Finite Group Databasebeta

← #30598211

Subgroups — #30598211

Order 15616.

1620 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461642C211yesmaximal, characteristic, complemented
2C2×#16840613C214yesmaximal, complemented
3#21460993C211yesmaximal, characteristic, complemented
4C22×#14519748C211yesmaximal, characteristic
5C22×C22⋊#6923—21no—
6C2×#14519834—21no—
7C2×#14519834C2212yescomplemented
8C2×C22⋊#92420—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yescomplemented
11#16840613—28no—
12C23×C22⋊D61—21no—
13C23×C4×D61C2211yescharacteristic
14C2×#14519748—28no—
15C2×#14519748—21no—
16C2×#14519748—21no—
17C2×#14519748C412yescomplemented
18C2×#14519748C2212yes—
19C22×C4⋊Dic61C411yescharacteristic, complemented
20C22×C22⋊Dic61—21no—
21C23×D4×C61C411yescharacteristic, complemented
22C23×#6923—22no—
23C22×#92420—22no—
24C2×C22⋊#6923—24no—
25C2×C22⋊#6923—42no—
26C2×C22⋊#6923—42no—
27#14519834—24no—
28#14519834—42no—
29C22⋊#92420—42no—
30(C2×#6923)⋊C4—24no—
31(C2×#6923)⋊C4—42no—
32C24×D61D411yescharacteristic
33C22×C22⋊D61—28no—
34C22×C22⋊D61—22no—
35C22×C22⋊D61—22no—
36C22×C22⋊D61—44no—
37C22×C22⋊D61—42no—
38C22×C22⋊D61—42no—
39C23×Dic61—21no—
40C23×Dic61D411yescharacteristic
41C22×C4×D61—24no—
42C22×C4×D61—42no—
43C22×C4×D61—42no—
44C22×C4×D61C2×C411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61D411yescharacteristic
47C22×C4×D61C2311yescharacteristic
48#14519748—28no—
49#14519748—28no—
50#14519748—24no—
51#14519748—24no—
52#14519748—24no—
53#14519748D418yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—24no—
56C2×C4⋊Dic61C2×C412yes—
57C2×C22⋊Dic61—28no—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—22no—
60C25×C61—21no—
61C22×D4×C61—24no—
62C22×D4×C61—21no—
63C22×D4×C61—42no—
64C22×D4×C61—42no—
65C22×D4×C61C2×C412yes—
66C23×C4×C61C2×C411yescharacteristic
67C24×C61—24no—
68C24×C61—22no—
69C24×C61—42no—
70C24×C61—42no—
71C24×C61—42no—
72C24×C61—42no—
73C24×C61—42no—
74C24×C61—42no—
75C24×C61C22⋊C411yescharacteristic
76C23×D61—22no—
77C23×D61—22no—
78C23×D61—22no—
79C23×D61—22no—
80C23×D61—41no—
81C23×D61—41no—
82C23×D61C22⋊C411yescharacteristic
83C23×D61D812yes—
84C23×D61SD1612yes—
85C23×D61C2×D411yescharacteristic
86C22×#6923—24no—
87C22×#6923—42no—
88C22×#6923—42no—
89C22×#6923—42no—
90C22×#6923—42no—
91C22×#6923—42no—
92C22×#6923—42no—
93C2×D4×C61—24no—
94C2×D4×C61—22no—
95C2×D4×C61—416no—
96C2×D4×C61—42no—
97C2×D4×C61—42no—
98C2×D4×C61—42no—
99C2×D4×C61—42no—
100C2×D4×C61C22⋊C414yescomplemented
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61C22⋊C411yescharacteristic
105C22×C4×C61C22×C411yescharacteristic
106C2×C22⋊D61—28no—
107C2×C22⋊D61—28no—
108C2×C22⋊D61—416no—
109C2×C22⋊D61—416no—
110C2×C22⋊D61—44no—
111C2×C22⋊D61—44no—
112C2×C22⋊D61—42no—
113C2×C22⋊D61—42no—
114C2×C22⋊D61—44no—
115C22×Dic61—24no—
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—41no—
120C22×Dic61—42no—
121C22×Dic61—42no—
122C22×Dic61C22⋊C411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—41no—
130C2×C4×D61—41no—
131C2×C4×D61—41no—
132C2×C4×D61—41no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61—416no—
136C2×C4×D61C2×D414yes—
137C22⋊#6923—48no—
138C2×#92420—42no—
139C2×#92420—42no—
140C2×#92420—42no—
141C22⋊Dic61—28no—
142C22⋊Dic61—28no—
143C4⋊Dic61—24no—
144C4⋊Dic61—22no—
145C4⋊Dic61C22⋊C414yescomplemented
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—21no—
149C23×C61—21no—
150C23×C61—24no—
151C23×C61—48no—
152C23×C61—48no—
153C23×C61—48no—
154C23×C61—42no—
155C23×C61—42no—
156C23×C61—42no—
157C23×C61—42no—
158C23×C61—42no—
159C23×C61—42no—
160C23×C61—84no—
161C23×C61—84no—
162C23×C61—84no—
163C23×C61—84no—
164C23×C61#9912yes—
165C23×C61#10212yes—
166C23×C61#10412yes—
167C23×C61C2×C22⋊C411yescharacteristic
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—24no—
173C22×D61—24no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—21no—
180C22×D61—24no—
181C22×D61—24no—
182C22×D61—42no—
183C22×D61—42no—
184C22×D61—44no—
185C22×D61—44no—
186C22×D61—44no—
187C22×D61—44no—
188C22×D61#10212yes—
189C22×D61#12012yes—
190C22×D61C2×D811yescharacteristic
191C22×D61C2×SD1611yescharacteristic
192C22×D61AGL(1,ℤ/8)12yes—
193C2×#6923—44no—
194C2×#6923—44no—
195C2×#6923—44no—
196C2×#6923—82no—
197C2×#6923—82no—
198C2×#6923—82no—
199C2×#6923—82no—
200D4×C61—44no—
201D4×C61—48no—
202D4×C61—44no—
203C2×C4×C61—28no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61#9912yes—
209C2×C4×C61C2×C22⋊C412yes—
210C22⋊D61—416no—
211C22⋊D61—48no—
212C22⋊D61—48no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—24no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—44no—
220C2×Dic61—48no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61#12012yes—
224C4×D61—44no—
225C4×D61—44no—
226C4×D61—44no—
227C4×D61—44no—
228#92420—82no—
229#8328—611nomaximal, Sylow
230C22×C61—22no—
231C22×C61—22no—
232C22×C61—22no—
233C22×C61—22no—
234C22×C61—22no—
235C22×C61—22no—
236C22×C61—44no—
237C22×C61—44no—
238C22×C61—44no—
239C22×C61—816no—
240C22×C61—82no—
241C22×C61—82no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61#32714yescomplemented
245C22×C61#37911yescharacteristic
246C22×C61C2×#9911yescharacteristic
247C22×C61C2×#9911yescharacteristic
248C22×C61C2×#10211yescharacteristic
249C22×C61#41811yescharacteristic
250C22×C61C2×#10411yescharacteristic
251C22×C61#42111yescharacteristic
252C2×D61—21no—
253C2×D61—21no—
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—48no—
264C2×D61—48no—
265C2×D61—41no—
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—41no—
269C2×D61#44712yes—
270C2×D61#45012yes—
271#6923—84no—
272C4×C61—22no—
273C4×C61—22no—
274Dic61—22no—
275Dic61—41no—
276Dic61—44no—
277C22×#121—611no—
278#1779—611no—
279C2×#327—614no—
280C22⋊#95—611no—
281C2×C61—24no—
282C2×C61—84no—
283C2×C61C2×#32711yescharacteristic
284C2×C61#150711yescharacteristic
285C2×C61#151011yescharacteristic
286C2×C61#151911yescharacteristic
287C2×C61#178111yescharacteristic
288C2×C61#178611yescharacteristic
289C2×C61#178911yescharacteristic
290D61—22no—
291D61—22no—
292D61—42no—
293C23×D4—1221no—
294C23×D4#692311yescharacteristic, complemented
295C24×C4—611no—
296C2×#121—612no—
297C2×#121—612no—
298C2×#121—1228no—
299C2×#121—1221no—
300C2×#121—1221no—
301C22×C4⋊C4—611no—
302C22×C22⋊C4—1221no—
303C22×C22⋊C4—1221no—
304C2×C22⋊C8—612no—
305C2×C22⋊C8—1221no—
306C2×C22⋊C8—1221no—
307C2×#95—612no—
308C2×#95—1221no—
309#327—1228no—
310C61#832811yescharacteristic, complemented, Sylow
311C25—21no—
312C25—611no—
313C22×D4—24no—
314C22×D4—21no—
315C22×D4—42no—
316C22×D4—42no—
317C22×D4—1222no—
318C22×D4—1222no—
319C22×D4—1228no—
320C22×D4—2442no—
321C22×D4—2442no—
322C22×D4—2444no—
323C22×D4C2×#692312yes—
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—611no—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—1224no—
330C23×C4—1221no—
331C23×C4—1222no—
332C23×C4—2442no—
333C23×C4—2442no—
334C23×C4C2×#692311yescharacteristic
335C22×C8—1222no—
336#121—618no—
337#121—1224no—
338#121—1224no—
339#121—1228no—
340#121—1228no—
341#121—1224no—
342C2×C4⋊C4—612no—
343C2×C4⋊C4—1224no—
344C2×C4⋊C4—1221no—
345C2×C22⋊C4—1222no—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4—1228no—
348C2×C22⋊C4—1224no—
349C2×C22⋊C4—2442no—
350C2×C22⋊C4—2442no—
351C22⋊C8—1224no—
352C22⋊C8—2442no—
353C22⋊C8—2442no—
354#95—1224no—
355#95—2442no—
356C24—24no—
357C24—22no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—611no—
365C24—611no—
366C24—612no—
367C24—612no—
368C24—1222no—
369C24—1222no—
370C24—1222no—
371C24—1222no—
372C24—2441no—
373C24—2441no—
374C24C22⋊#692311yescharacteristic
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—416no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—1228no—
383C2×D4—1228no—
384C2×D4—2442no—
385C2×D4—2442no—
386C2×D4—2444no—
387C2×D4—2444no—
388C2×D4—24416no—
389C2×D4—24416no—
390C2×D4—2444no—
391C2×D4C22⋊#692314yescomplemented
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—611no—
396C22×C4—614no—
397C22×C4—611no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1224no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1224no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4—2441no—
412C22×C4—2441no—
413C22×C4—2441no—
414C22×C4—24416no—
415C22×C4—2441no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4C22⋊#692311yescharacteristic
425C22×C4C22×#692311yescharacteristic
426C2×C8—2442no—
427C2×C8—2442no—
428C2×C8—2442no—
429C4⋊C4—614no—
430C4⋊C4—1224no—
431C4⋊C4—1222no—
432C22⋊C4—1228no—
433C22⋊C4—1228no—
434C22⋊C4—2448no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—24no—
440C23—48no—
441C23—48no—
442C23—48no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—84no—
450C23—84no—
451C23—84no—
452C23—84no—
453C23—611no—
454C23—611no—
455C23—612no—
456C23—612no—
457C23—612no—
458C23—1224no—
459C23—1224no—
460C23—1221no—
461C23—1221no—
462C23—1221no—
463C23—1221no—
464C23—1221no—
465C23—1221no—
466C23—1221no—
467C23—1221no—
468C23—1221no—
469C23—1224no—
470C23—1224no—
471C23—1221no—
472C23—2442no—
473C23—2442no—
474C23—2444no—
475C23—2444no—
476C23—2444no—
477C23—2444no—
478C23#1451983512yes—
479C23#1451983912yes—
480C23#1451984112yes—
481C23C2×C22⋊#692311yescharacteristic
482D4—44no—
483D4—48no—
484D4—44no—
485D4—2448no—
486D4—2448no—
487D4—24416no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—612no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1224no—
500C2×C4—4882no—
501C2×C4—4882no—
502C2×C4—4882no—
503C2×C4—4882no—
504C2×C4—2444no—
505C2×C4—2444no—
506C2×C4—2444no—
507C2×C4—2444no—
508C2×C4—2444no—
509C2×C4—2444no—
510C2×C4—2444no—
511C2×C4—2444no—
512C2×C4—2448no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4#1451983512yes—
516C2×C4C2×C22⋊#692312yes—
517C8—4882no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—816no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—612no—
533C22—612no—
534C22—1221no—
535C22—1221no—
536C22—1222no—
537C22—1222no—
538C22—1221no—
539C22—1221no—
540C22—1222no—
541C22—1222no—
542C22—1222no—
543C22—1222no—
544C22—1222no—
545C22—2448no—
546C22—2448no—
547C22—2441no—
548C22—2441no—
549C22—2441no—
550C22—2441no—
551C22#1684061314yescomplemented
552C22#1684070311yescharacteristic
553C22C2×#1451983511yescharacteristic
554C22C2×#1451983511yescharacteristic
555C22C2×#1451983911yescharacteristic
556C22#1684075411yescharacteristic
557C22C2×#1451984111yescharacteristic
558C22#1684076011yescharacteristic
559C4—22no—
560C4—22no—
561C4—1222no—
562C4—4884no—
563C4—2441no—
564C4—2444no—
565C2—24no—
566C2—84no—
567C2—1222no—
568C2—1222no—
569C2—2442no—
570C2C2×#1684061311yescharacteristic
571C2#2146121511yescharacteristic
572C2#2146122411yescharacteristic
573C2#2146125511yescharacteristic
574C2#2146164511yescharacteristic
575C2#2146165711yescharacteristic
576C2#2146166011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×D4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC22×#14519748C2
cosocleC22×C4×D61C23
hypercenterC23×D4#6923
hypocenterC61#8328
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #21461642C2semidirect
2. C2×#16840613C2semidirect
3. #21460993C2semidirect
7. C2×#14519834C22semidirect
10. C2×(C2×#6923)⋊C4C22semidirect
17. C2×#14519748C4semidirect
19. C22×C4⋊Dic61C4semidirect
21. C23×D4×C61C4semidirect
100. C2×D4×C61C22⋊C4semidirect
145. C4⋊Dic61C22⋊C4semidirect
244. C22×C61#327semidirect
294. C23×D4#6923semidirect
310. C61#8328semidirect
391. C2×D4C22⋊#6923semidirect
551. C22#16840613semidirect

Maximal subgroups

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

subgroupindex
#214616422
C2×#168406132
#214609932
C22×#145197482
#832861