Finite Group Databasebeta

← #11780642

Subgroups — #11780642

Order 1152.

461 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 70 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#10593985C211yesmaximal, characteristic
2#10592780C211yesmaximal, characteristic, complemented
3C3×#5089C211yesmaximal, characteristic, complemented
4C3×#5086C211yesmaximal, characteristic, complemented
5#10589089C211yesmaximal, characteristic, complemented
6#10589086C211yesmaximal, characteristic, complemented
7#10589086C211yesmaximal, characteristic, complemented
8#71331—31nomaximal
9#71245—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64079C2211yescharacteristic
12C2×C3×#216—21no—
13C2×C3×#215—21no—
14C3×#875C2211yescharacteristic
15C3×#875C2211yescharacteristic
16#63891—21no—
17#63891—21no—
18#63887—21no—
19#63887—21no—
20#63885—21no—
21#63885—21no—
22#63635—21no—
23#63634—21no—
24#63630C2211yescharacteristic
25C3×#810—21no—
26C3×#809—21no—
27C3×#803C2211yescharacteristic
28C3×#803C2211yescharacteristic
29#5684—31no—
30#5684—31no—
31C3×#497—31no—
32C3×#489—31no—
33#5089D311yescharacteristic, complemented
34#5086D311yescharacteristic, complemented
35#5081—31no—
36#5081—31no—
37#5081—61no—
38#4540—31no—
39#4540—31no—
40#4465—31no—
41#4465—31no—
42#4464—31no—
43#4464—31no—
44C2×C32×Q8—21no—
45C2×C3×D12D411yescharacteristic
46C2×C3×D12D411yescharacteristic
47C2×C3×Dic6D411yescharacteristic
48C2×C3×Dic6D411yescharacteristic
49C2×#625—21no—
50C2×#625—21no—
51C2×D3×Dic3—21no—
52C2×D3×Dic3—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4—21no—
55C32×C42C2311yescharacteristic
56C4⋊(C3⋊Dic3)D411yescharacteristic
57C4⋊(C3⋊Dic3)D411yescharacteristic
58#3750—21no—
59#3750—21no—
60C2×C3⋊(C3⋊C8)—21no—
61C3×#216—42no—
62C3×#215—42no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×(C2×D3)⋊C4—21no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8—21no—
69Dic3⋊Dic3—21no—
70Dic3⋊Dic3—21no—
71(C2×D3)⋊Dic3—21no—
72(C2×D3)⋊Dic3—21no—
73#1661—91noSylow
74C2×C3×Q16—61no—
75C2×C3×SD16—61no—
76C3×C4⋊Q8—21no—
77C3×C4⋊Q8C2×D311yescharacteristic
78C3×C4⋊Q8C2×D311yescharacteristic
79C3×#124—31no—
80C3×#124—31no—
81#946—61no—
82#946—61no—
83C2×#216—21no—
84C2×#215—21no—
85#939—61no—
86#939—61no—
87#886—61no—
88#886—61no—
89#884—61no—
90#884—61no—
91#883—61no—
92#883—61no—
93#878—61no—
94#878—61no—
95#875C2×D311yescharacteristic
96#875C2×D311yescharacteristic
97#870—31no—
98#870—31no—
99#870—61no—
100C3×#103—61no—
101C3×#102—61no—
102C3×#97—31no—
103C3×#97—31no—
104#810—21no—
105#809—21no—
106#808—61no—
107#808—61no—
108#808—62no—
109#807—61no—
110#807—61no—
111#807—62no—
112#803—31no—
113#803—31no—
114#803—61no—
115#803C2×D311yescharacteristic
116#803C2×D311yescharacteristic
117C22×C3×D3—21no—
118C22×C3×D3—21no—
119C32×Q8—41no—
120C32×Q8—41no—
121C2×C32×C4—41no—
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—21no—
129C3×D12—41no—
130C3×D12—41no—
131C3×Dic6—41no—
132C3×Dic6—41no—
133#625—81no—
134#625—81no—
135D3×Dic3—81no—
136D3×Dic3—81no—
137C3⋊(C3⋊C8)—22no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—41no—
140#562—91no—
141#501—91no—
142#497—31no—
143#489—31no—
144#330—91no—
145#329—91no—
146#329—91no—
147C2×C3×Q8—21no—
148C2×C3×Q8—21no—
149C2×C3×Q8—31no—
150C2×C3×Q8—31no—
151C2×C3×Q8—41no—
152C2×C3×D4—31no—
153C2×C3×D4—31no—
154C2×#67—61no—
155C2×#67—61no—
156C2×#67—61no—
157C2×#67—61no—
158C22×Dic3—61no—
159C22×Dic3—61no—
160C2×D12#6711yescharacteristic
161C2×D12#6711yescharacteristic
162C2×C4×D3—61no—
163C2×C4×D3—61no—
164C2×Dic6#6711yescharacteristic
165C2×Dic6#6711yescharacteristic
166C3×Q16—122no—
167C3×SD16—122no—
168C2×C3×C8—61no—
169C2×C3×C8—61no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—41no—
176C3×C22⋊C4—61no—
177C3×C22⋊C4—61no—
178C3×C22⋊C4—61no—
179C3×C22⋊C4—61no—
180C3×C42—21no—
181C3×C42C22×D311yescharacteristic
182C3×C42C22×D311yescharacteristic
183#217—61no—
184#217—61no—
185#216—42no—
186#215—42no—
187(C2×D3)⋊C4—21no—
188(C2×D3)⋊C4—21no—
189(C2×D3)⋊C4—21no—
190(C2×D3)⋊C4—21no—
191(C2×D3)⋊C4—61no—
192(C2×D3)⋊C4—61no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197C4⋊Dic3—61no—
198C4⋊Dic3—61no—
199C4⋊Dic3—61no—
200C4⋊Dic3—61no—
201Dic3⋊C4—61no—
202Dic3⋊C4—61no—
203Dic3⋊C4—61no—
204Dic3⋊C4—61no—
205Dic3⋊C4—61no—
206Dic3⋊C4—61no—
207Dic3⋊C4—121no—
208Dic3⋊C4—121no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—62no—
212C2×C3⋊C8—61no—
213C2×C3⋊C8—61no—
214C22×C32#12011yescharacteristic
215C2×C3×D3—41no—
216C2×C3×D3—41no—
217C2×C3×D3—41no—
218C2×C3×D3—41no—
219C2×C3×D3—41no—
220C2×C3×D3—41no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—81no—
225C3⋊Dic3—41no—
226C3⋊Dic3—41no—
227C3×Dic3—41no—
228C3×Dic3—41no—
229C2×Q16—61no—
230C2×SD16—61no—
231C4⋊Q8D3211yescharacteristic
232#125—91no—
233#124—31no—
234#124—31no—
235#123—181no—
236#123—181no—
237C22⋊Q8—181no—
238C22⋊Q8—181no—
239#121—181no—
240#121—181no—
241#107—181no—
242#106—181no—
243#103—61no—
244#102—61no—
245#97—31no—
246#97—31no—
247#97—91no—
248C23×C3—61no—
249C23×C3—61no—
250C22×D3—21no—
251C22×D3—21no—
252C3×Q8—41no—
253C3×Q8—41no—
254C3×Q8—41no—
255C3×Q8—41no—
256C3×Q8—81no—
257C3×Q8—81no—
258C3×Q8—121no—
259C3×Q8—121no—
260C3×D4—121no—
261C3×D4—121no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4D3×D411yescharacteristic
271C2×C3×C4D3×D411yescharacteristic
272C2×C3×C4D3×D411yescharacteristic
273C2×C3×C4D3×D411yescharacteristic
274C2×C3×C4C2×#6711yescharacteristic
275C2×C3×C4C2×#6711yescharacteristic
276#67—241no—
277#67—241no—
278#67—241no—
279#67—241no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—121no—
287C2×Dic3—121no—
288C2×Dic3—241no—
289C2×Dic3—241no—
290D12—41no—
291D12—41no—
292C4×D3—241no—
293C4×D3—241no—
294Dic6—41no—
295Dic6—41no—
296C3×C8—62no—
297C3×C8—121no—
298C3⋊C8—22no—
299C3⋊C8—41no—
300C3⋊C8—64no—
301C3⋊C8—62no—
302C3⋊C8—62no—
303C2×C32#45311yescharacteristic
304C2×C32#45511yescharacteristic
305C2×C32#45611yescharacteristic
306C3×D3—81no—
307C3×D3—81no—
308C2×Q8—21no—
309C2×Q8—31no—
310C2×Q8—31no—
311C2×D4—31no—
312C2×D4—31no—
313C2×D4—181no—
314C2×D4—181no—
315C22×C4—181no—
316C22×C4—181no—
317Q16—122no—
318SD16—122no—
319C2×C8—61no—
320C2×C8—61no—
321C2×C8—181no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—91no—
325C4⋊C4—91no—
326C4⋊C4—181no—
327C4⋊C4—181no—
328C4⋊C4—181no—
329C4⋊C4—181no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—181no—
335C22⋊C4—181no—
336C42C2×D3211yescharacteristic
337C22×C3—21no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3—121no—
341C22×C3—121no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3#93711yescharacteristic
345C22×C3#93711yescharacteristic
346C2×D3—41no—
347C2×D3—41no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—41no—
352C3×C4—22no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—82no—
362C3×C4—81no—
363C3×C4—81no—
364C3×C4—121no—
365C3×C4—121no—
366Dic3—41no—
367Dic3—41no—
368Dic3—121no—
369Dic3—121no—
370Dic3—121no—
371Dic3—121no—
372Dic3—241no—
373Dic3—241no—
374C32#166111yescharacteristic, complemented, Sylow
375C23—61no—
376C23—61no—
377Q8—41no—
378Q8—41no—
379Q8—121no—
380Q8—121no—
381D4—121no—
382D4—121no—
383D4—721no—
384D4—721no—
385C2×C4—41no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—721no—
391C2×C4—721no—
392C2×C4D3×#6711yescharacteristic
393C2×C4D3×#6711yescharacteristic
394C2×C4#381111yescharacteristic
395C8—62no—
396C8—121no—
397C8—182no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—241no—
402C2×C3—241no—
403C2×C3#506111yescharacteristic
404C2×C3#507711yescharacteristic
405C2×C3#519811yescharacteristic
406C2×C3#520211yescharacteristic
407C2×C3#520511yescharacteristic
408C2×C3#520511yescharacteristic
409D3—81no—
410D3—81no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22#6398311yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—81no—
422C4—121no—
423C4—121no—
424C4—361no—
425C4—361no—
426C3—21no—
427C3#7124511yescharacteristic, complemented
428C3#7133111yescharacteristic, complemented
429C2—241no—
430C2—241no—
431C2#1059152311yescharacteristic
432C2#1059165911yescharacteristic
433C2#1059166011yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radical#10593985C2
hypercenterC4⋊Q8D32
hypocenterC32#1661
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10592780C2semidirect
3. C3×#5089C2semidirect
4. C3×#5086C2semidirect
5. #10589089C2semidirect
6. #10589086C2semidirect
7. #10589086C2semidirect
33. #5089D3semidirect
34. #5086D3semidirect
374. C32#1661semidirect
427. C3#71245semidirect
428. C3#71331semidirect

Maximal subgroups

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

subgroupindex
#105939852
#105927802
C3×#50892
C3×#50862
#105890892
#105890862
#713313
#712453