Finite Group Databasebeta

← #11780657

Subgroups — #11780657

Order 1152.

526 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 72 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#10596155C211yesmaximal, characteristic
2C3×#5055C211yesmaximal, characteristic, complemented
3#10590374C211yesmaximal, characteristic, complemented
4#10590305C211yesmaximal, characteristic, complemented
5C3×#4540C211yesmaximal, characteristic, complemented
6#10589089C211yesmaximal, characteristic, complemented
7#10589087C211yesmaximal, characteristic, complemented
8#71245—31nomaximal
9#68357—31nomaximal
10C32×#124C2211yescharacteristic
11#64155—21no—
12#64148—21no—
13#64095—21no—
14#64093—21no—
15#64092—21no—
16#64087—21no—
17#64084C2211yescharacteristic
18#64079C2211yescharacteristic
19C3×#869C2211yescharacteristic
20C2×#3719—21no—
21C2×#3717—21no—
22C3×#835—21no—
23C3×#832—21no—
24C3×#803C2211yescharacteristic
25C3×#803C2211yescharacteristic
26C3×#803C2211yescharacteristic
27#63575—21no—
28#63573—21no—
29#5684—31no—
30#5684—31no—
31#5684—61no—
32C3×#489—31no—
33#5086—31no—
34#5081—31no—
35#5055D311yescharacteristic, complemented
36#4715—31no—
37#4702—31no—
38C3×#329—31no—
39#4540D311yescharacteristic, complemented
40#4465—31no—
41#4465—31no—
42#4464—31no—
43#4464—31no—
44C2×C32×Q8D411yescharacteristic
45C2×C32×D4D411yescharacteristic
46C2×#638—21no—
47C2×#638—21no—
48C22×C3⋊Dic3—21no—
49C2×C4⋊(C3⋊D3)D411yescharacteristic
50C2×C4×C3⋊D3—21no—
51C2×#634D411yescharacteristic
52C2×C3×Dic6—21no—
53C32×C22⋊C4—21no—
54C32×C22⋊C4—21no—
55C32×C42C2311yescharacteristic
56#3757—21no—
57#3752—21no—
58#3752—21no—
59#3752—21no—
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)D411yescharacteristic
62C4⋊(C3⋊Dic3)D411yescharacteristic
63#3750—21no—
64#3750—21no—
65#3750—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71#3719—42no—
72#3717—42no—
73#1661—91noSylow
74C3×C4⋊Q8—31no—
75C3×#124—21no—
76C3×#124C2×D311yescharacteristic
77C3×#124C2×D311yescharacteristic
78#946—62no—
79#946—61no—
80#946—61no—
81C2×#216—61no—
82C2×#215—61no—
83#939—62no—
84#939—61no—
85#939—61no—
86C2×Dic12—61no—
87C2×#204—61no—
88#886—61no—
89#886—61no—
90#886—62no—
91#884—61no—
92#884—61no—
93#884—62no—
94#883—61no—
95#883—61no—
96#883—62no—
97#878—61no—
98#878—61no—
99#878—62no—
100#875—31no—
101#875—31no—
102#875—61no—
103#870—31no—
104#870—31no—
105#870—61no—
106#869C2×D311yescharacteristic
107C3×#103—61no—
108C3×#102—61no—
109C3×#97—31no—
110C3×#97—31no—
111C3×#97—31no—
112#835—21no—
113#832—21no—
114#818—61no—
115#817—61no—
116#808—61no—
117#807—61no—
118#803C2×D311yescharacteristic
119#803C2×D311yescharacteristic
120#803C2×D311yescharacteristic
121C23×C32—21no—
122C22×C3⋊D3—21no—
123C32×Q8—41no—
124C32×D4—41no—
125C2×C32×C4—21no—
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129#638—81no—
130#638—81no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—81no—
135C4⋊(C3⋊D3)—41no—
136C4×C3⋊D3—81no—
137#634—41no—
138C2×C3×Dic3—41no—
139C3×Dic6—41no—
140C3×Dic6—41no—
141C3×C3⋊C8—22no—
142C3×C3⋊C8—22no—
143C3×C3⋊C8—41no—
144#562—91no—
145#501—91no—
146#497—91no—
147#489—31no—
148#330—91no—
149#329—31no—
150#329—91no—
151C2×C3×Q8—21no—
152C2×C3×Q8—61no—
153C2×C3×Q8D1211yescharacteristic
154C2×C3×Q8#6711yescharacteristic
155C2×C3×D4—21no—
156C2×C3×D4D1211yescharacteristic
157C2×C3×D4#6711yescharacteristic
158C2×#67—62no—
159C2×#67—61no—
160C2×#67—61no—
161C2×#67—62no—
162C2×#67—61no—
163C2×#67—61no—
164C22×Dic3—62no—
165C22×Dic3—61no—
166C22×Dic3—61no—
167C2×D12—31no—
168C2×D12—31no—
169C2×D12—61no—
170C2×C4×D3—62no—
171C2×C4×D3—61no—
172C2×C4×D3—61no—
173C2×Dic6—21no—
174C2×Dic6—31no—
175C2×Dic6—31no—
176C2×Dic6—61no—
177C2×C3×C8—61no—
178C2×C3×C8—61no—
179C2×C3×C8—61no—
180C3×C4⋊C4—61no—
181C3×C4⋊C4—61no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—21no—
185C3×C22⋊C4—21no—
186C3×C22⋊C4—21no—
187C3×C22⋊C4—21no—
188C3×C42—21no—
189C3×C42C22×D311yescharacteristic
190C3×C42C22×D311yescharacteristic
191#217—62no—
192#217—61no—
193#217—61no—
194#216—122no—
195#215—122no—
196(C2×D3)⋊C4—62no—
197(C2×D3)⋊C4—62no—
198(C2×D3)⋊C4—62no—
199(C2×D3)⋊C4—61no—
200(C2×D3)⋊C4—61no—
201(C2×D3)⋊C4—61no—
202(C2×D3)⋊C4—61no—
203(C2×D3)⋊C4—61no—
204(C2×D3)⋊C4—61no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—31no—
208C4⋊Dic3—31no—
209C4⋊Dic3—31no—
210C4⋊Dic3—31no—
211C4⋊Dic3—61no—
212C4⋊Dic3—61no—
213C4⋊Dic3—62no—
214C4⋊Dic3—61no—
215C4⋊Dic3—61no—
216Dic3⋊C4—62no—
217Dic3⋊C4—62no—
218Dic3⋊C4—62no—
219Dic3⋊C4—61no—
220Dic3⋊C4—61no—
221Dic3⋊C4—61no—
222Dic3⋊C4—61no—
223Dic3⋊C4—61no—
224Dic3⋊C4—61no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—21no—
228Dic12—122no—
229#204—122no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32#12011yescharacteristic
234C2×C3⋊D3—41no—
235C2×C3⋊D3—41no—
236C2×C3⋊D3—41no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—41no—
241C3⋊Dic3—41no—
242C3⋊Dic3—41no—
243C3⋊Dic3—41no—
244C3×Dic3—81no—
245C2×Q16—181no—
246C2×SD16—181no—
247C4⋊Q8—31no—
248#125—91no—
249#124—91no—
250#124D3211yescharacteristic
251#123—181no—
252#123—181no—
253C22⋊Q8—181no—
254C22⋊Q8—181no—
255#121—181no—
256#121—181no—
257#107—181no—
258#106—181no—
259#103—61no—
260#102—61no—
261#97—31no—
262#97—31no—
263#97—31no—
264C23×C3—22no—
265C23×C3—21no—
266C23×C3—21no—
267C22×D3—62no—
268C22×D3—61no—
269C22×D3—61no—
270C3×Q8—41no—
271C3×Q8—41no—
272C3×Q8—81no—
273C3×Q8—121no—
274C3×Q8—121no—
275C3×D4—41no—
276C3×D4—41no—
277C3×D4—81no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—121no—
285C2×C3×C4C2×D1211yescharacteristic
286C2×C3×C4D3×D411yescharacteristic
287C2×C3×C4D3×D411yescharacteristic
288C2×C3×C4D3×D411yescharacteristic
289C2×C3×C4D3×D411yescharacteristic
290C2×C3×C4C2×#6711yescharacteristic
291#67—242no—
292#67—242no—
293#67—241no—
294#67—241no—
295#67—241no—
296#67—241no—
297C2×Dic3—41no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300C2×Dic3—62no—
301C2×Dic3—61no—
302C2×Dic3—61no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—242no—
308C2×Dic3—241no—
309C2×Dic3—241no—
310D12—121no—
311D12—121no—
312D12—241no—
313C4×D3—242no—
314C4×D3—241no—
315C4×D3—241no—
316Dic6—41no—
317Dic6—41no—
318Dic6—121no—
319Dic6—121no—
320Dic6—241no—
321C3×C8—62no—
322C3×C8—62no—
323C3×C8—121no—
324C3⋊C8—22no—
325C3⋊C8—22no—
326C3⋊C8—41no—
327C2×C32—81no—
328C2×C32#45311yescharacteristic
329C2×C32#45511yescharacteristic
330C2×C32#45611yescharacteristic
331C3⋊D3—81no—
332C2×Q8—61no—
333C2×Q8—91no—
334C2×Q8#62611yescharacteristic
335C2×D4—91no—
336C2×D4—181no—
337C2×D4—181no—
338C2×D4#62611yescharacteristic
339C22×C4—181no—
340C22×C4—181no—
341Q16—362no—
342SD16—362no—
343C2×C8—61no—
344C2×C8—61no—
345C2×C8—61no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—91no—
349C4⋊C4—91no—
350C4⋊C4—181no—
351C4⋊C4—181no—
352C4⋊C4—181no—
353C4⋊C4—181no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—181no—
357C22⋊C4—181no—
358C22⋊C4—181no—
359C22⋊C4—181no—
360C42C2×D3211yescharacteristic
361C22×C3—21no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3#88211yescharacteristic
372C22×C3#93711yescharacteristic
373C2×D3—122no—
374C2×D3—122no—
375C2×D3—122no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—121no—
379C2×D3—121no—
380C2×D3—121no—
381C2×D3—121no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—42no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—241no—
395Dic3—81no—
396Dic3—122no—
397Dic3—122no—
398Dic3—122no—
399Dic3—121no—
400Dic3—121no—
401Dic3—121no—
402Dic3—121no—
403Dic3—121no—
404Dic3—121no—
405C32#166111yescharacteristic, complemented, Sylow
406C23—21no—
407C23—181no—
408Q8—41no—
409Q8—121no—
410Q8—121no—
411Q8—361no—
412D4—41no—
413D4—361no—
414D4—721no—
415D4—721no—
416C2×C4—21no—
417C2×C4—121no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4—721no—
422C2×C4—721no—
423C2×C4C2×#62611yescharacteristic
424C2×C4#381111yescharacteristic
425C2×C4#381111yescharacteristic
426C8—62no—
427C8—62no—
428C8—121no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—82no—
433C2×C3—81no—
434C2×C3—81no—
435C2×C3#474611yescharacteristic
436C2×C3#482211yescharacteristic
437C2×C3#482611yescharacteristic
438C2×C3#506111yescharacteristic
439C2×C3#519811yescharacteristic
440C2×C3#520511yescharacteristic
441D3—242no—
442D3—241no—
443D3—241no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—361no—
448C22—361no—
449C22—361no—
450C22#6398111yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—41no—
455C4—241no—
456C4—361no—
457C4—361no—
458C4—361no—
459C3—21no—
460C3#6835711yescharacteristic, complemented
461C3#7124511yescharacteristic, complemented
462C2—81no—
463C2—721no—
464C2#1059151411yescharacteristic
465C2#1059166711yescharacteristic
466C2#1059166811yescharacteristic
subgroupG/H
centerC22#63981
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×#124C22
socleC22×C32#120
prime radical#10596155C2
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. C3×#5055C2semidirect
3. #10590374C2semidirect
4. #10590305C2semidirect
5. C3×#4540C2semidirect
6. #10589089C2semidirect
7. #10589087C2semidirect
35. #5055D3semidirect
39. #4540D3semidirect
405. C32#1661semidirect
460. C3#68357semidirect
461. C3#71245semidirect

Maximal subgroups

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

subgroupindex
#105961552
C3×#50552
#105903742
#105903052
C3×#45402
#105890892
#105890872
#712453
#683573