Finite Group Databasebeta

← #11781304

Subgroups — #11781304

Order 1152.

368 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 96 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#10592720C211yesmaximal, characteristic, complemented
2C3×#5027C211yesmaximal, characteristic
3C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4#10590273C211yesmaximal, characteristic
5#10589097C211yesmaximal, characteristic, complemented
6#10589096C211yesmaximal, characteristic, complemented
7#10589090C211yesmaximal, characteristic
8#71022—31nomaximal
9#70940—31nomaximal
10C32×C4×Q8C2211yescharacteristic
11#64078C2211yescharacteristic
12C2×C3×#215—21no—
13C3×#874C2211yescharacteristic
14#63662—21no—
15#63634—21no—
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#809—21no—
18C3×C4⋊(C3⋊C8)—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8—21no—
22C3×C4×C3⋊C8C2211yescharacteristic
23#63563—21no—
24#63563—21no—
25C3×#460—31no—
26C3×C8×Q8—31no—
27#5070—31no—
28#5027D311yescharacteristic
29Q8×C3⋊C8D311yescharacteristic, complemented
30#5021—31no—
31#5021—31no—
32#5021—61no—
33#4694—31no—
34#4544—31no—
35#4485—31no—
36#4481—31no—
37#4481—31no—
38#4480—31no—
39#4457—31no—
40C2×C32×Q8D411yescharacteristic
41C2×#634—21no—
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4D411yescharacteristic
47C32×C42—21no—
48C32×C42C2311yescharacteristic
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊C8)—21no—
52C3×#215—42no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8D411yescharacteristic
56C2×C3×C3⋊C8D411yescharacteristic
57#1678—91noSylow
58C2×C3×SD16—61no—
59C3×#127—31no—
60C3×C4×Q8—21no—
61C3×C4×Q8C2×D311yescharacteristic
62C3×C4×Q8C2×D311yescharacteristic
63C2×#215—21no—
64#874C2×D311yescharacteristic
65#869—31no—
66#869—31no—
67#869—61no—
68C3×C4⋊C8—31no—
69C3×C4⋊C8—31no—
70C3×C4⋊C8—61no—
71C3×#102—61no—
72C3×C4×C8—31no—
73C3×C4×C8—61no—
74#835—62no—
75#835—61no—
76#835—61no—
77#832—61no—
78#832—61no—
79#821—61no—
80#821—61no—
81#809—21no—
82#807—62no—
83#807—61no—
84#807—61no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4×C3⋊C8—21no—
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3×D3—41no—
94C32×Q8—21no—
95C32×Q8SD1612yes—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4#4111yescharacteristic
101C2×C3⋊Dic3—41no—
102#634—81no—
103C3×D12—41no—
104C3×D12—41no—
105C3×D12—41no—
106C3×D12—41no—
107C3×D12—81no—
108C3⋊(C3⋊C8)—41no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8—41no—
113#486—91no—
114#477—91no—
115#460—31no—
116C8×Q8—31no—
117#334—91no—
118#333—91no—
119#331—91no—
120C2×C3×Q8—21no—
121C2×C3×Q8D1211yescharacteristic
122C2×C3×Q8#6711yescharacteristic
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×C3×D4—61no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×Dic6—61no—
130C2×Dic6—61no—
131C2×Dic6—121no—
132C3×SD16—122no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—61no—
136C2×C3×C8—61no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4D1211yescharacteristic
142C3×C4⋊C4#6711yescharacteristic
143C3×C42—21no—
144C3×C42—21no—
145C3×C42—21no—
146C3×C42—41no—
147C3×C42C22×D311yescharacteristic
148C3×C42C22×D311yescharacteristic
149#215—42no—
150C4⋊Dic3—62no—
151C4⋊Dic3—61no—
152C4⋊Dic3—61no—
153C4⋊Dic3—61no—
154C4⋊Dic3—61no—
155C4⋊Dic3—121no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—62no—
159C2×C3⋊C8—61no—
160C2×C3⋊C8—61no—
161C2×C3⋊C8#6711yescharacteristic
162C2×C3⋊C8#6711yescharacteristic
163C22×C32#12111yescharacteristic
164C2×C3×D3—81no—
165C2×C3×D3—81no—
166C2×C3×D3—81no—
167C32×C4—22no—
168C32×C4—21no—
169C32×C4—41no—
170C32×C4C2×SD1611yescharacteristic
171C32×C4#13511yescharacteristic
172C32×C4#13511yescharacteristic
173C32×C4#13711yescharacteristic
174C3⋊Dic3—81no—
175C2×SD16—61no—
176C4⋊Q8—91no—
177#127—31no—
178C4×Q8D3211yescharacteristic
179#107—181no—
180C4⋊C8—31no—
181C4⋊C8—31no—
182C4⋊C8—61no—
183C4⋊C8—91no—
184#103—181no—
185#102—61no—
186#102—181no—
187#102—181no—
188C4×C8—31no—
189C4×C8—61no—
190C23×C3—121no—
191C22×D3—41no—
192C3×Q8—21no—
193C3×Q8—21no—
194C3×Q8—22no—
195C3×Q8—41no—
196C3×Q8#20412yes—
197C3×Q8#21412yes—
198C3×D4—121no—
199C3×D4—121no—
200C3×D4—121no—
201C3×D4—121no—
202C3×D4—241no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4C2×D1211yescharacteristic
213C2×C3×C4#23511yescharacteristic
214C2×C3×C4D3×D411yescharacteristic
215C2×C3×C4#23711yescharacteristic
216C2×C3×C4C2×#6711yescharacteristic
217C2×C3×C4C2×#6711yescharacteristic
218C2×Dic3—122no—
219C2×Dic3—121no—
220C2×Dic3—121no—
221D12—41no—
222D12—41no—
223D12—41no—
224D12—41no—
225D12—81no—
226Dic6—242no—
227Dic6—241no—
228Dic6—241no—
229C3×C8—61no—
230C3×C8—61no—
231C3×C8—121no—
232C3×C8—121no—
233C3⋊C8—21no—
234C3⋊C8—21no—
235C3⋊C8—41no—
236C3⋊C8—41no—
237C3⋊C8—122no—
238C3⋊C8—121no—
239C3⋊C8—121no—
240C2×C32#46411yescharacteristic
241C2×C32#46511yescharacteristic
242C2×C32#47211yescharacteristic
243C3×D3—161no—
244C2×Q8—181no—
245C2×Q8#62611yescharacteristic
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249SD16—122no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—181no—
255C4⋊C4—21no—
256C4⋊C4—181no—
257C4⋊C4—181no—
258C4⋊C4#62611yescharacteristic
259C42—21no—
260C42C2×D3211yescharacteristic
261C22×C3—21no—
262C22×C3—241no—
263C22×C3—241no—
264C22×C3—241no—
265C22×C3#93011yescharacteristic
266C22×C3#93911yescharacteristic
267C2×D3—81no—
268C2×D3—81no—
269C2×D3—81no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—42no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—81no—
283C3×C4C2×#20411yescharacteristic
284C3×C4#90411yescharacteristic
285C3×C4#91511yescharacteristic
286C3×C4#91611yescharacteristic
287C3×C4C2×#21411yescharacteristic
288C3×C4#94211yescharacteristic
289C3×C4#95011yescharacteristic
290C3×C4#95011yescharacteristic
291Dic3—242no—
292Dic3—241no—
293Dic3—241no—
294C32#167811yescharacteristic, complemented, Sylow
295C23—121no—
296Q8—21no—
297Q8—721no—
298Q8#371612yes—
299D4—121no—
300D4—121no—
301D4—121no—
302D4—121no—
303D4—241no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—361no—
307C2×C4#380411yescharacteristic
308C2×C4C2×#62611yescharacteristic
309C2×C4D3×#6711yescharacteristic
310C8—61no—
311C8—61no—
312C8—121no—
313C8—121no—
314C8—361no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—481no—
319C2×C3#502811yescharacteristic
320C2×C3#504111yescharacteristic
321C2×C3#511411yescharacteristic
322C2×C3#514211yescharacteristic
323C2×C3#516611yescharacteristic
324C2×C3#519911yescharacteristic
325D3—161no—
326C22—241no—
327C22—241no—
328C22—241no—
329C22#6397811yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—721no—
334C4C2×#371611yescharacteristic
335C4#6382911yescharacteristic
336C4#6394611yescharacteristic
337C4#6394711yescharacteristic
338C3—21no—
339C3#7094011yescharacteristic, complemented
340C3#7102211yescharacteristic, complemented
341C2—481no—
342C2#1059117511yescharacteristic
343C2#1059160411yescharacteristic
344C2#1059167711yescharacteristic
subgroupG/H
centerC22#63978
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32#121
prime radicalC3×#874C22
hypercenterC4×Q8D32
hypocenterC32#1678
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
1. #10592720C2semidirect
3. C3×Q8×C3⋊C8C2semidirect
5. #10589097C2semidirect
6. #10589096C2semidirect
29. Q8×C3⋊C8D3semidirect
294. C32#1678semidirect
339. C3#70940semidirect
340. C3#71022semidirect

Maximal subgroups

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

subgroupindex
#105927202
C3×#50272
C3×Q8×C3⋊C82
#105902732
#105890972
#105890962
#105890902
#710223
#709403