Finite Group Databasebeta

← #11781492

Subgroups — #11781492

Order 1152.

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

normal subgroups only

This group has 391 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#10592787C211yesmaximal, characteristic
2C3×#5090C211yesmaximal, characteristic, complemented
3#10590367C211yesmaximal, characteristic
4#10590250C211yesmaximal, characteristic, complemented
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589100C211yesmaximal, characteristic, complemented
7#10589091C211yesmaximal, characteristic
8#71055—31nomaximal
9#68470—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64083C2211yescharacteristic
12C2×C3×#216—21no—
13C3×C4×Dic6C2211yescharacteristic
14C2×#3717—21no—
15#63636—21no—
16#63636—21no—
17C4×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#835—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22Dic3⋊(C3⋊C8)—21no—
23#63564—21no—
24Dic3×C3⋊C8—21no—
25C3×#462—31no—
26#5090D311yescharacteristic, complemented
27#5088—31no—
28#5088—31no—
29#5088—61no—
30#5027—31no—
31Q8×C3⋊C8—31no—
32#4848—31no—
33#4830—31no—
34C3×#332—31no—
35#4547D311yescharacteristic, complemented
36#4485—31no—
37#4485—31no—
38#4479—31no—
39#4479—31no—
40C2×C32×Q8—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)—21no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C3×Dic6D411yescharacteristic
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C42C2311yescharacteristic
48C2×C3⋊(C3⋊C8)D411yescharacteristic
49C2×C3⋊(C3⋊C8)D411yescharacteristic
50C3×#216—42no—
51C3×C4⋊Dic3D411yescharacteristic
52C3×Dic3⋊C4—21no—
53C3×C4×Dic3—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8—21no—
57#3717—42no—
58#1682—91noSylow
59C2×C3×Q16—61no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×C4×Q8—31no—
64C2×#216—21no—
65C2×#215—61no—
66C2×#204—61no—
67#874—31no—
68#874—31no—
69#874—61no—
70C4×Dic6C2×D311yescharacteristic
71C3×C4⋊C8—31no—
72C3×C4⋊C8—31no—
73C3×C4⋊C8—31no—
74C3×#103—61no—
75#835—21no—
76#821—61no—
77Dic3⋊C8—61no—
78C8×Dic3—61no—
79#809—61no—
80#809—61no—
81#809—61no—
82#809—61no—
83#809—61no—
84#809—62no—
85#809—62no—
86C4⋊(C3⋊C8)—61no—
87C4⋊(C3⋊C8)C2×D311yescharacteristic
88C4⋊(C3⋊C8)C2×D311yescharacteristic
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4×C3⋊C8—31no—
91C4×C3⋊C8—31no—
92C4×C3⋊C8—61no—
93C4×C3⋊C8—61no—
94C22×C3⋊D3—41no—
95C32×Q8—41no—
96C32×Q8—41no—
97C2×C32×C4—41no—
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—21no—
107C2×C3×Dic3—21no—
108C3×Dic6—21no—
109C3×Dic6D812yes—
110C3⋊(C3⋊C8)—21no—
111C3⋊(C3⋊C8)—21no—
112C3×C3⋊C8—41no—
113C3×C3⋊C8—41no—
114C3×C3⋊C8—41no—
115#486—91no—
116#462—31no—
117#460—91no—
118C8×Q8—91no—
119#335—91no—
120#332—31no—
121#331—91no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—31no—
125C2×C3×Q8—41no—
126C2×D12—62no—
127C2×D12—61no—
128C2×D12—61no—
129C2×D12—61no—
130C2×D12—61no—
131C2×D12—61no—
132C2×D12—61no—
133C2×D12—121no—
134C2×D12—121no—
135C2×Dic6#6711yescharacteristic
136C3×Q16—122no—
137C2×C3×C8—61no—
138C2×C3×C8—61no—
139C2×C3×C8—61no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—61no—
148C3×C42—21no—
149C3×C42—61no—
150C3×C42C22×D311yescharacteristic
151C3×C42C22×D311yescharacteristic
152#216—42no—
153#215—122no—
154C4⋊Dic3#6711yescharacteristic
155Dic3⋊C4—21no—
156C4×Dic3—21no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—31no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—61no—
165C2×C3⋊C8—61no—
166#204—122no—
167C22×C32#12111yescharacteristic
168C2×C3⋊D3—81no—
169C2×C3⋊D3—81no—
170C2×C3⋊D3—81no—
171C32×C4—21no—
172C32×C4—81no—
173C32×C4C2×D811yescharacteristic
174C32×C4#13511yescharacteristic
175C32×C4#13511yescharacteristic
176C32×C4#13711yescharacteristic
177C3×Dic3—22no—
178C3×Dic3—41no—
179C2×Q16—61no—
180C2×SD16—181no—
181C4⋊Q8D3211yescharacteristic
182#127—91no—
183C4×Q8—31no—
184C4⋊C8—31no—
185C4⋊C8—31no—
186C4⋊C8—31no—
187C4⋊C8—181no—
188#103—61no—
189#102—181no—
190#102—181no—
191#102—181no—
192C4×C8—91no—
193C4×C8—181no—
194C22×D3—122no—
195C22×D3—121no—
196C22×D3—121no—
197C3×Q8—32no—
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—61no—
203C3×Q8—81no—
204C3×Q8—81no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—42no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4#23511yescharacteristic
214C2×C3×C4D3×D411yescharacteristic
215C2×C3×C4D3×D411yescharacteristic
216C2×C3×C4D3×D411yescharacteristic
217C2×C3×C4#23711yescharacteristic
218C2×C3×C4C2×#6711yescharacteristic
219C2×Dic3—21no—
220C2×Dic3—21no—
221D12—122no—
222D12—122no—
223D12—121no—
224D12—121no—
225D12—121no—
226D12—121no—
227D12—121no—
228D12—121no—
229D12—121no—
230D12—121no—
231D12—241no—
232D12—241no—
233D12—241no—
234D12—241no—
235D12—242no—
236Dic6—21no—
237Dic6#21312yes—
238C3×C8—121no—
239C3×C8—121no—
240C3×C8—121no—
241C3⋊C8—41no—
242C3⋊C8—41no—
243C3⋊C8—41no—
244C3⋊C8—61no—
245C3⋊C8—61no—
246C3⋊C8—61no—
247C3⋊C8—61no—
248C3⋊C8—121no—
249C3⋊C8—121no—
250C2×C32#46411yescharacteristic
251C2×C32#46611yescharacteristic
252C2×C32#47111yescharacteristic
253C3⋊D3—161no—
254C2×Q8—21no—
255C2×Q8—31no—
256C2×D4—181no—
257C2×D4—181no—
258C2×D4—181no—
259Q16—122no—
260SD16—362no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—91no—
265C2×C8—91no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—31no—
269C4⋊C4—61no—
270C42—61no—
271C42C2×D3211yescharacteristic
272C22×C3—21no—
273C22×C3#88411yescharacteristic
274C22×C3#93911yescharacteristic
275C2×D3—242no—
276C2×D3—242no—
277C2×D3—242no—
278C2×D3—241no—
279C2×D3—241no—
280C2×D3—241no—
281C2×D3—241no—
282C2×D3—241no—
283C2×D3—241no—
284C3×C4—22no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—62no—
292C3×C4—82no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4—121no—
296C3×C4D3×D811yescharacteristic
297C3×C4#91611yescharacteristic
298C3×C4#91611yescharacteristic
299C3×C4#91811yescharacteristic
300C3×C4#91811yescharacteristic
301C3×C4#91911yescharacteristic
302C3×C4C2×#21311yescharacteristic
303C3×C4#95011yescharacteristic
304Dic3—22no—
305Dic3—41no—
306C32#168211yescharacteristic, complemented, Sylow
307C23—361no—
308Q8—32no—
309Q8—41no—
310Q8—41no—
311Q8—61no—
312D4—361no—
313D4—361no—
314D4—361no—
315D4—361no—
316D4—721no—
317C2×C4—41no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4#380411yescharacteristic
321C2×C4D3×#6711yescharacteristic
322C2×C4#381111yescharacteristic
323C8—121no—
324C8—121no—
325C8—121no—
326C8—181no—
327C8—181no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3#475411yescharacteristic
332C2×C3#477511yescharacteristic
333C2×C3#479811yescharacteristic
334C2×C3#503811yescharacteristic
335C2×C3#518511yescharacteristic
336C2×C3#520311yescharacteristic
337D3—482no—
338D3—481no—
339D3—481no—
340C22—721no—
341C22—721no—
342C22—721no—
343C22#6397111yescharacteristic
344C4—21no—
345C4—62no—
346C4—81no—
347C4—121no—
348C4D3×#21311yescharacteristic
349C4#6393511yescharacteristic
350C4#6394511yescharacteristic
351C4#6395011yescharacteristic
352C3—21no—
353C3#6847011yescharacteristic, complemented
354C3#7105511yescharacteristic, complemented
355C2—1441no—
356C2#1059144911yescharacteristic
357C2#1059157911yescharacteristic
358C2#1059164311yescharacteristic
subgroupG/H
centerC22#63971
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radical#64083C22
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
2. C3×#5090C2semidirect
4. #10590250C2semidirect
5. C3×#4547C2semidirect
6. #10589100C2semidirect
26. #5090D3semidirect
35. #4547D3semidirect
306. C32#1682semidirect
353. C3#68470semidirect
354. C3#71055semidirect

Maximal subgroups

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

subgroupindex
#105927872
C3×#50902
#105903672
#105902502
C3×#45472
#105891002
#105890912
#710553
#684703