Finite Group Databasebeta

← #11781491

Subgroups — #11781491

Order 1152.

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

normal subgroups only

This group has 386 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#10592730C211yesmaximal, characteristic, complemented
2C3×#5027C211yesmaximal, characteristic
3C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4#10590273C211yesmaximal, characteristic
5#10589099C211yesmaximal, characteristic, complemented
6#10589094C211yesmaximal, characteristic, complemented
7#10589090C211yesmaximal, characteristic
8#71023—31nomaximal
9#70941—31nomaximal
10C32×C4×Q8C2211yescharacteristic
11C2×#3756—21no—
12#64078C2211yescharacteristic
13C2×C3×#215—21no—
14C3×#874C2211yescharacteristic
15#63637—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#5031—31no—
29#5031—31no—
30#5031—61no—
31#5027D311yescharacteristic
32Q8×C3⋊C8D311yescharacteristic, complemented
33#4694—31no—
34#4544—31no—
35#4486—31no—
36#4485—31no—
37#4482—31no—
38#4479—31no—
39#4458—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
49#3756—42no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)—21no—
52C2×C3⋊(C3⋊C8)—21no—
53C3×#215—42no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8D411yescharacteristic
57C2×C3×C3⋊C8D411yescharacteristic
58#1682—91noSylow
59C2×C3×SD16—61no—
60C3×#127—31no—
61C3×C4×Q8—21no—
62C3×C4×Q8C2×D311yescharacteristic
63C3×C4×Q8C2×D311yescharacteristic
64C2×#216—62no—
65C2×#216—61no—
66C2×#216—61no—
67C2×#215—21no—
68#874C2×D311yescharacteristic
69#869—31no—
70#869—31no—
71#869—61no—
72C3×C4⋊C8—31no—
73C3×C4⋊C8—31no—
74C3×C4⋊C8—61no—
75C3×#102—61no—
76C3×C4×C8—31no—
77C3×C4×C8—61no—
78#832—61no—
79#832—61no—
80#821—61no—
81#821—61no—
82#810—62no—
83#810—61no—
84#810—61no—
85#809—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8—21no—
93C4×C3⋊C8C2×D311yescharacteristic
94C22×C3×D3—41no—
95C32×Q8—21no—
96C32×Q8D812yes—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4#4111yescharacteristic
102C2×C3⋊Dic3—41no—
103#634—41no—
104#634—41no—
105C3×D12—41no—
106C3×D12—41no—
107C3×D12—41no—
108C3×D12—41no—
109C3×D12—81no—
110C3⋊(C3⋊C8)—41no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—21no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8—41no—
115#486—91no—
116#462—91no—
117#460—31no—
118C8×Q8—31no—
119#335—91no—
120#332—91no—
121#331—91no—
122C2×C3×Q8—21no—
123C2×C3×Q8D1211yescharacteristic
124C2×C3×Q8#6711yescharacteristic
125C2×C3×D4—61no—
126C2×C3×D4—61no—
127C2×C3×D4—61no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×Dic6—62no—
132C2×Dic6—61no—
133C2×Dic6—61no—
134C3×SD16—122no—
135C2×C3×C8—31no—
136C2×C3×C8—31no—
137C2×C3×C8—61no—
138C2×C3×C8—61no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4D1211yescharacteristic
144C3×C4⋊C4#6711yescharacteristic
145C3×C42—21no—
146C3×C42—21no—
147C3×C42—21no—
148C3×C42—41no—
149C3×C42C22×D311yescharacteristic
150C3×C42C22×D311yescharacteristic
151#216—124no—
152#216—122no—
153#216—122no—
154#215—42no—
155C4⋊Dic3—61no—
156C4⋊Dic3—61no—
157C4⋊Dic3—61no—
158C4⋊Dic3—61no—
159C4⋊Dic3—121no—
160C4⋊Dic3—121no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—62no—
164C2×C3⋊C8—61no—
165C2×C3⋊C8—61no—
166C2×C3⋊C8#6711yescharacteristic
167C2×C3⋊C8#6711yescharacteristic
168C22×C32#12111yescharacteristic
169C2×C3×D3—81no—
170C2×C3×D3—81no—
171C2×C3×D3—81no—
172C32×C4—22no—
173C32×C4—21no—
174C32×C4—41no—
175C32×C4C2×D811yescharacteristic
176C32×C4#13511yescharacteristic
177C32×C4#13511yescharacteristic
178C32×C4#13711yescharacteristic
179C3⋊Dic3—81no—
180C2×Q16—181no—
181C2×SD16—61no—
182C4⋊Q8—91no—
183#127—31no—
184C4×Q8D3211yescharacteristic
185C4⋊C8—31no—
186C4⋊C8—31no—
187C4⋊C8—61no—
188C4⋊C8—91no—
189#103—181no—
190#102—61no—
191#102—181no—
192#102—181no—
193C4×C8—31no—
194C4×C8—61no—
195C23×C3—121no—
196C22×D3—41no—
197C3×Q8—21no—
198C3×Q8—21no—
199C3×Q8—22no—
200C3×Q8—41no—
201C3×Q8D2412yes—
202C3×Q8#21312yes—
203C3×D4—121no—
204C3×D4—121no—
205C3×D4—121no—
206C3×D4—121no—
207C3×D4—241no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4C2×D1211yescharacteristic
218C2×C3×C4#23511yescharacteristic
219C2×C3×C4D3×D411yescharacteristic
220C2×C3×C4#23711yescharacteristic
221C2×C3×C4C2×#6711yescharacteristic
222C2×C3×C4C2×#6711yescharacteristic
223C2×Dic3—122no—
224C2×Dic3—121no—
225C2×Dic3—121no—
226D12—41no—
227D12—41no—
228D12—41no—
229D12—41no—
230D12—81no—
231Dic6—122no—
232Dic6—122no—
233Dic6—121no—
234Dic6—121no—
235Dic6—121no—
236Dic6—121no—
237C3×C8—61no—
238C3×C8—61no—
239C3×C8—121no—
240C3×C8—121no—
241C3⋊C8—21no—
242C3⋊C8—21no—
243C3⋊C8—41no—
244C3⋊C8—41no—
245C3⋊C8—122no—
246C3⋊C8—121no—
247C3⋊C8—121no—
248C2×C32#46411yescharacteristic
249C2×C32#46611yescharacteristic
250C2×C32#47111yescharacteristic
251C3×D3—161no—
252C2×Q8—181no—
253C2×Q8#62611yescharacteristic
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257Q16—362no—
258SD16—122no—
259C2×C8—31no—
260C2×C8—31no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—181no—
264C4⋊C4—21no—
265C4⋊C4—181no—
266C4⋊C4—181no—
267C4⋊C4#62611yescharacteristic
268C42—21no—
269C42C2×D3211yescharacteristic
270C22×C3—21no—
271C22×C3—241no—
272C22×C3—241no—
273C22×C3—241no—
274C22×C3#93011yescharacteristic
275C22×C3#93911yescharacteristic
276C2×D3—81no—
277C2×D3—81no—
278C2×D3—81no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—42no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—81no—
292C3×C4C2×D2411yescharacteristic
293C3×C4#90411yescharacteristic
294C3×C4#91511yescharacteristic
295C3×C4#91611yescharacteristic
296C3×C4C2×#21311yescharacteristic
297C3×C4#94211yescharacteristic
298C3×C4#95011yescharacteristic
299C3×C4#95011yescharacteristic
300Dic3—242no—
301Dic3—241no—
302Dic3—241no—
303C32#168211yescharacteristic, complemented, Sylow
304C23—121no—
305Q8—21no—
306Q8—361no—
307Q8—361no—
308Q8#371412yes—
309D4—121no—
310D4—121no—
311D4—121no—
312D4—121no—
313D4—241no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—361no—
317C2×C4#380411yescharacteristic
318C2×C4C2×#62611yescharacteristic
319C2×C4D3×#6711yescharacteristic
320C8—61no—
321C8—61no—
322C8—121no—
323C8—121no—
324C8—361no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—481no—
329C2×C3#502811yescharacteristic
330C2×C3#503811yescharacteristic
331C2×C3#511511yescharacteristic
332C2×C3#514111yescharacteristic
333C2×C3#516611yescharacteristic
334C2×C3#520011yescharacteristic
335D3—161no—
336C22—241no—
337C22—241no—
338C22—241no—
339C22#6397811yescharacteristic
340C4—22no—
341C4—21no—
342C4—41no—
343C4—721no—
344C4C2×#371411yescharacteristic
345C4#6382911yescharacteristic
346C4#6394611yescharacteristic
347C4#6394711yescharacteristic
348C3—21no—
349C3#7094111yescharacteristic, complemented
350C3#7102311yescharacteristic, complemented
351C2—481no—
352C2#1059116411yescharacteristic
353C2#1059160411yescharacteristic
354C2#1059167911yescharacteristic
subgroupG/H
centerC22#63978
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32#121
prime radicalC3×#874C22
hypercenterC4×Q8D32
hypocenterC32#1682
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. #10592730C2semidirect
3. C3×Q8×C3⋊C8C2semidirect
5. #10589099C2semidirect
6. #10589094C2semidirect
32. Q8×C3⋊C8D3semidirect
303. C32#1682semidirect
349. C3#70941semidirect
350. C3#71023semidirect

Maximal subgroups

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

subgroupindex
#105927302
C3×#50272
C3×Q8×C3⋊C82
#105902732
#105890992
#105890942
#105890902
#710233
#709413