Finite Group Databasebeta

← #11780656

Subgroups — #11780656

Order 1152.

466 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 76 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×#5684C211yesmaximal, characteristic
2#10592754C211yesmaximal, characteristic, complemented
3#10590372C211yesmaximal, characteristic, complemented
4#10590306C211yesmaximal, characteristic, complemented
5C3×#4540C211yesmaximal, characteristic, complemented
6#10589088C211yesmaximal, characteristic, complemented
7#10589086C211yesmaximal, characteristic, complemented
8#71953—31nomaximal
9#68357—31nomaximal
10C32×#124C2211yescharacteristic
11#64078C2211yescharacteristic
12C3×#946—21no—
13C3×#939—21no—
14C3×#886—21no—
15C3×#884—21no—
16C3×#883—21no—
17C3×#878—21no—
18C3×#875C2211yescharacteristic
19C3×#870C2211yescharacteristic
20C2×#3719—21no—
21C2×#3716—21no—
22#63662—21no—
23#63659—21no—
24#63630C2211yescharacteristic
25C3×#803C2211yescharacteristic
26C3×#803C2211yescharacteristic
27#63575—21no—
28#63573—21no—
29C3×#562—31no—
30#5684D311yescharacteristic
31#5064—31no—
32#5060—31no—
33#5055—31no—
34#5055—31no—
35#5055—61no—
36#4715—31no—
37#4702—31no—
38C3×#329—31no—
39#4543—31no—
40#4540—31no—
41#4540D311yescharacteristic, complemented
42#4465—31no—
43#4464—31no—
44C2×C32×Q8D411yescharacteristic
45C2×C32×D4D411yescharacteristic
46C2×#634—21no—
47C2×C3×#67—21no—
48C2×C3×#67—21no—
49C22×C3×Dic3—21no—
50C2×C3×D12D411yescharacteristic
51C2×C3×C4×D3—21no—
52C2×C3×Dic6D411yescharacteristic
53C32×C22⋊C4—21no—
54C32×C22⋊C4—21no—
55C32×C42C2311yescharacteristic
56C4⋊(C3⋊Dic3)—21no—
57C4⋊(C3⋊Dic3)—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C3×#217—21no—
60C3×(C2×D3)⋊C4—21no—
61C3×(C2×D3)⋊C4—21no—
62C3×(C2×D3)⋊C4—21no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3D411yescharacteristic
65C3×C4⋊Dic3D411yescharacteristic
66C3×Dic3⋊C4—21no—
67C3×Dic3⋊C4—21no—
68C3×Dic3⋊C4—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71#3719—42no—
72#3716—42no—
73#1661—91noSylow
74C3×#125—31no—
75C3×#124—21no—
76C3×#124—31no—
77C3×#124C2×D311yescharacteristic
78C3×#124C2×D311yescharacteristic
79C3×#123—61no—
80C3×#123—61no—
81C3×C22⋊Q8—61no—
82C3×C22⋊Q8—61no—
83C3×#121—61no—
84C3×#121—61no—
85#946—21no—
86C2×#216—61no—
87#939—21no—
88C2×#214—61no—
89C2×Dic12—61no—
90C2×#204—61no—
91#886—21no—
92#884—21no—
93#883—21no—
94#878—21no—
95#875C2×D311yescharacteristic
96#870C2×D311yescharacteristic
97#869—31no—
98#869—31no—
99#869—61no—
100C3×#97—31no—
101C3×#97—31no—
102#835—62no—
103#835—61no—
104#835—61no—
105#832—62no—
106#832—61no—
107#832—61no—
108#818—61no—
109#817—61no—
110#808—61no—
111#807—61no—
112#803—31no—
113#803—31no—
114#803—61no—
115#803C2×D311yescharacteristic
116#803C2×D311yescharacteristic
117C23×C32—21no—
118C22×C3×D3—21no—
119C32×Q8—41no—
120C32×D4—41no—
121C2×C32×C4—21no—
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C3⋊Dic3—41no—
126#634—41no—
127#634—41no—
128C3×#67—81no—
129C3×#67—81no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—81no—
134C3×D12—41no—
135C3×C4×D3—81no—
136C3×Dic6—41no—
137C3⋊(C3⋊C8)—41no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—22no—
140#562—31no—
141#501—91no—
142#497—91no—
143#489—91no—
144#330—91no—
145#329—31no—
146#329—91no—
147C2×C3×Q8—21no—
148C2×C3×Q8—31no—
149C2×C3×Q8D1211yescharacteristic
150C2×C3×Q8#6711yescharacteristic
151C2×C3×D4—21no—
152C2×C3×D4—31no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C2×C3×D4D1211yescharacteristic
156C2×C3×D4#6711yescharacteristic
157C22×C3×C4—61no—
158C22×C3×C4—61no—
159C2×#67—21no—
160C2×#67—21no—
161C22×Dic3—21no—
162C2×D12#6711yescharacteristic
163C2×C4×D3—21no—
164C2×Dic6—61no—
165C2×Dic6—61no—
166C2×Dic6—121no—
167C2×Dic6#6711yescharacteristic
168C2×C3×C8—61no—
169C2×C3×C8—61no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—61no—
175C3×C4⋊C4—61no—
176C3×C22⋊C4—21no—
177C3×C22⋊C4—21no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—21no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—41no—
182C3×C22⋊C4—61no—
183C3×C22⋊C4—61no—
184C3×C22⋊C4—61no—
185C3×C22⋊C4—61no—
186C3×C42—21no—
187C3×C42C22×D311yescharacteristic
188C3×C42C22×D311yescharacteristic
189#217—21no—
190#216—122no—
191#214—122no—
192(C2×D3)⋊C4—21no—
193(C2×D3)⋊C4—21no—
194(C2×D3)⋊C4—21no—
195C4⋊Dic3—21no—
196C4⋊Dic3—62no—
197C4⋊Dic3—61no—
198C4⋊Dic3—61no—
199C4⋊Dic3—61no—
200C4⋊Dic3—61no—
201C4⋊Dic3—121no—
202C4⋊Dic3#6711yescharacteristic
203C4⋊Dic3#6711yescharacteristic
204Dic3⋊C4—21no—
205Dic3⋊C4—21no—
206Dic3⋊C4—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—62no—
210C2×C3⋊C8—61no—
211C2×C3⋊C8—61no—
212Dic12—122no—
213#204—122no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32#12011yescharacteristic
218C2×C3×D3—41no—
219C2×C3×D3—41no—
220C2×C3×D3—41no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—41no—
225C3⋊Dic3—81no—
226C3×Dic3—41no—
227C3×Dic3—41no—
228C3×Dic3—41no—
229C2×Q16—181no—
230C2×SD16—181no—
231C4⋊Q8—91no—
232#125—31no—
233#124—31no—
234#124D3211yescharacteristic
235#123—61no—
236#123—61no—
237C22⋊Q8—61no—
238C22⋊Q8—61no—
239#121—61no—
240#121—61no—
241#107—181no—
242#106—181no—
243#103—181no—
244#102—181no—
245#97—31no—
246#97—31no—
247#97—91no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—41no—
251C23×C3—61no—
252C22×D3—21no—
253C3×Q8—41no—
254C3×Q8—41no—
255C3×Q8—42no—
256C3×Q8—121no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—42no—
260C3×D4—121no—
261C3×D4—241no—
262C3×D4—241no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—41no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—241no—
273C2×C3×C4—241no—
274C2×C3×C4C2×D1211yescharacteristic
275C2×C3×C4D3×D411yescharacteristic
276C2×C3×C4D3×D411yescharacteristic
277C2×C3×C4C2×#6711yescharacteristic
278C2×C3×C4C2×#6711yescharacteristic
279C2×C3×C4C2×#6711yescharacteristic
280#67—81no—
281#67—81no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—81no—
286C2×Dic3—122no—
287C2×Dic3—121no—
288C2×Dic3—121no—
289D12—41no—
290C4×D3—81no—
291Dic6—41no—
292Dic6—121no—
293Dic6—121no—
294Dic6—121no—
295Dic6—121no—
296Dic6—241no—
297Dic6—241no—
298C3×C8—62no—
299C3×C8—62no—
300C3⋊C8—22no—
301C3⋊C8—22no—
302C3⋊C8—122no—
303C3⋊C8—121no—
304C3⋊C8—121no—
305C2×C32—81no—
306C2×C32#45311yescharacteristic
307C2×C32#45511yescharacteristic
308C2×C32#45611yescharacteristic
309C3×D3—81no—
310C2×Q8—31no—
311C2×Q8—181no—
312C2×Q8#62611yescharacteristic
313C2×D4—31no—
314C2×D4—61no—
315C2×D4—61no—
316C2×D4#62611yescharacteristic
317C22×C4—61no—
318C22×C4—61no—
319Q16—362no—
320SD16—362no—
321C2×C8—61no—
322C2×C8—61no—
323C2×C8—181no—
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—181no—
331C4⋊C4—181no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—61no—
338C42C2×D3211yescharacteristic
339C22×C3—21no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3#88211yescharacteristic
353C22×C3#95311yescharacteristic
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C3×C4—22no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—81no—
369C3×C4—121no—
370C3×C4—121no—
371C3×C4—121no—
372Dic3—41no—
373Dic3—41no—
374Dic3—41no—
375Dic3—242no—
376Dic3—241no—
377Dic3—241no—
378C32#166111yescharacteristic, complemented, Sylow
379C23—21no—
380C23—61no—
381Q8—41no—
382Q8—121no—
383Q8—361no—
384Q8—361no—
385D4—41no—
386D4—121no—
387D4—241no—
388D4—241no—
389C2×C4—21no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—241no—
394C2×C4—241no—
395C2×C4—361no—
396C2×C4C2×#62611yescharacteristic
397C2×C4D3×#6711yescharacteristic
398C2×C4D3×#6711yescharacteristic
399C8—62no—
400C8—62no—
401C8—361no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—81no—
406C2×C3—81no—
407C2×C3—161no—
408C2×C3—241no—
409C2×C3#474611yescharacteristic
410C2×C3#482211yescharacteristic
411C2×C3#482611yescharacteristic
412C2×C3#524111yescharacteristic
413C2×C3#524211yescharacteristic
414C2×C3#524611yescharacteristic
415D3—81no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22#6397711yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—41no—
427C4—121no—
428C4—121no—
429C4—121no—
430C4—721no—
431C3—21no—
432C3#6835711yescharacteristic, complemented
433C3#7195311yescharacteristic, complemented
434C2—81no—
435C2—241no—
436C2#1059151311yescharacteristic
437C2#1059166611yescharacteristic
438C2#1059168911yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×#124C22
socleC22×C32#120
prime radicalC3×#5684C2
hypercenter#124D32
hypocenterC32#1661
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #10592754C2semidirect
3. #10590372C2semidirect
4. #10590306C2semidirect
5. C3×#4540C2semidirect
6. #10589088C2semidirect
7. #10589086C2semidirect
41. #4540D3semidirect
378. C32#1661semidirect
432. C3#68357semidirect
433. C3#71953semidirect

Maximal subgroups

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

subgroupindex
C3×#56842
#105927542
#105903722
#105903062
C3×#45402
#105890882
#105890862
#719533
#683573