Finite Group Databasebeta

← #30599659

Subgroups — #30599659

Order 15616.

574 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462312C211yesmaximal, characteristic
2#21461804C211yesmaximal, characteristic
3#21461769C211yesmaximal, characteristic
4#21461005C211yesmaximal, characteristic
5#21460693C211yesmaximal, characteristic
6#21460656C211yesmaximal, characteristic
7#328×D61C211yesmaximal, characteristic
8C22×C4⋊#6923C2211yescharacteristic
9C2×#14519845—21no—
10C2×#14519840—21no—
11D61⋊(C22⋊C8)C2211yescharacteristic
12(C2×#6923)⋊Q8—21no—
13#16840713—21no—
14#16840711—21no—
15#16840711—21no—
16#16840709—21no—
17#16840704C2211yescharacteristic
18C22⋊C4×#6923—21no—
19C2×(C2×#6923)⋊C4—21no—
20C2×(C2×#6923)⋊C4—21no—
21C2×(C2×#6923)⋊C4C2211yescharacteristic
22#16840636—21no—
23#16840635—21no—
24C22⋊Q8×D61C2211yescharacteristic
25C22⋊C8×D61C2211yescharacteristic
26#95×D61C2211yescharacteristic
27#16839500C411yescharacteristic, complemented
28#16839499C411yescharacteristic, complemented
29#16839498C411yescharacteristic, complemented
30C61×#328C411yescharacteristic, complemented
31C23×#6923—21no—
32C23×#6923D411yescharacteristic
33C23×#6923D411yescharacteristic
34C2×D61⋊C8—21no—
35C2×C4⋊#6923—21no—
36C2×C4⋊#6923—21no—
37C2×C4⋊#6923—21no—
38C2×C4⋊#6923—21no—
39C2×C4⋊#6923—42no—
40C2×C4⋊#6923D411yescharacteristic
41C2×C4⋊#6923D411yescharacteristic
42C2×C22⋊#6923—21no—
43C2×C4×#6923—21no—
44C2×C4×#6923—21no—
45#14519845—42no—
46#14519840—42no—
47#14519834—21no—
48C22⋊#92420—21no—
49(C2×#6923)⋊C4—21no—
50(C2×#6923)⋊C4—21no—
51(C2×#6923)⋊C4—21no—
52(C2×#6923)⋊C4—21no—
53(C2×#6923)⋊C4—21no—
54(C2×#6923)⋊C4—21no—
55(C2×#6923)⋊C4—21no—
56(C2×#6923)⋊C4—42no—
57(C2×#6923)⋊C4—42no—
58C2×Q8×D61D411yescharacteristic
59C22×C4×D61—21no—
60C22×C4×D61C2311yescharacteristic
61C2×C8×D61—21no—
62#14519757Q811yescharacteristic
63#14519756C2×C411yescharacteristic
64#14519755—21no—
65#14519754—21no—
66#14519753Q811yescharacteristic
67C4⋊C4×D61—21no—
68C4⋊C4×D61D411yescharacteristic
69C22⋊C4×D61—21no—
70C22⋊#92418C2×C411yescharacteristic
71#14519689—21no—
72#14519684—21no—
73#14519684C2×C411yescharacteristic
74C22⋊Q8×C61C2×C411yescharacteristic
75C22⋊C8×C61C2×C411yescharacteristic
76#95×C61C2×C411yescharacteristic
77C23×D61C2×D411yescharacteristic
78C22×#6923—21no—
79C22×#6923—21no—
80C22×#6923—22no—
81C22×#6923—21no—
82C22×#6923—21no—
83C22×#6923—21no—
84C22×#6923—21no—
85C22×#6923—21no—
86C22×#6923—41no—
87C22×#6923—41no—
88C22×#6923—41no—
89C22×#6923—41no—
90C22×#6923—41no—
91C22×#6923—41no—
92C22×#6923—41no—
93C22×#6923—41no—
94C22×#6923—41no—
95C22×#6923—41no—
96C22×#6923SD1612yes—
97C2×Q8×C61C4⋊C411yescharacteristic
98C22×C4×C61—21no—
99C22×C4×C61C22×C411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61#4111yescharacteristic
102Q8×D61—41no—
103Q8×D61—41no—
104C2×C4×D61—21no—
105C2×C4×D61—21no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61—41no—
112C2×C4×D61—41no—
113C2×C4×D61—41no—
114C2×C4×D61C2×D411yescharacteristic
115C2×C4×D61C2×D411yescharacteristic
116C2×C4×D61C2×Q811yescharacteristic
117C2×C4×D61#4111yescharacteristic
118C2×Dic122—21no—
119C2×Dic122C4⋊C411yescharacteristic
120C22⋊#6923—41no—
121C22⋊#6923—41no—
122C2×#92420—41no—
123C4⋊#6923—22no—
124C4⋊#6923—22no—
125C4⋊#6923—41no—
126C4⋊#6923—41no—
127C4⋊#6923—41no—
128C4⋊#6923—41no—
129C4⋊#6923—81no—
130C4⋊#6923—81no—
131C4×#6923—81no—
132C4×#6923—81no—
133D61⋊C8—41no—
134D61⋊C8—41no—
135C2×C8×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61C4⋊C411yescharacteristic
138C22⋊C4×C61—21no—
139C22⋊Dic61—21no—
140#11758198—21no—
141#11758198—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61C4⋊C411yescharacteristic
144#11758196—21no—
145#11758196—21no—
146#11758196—21no—
147C2×#92418—21no—
148C8×D61—41no—
149C8×D61—41no—
150C23×C61C4×D411yescharacteristic
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61#12011yescharacteristic
156C22×D61C22⋊Q811yescharacteristic
157C22×D61C2×SD1611yescharacteristic
158C22×D61#13511yescharacteristic
159C22×D61#13711yescharacteristic
160C22×D61#13711yescharacteristic
161C2×#6923—22no—
162C2×#6923—41no—
163C2×#6923—41no—
164C2×#6923—42no—
165C2×#6923—42no—
166C2×#6923—42no—
167C2×#6923—42no—
168C2×#6923—41no—
169C2×#6923—41no—
170C2×#6923—41no—
171C2×#6923—41no—
172C2×#6923—41no—
173C2×#6923—41no—
174C2×#6923—42no—
175C2×#6923—41no—
176C2×#6923—41no—
177C2×#6923—41no—
178C2×#6923—81no—
179C2×#6923—81no—
180C2×#6923—81no—
181C2×#6923—81no—
182C2×#6923—81no—
183C2×#6923—81no—
184Q8×C61—41no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61C2×C4⋊C411yescharacteristic
192C2×C4×C61C4×D411yescharacteristic
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61C22⋊Q811yescharacteristic
200C2×Dic61#12311yescharacteristic
201C4×D61—21no—
202C4×D61—21no—
203C4×D61—21no—
204C4×D61—21no—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61—41no—
208C4×D61—41no—
209C4×D61—41no—
210C4×D61—41no—
211C4×D61—81no—
212C4×D61—81no—
213C4×D61—81no—
214C4×D61—81no—
215Dic122—41no—
216Dic122—81no—
217#92420—81no—
218C8×C61—41no—
219#92418—41no—
220#8872—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61#38511yescharacteristic
224C22×C61C4×SD1611yescharacteristic
225C22×C61#44111yescharacteristic
226C2×D61—21no—
227C2×D61—21no—
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—21no—
233C2×D61—21no—
234C2×D61—21no—
235C2×D61—21no—
236C2×D61#44811yescharacteristic
237C2×D61#45211yescharacteristic
238C2×D61#45411yescharacteristic
239C2×D61#45511yescharacteristic
240C2×D61B(4,𝔽2)11yescharacteristic
241C2×D61#45811yescharacteristic
242C2×D61#47511yescharacteristic
243C2×D61#47911yescharacteristic
244#6923—42no—
245#6923—82no—
246#6923—81no—
247#6923—81no—
248#6923—81no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61—81no—
254Dic61—21no—
255Dic61—21no—
256Dic61—41no—
257Dic61—41no—
258Dic61—81no—
259#2329—611no—
260#1872—611no—
261#1853—611no—
262C2×#328—611no—
263#1442—611no—
264#1294—611no—
265#1280—611no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61#186011yescharacteristic
269C2×C61#186511yescharacteristic
270C2×C61#186711yescharacteristic
271D61—21no—
272D61—21no—
273D61#159911yescharacteristic
274D61#160611yescharacteristic
275D61#160811yescharacteristic
276D61#161511yescharacteristic
277C2×C22⋊Q8—611no—
278C22×C4⋊C4—611no—
279C2×#106—1221no—
280C2×#103—1221no—
281C2×C22⋊C8—611no—
282C2×C22⋊C8—611no—
283#391—1221no—
284#389—1221no—
285#387—1221no—
286#387—1221no—
287#385—1221no—
288#380—611no—
289C4×C22⋊C4—1221no—
290C2×#95—611no—
291C2×#95—611no—
292C2×#95—1221no—
293C2×#95—1221no—
294#340—1221no—
295#340—1221no—
296#328—611no—
297#328—611no—
298#328—611no—
299#328#692311yescharacteristic, complemented
300C61#887211yescharacteristic, complemented, Sylow
301C22×Q8—611no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—611no—
305C23×C4—1221no—
306C23×C4—1221no—
307C22×C8—1221no—
308C22×C8—1221no—
309C22⋊Q8—611no—
310C22⋊Q8—611no—
311C22⋊Q8—611no—
312C22⋊Q8—1221no—
313C22⋊Q8—1221no—
314C22⋊Q8C2×#692311yescharacteristic
315C2×C4⋊C4—611no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—2442no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C42—1221no—
327C2×C42—1221no—
328#106—2442no—
329#103—2442no—
330C22⋊C8—611no—
331C22⋊C8—1221no—
332C22⋊C8—1221no—
333C22⋊C8—1221no—
334C22⋊C8C2×#692311yescharacteristic
335#95—611no—
336#95—1221no—
337#95—1221no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341#95—1221no—
342#95—1221no—
343#95—1221no—
344#95—2442no—
345#95—2442no—
346#95C2×#692311yescharacteristic
347C24—611no—
348C2×Q8—611no—
349C2×Q8—1221no—
350C2×Q8—2441no—
351C2×Q8—2441no—
352C2×Q8C4⋊#692311yescharacteristic
353C22×C4—21no—
354C22×C4—611no—
355C22×C4—611no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—611no—
359C22×C4—612no—
360C22×C4—1221no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1222no—
370C22×C4—1221no—
371C22×C4—2441no—
372C22×C4—2441no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4—2441no—
377C22×C4—2441no—
378C22×C4—2441no—
379C22×C4—2441no—
380C22×C4—2441no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4—2441no—
385C22×C4—2441no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4C22×#692311yescharacteristic
390C2×C8—21no—
391C2×C8—1221no—
392C2×C8—2441no—
393C2×C8—2441no—
394C2×C8—2441no—
395C2×C8—2441no—
396C2×C8—2441no—
397C4⋊C4—21no—
398C4⋊C4—611no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—1222no—
404C4⋊C4—1222no—
405C4⋊C4—4881no—
406C4⋊C4—4881no—
407C4⋊C4—2441no—
408C4⋊C4—2441no—
409C4⋊C4—2441no—
410C4⋊C4—2441no—
411C4⋊C4C4⋊#692311yescharacteristic
412C22⋊C4—21no—
413C22⋊C4—1221no—
414C22⋊C4—1221no—
415C22⋊C4—1221no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C42—4881no—
419C42—4881no—
420C23—611no—
421C23—611no—
422C23—611no—
423C23—611no—
424C23—611no—
425C23—611no—
426C23—1221no—
427C23—1221no—
428C23—1221no—
429C23—1221no—
430C23D4×#692311yescharacteristic
431Q8—41no—
432Q8—4881no—
433Q8—2441no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—611no—
441C2×C4—611no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1222no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—4881no—
450C2×C4—4881no—
451C2×C4—4881no—
452C2×C4—4881no—
453C2×C4—4881no—
454C2×C4—4881no—
455C2×C4—4881no—
456C2×C4—4881no—
457C2×C4—4881no—
458C2×C4—4881no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2442no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4C2×C4⋊#692311yescharacteristic
486C2×C4D4×#692311yescharacteristic
487C8—41no—
488C8—4881no—
489C8—2441no—
490C22—21no—
491C22—21no—
492C22—611no—
493C22—611no—
494C22—611no—
495C22—611no—
496C22—611no—
497C22—611no—
498C22—611no—
499C22—611no—
500C22—1221no—
501C22—1221no—
502C22—1221no—
503C22—1221no—
504C22—1221no—
505C22—1221no—
506C22—1221no—
507C22—1221no—
508C22—1221no—
509C22—1221no—
510C22#1684070911yescharacteristic
511C22SD16×#692311yescharacteristic
512C22#1684080211yescharacteristic
513C4—21no—
514C4—21no—
515C4—41no—
516C4—41no—
517C4—81no—
518C4—1221no—
519C4—1221no—
520C4—4881no—
521C4—4882no—
522C4—4881no—
523C4—4881no—
524C4—4881no—
525C4—2442no—
526C4—2441no—
527C4—2441no—
528C2—21no—
529C2—21no—
530C2—611no—
531C2—611no—
532C2—611no—
533C2—611no—
534C2—1221no—
535C2—1221no—
536C2#2146177911yescharacteristic
537C2#2146178511yescharacteristic
538C2#2146179011yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#328C4
socleC22×C61#385
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#328#6923
hypocenterC61#8872
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
27. #16839500C4semidirect
28. #16839499C4semidirect
29. #16839498C4semidirect
30. C61×#328C4semidirect
299. #328#6923semidirect
300. C61#8872semidirect

Maximal subgroups

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

subgroupindex
#214623122
#214618042
#214617692
#214610052
#214606932
#214606562
#328×D612
#887261