Finite Group Databasebeta

← #30609409

Subgroups — #30609409

Order 15616.

1210 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461009C212yesmaximal
2#21461006C212yesmaximal
3#21461005C212yesmaximal
4#21460999C212yesmaximal
5#21460998C212yesmaximal
6#21460997C211yesmaximal, characteristic
7#21460995C211yesmaximal, characteristic
8#21460994C212yesmaximal
9#391×D61C211yesmaximal, characteristic
10C22×C4⋊#6923C2212yes—
11C22×C4×#6923C2212yes—
12C2×(C2×#6923)⋊C4C2212yes—
13C2×(C2×#6923)⋊C4C2212yes—
14C2×(C2×#6923)⋊C4C2211yescharacteristic
15C2×(C2×#6923)⋊C4C2212yes—
16C2×(C2×#6923)⋊C4C2211yescharacteristic
17C2×(C2×#6923)⋊C4C2212yes—
18C2×(C2×#6923)⋊C4C2212yes—
19C2×(C2×#6923)⋊C4C2212yes—
20C2×(C2×#6923)⋊C4C2211yescharacteristic
21C2×(C2×#6923)⋊C4C2212yes—
22C2×(C2×#6923)⋊C4C2212yes—
23C2×(C2×#6923)⋊C4C2212yes—
24C2×(C2×#6923)⋊C4C2212yes—
25C2×(C2×#6923)⋊C4C2211yescharacteristic
26C22×Q8×D61C2211yescharacteristic
27C2×C4⋊C4×D61C2211yescharacteristic
28C2×C42×D61C2211yescharacteristic
29#16839669C411yescharacteristic, complemented
30#16839668C411yescharacteristic, complemented
31#16839667C411yescharacteristic, complemented
32#16839666C412yescomplemented
33#16839666C412yescomplemented
34#95×D61C2212yes—
35#95×D61C2212yes—
36C61×#391C411yescharacteristic, complemented
37C23×#6923C2312yes—
38C23×#6923C2312yes—
39C23×#6923C2312yes—
40C23×#6923C2312yes—
41C2×C4⋊#6923—24no—
42C2×C4⋊#6923—24no—
43C2×C4⋊#6923—42no—
44C2×C4×#6923—24no—
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—22no—
51(C2×#6923)⋊C4—42no—
52(C2×#6923)⋊C4—42no—
53(C2×#6923)⋊C4—42no—
54(C2×#6923)⋊C4—42no—
55(C2×#6923)⋊C4—41no—
56(C2×#6923)⋊C4—41no—
57(C2×#6923)⋊C4—42no—
58(C2×#6923)⋊C4—42no—
59(C2×#6923)⋊C4—42no—
60(C2×#6923)⋊C4Q814yescomplemented
61C22×Dic122C2×C412yes—
62C22×Dic122C2×C411yescharacteristic
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—42no—
66C2×Q8×D61—42no—
67C22×C4×D61C2311yescharacteristic
68C22×C4×D61C2312yes—
69C22×C4×D61C2312yes—
70C22×C4×D61C2311yescharacteristic
71C22×C4×D61C2311yescharacteristic
72C2×C4⋊Dic61C2×C411yescharacteristic
73C2×#11758196C2×C412yes—
74C4⋊C4×D61—42no—
75C2×C4×Dic61C2×C411yescharacteristic
76C2×C4×Dic61C2×C411yescharacteristic
77C2×C4×Dic61C2×C411yescharacteristic
78C42×D61—42no—
79#14519684C2×C412yes—
80#14519684C2×C412yes—
81#14519684C2×C412yes—
82#14519684C2×C412yes—
83#14519684C2×C412yes—
84#14519684C2×C412yes—
85C22×Q8×C61C2×C411yescharacteristic
86C2×C4⋊C4×C61C2×C411yescharacteristic
87C2×C42×C61C2×C411yescharacteristic
88#95×C61C2×C412yes—
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—
119C2×Q8×C61—22no—
120C2×Q8×C61—41no—
121C2×Q8×C61—41no—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C411yescharacteristic
129C22×Dic61C22×C412yes—
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C411yescharacteristic
132Q8×D61—48no—
133Q8×D61—48no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
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—21no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—21no—
154C2×C4×D61C2×Q812yes—
155C2×C4×D61#4112yes—
156C2×C4×D61#4112yes—
157C2×C4×D61#4112yes—
158C2×Dic122—22no—
159C2×Dic122—24no—
160C2×Dic122—42no—
161C2×Dic122—42no—
162C2×Dic122—41no—
163C2×Dic122—41no—
164C4⋊#6923—48no—
165C4×#6923—48no—
166C4⋊C4×C61—41no—
167C42×C61—41no—
168C4⋊Dic61—41no—
169#11758196—42no—
170C4×Dic61—41no—
171C4×Dic61—41no—
172C4×Dic61—41no—
173C23×C61C23×C411yescharacteristic
174C22×D61C22×Q811yescharacteristic
175C22×D61C2×#4111yescharacteristic
176C22×D61C2×#4111yescharacteristic
177C22×D61C2×#4112yes—
178C22×D61C2×#4111yescharacteristic
179C22×D61H(5,𝔽2)12yes—
180C22×D61H(5,𝔽2)12yes—
181C22×D61H(5,𝔽2)11yescharacteristic
182C22×D612−1+411yescharacteristic
183C22×D612−1+412yes—
184C2×#6923—24no—
185C2×#6923—24no—
186C2×#6923—24no—
187C2×#6923—24no—
188C2×#6923—24no—
189C2×#6923—24no—
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×#6923—42no—
209C2×#6923—42no—
210C2×#6923—42no—
211C2×#6923—42no—
212Q8×C61—44no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61#11712yes—
227C2×C4×C61C4×Q812yes—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239C2×Dic61—22no—
240C2×Dic61—21no—
241C2×Dic61#11712yes—
242C2×Dic61C4×Q812yes—
243C4×D61—24no—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—42no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61—42no—
256C4×D61—42no—
257Dic122—48no—
258Dic122—44no—
259#14469—611nomaximal, Sylow
260C22×C61C2×#11711yescharacteristic
261C22×C61C2×C4×Q811yescharacteristic
262C22×C61C4×#4111yescharacteristic
263C22×C61#51811yescharacteristic
264C22×C61#51811yescharacteristic
265C22×C61#51911yescharacteristic
266C22×C61#52011yescharacteristic
267C2×D61#53812yes—
268C2×D61#53812yes—
269C2×D61#53812yes—
270C2×D61#54212yes—
271C2×D61#54212yes—
272C2×D61#54412yes—
273C2×D61#55112yes—
274C2×D61#55212yes—
275C2×D61#55412yes—
276C2×D61#55411yescharacteristic
277C2×D61#55611yescharacteristic
278C2×D61#55612yes—
279C2×D61#55711yescharacteristic
280C2×D61#55911yescharacteristic
281C2×D61#56212yes—
282C2×D61#56312yes—
283#6923—44no—
284#6923—44no—
285#6923—82no—
286#6923—82no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61—41no—
292Dic61—22no—
293Dic61—22no—
294Dic61—42no—
295Dic61—42no—
296Dic61—41no—
297C2×#391—611no—
298#1446—612no—
299#1443—612no—
300#1442—612no—
301#1436—612no—
302#1436—612no—
303C4⋊#95—611no—
304#1434—612no—
305#1434—611no—
306C2×C61#231211yescharacteristic
307C2×C61#233111yescharacteristic
308C2×C61#233711yescharacteristic
309C2×C61#235511yescharacteristic
310C2×C61#235611yescharacteristic
311C2×C61#236811yescharacteristic
312C2×C61#237811yescharacteristic
313D61#275112yes—
314D61#275912yes—
315D61#277412yes—
316D61#278812yes—
317C23×Q8—611no—
318C22×C4⋊C4—611no—
319C22×C4⋊C4—612no—
320C22×C42—611no—
321C22×C42—612no—
322#391—611no—
323#391—611no—
324#391—611no—
325#391—612no—
326#391—612no—
327#391#692311yescharacteristic, complemented
328C2×#95—611no—
329C2×#95—612no—
330C2×#95—612no—
331C2×#95—612no—
332C2×#95—612no—
333C2×#95—611no—
334C2×#95—612no—
335C2×#95—611no—
336C2×#95—612no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—611no—
340C2×#95—612no—
341C2×#95—612no—
342C2×#95—612no—
343C2×#95—612no—
344C61#1446911yescharacteristic, complemented, Sylow
345C22×Q8—612no—
346C22×Q8—611no—
347C22×Q8—1222no—
348C22×Q8—1222no—
349C22×Q8—2442no—
350C22×Q8—2442no—
351C22×Q8C2×#692311yescharacteristic
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—612no—
355C23×C4—612no—
356C23×C4—612no—
357C23×C4—612no—
358C23×C4—611no—
359C23×C4—612no—
360C23×C4—612no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—1224no—
364C2×C4⋊C4—1224no—
365C2×C4⋊C4—2442no—
366C2×C4⋊C4—2442no—
367C2×C4⋊C4C2×#692311yescharacteristic
368C2×C42—611no—
369C2×C42—611no—
370C2×C42—611no—
371C2×C42—1224no—
372C2×C42—2442no—
373C2×C42—2442no—
374C2×C42—2442no—
375C2×C42C2×#692311yescharacteristic
376#95—614no—
377#95—612no—
378#95—612no—
379#95—612no—
380#95—612no—
381#95—612no—
382#95—612no—
383#95—1224no—
384#95—1224no—
385#95—1222no—
386#95—1224no—
387#95—2441no—
388#95—2441no—
389#95—2442no—
390#95—2442no—
391#95—2442no—
392#95—2442no—
393#95—2442no—
394#95—2442no—
395#95—2442no—
396#95C2×#692312yes—
397#95C2×#692312yes—
398C24—611no—
399C2×Q8—22no—
400C2×Q8—41no—
401C2×Q8—41no—
402C2×Q8—1224no—
403C2×Q8—1222no—
404C2×Q8—2441no—
405C2×Q8—2441no—
406C2×Q8—2442no—
407C2×Q8—2442no—
408C2×Q8—2448no—
409C2×Q8—2448no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—614no—
417C22×C4—614no—
418C22×C4—611no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1221no—
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—1221no—
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×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4C22×#692311yescharacteristic
468C22×C4C22×#692312yes—
469C22×C4C22×#692312yes—
470C22×C4C22×#692311yescharacteristic
471C22×C4C22×#692311yescharacteristic
472C4⋊C4—41no—
473C4⋊C4—2442no—
474C4⋊C4—2441no—
475C4⋊C4—2448no—
476C42—41no—
477C42—2441no—
478C42—2441no—
479C42—2441no—
480C42—2448no—
481C23—611no—
482C23—611no—
483C23—611no—
484C23—611no—
485C23—612no—
486C23—612no—
487C23—612no—
488C23—612no—
489C23—611no—
490C23—611no—
491C23C23×#692311yescharacteristic
492Q8—44no—
493Q8—2444no—
494Q8—2448no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—1224no—
513C2×C4—1224no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1224no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1221no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1221no—
531C2×C4—1222no—
532C2×C4—1221no—
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—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4#1451985612yes—
566C2×C4Q8×#692312yes—
567C22—612no—
568C22—612no—
569C22—612no—
570C22—612no—
571C22—612no—
572C22—612no—
573C22—611no—
574C22—611no—
575C22—611no—
576C22—611no—
577C22—612no—
578C22—612no—
579C22—612no—
580C22—612no—
581C22—612no—
582C22—612no—
583C22C2×#1451985611yescharacteristic
584C22C2×Q8×#692311yescharacteristic
585C22#41×#692311yescharacteristic
586C22#1684082911yescharacteristic
587C22#1684082911yescharacteristic
588C22#1684083011yescharacteristic
589C22#1684083111yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—1222no—
596C4—1222no—
597C4—4882no—
598C4—4882no—
599C4—2444no—
600C4—2444no—
601C4—2442no—
602C4—2442no—
603C4—2441no—
604C2—612no—
605C2—612no—
606C2—612no—
607C2—612no—
608C2#2146229511yescharacteristic
609C2#2146231411yescharacteristic
610C2#2146232011yescharacteristic
611C2#2146233811yescharacteristic
612C2#2146233911yescharacteristic
613C2#2146235111yescharacteristic
614C2#2146236111yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#391C4
prime radical, cosocleC23×D61C24
hypercenter#391#6923
hypocenterC61#14469
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
29. #16839669C4semidirect
30. #16839668C4semidirect
31. #16839667C4semidirect
32. #16839666C4semidirect
33. #16839666C4semidirect
36. C61×#391C4semidirect
60. (C2×#6923)⋊C4Q8semidirect
327. #391#6923semidirect
344. C61#14469semidirect

Maximal subgroups

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

subgroupindex
#214610092
#214610062
#214610052
#214609992
#214609982
#214609972
#214609952
#214609942
#391×D612
#1446961