Finite Group Databasebeta

← #11781302

Subgroups — #11781302

Order 1152.

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

normal subgroups only

This group has 384 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#10592726C211yesmaximal, characteristic
2C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
3C3×#5021C211yesmaximal, characteristic, complemented
4#10590281C211yesmaximal, characteristic
5#10589097C211yesmaximal, characteristic, complemented
6#10589095C211yesmaximal, characteristic, complemented
7#10589091C211yesmaximal, characteristic
8#71150—31nomaximal
9#70940—31nomaximal
10C32×C4×Q8C2211yescharacteristic
11C2×#3755—21no—
12#64083C2211yescharacteristic
13C3×#869C2211yescharacteristic
14#63636—21no—
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#835—21no—
17C3×#807—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#63564—21no—
24#63564—21no—
25C3×#477—31no—
26C3×C8×Q8—31no—
27#5088—31no—
28#5027—31no—
29#5027—31no—
30#5027—61no—
31Q8×C3⋊C8D311yescharacteristic, complemented
32#5021D311yescharacteristic, complemented
33#4694—31no—
34#4546—31no—
35#4485—31no—
36#4485—31no—
37#4481—31no—
38#4480—31no—
39#4457—31no—
40C2×C32×Q8D411yescharacteristic
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)—21no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C3×Dic6—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4D411yescharacteristic
47C32×C42—21no—
48C32×C42C2311yescharacteristic
49#3755—42no—
50C2×C3⋊(C3⋊C8)—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8D411yescharacteristic
56C2×C3×C3⋊C8D411yescharacteristic
57#1678—91noSylow
58C3×C4⋊Q8—31no—
59C3×C4×Q8—21no—
60C3×C4×Q8C2×D311yescharacteristic
61C3×C4×Q8C2×D311yescharacteristic
62C2×#215—62no—
63C2×#215—61no—
64C2×#215—61no—
65#874—31no—
66#874—31no—
67#874—61no—
68#869C2×D311yescharacteristic
69C3×#107—61no—
70C3×C4⋊C8—31no—
71C3×C4⋊C8—31no—
72C3×C4⋊C8—61no—
73C3×#103—61no—
74C3×C4×C8—31no—
75C3×C4×C8—61no—
76#835—21no—
77#821—61no—
78#821—61no—
79#809—62no—
80#809—61no—
81#809—61no—
82#809—61no—
83#809—61no—
84#807—21no—
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
101C4⋊(C3⋊D3)—41no—
102C4⋊(C3⋊D3)—41no—
103C4⋊(C3⋊D3)—41no—
104C4⋊(C3⋊D3)—41no—
105C4⋊(C3⋊D3)—81no—
106C2×C3×Dic3—41no—
107C3×Dic6—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—31no—
115#460—91no—
116C8×Q8—31no—
117#334—91no—
118#333—91no—
119#331—91no—
120C2×C3×Q8—21no—
121C2×C3×Q8—61no—
122C2×C3×Q8D1211yescharacteristic
123C2×C3×Q8#6711yescharacteristic
124C2×D12—62no—
125C2×D12—61no—
126C2×D12—61no—
127C2×D12—61no—
128C2×D12—61no—
129C2×D12—61no—
130C2×D12—61no—
131C2×D12—121no—
132C2×D12—121no—
133C2×Dic6—21no—
134C2×C3×C8—31no—
135C2×C3×C8—31no—
136C2×C3×C8—61no—
137C2×C3×C8—61no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—61no—
143C3×C4⋊C4—61no—
144C3×C4⋊C4D1211yescharacteristic
145C3×C4⋊C4#6711yescharacteristic
146C3×C42—21no—
147C3×C42—21no—
148C3×C42—21no—
149C3×C42—41no—
150C3×C42C22×D311yescharacteristic
151C3×C42C22×D311yescharacteristic
152#215—124no—
153#215—122no—
154#215—122no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—62no—
160C2×C3⋊C8—61no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8#6711yescharacteristic
163C2×C3⋊C8#6711yescharacteristic
164C22×C32#12111yescharacteristic
165C2×C3⋊D3—81no—
166C2×C3⋊D3—81no—
167C2×C3⋊D3—81no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4—41no—
171C32×C4C2×SD1611yescharacteristic
172C32×C4#13511yescharacteristic
173C32×C4#13511yescharacteristic
174C32×C4#13711yescharacteristic
175C3×Dic3—81no—
176C2×SD16—181no—
177C4⋊Q8—31no—
178#127—91no—
179C4×Q8D3211yescharacteristic
180#107—61no—
181C4⋊C8—31no—
182C4⋊C8—31no—
183C4⋊C8—61no—
184C4⋊C8—91no—
185#103—61no—
186#102—181no—
187#102—181no—
188#102—181no—
189C4×C8—31no—
190C4×C8—61no—
191C22×D3—122no—
192C22×D3—121no—
193C22×D3—121no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×Q8—22no—
197C3×Q8—41no—
198C3×Q8—241no—
199C3×Q8#20412yes—
200C3×Q8#21512yes—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—121no—
211C2×C3×C4C2×D1211yescharacteristic
212C2×C3×C4#23511yescharacteristic
213C2×C3×C4D3×D411yescharacteristic
214C2×C3×C4#23711yescharacteristic
215C2×C3×C4C2×#6711yescharacteristic
216C2×C3×C4C2×#6711yescharacteristic
217C2×Dic3—41no—
218D12—122no—
219D12—122no—
220D12—121no—
221D12—121no—
222D12—121no—
223D12—121no—
224D12—121no—
225D12—121no—
226D12—121no—
227D12—121no—
228D12—241no—
229D12—241no—
230D12—241no—
231D12—241no—
232D12—242no—
233Dic6—81no—
234C3×C8—61no—
235C3×C8—61no—
236C3×C8—121no—
237C3×C8—121no—
238C3⋊C8—21no—
239C3⋊C8—21no—
240C3⋊C8—41no—
241C3⋊C8—41no—
242C3⋊C8—122no—
243C3⋊C8—121no—
244C3⋊C8—121no—
245C2×C32#46411yescharacteristic
246C2×C32#46511yescharacteristic
247C2×C32#47211yescharacteristic
248C3⋊D3—161no—
249C2×Q8—61no—
250C2×Q8#62611yescharacteristic
251C2×D4—181no—
252C2×D4—181no—
253C2×D4—181no—
254SD16—362no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8—61no—
258C2×C8—61no—
259C2×C8—181no—
260C4⋊C4—21no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4#62611yescharacteristic
264C42—21no—
265C42C2×D3211yescharacteristic
266C22×C3—21no—
267C22×C3#93011yescharacteristic
268C22×C3#93911yescharacteristic
269C2×D3—242no—
270C2×D3—242no—
271C2×D3—242no—
272C2×D3—241no—
273C2×D3—241no—
274C2×D3—241no—
275C2×D3—241no—
276C2×D3—241no—
277C2×D3—241no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—42no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—81no—
291C3×C4—241no—
292C3×C4C2×#20411yescharacteristic
293C3×C4#90411yescharacteristic
294C3×C4#91811yescharacteristic
295C3×C4#91911yescharacteristic
296C3×C4C2×#21511yescharacteristic
297C3×C4#94211yescharacteristic
298C3×C4#95011yescharacteristic
299C3×C4#95011yescharacteristic
300Dic3—81no—
301C32#167811yescharacteristic, complemented, Sylow
302C23—361no—
303Q8—21no—
304Q8—241no—
305Q8#371712yes—
306D4—361no—
307D4—361no—
308D4—361no—
309D4—361no—
310D4—721no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—121no—
314C2×C4#380411yescharacteristic
315C2×C4C2×#62611yescharacteristic
316C2×C4D3×#6711yescharacteristic
317C8—61no—
318C8—61no—
319C8—121no—
320C8—121no—
321C8—361no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3#502811yescharacteristic
326C2×C3#504811yescharacteristic
327C2×C3#511411yescharacteristic
328C2×C3#514211yescharacteristic
329C2×C3#518511yescharacteristic
330C2×C3#519911yescharacteristic
331D3—482no—
332D3—481no—
333D3—481no—
334C22—721no—
335C22—721no—
336C22—721no—
337C22#6397811yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—241no—
342C4#6382911yescharacteristic
343C4C2×#371711yescharacteristic
344C4#6394811yescharacteristic
345C4#6394911yescharacteristic
346C3—21no—
347C3#7094011yescharacteristic, complemented
348C3#7115011yescharacteristic, complemented
349C2—1441no—
350C2#1059117711yescharacteristic
351C2#1059160711yescharacteristic
352C2#1059167711yescharacteristic
subgroupG/H
centerC22#63978
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32#121
prime radical#64083C22
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
2. C3×Q8×C3⋊C8C2semidirect
3. C3×#5021C2semidirect
5. #10589097C2semidirect
6. #10589095C2semidirect
31. Q8×C3⋊C8D3semidirect
32. #5021D3semidirect
301. C32#1678semidirect
347. C3#70940semidirect
348. C3#71150semidirect

Maximal subgroups

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

subgroupindex
#105927262
C3×Q8×C3⋊C82
C3×#50212
#105902812
#105890972
#105890952
#105890912
#711503
#709403