56(C2×#6923)⋊C4—42no—
57(C2×#6923)⋊C4—42no—
58(C2×#6923)⋊C4—42no—
59(C2×#6923)⋊C4—42no—
60(C2×#6923)⋊C4Q814yes—
61C22×C4×D61C2311yescharacteristic
62C22×C4×D61C2312yes—
63C22×C4×D61C2312yes—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C2×C4⋊Dic61C2×C411yescharacteristic
67C2×C4⋊Dic61C2×C412yes—
68C2×C4⋊Dic61C2×C411yescharacteristic
69C2×#11758196C2×C412yes—
70C2×#11758196C2×C412yes—
71C2×#11758196C2×C412yes—
72C2×#11758196C2×C412yes—
73C4⋊C4×D61—22no—
74C4⋊C4×D61—22no—
75C4⋊C4×D61—42no—
76C4⋊C4×D61—42no—
77C4⋊C4×D61—42no—
78C2×C4×Dic61C2×C411yescharacteristic
79C2×C4×Dic61C2×C411yescharacteristic
80C2×C4×Dic61C2×C411yescharacteristic
81C42×D61—42no—
82#14519684C2×C412yes—
83#14519684C2×C412yes—
84#14519684C2×C412yes—
85C2×C4⋊C4×C61C2×C411yescharacteristic
86C2×C4⋊C4×C61C2×C412yes—
87C2×C4⋊C4×C61C2×C411yescharacteristic
88C2×C42×C61C2×C411yescharacteristic
89#95×C61C2×C412yes—
90C23×D61C2411yescharacteristic
91C22×#6923—22no—
92C22×#6923—22no—
93C22×#6923—22no—
94C22×#6923—22no—
95C22×#6923—22no—
96C22×#6923—22no—
97C22×#6923—22no—
98C22×#6923—22no—
99C22×#6923—22no—
100C22×#6923—22no—
101C22×#6923—22no—
102C22×#6923—22no—
103C22×#6923—22no—
104C22×#6923—22no—
105C22×#6923—22no—
106C22×#6923—22no—
107C22×#6923—22no—
108C22×#6923—22no—
109C22×#6923—22no—
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923—22no—
113C22×#6923—22no—
114C22×#6923—22no—
115C22×#6923—22no—
116C22×#6923—22no—
117C22×#6923C2×Q814yes—
118C22×#6923#4114yes—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×Dic61C22×C412yes—
125C22×Dic61C22×C412yes—
126C22×Dic61C22×C411yescharacteristic
127C22×Dic61C22×C411yescharacteristic
128C22×Dic61C22×C411yescharacteristic
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—21no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—21no—
148C2×C4×D61—22no—
149C2×C4×D61C2×Q812yes—
150C2×C4×D61#4112yes—
151C2×C4×D61#4112yes—
152C2×C4×D61#4112yes—
153C4⋊#6923—48no—
154C4×#6923—48no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—41no—
157C4⋊C4×C61—42no—
158C42×C61—41no—
159C4⋊Dic61—22no—
160C4⋊Dic61—42no—
161C4⋊Dic61—41no—
162#11758196—24no—
163#11758196—42no—
164#11758196—42no—
165#11758196—42no—
166C4×Dic61—41no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169C23×C61C23×C411yescharacteristic
170C22×D61C22×Q811yescharacteristic
171C22×D61C2×#4111yescharacteristic
172C22×D61C2×#4111yescharacteristic
173C22×D61C2×#4112yes—
174C22×D61C2×#4111yescharacteristic
175C22×D61H(5,𝔽2)12yes—
176C22×D61H(5,𝔽2)11yescharacteristic
177C22×D61H(5,𝔽2)12yes—
178C22×D612−1+412yes—
179C22×D612−1+411yescharacteristic
180C2×#6923—24no—
181C2×#6923—24no—
182C2×#6923—24no—
183C2×#6923—24no—
184C2×#6923—24no—
185C2×#6923—24no—
186C2×#6923—42no—
187C2×#6923—42no—
188C2×#6923—42no—
189C2×#6923—42no—
190C2×#6923—42no—
191C2×#6923—42no—
192C2×#6923—42no—
193C2×#6923—42no—
194C2×#6923—42no—
195C2×#6923—42no—
196C2×#6923—42no—
197C2×#6923—42no—
198C2×#6923—42no—
199C2×#6923—42no—
200C2×#6923—42no—
201C2×#6923—42no—
202C2×#6923—42no—
203C2×#6923—42no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—42no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61#11712yes—
222C2×C4×C61C4×Q812yes—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61#11712yes—
237C2×Dic61C4×Q812yes—
238C4×D61—24no—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61—42no—
245C4×D61—42no—
246C4×D61—42no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—42no—
251C4×D61—42no—
252#14470—611nomaximal, Sylow
253C22×C61C2×#11711yescharacteristic
254C22×C61C2×C4×Q811yescharacteristic
255C22×C61C4×#4111yescharacteristic
256C22×C61#51811yescharacteristic
257C22×C61#51811yescharacteristic
258C22×C61#51911yescharacteristic
259C22×C61#52011yescharacteristic
260C2×D61#53912yes—
261C2×D61#53912yes—
262C2×D61#53912yes—
263C2×D61#54112yes—
264C2×D61#54212yes—
265C2×D61#54412yes—
266C2×D61#55112yes—
267C2×D61#55312yes—
268C2×D61#55412yes—
269C2×D61#55411yescharacteristic
270C2×D61#55611yescharacteristic
271C2×D61#55711yescharacteristic
272C2×D61#55911yescharacteristic
273C2×D61#55912yes—
274C2×D61#56012yes—
275C2×D61#56212yes—
276#6923—44no—
277#6923—44no—
278#6923—82no—
279#6923—82no—
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—41no—
285Dic61—22no—
286Dic61—22no—
287Dic61—42no—
288Dic61—42no—
289Dic61—41no—
290C2×#389—611no—
291#1445—612no—
292#1445—612no—
293#1442—612no—
294#1436—612no—
295#1436—612no—
296#1436—612no—
297C4⋊#95—611no—
298#1434—611no—
299C2×C61#231311yescharacteristic
300C2×C61#233111yescharacteristic
301C2×C61#233611yescharacteristic
302C2×C61#235511yescharacteristic
303C2×C61#235711yescharacteristic
304C2×C61#236811yescharacteristic
305C2×C61#237711yescharacteristic
306D61#279612yes—
307D61#279812yes—
308D61#280212yes—
309D61#280712yes—
310C22×C4⋊C4—611no—
311C22×C4⋊C4—611no—
312C22×C4⋊C4—612no—
313C22×C4⋊C4—612no—
314C22×C42—611no—
315C22×C42—612no—
316#389—612no—
317#389—611no—
318#389—611no—
319#389—611no—
320#389—612no—
321#389#692311yescharacteristic, complemented
322C2×#95—611no—
323C2×#95—612no—
324C2×#95—612no—
325C2×#95—611no—
326C2×#95—612no—
327C2×#95—612no—
328C2×#95—612no—
329C2×#95—612no—
330C2×#95—611no—
331C2×#95—611no—
332C2×#95—612no—
333C2×#95—612no—
334C2×#95—612no—
335C2×#95—612no—
336C2×#95—612no—
337C61#1447011yescharacteristic, complemented, Sylow
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—612no—
341C23×C4—612no—
342C23×C4—612no—
343C23×C4—612no—
344C23×C4—611no—
345C23×C4—612no—
346C23×C4—612no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—612no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1224no—
358C2×C4⋊C4—2442no—
359C2×C4⋊C4—2442no—
360C2×C4⋊C4—2442no—
361C2×C4⋊C4—2442no—
362C2×C4⋊C4C2×#692311yescharacteristic
363C2×C4⋊C4C2×#692312yes—
364C2×C4⋊C4C2×#692311yescharacteristic
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—611no—
368C2×C42—2442no—
369C2×C42—2442no—
370C2×C42—2442no—
371C2×C42—2442no—
372C2×C42C2×#692311yescharacteristic
373#95—614no—
374#95—612no—
375#95—612no—
376#95—612no—
377#95—1224no—
378#95—1224no—
379#95—1222no—
380#95—1224no—
381#95—1224no—
382#95—2441no—
383#95—2441no—
384#95—2442no—
385#95—2442no—
386#95—2442no—
387#95—2442no—
388#95—2442no—
389#95—2442no—
390#95C2×#692312yes—
391C24—611no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—612no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—614no—
399C22×C4—614no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—611no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1221no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4C22×#692312yes—
450C22×C4C22×#692312yes—
451C22×C4C22×#692311yescharacteristic
452C22×C4C22×#692311yescharacteristic
453C22×C4C22×#692311yescharacteristic
454C4⋊C4—22no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—1224no—
458C4⋊C4—1222no—
459C4⋊C4—2442no—
460C4⋊C4—2442no—
461C4⋊C4—2442no—
462C4⋊C4—2441no—
463C4⋊C4—2442no—
464C4⋊C4—2448no—
465C42—41no—
466C42—2441no—
467C42—2441no—
468C42—2441no—
469C42—2448no—
470C23—611no—
471C23—611no—
472C23—611no—
473C23—611no—
474C23—612no—
475C23—612no—
476C23—612no—
477C23—612no—
478C23—611no—
479C23—611no—
480C23C23×#692311yescharacteristic
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—612no—
495C2×C4—612no—
496C2×C4—1224no—
497C2×C4—1224no—
498C2×C4—1224no—
499C2×C4—1224no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1224no—
505C2×C4—1224no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1221no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4#1451985612yes—
552C2×C4Q8×#692312yes—
553C22—612no—
554C22—612no—
555C22—612no—
556C22—612no—
557C22—612no—
558C22—612no—
559C22—611no—
560C22—611no—
561C22—611no—
562C22—611no—
563C22—612no—
564C22—612no—
565C22—612no—
566C22—612no—
567C22—612no—
568C22—612no—
569C22C2×#1451985611yescharacteristic
570C22C2×Q8×#692311yescharacteristic
571C22#41×#692311yescharacteristic
572C22#1684082911yescharacteristic
573C22#1684082911yescharacteristic
574C22#1684083011yescharacteristic
575C22#1684083111yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—1222no—
582C4—1222no—
583C4—4882no—
584C4—4882no—
585C4—2444no—
586C4—2444no—
587C4—2442no—
588C4—2441no—
589C4—2442no—
590C2—612no—
591C2—612no—
592C2—612no—
593C2—612no—
594C2#2146229611yescharacteristic
595C2#2146231411yescharacteristic
596C2#2146231911yescharacteristic
597C2#2146233811yescharacteristic
598C2#2146234011yescharacteristic
599C2#2146235111yescharacteristic
600C2#2146236011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#389C4
prime radical, cosocleC23×D61C24
hypercenter#389#6923
hypocenterC61#14470
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
30. #16839661C4semidirect
31. #16839660C4semidirect
32. #16839659C4semidirect
33. #16839658C4semidirect
34. #16839657C4semidirect
36. C61×#389C4semidirect
321. #389#6923semidirect
337. C61#14470semidirect

Maximal subgroups

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

subgroupindex
#214610082
#214610052
#214609992
#214609982
#214609972
#214609952
#389×D612
#1447061