Finite Group Databasebeta

← #11781686

Subgroups — #11781686

Order 1152.

384 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 86 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#10592550C211yesmaximal, characteristic, complemented
2#10590377C211yesmaximal, characteristic
3#10590369C211yesmaximal, characteristic
4#10590296C211yesmaximal, characteristic, complemented
5C3×#4485C211yesmaximal, characteristic
6C3×#4479C211yesmaximal, characteristic, complemented
7#10589094C211yesmaximal, characteristic, complemented
8#68592—31nomaximal
9#68412—31nomaximal
10#64078C2211yescharacteristic
11C4×#634C2211yescharacteristic
12C3×#874C2211yescharacteristic
13C2×#3716—21no—
14C2×#3716—21no—
15C2×#3716—21no—
16C32×C4⋊C8C2211yescharacteristic
17#63644—21no—
18#63643—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63572—21no—
23#63563—21no—
24#63555—21no—
25#5076—31no—
26#5075—31no—
27#4851—31no—
28#4851—31no—
29#4851—61no—
30#4848—31no—
31#4831—31no—
32#4693—31no—
33#4679—31no—
34C3×#335—31no—
35C3×#331—31no—
36#4486—31no—
37#4485D311yescharacteristic
38#4482—31no—
39#4479D311yescharacteristic, complemented
40C2×#634—21no—
41C2×#634D411yescharacteristic
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C2×C32×C8—21no—
46C32×C42C2311yescharacteristic
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)D411yescharacteristic
50#3750—21no—
51C4×C3⋊Dic3—21no—
52C2×C3×C3⋊C8—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3716—42no—
57#3716—42no—
58#3716—42no—
59#1686—91noSylow
60C3×#127—31no—
61C2×#214—61no—
62C2×#214—61no—
63C2×#214—61no—
64C2×#204—61no—
65C2×#204—61no—
66C2×#204—61no—
67#874C2×D311yescharacteristic
68#869—31no—
69#869—31no—
70#869—61no—
71C4×Dic6—31no—
72C4×Dic6—31no—
73C4×Dic6—61no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—31no—
76C3×C4⋊C8—31no—
77C3×C4⋊C8C2×D311yescharacteristic
78C3×C4⋊C8C2×D311yescharacteristic
79C3×C4×C8—31no—
80#832—61no—
81#821—61no—
82#817—61no—
83#817—61no—
84#817—62no—
85#816—61no—
86#816—61no—
87#816—62no—
88#815—61no—
89#815—61no—
90Dic3⋊C8—61no—
91Dic3⋊C8—61no—
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4⋊(C3⋊C8)C2×D311yescharacteristic
94C4×C3⋊C8C2×D311yescharacteristic
95C22×C3×D3—41no—
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—41no—
102#634—21no—
103#634—21no—
104#634—41no—
105#634—41no—
106C3×D12—41no—
107C3×D12—41no—
108C3×D12—41no—
109C3×D12—41no—
110C3×D12—81no—
111C32×C8—41no—
112C3×C3⋊C8—21no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8—41no—
115C3×C3⋊C8SD1612yescomplemented
116#492—91no—
117#475—91no—
118#460—91no—
119#446—91no—
120#335—31no—
121#332—91no—
122#331—31no—
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×C3×D4—61no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×Dic6—31no—
130C2×Dic6—31no—
131C2×Dic6—61no—
132C2×Dic6—61no—
133C2×Dic6—61no—
134C2×Dic6—121no—
135C2×C3×C8—22no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—31no—
139C2×C3×C8—31no—
140C2×C3×C8—61no—
141C2×C3×C8—61no—
142C3×C42—21no—
143C3×C42C22×D311yescharacteristic
144C3×C42C22×D311yescharacteristic
145#214—122no—
146#214—122no—
147#214—122no—
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150C4⋊Dic3—61no—
151C4⋊Dic3—62no—
152C4⋊Dic3—61no—
153C4⋊Dic3—61no—
154C4⋊Dic3—61no—
155C4⋊Dic3—61no—
156C4⋊Dic3—121no—
157Dic3⋊C4—61no—
158Dic3⋊C4—61no—
159Dic3⋊C4—121no—
160C4×Dic3—61no—
161C4×Dic3—61no—
162C4×Dic3—121no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8D1211yescharacteristic
166C2×C3⋊C8D1211yescharacteristic
167#204—122no—
168#204—122no—
169#204—122no—
170C22×C32#12111yescharacteristic
171C2×C3×D3—81no—
172C2×C3×D3—81no—
173C2×C3×D3—81no—
174C32×C4—21no—
175C32×C4C2×SD1611yescharacteristic
176C32×C4AGL(1,ℤ/8)11yescharacteristic
177C32×C4AGL(1,ℤ/8)11yescharacteristic
178C32×C4#13711yescharacteristic
179C3⋊Dic3—41no—
180C3⋊Dic3—41no—
181C3⋊Dic3—81no—
182C2×SD16—181no—
183C2×SD16—181no—
184C2×SD16—181no—
185C4⋊Q8—91no—
186#127—31no—
187C4×Q8—91no—
188#106—181no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8—181no—
192C4⋊C8D3211yescharacteristic
193#103—181no—
194#102—181no—
195#97—181no—
196C4×C8—31no—
197C23×C3—121no—
198C22×D3—41no—
199C3×D4—121no—
200C3×D4—121no—
201C3×D4—121no—
202C3×D4—121no—
203C3×D4—241no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4C2×D1211yescharacteristic
208C2×C3×C4#23511yescharacteristic
209C2×C3×C4D3×D411yescharacteristic
210C2×C3×C4D3×D411yescharacteristic
211C2×C3×C4D3×D411yescharacteristic
212C2×C3×C4#23911yescharacteristic
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—121no—
218C2×Dic3—121no—
219C2×Dic3—122no—
220C2×Dic3—121no—
221C2×Dic3—121no—
222D12—41no—
223D12—41no—
224D12—41no—
225D12—41no—
226D12—81no—
227Dic6—62no—
228Dic6—61no—
229Dic6—61no—
230Dic6—61no—
231Dic6—61no—
232Dic6—121no—
233Dic6—121no—
234Dic6—121no—
235Dic6—121no—
236Dic6—121no—
237Dic6—241no—
238Dic6—241no—
239C3×C8—32no—
240C3×C8—42no—
241C3×C8—41no—
242C3×C8—41no—
243C3×C8—61no—
244C3×C8—121no—
245C3×C8—121no—
246C3⋊C8—21no—
247C3⋊C8—41no—
248C3⋊C8—41no—
249C3⋊C8#20412yescomplemented
250C2×C32#46011yescharacteristic
251C2×C32#46811yescharacteristic
252C2×C32#47111yescharacteristic
253C3×D3—161no—
254C2×Q8—91no—
255C2×Q8—181no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259SD16—362no—
260SD16—362no—
261SD16—362no—
262C2×C8—21no—
263C2×C8—31no—
264C2×C8—31no—
265C2×C8—61no—
266C2×C8—61no—
267C4⋊C4—91no—
268C4⋊C4—181no—
269C4⋊C4—181no—
270C4⋊C4—181no—
271C42—181no—
272C42C2×D3211yescharacteristic
273C22×C3—21no—
274C22×C3—241no—
275C22×C3—241no—
276C22×C3—241no—
277C22×C3#88411yescharacteristic
278C22×C3#89511yescharacteristic
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4C2×#20411yescharacteristic
290C3×C4#90811yescharacteristic
291C3×C4#91111yescharacteristic
292C3×C4#91111yescharacteristic
293C3×C4#91111yescharacteristic
294C3×C4D3×SD1611yescharacteristic
295C3×C4#91511yescharacteristic
296C3×C4#91811yescharacteristic
297Dic3—122no—
298Dic3—121no—
299Dic3—121no—
300Dic3—121no—
301Dic3—121no—
302Dic3—241no—
303Dic3—242no—
304Dic3—241no—
305Dic3—241no—
306C32#168611yescharacteristic, complemented, Sylow
307C23—121no—
308Q8—181no—
309Q8—181no—
310Q8—361no—
311Q8—361no—
312D4—121no—
313D4—121no—
314D4—121no—
315D4—121no—
316D4—241no—
317C2×C4—181no—
318C2×C4—181no—
319C2×C4—361no—
320C2×C4#379911yescharacteristic
321C2×C4D3×D1211yescharacteristic
322C2×C4#381111yescharacteristic
323C8—32no—
324C8—41no—
325C8—61no—
326C8—121no—
327C8—121no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—481no—
332C2×C3#474811yescharacteristic
333C2×C3#476211yescharacteristic
334C2×C3#478011yescharacteristic
335C2×C3#484811yescharacteristic
336C2×C3#488611yescharacteristic
337C2×C3#489811yescharacteristic
338D3—161no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22#6391011yescharacteristic
343C4—21no—
344C4—361no—
345C4—361no—
346C4—721no—
347C4D3×#20411yescharacteristic
348C4#6379511yescharacteristic
349C4#6392711yescharacteristic
350C4#6395111yescharacteristic
351C3—21no—
352C3#6841211yescharacteristic, complemented
353C3#6859211yescharacteristic, complemented
354C2—481no—
355C2#1059050611yescharacteristic
356C2#1059087911yescharacteristic
357C2#1059098911yescharacteristic
subgroupG/H
centerC22#63910
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC3×#874C22
hypercenterC4⋊C8D32
hypocenterC32#1686
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. #10592550C2semidirect
4. #10590296C2semidirect
6. C3×#4479C2semidirect
7. #10589094C2semidirect
39. #4479D3semidirect
115. C3×C3⋊C8SD16semidirect
249. C3⋊C8#204semidirect
306. C32#1686semidirect
352. C3#68412semidirect
353. C3#68592semidirect

Maximal subgroups

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

subgroupindex
#105925502
#105903772
#105903692
#105902962
C3×#44852
C3×#44792
#105890942
#685923
#684123