Finite Group Databasebeta

← #11778077

Subgroups — #11778077

Order 1152.

638 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C3×#5645C211yesmaximal, characteristic, complemented
2#10592745C211yesmaximal, characteristic, complemented
3#10591321C211yesmaximal, characteristic, complemented
4#10591168C211yesmaximal, characteristic
5C3×#4538C211yesmaximal, characteristic, complemented
6#10589149C211yesmaximal, characteristic, complemented
7#10589147C211yesmaximal, characteristic, complemented
8#71888—31nomaximal
9#68324—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C32×C22⋊Q8C2211yescharacteristic
12C2×#3755—21no—
13C3×#946—21no—
14C3×#930C2211yescharacteristic
15C2×C3×Dic3⋊C4C2211yescharacteristic
16C3×#898—21no—
17C3×#894—21no—
18C3×#890—21no—
19C3×#886—21no—
20C3×#67⋊C4—21no—
21C2×#3722—21no—
22C2×#3722—21no—
23C2×#3714—21no—
24#63657C2211yescharacteristic
25#63636—21no—
26C3×C3⋊#95C2211yescharacteristic
27C3×#830C2211yescharacteristic
28#63564—21no—
29C3×#539—31no—
30#5645D311yescharacteristic, complemented
31#5046—31no—
32#5046—31no—
33#5046—61no—
34#5036—31no—
35#4991—31no—
36#4732—31no—
37#4672—31no—
38C3×#328—31no—
39#4538D311yescharacteristic, complemented
40#4537—31no—
41#4528—31no—
42#4469—31no—
43#4451—31no—
44C23×C3⋊D3—21no—
45C2×C32×Q8D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—42no—
50C2×C3×#67—21no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3D411yescharacteristic
53C22×C3×Dic3D411yescharacteristic
54C2×C3×D12D411yescharacteristic
55C2×C3×C4×D3—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4D411yescharacteristic
58C32×C22⋊C4—21no—
59#3755—42no—
60C2×C3⋊(C3⋊C8)—21no—
61C3×(C2×D3)⋊C4—21no—
62C3×(C2×D3)⋊C4—21no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3D411yescharacteristic
68C3×Dic3⋊C4—21no—
69C3×Dic3⋊C4—21no—
70C3×Dic3⋊C4—21no—
71C3×C4×Dic3—21no—
72C2×C3×C3⋊C8—21no—
73#3722—42no—
74#3722—42no—
75#3714—42no—
76#1597—91noSylow
77C22×D12—31no—
78C22×D12—31no—
79C22×D12—61no—
80C3×#126—61no—
81C3×#125—61no—
82C3×#123—61no—
83C3×#123—61no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8—61no—
86C3×C22⋊Q8C2×D311yescharacteristic
87C3×C22⋊Q8C2×D311yescharacteristic
88C3×#121—31no—
89C3×C4×D4—61no—
90C2×C3×C4⋊C4—31no—
91#946—21no—
92C2×#215—62no—
93C2×#215—61no—
94C2×#215—61no—
95C2×#213—61no—
96#930C2×D311yescharacteristic
97C2×(C2×D3)⋊C4—61no—
98C2×(C2×D3)⋊C4—61no—
99C2×(C2×D3)⋊C4—61no—
100C2×(C2×D3)⋊C4—61no—
101C2×Dic3⋊C4C2×D311yescharacteristic
102C2×D24—61no—
103#898—21no—
104#894—21no—
105#890—21no—
106#886—21no—
107#67⋊C4—21no—
108C3×C22⋊C8—31no—
109C3×#95—31no—
110C3⋊#95C2×D311yescharacteristic
111#830—31no—
112#830—31no—
113#830—61no—
114#830C2×D311yescharacteristic
115#821—61no—
116#809—61no—
117#809—62no—
118#809—61no—
119#809—61no—
120C23×C32C2×D411yescharacteristic
121C22×C3⋊D3—22no—
122C22×C3⋊D3—41no—
123C22×C3⋊D3—41no—
124C22×C3⋊D3—41no—
125C22×C3⋊D3—41no—
126C22×C3⋊D3—41no—
127C22×C3⋊D3—41no—
128C22×C3×D3—21no—
129C32×Q8—41no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C4⋊(C3⋊D3)—22no—
136C4⋊(C3⋊D3)—22no—
137C4⋊(C3⋊D3)—41no—
138C4⋊(C3⋊D3)—41no—
139C3×#67—41no—
140C3×#67—41no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3—41no—
148C2×C3×Dic3—41no—
149C2×C3×Dic3—41no—
150C3×D12—41no—
151C3×C4×D3—81no—
152C3⋊(C3⋊C8)—41no—
153C3×C3⋊C8—41no—
154#539—31no—
155#450—91no—
156#447—91no—
157#394—91no—
158#328—31no—
159#327—91no—
160#324—91no—
161C23×D3—61no—
162C23×D3—61no—
163C23×D3—121no—
164C2×C3×Q8—21no—
165C2×C3×Q8D1211yescharacteristic
166C2×C3×Q8#6711yescharacteristic
167C2×C3×D4—31no—
168C2×C3×D4—61no—
169C22×C3×C4—21no—
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—61no—
173C22×C3×C4—61no—
174C22×C3×C4C22×D311yescharacteristic
175C22×C3×C4C22×D311yescharacteristic
176C2×#67—21no—
177C22×Dic3—21no—
178C22×Dic3#6711yescharacteristic
179C22×Dic3#6711yescharacteristic
180C2×D12—62no—
181C2×D12—61no—
182C2×D12—61no—
183C2×D12—61no—
184C2×D12—61no—
185C2×D12—121no—
186C2×D12—122no—
187C2×D12—122no—
188C2×D12—124no—
189C2×D12#6711yescharacteristic
190C2×C4×D3—21no—
191C2×C3×C8—61no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—31no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4D1211yescharacteristic
203C3×C4⋊C4#6711yescharacteristic
204C3×C22⋊C4—21no—
205C3×C22⋊C4—21no—
206C3×C22⋊C4—41no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—61no—
210C3×C22⋊C4—61no—
211C3×C42—61no—
212#215—122no—
213#215—122no—
214#215—124no—
215#213—122no—
216(C2×D3)⋊C4—21no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—21no—
220(C2×D3)⋊C4—122no—
221(C2×D3)⋊C4—122no—
222(C2×D3)⋊C4—122no—
223(C2×D3)⋊C4—122no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3#6711yescharacteristic
227Dic3⋊C4—21no—
228Dic3⋊C4—21no—
229Dic3⋊C4—21no—
230C4×Dic3—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—62no—
233C2×C3⋊C8—61no—
234C2×C3⋊C8—61no—
235D24—122no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32#12011yescharacteristic
239C22×C32#13511yescharacteristic
240C22×C32AGL(1,ℤ/8)11yescharacteristic
241C2×C3⋊D3—42no—
242C2×C3⋊D3—42no—
243C2×C3⋊D3—42no—
244C2×C3⋊D3—81no—
245C2×C3⋊D3—81no—
246C2×C3⋊D3—81no—
247C2×C3⋊D3—81no—
248C2×C3×D3—41no—
249C2×C3×D3—41no—
250C2×C3×D3—41no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—41no—
254C32×C4—41no—
255C3×Dic3—41no—
256C3×Dic3—41no—
257C3×Dic3—41no—
258C3×Dic3—81no—
259C22×D4—91no—
260C2×SD16—181no—
261C2×D8—181no—
262#126—61no—
263#125—61no—
264#123—61no—
265#123—61no—
266C22⋊Q8—61no—
267C22⋊Q8D3211yescharacteristic
268#121—31no—
269C4×D4—61no—
270C2×C4⋊C4—31no—
271C2×C22⋊C4—181no—
272C2×C22⋊C4—181no—
273#102—181no—
274#102—181no—
275C22⋊C8—31no—
276C22⋊C8—91no—
277#95—31no—
278C23×C3—21no—
279C23×C3—61no—
280C23×C3D3×D411yescharacteristic
281C23×C3C2×#6711yescharacteristic
282C22×D3—21no—
283C22×D3—62no—
284C22×D3—62no—
285C22×D3—122no—
286C22×D3—122no—
287C22×D3—121no—
288C22×D3—121no—
289C22×D3—121no—
290C22×D3—121no—
291C22×D3—121no—
292C22×D3—121no—
293C22×D3—122no—
294C22×D3—121no—
295C22×D3—121no—
296C22×D3—121no—
297C22×D3—121no—
298C22×D3—121no—
299C22×D3—121no—
300C22×D3—241no—
301C22×D3—241no—
302C22×D3—241no—
303C22×D3—241no—
304C3×Q8—41no—
305C3×Q8—41no—
306C3×Q8—42no—
307C3×D4—121no—
308C3×D4—121no—
309C3×D4—121no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—121no—
327C2×C3×C4—121no—
328C2×C3×C4—121no—
329C2×C3×C4—121no—
330C2×C3×C4—241no—
331C2×C3×C4C2×D1211yescharacteristic
332C2×C3×C4D3×D411yescharacteristic
333C2×C3×C4C2×#6711yescharacteristic
334C2×C3×C4C2×#6711yescharacteristic
335#67—41no—
336#67—41no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346D12—41no—
347D12—62no—
348D12—62no—
349D12—64no—
350D12—62no—
351D12—62no—
352D12—122no—
353D12—122no—
354D12—121no—
355D12—121no—
356D12—121no—
357D12—121no—
358D12—122no—
359C4×D3—81no—
360C3×C8—121no—
361C3⋊C8—41no—
362C3⋊C8—122no—
363C3⋊C8—121no—
364C3⋊C8—121no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32#44911yescharacteristic
368C2×C32#45411yescharacteristic
369C2×C32B(4,𝔽2)11yescharacteristic
370C3⋊D3—82no—
371C3×D3—81no—
372C24—181no—
373C2×Q8#62611yescharacteristic
374C2×D4—31no—
375C2×D4—61no—
376C2×D4—181no—
377C2×D4—181no—
378C2×D4—362no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—61no—
382C22×C4—61no—
383C22×C4C2×D3211yescharacteristic
384SD16—362no—
385D8—362no—
386C2×C8—61no—
387C2×C8—181no—
388C4⋊C4—21no—
389C4⋊C4—31no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4#62611yescharacteristic
396C22⋊C4—21no—
397C22⋊C4—61no—
398C22⋊C4—61no—
399C22⋊C4—61no—
400C22⋊C4—61no—
401C22⋊C4—362no—
402C22⋊C4—362no—
403C42—61no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—121no—
414C22×C3—121no—
415C22×C3—121no—
416C22×C3#88211yescharacteristic
417C22×C3#91411yescharacteristic
418C22×C3#91911yescharacteristic
419C22×C3#94911yescharacteristic
420C22×C3#95011yescharacteristic
421C22×C3#95311yescharacteristic
422C2×D3—41no—
423C2×D3—41no—
424C2×D3—41no—
425C2×D3—124no—
426C2×D3—122no—
427C2×D3—122no—
428C2×D3—122no—
429C2×D3—122no—
430C2×D3—122no—
431C2×D3—122no—
432C2×D3—242no—
433C2×D3—242no—
434C2×D3—241no—
435C2×D3—241no—
436C2×D3—241no—
437C2×D3—241no—
438C2×D3—242no—
439C2×D3—242no—
440C2×D3—242no—
441C2×D3—242no—
442C2×D3—241no—
443C2×D3—241no—
444C2×D3—241no—
445C2×D3—241no—
446C3×C4—22no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—81no—
457C3×C4—81no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—241no—
462Dic3—41no—
463Dic3—41no—
464Dic3—41no—
465Dic3—81no—
466C32#159711yescharacteristic, complemented, Sylow
467C23—61no—
468C23—182no—
469C23—361no—
470C23—361no—
471C23—361no—
472C23—361no—
473C23—361no—
474C23—361no—
475C23D3×#6711yescharacteristic
476Q8—41no—
477D4—121no—
478D4—121no—
479D4—121no—
480D4—182no—
481D4—182no—
482D4—361no—
483D4—361no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—41no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—241no—
497C2×C4C2×#62611yescharacteristic
498C2×C4D3×#6711yescharacteristic
499C8—121no—
500C8—361no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—41no—
509C2×C3—41no—
510C2×C3—241no—
511C2×C3#474111yescharacteristic
512C2×C3#481011yescharacteristic
513C2×C3#482711yescharacteristic
514C2×C3#523811yescharacteristic
515C2×C3#524411yescharacteristic
516C2×C3#524511yescharacteristic
517D3—81no—
518D3—244no—
519D3—242no—
520D3—242no—
521C22—21no—
522C22—21no—
523C22—121no—
524C22—121no—
525C22—121no—
526C22—362no—
527C22—362no—
528C22—362no—
529C22—721no—
530C22—721no—
531C22—721no—
532C22—721no—
533C22#6393911yescharacteristic
534C22#6394911yescharacteristic
535C22#6397711yescharacteristic
536C4—21no—
537C4—21no—
538C4—41no—
539C4—41no—
540C4—121no—
541C4—121no—
542C4—121no—
543C4—241no—
544C3—21no—
545C3#6832411yescharacteristic, complemented
546C3#7188811yescharacteristic, complemented
547C2—21no—
548C2—21no—
549C2—241no—
550C2—722no—
551C2#1059153111yescharacteristic
552C2#1059158811yescharacteristic
553C2#1059168511yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
prime radical#10591168C2
hypercenterC22⋊Q8D32
hypocenterC32#1597
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5645C2semidirect
2. #10592745C2semidirect
3. #10591321C2semidirect
5. C3×#4538C2semidirect
6. #10589149C2semidirect
7. #10589147C2semidirect
30. #5645D3semidirect
39. #4538D3semidirect
466. C32#1597semidirect
545. C3#68324semidirect
546. C3#71888semidirect

Maximal subgroups

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

subgroupindex
C3×#56452
#105927452
#105913212
#105911682
C3×#45382
#105891492
#105891472
#718883
#683243