Finite Group Databasebeta

← #30599658

Subgroups — #30599658

Order 15616.

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

normal subgroups only

This group has 618 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#21461007C211yesmaximal, characteristic
5#21460693C211yesmaximal, characteristic
6#21460656C211yesmaximal, characteristic
7#328×D61C211yesmaximal, characteristic
8C22×C4⋊#6923—21no—
9C22×C4⋊#6923C2211yescharacteristic
10C2×#14519845—21no—
11C2×#14519840—21no—
12D61⋊(C22⋊C8)C2211yescharacteristic
13(C2×#6923)⋊Q8—21no—
14#16840713—21no—
15#16840711—21no—
16#16840711—21no—
17#16840709—21no—
18#16840704C2211yescharacteristic
19C22⋊C4×#6923—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⋊#6923—42no—
41C2×C4⋊#6923—42no—
42C2×C4⋊#6923—42no—
43C2×C4⋊#6923D411yescharacteristic
44C2×C4⋊#6923D411yescharacteristic
45C2×C22⋊#6923—21no—
46C2×C4×#6923—21no—
47C2×C4×#6923—21no—
48#14519845—42no—
49#14519840—42no—
50#14519834—21no—
51C22⋊#92420—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—21no—
57(C2×#6923)⋊C4—21no—
58(C2×#6923)⋊C4—21no—
59(C2×#6923)⋊C4—42no—
60C2×Q8×D61D411yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×C8×D61—21no—
64#14519757Q811yescharacteristic
65#14519756C2×C411yescharacteristic
66#14519755—21no—
67#14519754—21no—
68#14519753Q811yescharacteristic
69C4⋊C4×D61—21no—
70C4⋊C4×D61D411yescharacteristic
71C22⋊C4×D61—21no—
72C22⋊#92418C2×C411yescharacteristic
73#14519689—21no—
74#14519684—21no—
75#14519684C2×C411yescharacteristic
76C22⋊Q8×C61C2×C411yescharacteristic
77C22⋊C8×C61C2×C411yescharacteristic
78#95×C61C2×C411yescharacteristic
79C23×D61C2×D411yescharacteristic
80C22×#6923—21no—
81C22×#6923—21no—
82C22×#6923—22no—
83C22×#6923—21no—
84C22×#6923—21no—
85C22×#6923—21no—
86C22×#6923—21no—
87C22×#6923—21no—
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×#6923—41no—
97C22×#6923—41no—
98C22×#6923SD1612yes—
99C2×Q8×C61C4⋊C411yescharacteristic, complemented
100C22×C4×C61—21no—
101C22×C4×C61C22×C411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61#4111yescharacteristic
104Q8×D61—41no—
105Q8×D61—41no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—41no—
111C2×C4×D61—41no—
112C2×C4×D61—41no—
113C2×C4×D61—41no—
114C2×C4×D61—41no—
115C2×C4×D61—41no—
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61C2×Q811yescharacteristic
119C2×C4×D61#4111yescharacteristic
120C2×Dic122—21no—
121C2×Dic122C4⋊C411yescharacteristic, complemented
122C22⋊#6923—41no—
123C22⋊#6923—41no—
124C2×#92420—41no—
125C4⋊#6923—22no—
126C4⋊#6923—22no—
127C4⋊#6923—41no—
128C4⋊#6923—41no—
129C4⋊#6923—41no—
130C4⋊#6923—41no—
131C4⋊#6923—48no—
132C4⋊#6923—81no—
133C4⋊#6923—81no—
134C4×#6923—81no—
135C4×#6923—81no—
136D61⋊C8—41no—
137D61⋊C8—41no—
138C2×C8×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61C4⋊C411yescharacteristic, complemented
141C22⋊C4×C61—21no—
142C22⋊Dic61—21no—
143#11758198—21no—
144#11758198—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61C4⋊C411yescharacteristic, complemented
147#11758196—21no—
148#11758196—21no—
149#11758196—21no—
150C2×#92418—21no—
151C8×D61—41no—
152C8×D61—41no—
153C23×C61C4×D411yescharacteristic
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—21no—
158C22×D61#12011yescharacteristic
159C22×D61C22⋊Q811yescharacteristic
160C22×D61C2×SD1611yescharacteristic
161C22×D61#13511yescharacteristic
162C22×D61#13711yescharacteristic
163C22×D61#13711yescharacteristic
164C2×#6923—22no—
165C2×#6923—41no—
166C2×#6923—41no—
167C2×#6923—42no—
168C2×#6923—42no—
169C2×#6923—42no—
170C2×#6923—42no—
171C2×#6923—41no—
172C2×#6923—41no—
173C2×#6923—41no—
174C2×#6923—41no—
175C2×#6923—41no—
176C2×#6923—41no—
177C2×#6923—41no—
178C2×#6923—42no—
179C2×#6923—41no—
180C2×#6923—41no—
181C2×#6923—81no—
182C2×#6923—81no—
183C2×#6923—81no—
184C2×#6923—81no—
185C2×#6923—81no—
186C2×#6923—81no—
187Q8×C61—41no—
188C2×C4×C61—22no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61C2×C4⋊C411yescharacteristic
195C2×C4×C61C4×D411yescharacteristic
196C2×Dic61—22no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61C22⋊Q811yescharacteristic
203C2×Dic61#12311yescharacteristic
204C4×D61—21no—
205C4×D61—21no—
206C4×D61—21no—
207C4×D61—21no—
208C4×D61—41no—
209C4×D61—41no—
210C4×D61—42no—
211C4×D61—42no—
212C4×D61—42no—
213C4×D61—42no—
214C4×D61—41no—
215C4×D61—41no—
216C4×D61—41no—
217C4×D61—41no—
218Dic122—41no—
219Dic122—81no—
220#92420—81no—
221C8×C61—41no—
222#92418—41no—
223#8871—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61#38511yescharacteristic
227C22×C61C4×SD1611yescharacteristic
228C22×C61#44111yescharacteristic
229C2×D61—21no—
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—21no—
233C2×D61—21no—
234C2×D61—21no—
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—21no—
238C2×D61—21no—
239C2×D61#44811yescharacteristic
240C2×D61#45211yescharacteristic
241C2×D61#45311yescharacteristic
242C2×D61#45611yescharacteristic
243C2×D61B(4,𝔽2)11yescharacteristic
244C2×D61#45811yescharacteristic
245C2×D61#47511yescharacteristic
246C2×D61#47911yescharacteristic
247#6923—42no—
248#6923—82no—
249#6923—81no—
250#6923—81no—
251#6923—81no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—42no—
255C4×C61—41no—
256C4×C61—41no—
257Dic61—21no—
258Dic61—21no—
259Dic61—42no—
260Dic61—41no—
261Dic61—41no—
262#2329—611no—
263#1872—611no—
264#1853—611no—
265C2×#328—611no—
266#1444—611no—
267#1294—611no—
268#1280—611no—
269C2×C61—21no—
270C2×C61—21no—
271C2×C61#185911yescharacteristic
272C2×C61#186511yescharacteristic
273C2×C61#186711yescharacteristic
274D61—21no—
275D61—21no—
276D61#159611yescharacteristic
277D61#160311yescharacteristic
278D61#161111yescharacteristic
279D61#161811yescharacteristic
280C2×C22⋊Q8—611no—
281C22×C4⋊C4—611no—
282C22×C4⋊C4—1221no—
283C2×#106—1221no—
284C2×#103—1221no—
285C2×C22⋊C8—611no—
286C2×C22⋊C8—611no—
287#391—1221no—
288#389—1221no—
289#387—1221no—
290#387—1221no—
291#385—1221no—
292#380—611no—
293C4×C22⋊C4—1221no—
294C2×#95—611no—
295C2×#95—611no—
296C2×#95—1221no—
297#340—1221no—
298#340—1221no—
299#328—611no—
300#328—611no—
301#328—611no—
302#328#692311yescharacteristic, complemented
303C61#887111yescharacteristic, complemented, Sylow
304C22×Q8—611no—
305C23×C4—611no—
306C23×C4—611no—
307C23×C4—611no—
308C23×C4—1221no—
309C23×C4—1221no—
310C22×C8—1221no—
311C22×C8—1221no—
312C22⋊Q8—611no—
313C22⋊Q8—611no—
314C22⋊Q8—611no—
315C22⋊Q8—1221no—
316C22⋊Q8—1221no—
317C22⋊Q8C2×#692311yescharacteristic
318C2×C4⋊C4—611no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1221no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—2442no—
327C2×C4⋊C4—2442no—
328C2×C4⋊C4—2442no—
329C2×C4⋊C4—2442no—
330C2×C22⋊C4—1221no—
331C2×C22⋊C4—1221no—
332C2×C42—1221no—
333C2×C42—1221no—
334#106—2442no—
335#103—2442no—
336C22⋊C8—611no—
337C22⋊C8—1221no—
338C22⋊C8—1221no—
339C22⋊C8—1221no—
340C22⋊C8C2×#692311yescharacteristic
341#95—611no—
342#95—1221no—
343#95—1221no—
344#95—1221no—
345#95—1221no—
346#95—1221no—
347#95—1221no—
348#95—1221no—
349#95—1221no—
350#95—2442no—
351#95C2×#692311yescharacteristic
352C24—611no—
353C2×Q8—611no—
354C2×Q8—1221no—
355C2×Q8—2441no—
356C2×Q8—2441no—
357C2×Q8C4⋊#692311yescharacteristic, complemented
358C22×C4—21no—
359C22×C4—611no—
360C22×C4—611no—
361C22×C4—611no—
362C22×C4—611no—
363C22×C4—611no—
364C22×C4—612no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—1222no—
368C22×C4—1222no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4—1221no—
377C22×C4—1222no—
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×C4—2441no—
390C22×C4—2441no—
391C22×C4—2441no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4C22×#692311yescharacteristic
395C2×C8—21no—
396C2×C8—1221no—
397C2×C8—2441no—
398C2×C8—2441no—
399C2×C8—2441no—
400C2×C8—2441no—
401C2×C8—2441no—
402C4⋊C4—21no—
403C4⋊C4—611no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—4881no—
411C4⋊C4—4881no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4—2448no—
417C4⋊C4C4⋊#692311yescharacteristic, complemented
418C22⋊C4—21no—
419C22⋊C4—1221no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C42—4881no—
425C42—4881no—
426C23—611no—
427C23—611no—
428C23—611no—
429C23—611no—
430C23—611no—
431C23—611no—
432C23—1221no—
433C23—1221no—
434C23—1221no—
435C23—1221no—
436C23D4×#692311yescharacteristic
437Q8—41no—
438Q8—4881no—
439Q8—2441no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—611no—
447C2×C4—611no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—4881no—
457C2×C4—4881no—
458C2×C4—4881no—
459C2×C4—4881no—
460C2×C4—4881no—
461C2×C4—4881no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2442no—
491C2×C4C2×C4⋊#692311yescharacteristic
492C2×C4D4×#692311yescharacteristic
493C8—41no—
494C8—4881no—
495C8—2441no—
496C22—21no—
497C22—21no—
498C22—611no—
499C22—611no—
500C22—611no—
501C22—611no—
502C22—611no—
503C22—611no—
504C22—611no—
505C22—611no—
506C22—1221no—
507C22—1221no—
508C22—1221no—
509C22—1221no—
510C22—1221no—
511C22—1221no—
512C22—1221no—
513C22—1221no—
514C22—1221no—
515C22—1221no—
516C22#1684070911yescharacteristic
517C22SD16×#692311yescharacteristic
518C22#1684080211yescharacteristic
519C4—21no—
520C4—21no—
521C4—42no—
522C4—41no—
523C4—41no—
524C4—1221no—
525C4—1221no—
526C4—4882no—
527C4—4881no—
528C4—4881no—
529C4—4881no—
530C4—2442no—
531C4—2442no—
532C4—2441no—
533C4—2441no—
534C2—21no—
535C2—21no—
536C2—611no—
537C2—611no—
538C2—611no—
539C2—611no—
540C2—1221no—
541C2—1221no—
542C2#2146177811yescharacteristic
543C2#2146178511yescharacteristic
544C2#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#8871
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
99. C2×Q8×C61C4⋊C4semidirect
121. C2×Dic122C4⋊C4semidirect
140. C4⋊C4×C61C4⋊C4semidirect
146. C4⋊Dic61C4⋊C4semidirect
302. #328#6923semidirect
303. C61#8871semidirect
357. C2×Q8C4⋊#6923semidirect
417. C4⋊C4C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
#214623122
#214618042
#214617692
#214610072
#214606932
#214606562
#328×D612
#887161