Finite Group Databasebeta

← #11779376

Subgroups — #11779376

Order 1152.

566 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 90 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#10593900C211yesmaximal, characteristic
2#10592783C211yesmaximal, characteristic, complemented
3C3×#5091C211yesmaximal, characteristic, complemented
4C3×#5085C211yesmaximal, characteristic, complemented
5#10589090C211yesmaximal, characteristic, complemented
6#10589084C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic, complemented
8#71271—31nomaximal
9#71160—31nomaximal
10C2×#3796—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#215—22no—
14C3×#874C2211yescharacteristic
15C3×C4×Dic6C2211yescharacteristic
16#63916—21no—
17#63911—21no—
18Dic3×D12—21no—
19#63886—22no—
20#63636—21no—
21#63635—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C3×#810—21no—
24C3×#807—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5718—31no—
28#5577—31no—
29C3×#492—31no—
30C3×#477—31no—
31#5091D311yescharacteristic, complemented
32#5085D311yescharacteristic, complemented
33#5084—31no—
34#5084—31no—
35#5084—61no—
36#4544—31no—
37#4535—31no—
38#4485—31no—
39#4483—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×Q8—21no—
43C2×C4⋊(C3⋊D3)D411yescharacteristic
44C2×C4×C3⋊D3—21no—
45C2×C3×D12—21no—
46C2×C3×D12—21no—
47C2×C3×D12D412yes—
48C2×C3×Dic6D411yescharacteristic
49C2×#626—22no—
50C2×#626—22no—
51C2×D3×Dic3—22no—
52#3796—42no—
53#3796—42no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C42C2311yescharacteristic
57#3752—21no—
58C4⋊(C3⋊Dic3)D411yescharacteristic
59C2×C3⋊(C3⋊C8)—21no—
60C3×#215—44no—
61C3×C4⋊Dic3D411yescharacteristic
62C3×Dic3⋊C4—21no—
63C3×C4×Dic3—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66(C2×D3)⋊Dic3—22no—
67#1631—91noSylow
68C2×#239—61no—
69C2×#237—61no—
70C2×C3×SD16—62no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×#127—31no—
75C3×C4×Q8—31no—
76C2×#215—22no—
77#940—61no—
78#939—62no—
79#938—61no—
80D4×Dic3—61no—
81#895—61no—
82#895—62no—
83#874—61no—
84#874C2×D311yescharacteristic
85C4×D12—31no—
86C4×D12—31no—
87C4×D12—61no—
88C4×D12—61no—
89C4×Dic6C2×D311yescharacteristic
90C3×#107—61no—
91C3×C4⋊C8—31no—
92C3×#103—61no—
93C3×C4×C8—31no—
94#810—21no—
95#809—61no—
96#809—61no—
97#809—62no—
98#808—61no—
99#808—61no—
100#808—62no—
101#807—21no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—31no—
104C4⋊(C3⋊C8)—61no—
105C4⋊(C3⋊C8)C2×D311yescharacteristic
106C4×C3⋊C8C2×D311yescharacteristic
107C22×C3⋊D3—21no—
108C22×C3×D3—22no—
109C32×Q8—42no—
110C2×C32×C4—41no—
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C3⋊Dic3—21no—
115C4⋊(C3⋊D3)—21no—
116C4⋊(C3⋊D3)—21no—
117C4×C3⋊D3—41no—
118C4×C3⋊D3—41no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C3×D12—42no—
122C3×D12—41no—
123C3×D12—41no—
124C3×D12—41no—
125C3×D12—41no—
126C3×Dic6—21no—
127C3×Dic6SD1612yescomplemented
128#626—44no—
129#626—82no—
130D3×Dic3—44no—
131C3⋊(C3⋊C8)—41no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—41no—
134#550—91no—
135#492—31no—
136#477—31no—
137#476—91no—
138#331—91no—
139Q8⋊C8—91no—
140D4⋊C8—91no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—31no—
144C2×C3×Q8—41no—
145C2×C3×D4—32no—
146C2×C3×D4—61no—
147C2×C3×D4—61no—
148C2×#67—62no—
149C2×#67—62no—
150C22×Dic3—62no—
151#239—122no—
152#239—122no—
153#237—122no—
154#237—122no—
155C2×D12—21no—
156C2×D12—21no—
157C2×D12—31no—
158C2×D12—31no—
159C2×D12—61no—
160C2×D12—62no—
161C2×D12—62no—
162C2×D12#6712yes—
163C2×C4×D3—62no—
164C2×C4×D3—61no—
165C2×C4×D3—61no—
166C2×C4×D3—121no—
167C2×Dic6#6711yescharacteristic
168C3×SD16—124no—
169C2×C3×C8—61no—
170C2×C3×C8—61no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—31no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—61no—
179C3×C42—21no—
180C3×C42—61no—
181C3×C42C22×D311yescharacteristic
182C3×C42C22×D311yescharacteristic
183#217—62no—
184#215—44no—
185(C2×D3)⋊C4—62no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—121no—
189C4⋊Dic3—31no—
190C4⋊Dic3—31no—
191C4⋊Dic3—61no—
192C4⋊Dic3#6711yescharacteristic
193Dic3⋊C4—21no—
194C4×Dic3—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—62no—
198C2×C3⋊C8—61no—
199C2×C3⋊C8—61no—
200C22×C32#12011yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3×D3—42no—
205C2×C3×D3—42no—
206C2×C3×D3—42no—
207C32×C4—21no—
208C32×C4—81no—
209C32×C4C2×SD1611yescharacteristic
210C32×C4#13511yescharacteristic
211C32×C4AGL(1,ℤ/8)11yescharacteristic
212C32×C4#13711yescharacteristic
213C3⋊Dic3—41no—
214C3×Dic3—22no—
215C3×Dic3—41no—
216C2×#41—181no—
217C2×SD16—62no—
218C4⋊Q8D3211yescharacteristic
219#127—31no—
220#127—181no—
221#121—181no—
222#121—182no—
223C4×Q8—31no—
224C4×D4—91no—
225C4×D4—181no—
226#107—61no—
227#106—181no—
228C4⋊C8—31no—
229C4⋊C8—91no—
230#103—61no—
231#102—181no—
232C4×C8—31no—
233C23×C3—62no—
234C22×D3—22no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—121no—
238C3×Q8—32no—
239C3×Q8—42no—
240C3×Q8—42no—
241C3×Q8—61no—
242C3×Q8—82no—
243C3×D4—122no—
244C3×D4—121no—
245C3×D4—121no—
246C3×D4—121no—
247C3×D4—121no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4D3×D411yescharacteristic
257C2×C3×C4D3×D411yescharacteristic
258C2×C3×C4D3×D411yescharacteristic
259C2×C3×C4D3×D411yescharacteristic
260C2×C3×C4C2×#6711yescharacteristic
261C2×C3×C4C2×#6711yescharacteristic
262#67—124no—
263#67—242no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—121no—
269C2×Dic3—124no—
270D12—42no—
271D12—41no—
272D12—41no—
273D12—41no—
274D12—41no—
275D12—62no—
276D12—61no—
277D12—61no—
278D12—61no—
279D12—61no—
280D12—121no—
281D12—124no—
282D12—242no—
283C4×D3—124no—
284C4×D3—121no—
285C4×D3—121no—
286C4×D3—121no—
287C4×D3—121no—
288C4×D3—241no—
289C4×D3—241no—
290Dic6—21no—
291Dic6#21512yescomplemented
292C3×C8—62no—
293C3×C8—121no—
294C3⋊C8—22no—
295C3⋊C8—41no—
296C3⋊C8—122no—
297C3⋊C8—121no—
298C3⋊C8—121no—
299C2×C32#45011yescharacteristic
300C2×C32#45211yescharacteristic
301C2×C32#45311yescharacteristic
302C3⋊D3—81no—
303C3×D3—82no—
304#41—362no—
305#41—362no—
306C2×Q8—21no—
307C2×Q8—31no—
308C2×D4—32no—
309C2×D4—61no—
310C2×D4—61no—
311C2×D4—91no—
312C2×D4—182no—
313C2×D4—182no—
314C22×C4—182no—
315C22×C4—181no—
316SD16—124no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—181no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—31no—
323C4⋊C4—61no—
324C4⋊C4—91no—
325C22⋊C4—182no—
326C22⋊C4—181no—
327C42—61no—
328C42C2×D3211yescharacteristic
329C22×C3—21no—
330C22×C3—122no—
331C22×C3—122no—
332C22×C3—122no—
333C22×C3#93711yescharacteristic
334C22×C3#93711yescharacteristic
335C2×D3—42no—
336C2×D3—42no—
337C2×D3—42no—
338C2×D3—122no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—121no—
345C2×D3—241no—
346C2×D3—241no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—41no—
354C3×C4—62no—
355C3×C4—82no—
356C3×C4—81no—
357C3×C4—81no—
358C3×C4—121no—
359C3×C4D3×SD1611yescharacteristic
360C3×C4#91411yescharacteristic
361C3×C4#91511yescharacteristic
362C3×C4#91611yescharacteristic
363C3×C4#91811yescharacteristic
364C3×C4#91911yescharacteristic
365C3×C4C2×#21511yescharacteristic
366C3×C4#94911yescharacteristic
367Dic3—22no—
368Dic3—41no—
369Dic3—121no—
370Dic3—121no—
371Dic3—241no—
372C32#163111yescharacteristic, complemented, Sylow
373C23—62no—
374C23—181no—
375Q8—32no—
376Q8—42no—
377Q8—61no—
378D4—122no—
379D4—121no—
380D4—121no—
381D4—121no—
382D4—121no—
383D4—181no—
384D4—181no—
385D4—364no—
386D4—722no—
387C2×C4—41no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—181no—
391C2×C4—364no—
392C2×C4—361no—
393C2×C4—361no—
394C2×C4D3×#6711yescharacteristic
395C2×C4D3×#6711yescharacteristic
396C2×C4#381111yescharacteristic
397C8—62no—
398C8—121no—
399C8—361no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—242no—
404C2×C3#504611yescharacteristic
405C2×C3#507711yescharacteristic
406C2×C3#516911yescharacteristic
407C2×C3#517311yescharacteristic
408C2×C3#518711yescharacteristic
409C2×C3#519811yescharacteristic
410D3—82no—
411D3—242no—
412D3—241no—
413D3—241no—
414C22—122no—
415C22—122no—
416C22—122no—
417C22—361no—
418C22—361no—
419C22—361no—
420C22#6398311yescharacteristic
421C4—21no—
422C4—62no—
423C4—81no—
424C4—121no—
425C4—361no—
426C4D3×#21511yescharacteristic
427C4#6393911yescharacteristic
428C4#6395011yescharacteristic
429C4#6395111yescharacteristic
430C3—21no—
431C3#7116011yescharacteristic, complemented
432C3#7127111yescharacteristic, complemented
433C2—242no—
434C2—721no—
435C2#1059152311yescharacteristic
436C2#1059165111yescharacteristic
437C2#1059165611yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radical#10593900C2
hypercenterC4⋊Q8D32
hypocenterC32#1631
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. #10592783C2semidirect
3. C3×#5091C2semidirect
4. C3×#5085C2semidirect
5. #10589090C2semidirect
6. #10589084C2semidirect
7. #10589082C2semidirect
31. #5091D3semidirect
32. #5085D3semidirect
127. C3×Dic6SD16semidirect
291. Dic6#215semidirect
372. C32#1631semidirect
431. C3#71160semidirect
432. C3#71271semidirect

Maximal subgroups

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

subgroupindex
#105939002
#105927832
C3×#50912
C3×#50852
#105890902
#105890842
#105890822
#712713
#711603