Finite Group Databasebeta

← #11780855

Subgroups — #11780855

Order 1152.

415 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 88 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#10592784C211yesmaximal, characteristic
2C3×#5094C211yesmaximal, characteristic, complemented
3#10590367C211yesmaximal, characteristic
4#10590250C211yesmaximal, characteristic, complemented
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589098C211yesmaximal, characteristic, complemented
7#10589091C211yesmaximal, characteristic
8#71116—31nomaximal
9#68464—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×#3755—21no—
12C2×#3755—21no—
13#64083C2211yescharacteristic
14C3×C4×Dic6C2211yescharacteristic
15C2×#3717—21no—
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#810—21no—
18C3×#808—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×#475—31no—
26#5094D311yescharacteristic, complemented
27#5085—31no—
28#5085—31no—
29#5085—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#4480—31no—
39#4480—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
48#3755—42no—
49#3755—42no—
50C2×C3⋊(C3⋊C8)D411yescharacteristic
51C2×C3⋊(C3⋊C8)D411yescharacteristic
52C3×C4⋊Dic3D411yescharacteristic
53C3×Dic3⋊C4—21no—
54C3×C4×Dic3—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8—21no—
57C2×C3×C3⋊C8—21no—
58#3717—42no—
59#1666—91noSylow
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×C4×Q8—31no—
64C2×#215—61no—
65C2×#215—61no—
66C2×#215—61no—
67C2×#215—61no—
68C2×#215—61no—
69C2×#215—62no—
70C2×#215—62no—
71C2×#204—61no—
72#874—31no—
73#874—31no—
74#874—61no—
75C4×Dic6C2×D311yescharacteristic
76C3×#106—61no—
77C3×C4⋊C8—31no—
78C3×C4⋊C8—31no—
79C3×C4⋊C8—31no—
80C3×#103—61no—
81#821—61no—
82Dic3⋊C8—61no—
83C8×Dic3—61no—
84#810—21no—
85#809—61no—
86#808—21no—
87C4⋊(C3⋊C8)—61no—
88C4⋊(C3⋊C8)C2×D311yescharacteristic
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4×C3⋊C8—31no—
92C4×C3⋊C8—31no—
93C4×C3⋊C8—61no—
94C4×C3⋊C8—61no—
95C22×C3⋊D3—41no—
96C32×Q8—41no—
97C32×Q8—41no—
98C2×C32×C4—41no—
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4#4111yescharacteristic
102C4⋊(C3⋊D3)—41no—
103C4⋊(C3⋊D3)—41no—
104C4⋊(C3⋊D3)—41no—
105C4⋊(C3⋊D3)—41no—
106C4⋊(C3⋊D3)—81no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C3×Dic6—21no—
110C3×Dic6SD1612yescomplemented
111C3⋊(C3⋊C8)—21no—
112C3⋊(C3⋊C8)SD1612yescomplemented
113C3×C3⋊C8—41no—
114C3×C3⋊C8—41no—
115C3×C3⋊C8—41no—
116#492—91no—
117#475—31no—
118#460—91no—
119C8×Q8—91no—
120#334—91no—
121#332—31no—
122#331—91no—
123C2×C3×Q8—22no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—31no—
127C2×D12—62no—
128C2×D12—61no—
129C2×D12—61no—
130C2×D12—61no—
131C2×D12—61no—
132C2×D12—61no—
133C2×D12—61no—
134C2×D12—121no—
135C2×D12—121no—
136C2×Dic6#6711yescharacteristic
137C2×C3×C8—61no—
138C2×C3×C8—61no—
139C2×C3×C8—61no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—61no—
148C3×C42—21no—
149C3×C42—61no—
150C3×C42C22×D311yescharacteristic
151C3×C42C22×D311yescharacteristic
152#215—122no—
153#215—122no—
154#215—122no—
155#215—124no—
156#215—124no—
157#215—122no—
158#215—122no—
159C4⋊Dic3#6711yescharacteristic
160Dic3⋊C4—21no—
161C4×Dic3—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8—31no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8—61no—
170C2×C3⋊C8—61no—
171#204—122no—
172C22×C32#12111yescharacteristic
173C2×C3⋊D3—81no—
174C2×C3⋊D3—81no—
175C2×C3⋊D3—81no—
176C32×C4—21no—
177C32×C4—81no—
178C32×C4C2×SD1611yescharacteristic
179C32×C4C2×SD1611yescharacteristic
180C32×C4#13511yescharacteristic
181C32×C4AGL(1,ℤ/8)11yescharacteristic
182C3×Dic3—22no—
183C3×Dic3—41no—
184C2×SD16—181no—
185C2×SD16—181no—
186C2×SD16—181no—
187C4⋊Q8D3211yescharacteristic
188#127—91no—
189C4×Q8—31no—
190#106—61no—
191C4⋊C8—31no—
192C4⋊C8—31no—
193C4⋊C8—31no—
194C4⋊C8—181no—
195#103—61no—
196#102—181no—
197C4×C8—91no—
198C4×C8—181no—
199C22×D3—122no—
200C22×D3—121no—
201C22×D3—121no—
202C3×Q8—32no—
203C3×Q8—42no—
204C3×Q8—42no—
205C3×Q8—41no—
206C3×Q8—41no—
207C3×Q8—41no—
208C3×Q8—41no—
209C3×Q8—61no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4#23511yescharacteristic
219C2×C3×C4D3×D411yescharacteristic
220C2×C3×C4D3×D411yescharacteristic
221C2×C3×C4D3×D411yescharacteristic
222C2×C3×C4#23711yescharacteristic
223C2×C3×C4C2×#6711yescharacteristic
224C2×Dic3—21no—
225C2×Dic3—21no—
226D12—122no—
227D12—122no—
228D12—121no—
229D12—121no—
230D12—121no—
231D12—121no—
232D12—121no—
233D12—121no—
234D12—121no—
235D12—121no—
236D12—241no—
237D12—241no—
238D12—241no—
239D12—241no—
240D12—242no—
241Dic6—21no—
242Dic6#21512yescomplemented
243C3×C8—121no—
244C3×C8—121no—
245C3×C8—121no—
246C3⋊C8—32no—
247C3⋊C8—32no—
248C3⋊C8—41no—
249C3⋊C8—41no—
250C3⋊C8—41no—
251C3⋊C8—62no—
252C3⋊C8—62no—
253C3⋊C8—61no—
254C3⋊C8—61no—
255C2×C32#46011yescharacteristic
256C2×C32#46511yescharacteristic
257C2×C32#47111yescharacteristic
258C3⋊D3—161no—
259C2×Q8—21no—
260C2×Q8—31no—
261C2×D4—181no—
262C2×D4—181no—
263C2×D4—181no—
264SD16—362no—
265SD16—362no—
266SD16—362no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—91no—
271C2×C8—91no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—31no—
275C4⋊C4—61no—
276C42—61no—
277C42C2×D3211yescharacteristic
278C22×C3—21no—
279C22×C3#88411yescharacteristic
280C22×C3#93911yescharacteristic
281C2×D3—242no—
282C2×D3—242no—
283C2×D3—242no—
284C2×D3—241no—
285C2×D3—241no—
286C2×D3—241no—
287C2×D3—241no—
288C2×D3—241no—
289C2×D3—241no—
290C3×C4—22no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—62no—
298C3×C4—82no—
299C3×C4—81no—
300C3×C4—81no—
301C3×C4—121no—
302C3×C4#91111yescharacteristic
303C3×C4D3×SD1611yescharacteristic
304C3×C4D3×SD1611yescharacteristic
305C3×C4D3×SD1611yescharacteristic
306C3×C4#91411yescharacteristic
307C3×C4#91911yescharacteristic
308C3×C4C2×#21511yescharacteristic
309C3×C4#95011yescharacteristic
310Dic3—22no—
311Dic3—41no—
312C32#166611yescharacteristic, complemented, Sylow
313C23—361no—
314Q8—32no—
315Q8—41no—
316Q8—41no—
317Q8—61no—
318D4—361no—
319D4—361no—
320D4—361no—
321D4—361no—
322D4—721no—
323C2×C4—41no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4#380411yescharacteristic
327C2×C4D3×#6711yescharacteristic
328C2×C4#381111yescharacteristic
329C8—92no—
330C8—121no—
331C8—121no—
332C8—121no—
333C8—181no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3#475411yescharacteristic
338C2×C3#476211yescharacteristic
339C2×C3#480711yescharacteristic
340C2×C3#504811yescharacteristic
341C2×C3#516311yescharacteristic
342C2×C3#520311yescharacteristic
343D3—482no—
344D3—481no—
345D3—481no—
346C22—721no—
347C22—721no—
348C22—721no—
349C22#6397111yescharacteristic
350C4—21no—
351C4—62no—
352C4—81no—
353C4—121no—
354C4#6393011yescharacteristic
355C4#6393711yescharacteristic
356C4D3×#21511yescharacteristic
357C4#6394911yescharacteristic
358C3—21no—
359C3#6846411yescharacteristic, complemented
360C3#7111611yescharacteristic, complemented
361C2—1441no—
362C2#1059144411yescharacteristic
363C2#1059158511yescharacteristic
364C2#1059164311yescharacteristic
subgroupG/H
centerC22#63971
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radical#64083C22
hypercenterC4⋊Q8D32
hypocenterC32#1666
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×#5094C2semidirect
4. #10590250C2semidirect
5. C3×#4547C2semidirect
6. #10589098C2semidirect
26. #5094D3semidirect
35. #4547D3semidirect
110. C3×Dic6SD16semidirect
112. C3⋊(C3⋊C8)SD16semidirect
242. Dic6#215semidirect
312. C32#1666semidirect
359. C3#68464semidirect
360. C3#71116semidirect

Maximal subgroups

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

subgroupindex
#105927842
C3×#50942
#105903672
#105902502
C3×#45472
#105890982
#105890912
#711163
#684643