Finite Group Databasebeta

← #11898318

Subgroups — #11898318

Order 1152.

964 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596048C211yesmaximal, characteristic
2C3×#5570C211yesmaximal, characteristic, complemented
3C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4#10590379C213yesmaximal, complemented
5#10590373C213yesmaximal, complemented
6#10590367C213yesmaximal, complemented
7C4×#3717C213yesmaximal, complemented
8#83685—31nomaximal
9#79911—31nomaximal
10C2×#3832—21no—
11C32×C4×Q8C2211yescharacteristic
12#64104—23no—
13#64083C2213yes—
14C4×C4⋊(C3⋊D3)C2213yes—
15C3×Q8×Dic3—21no—
16C3×#890—23no—
17C3×#869C2213yes—
18C3×C4×Dic6C2213yes—
19C2×#3717C2213yes—
20C2×#3717C2216yes—
21C3×C4⋊(C3⋊C8)C2213yes—
22C3×C4×C3⋊C8C2213yes—
23#63573C2211yescharacteristic
24#63566C2213yes—
25#63564C2213yes—
26C3×#557—31no—
27#5577—31no—
28#5577—31no—
29#5577—61no—
30#5570D311yescharacteristic, complemented
31C3×C8×Q8—31no—
32#5085—33no—
33#5083—33no—
34#5027—33no—
35C4×#215—33no—
36Q8×C3⋊C8D311yescharacteristic, complemented
37#4848—33no—
38#4841—33no—
39#4693—33no—
40C4×#204—33no—
41C2×C32×Q8D411yescharacteristic
42#3832—44no—
43C2×C4⋊(C3⋊D3)—23no—
44C2×C4⋊(C3⋊D3)—23no—
45C2×C4⋊(C3⋊D3)C2313yes—
46C2×C4×C3⋊D3—23no—
47C2×C3×Dic6C2313yes—
48C32×C4⋊C4D413yes—
49C32×C42C2313yes—
50#3752—23no—
51C4⋊(C3⋊Dic3)C2311yescharacteristic
52C3×C4⋊Dic3—23no—
53C3×C4⋊Dic3—23no—
54C3×C4⋊Dic3C2311yescharacteristic
55C3×Dic3⋊C4—23no—
56C3×C4×Dic3—23no—
57C2×C3×C3⋊C8C2313yes—
58C2×C3×C3⋊C8C2311yescharacteristic
59#3717—26no—
60#3717—46no—
61#3392—91noSylow
62C2×#239—62no—
63C2×#239—61no—
64C2×#239—61no—
65C3×C4⋊Q8—33no—
66C3×#125—63no—
67C3×C4×Q8—21no—
68C3×C4×Q8—33no—
69C3×C4×Q8—61no—
70C3×C4×Q8C2×D311yescharacteristic
71C3×C4×Q8C2×D311yescharacteristic
72Q8×Dic3—21no—
73C2×#215—33no—
74C2×#215—36no—
75C2×#204—33no—
76C2×#204—36no—
77#895—66no—
78#895—63no—
79#895—63no—
80#890—23no—
81#874—33no—
82#874—33no—
83#874—63no—
84C4×D12—33no—
85C4×D12—33no—
86C4×D12—63no—
87#869C2×D313yes—
88C4×Dic6C2×D313yes—
89C3×C4⋊C8—33no—
90C3×C4×C8—33no—
91#835—33no—
92#821—33no—
93#817—31no—
94#816—33no—
95#809—33no—
96#808—31no—
97C4⋊(C3⋊C8)C2×D313yes—
98C4×C3⋊C8C2×D313yes—
99C22×C3⋊D3—23no—
100C32×Q8SD1614yescomplemented
101C2×C32×C4C2×D413yes—
102C2×C32×C4C2×D413yes—
103C2×C32×C4C2411yescharacteristic
104C2×C3⋊Dic3—21no—
105C4⋊(C3⋊D3)—23no—
106C4⋊(C3⋊D3)—23no—
107C4⋊(C3⋊D3)—46no—
108C4⋊(C3⋊D3)—83no—
109C4×C3⋊D3—46no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—23no—
112C3×Dic6—23no—
113C3×Dic6—23no—
114C3×C3⋊C8—23no—
115C3×C3⋊C8#4112yescomplemented
116#557—31no—
117#550—91no—
118#492—93no—
119#461—93no—
120#460—93no—
121C8×Q8—31no—
122C4×SD16—93no—
123C2×C3×Q8—21no—
124C2×C3×Q8—33no—
125C2×C3×Q8D1211yescharacteristic
126C2×C3×Q8#6711yescharacteristic
127#239—128no—
128#239—124no—
129#239—124no—
130C2×D12—33no—
131C2×D12—33no—
132C2×D12—63no—
133C2×D12—66no—
134C2×D12—63no—
135C2×D12—63no—
136C2×D12—66no—
137C2×D12—63no—
138C2×D12—63no—
139C2×C4×D3—66no—
140C2×C4×D3—63no—
141C2×C4×D3—63no—
142C2×Dic6C22×D313yes—
143C2×C3×C8—31no—
144C2×C3×C8—33no—
145C3×C4⋊C4—23no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—63no—
148C3×C4⋊C4—63no—
149C3×C4⋊C4—63no—
150C3×C4⋊C4D1213yes—
151C3×C4⋊C4#6713yes—
152C3×C42—23no—
153C3×C42—63no—
154C3×C42C22×D313yes—
155C3×C42C22×D313yes—
156#215—66no—
157#215—126no—
158(C2×D3)⋊C4—66no—
159(C2×D3)⋊C4—63no—
160(C2×D3)⋊C4—63no—
161C4⋊Dic3—23no—
162C4⋊Dic3—23no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—61no—
166C4⋊Dic3C22×D311yescharacteristic
167Dic3⋊C4—23no—
168C4×Dic3—23no—
169C2×C3⋊C8C22×D313yes—
170C2×C3⋊C8C22×D311yescharacteristic
171#204—66no—
172#204—126no—
173C22×C32C22×D411yescharacteristic
174C2×C3⋊D3—43no—
175C2×C3⋊D3—43no—
176C2×C3⋊D3—43no—
177C32×C4—23no—
178C32×C4C2×SD1616yes—
179C32×C4C2×#4111yescharacteristic
180C32×C4H(5,𝔽2)11yescharacteristic
181C3⋊Dic3—41no—
182C3×Dic3—43no—
183C3×Dic3—41no—
184C2×#41—181no—
185C2×SD16—93no—
186C2×SD16—96no—
187C4⋊Q8—33no—
188#127—93no—
189#125—63no—
190#121—183no—
191C4×Q8—33no—
192C4×Q8—61no—
193C4×Q8D3211yescharacteristic
194C4×D4—93no—
195#106—91no—
196C4⋊C8—33no—
197#103—93no—
198#102—93no—
199C4×C8—33no—
200C22×D3—66no—
201C22×D3—63no—
202C22×D3—63no—
203C3×Q8—24no—
204C3×Q8—63no—
205C3×Q8—63no—
206C3×Q8#20414yescomplemented
207C3×Q8#21514yescomplemented
208C2×C3×C4—21no—
209C2×C3×C4—23no—
210C2×C3×C4—23no—
211C2×C3×C4—61no—
212C2×C3×C4—63no—
213C2×C3×C4C2×D1213yes—
214C2×C3×C4C2×D1213yes—
215C2×C3×C4C2×#6713yes—
216C2×C3×C4C2×#6713yes—
217C2×C3×C4C23×D311yescharacteristic
218C2×C3×C4C23×D311yescharacteristic
219C2×Dic3—21no—
220C2×Dic3—23no—
221C2×Dic3—62no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224D12—63no—
225D12—63no—
226D12—63no—
227D12—63no—
228D12—123no—
229D12—123no—
230D12—1212no—
231D12—126no—
232D12—126no—
233D12—243no—
234D12—243no—
235D12—246no—
236C4×D3—1212no—
237C4×D3—126no—
238C4×D3—126no—
239Dic6—23no—
240Dic6—23no—
241C3×C8—32no—
242C3×C8—63no—
243C3⋊C8—23no—
244C3⋊C8#23912yescomplemented
245C2×C32#55011yescharacteristic
246C2×C32C22×SD1611yescharacteristic
247C2×C32#57711yescharacteristic
248C3⋊D3—83no—
249#41—364no—
250C2×Q8—33no—
251C2×Q8#62611yescharacteristic
252C2×D4—93no—
253C2×D4—183no—
254C2×D4—183no—
255C22×C4—183no—
256SD16—186no—
257SD16—366no—
258C2×C8—31no—
259C2×C8—33no—
260C4⋊C4—31no—
261C4⋊C4—63no—
262C4⋊C4—63no—
263C4⋊C4—63no—
264C4⋊C4—91no—
265C4⋊C4#62613yes—
266C22⋊C4—183no—
267C42—63no—
268C42C2×D3213yes—
269C22×C3—21no—
270C22×C3C22×D1211yescharacteristic
271C22×C3C22×#6711yescharacteristic
272C2×D3—126no—
273C2×D3—126no—
274C2×D3—126no—
275C2×D3—123no—
276C2×D3—123no—
277C2×D3—123no—
278C2×D3—123no—
279C2×D3—123no—
280C2×D3—123no—
281C3×C4—23no—
282C3×C4—23no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—26no—
286C3×C4—43no—
287C3×C4—121no—
288C3×C4—123no—
289C3×C4C2×#20416yes—
290C3×C4C2×#21516yes—
291C3×C4C2×#23911yescharacteristic
292C3×C4D3×#4111yescharacteristic
293C3×C4#100911yescharacteristic
294C3×C4#100911yescharacteristic
295Dic3—43no—
296Dic3—41no—
297Dic3—122no—
298Dic3—121no—
299Dic3—121no—
300C32#339211yescharacteristic, complemented, Sylow
301C23—183no—
302Q8—63no—
303Q8—63no—
304Q8#371714yescomplemented
305D4—183no—
306D4—183no—
307D4—366no—
308D4—723no—
309C2×C4—61no—
310C2×C4—63no—
311C2×C4—181no—
312C2×C4—366no—
313C2×C4C2×#62613yes—
314C2×C4C2×#62613yes—
315C2×C4C22×D3211yescharacteristic
316C8—32no—
317C8—63no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3#557711yescharacteristic
322C2×C3C22×#20411yescharacteristic
323C2×C3#575111yescharacteristic
324C2×C3C22×#21511yescharacteristic
325C2×C3#583411yescharacteristic
326C2×C3#583711yescharacteristic
327D3—246no—
328D3—243no—
329D3—243no—
330C22—363no—
331C22—363no—
332C22—363no—
333C22C22×#62611yescharacteristic
334C4—23no—
335C4—123no—
336C4—121no—
337C4—361no—
338C4C2×#371716yes—
339C4D3×#23911yescharacteristic
340C4#6432011yescharacteristic
341C3—21no—
342C3#7991111yescharacteristic, complemented
343C3#8368511yescharacteristic, complemented
344C2—723no—
345C2C22×#371711yescharacteristic
346C2#1059541011yescharacteristic
347C2#1059544811yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radical#10596048C2
hypercenterC4×Q8D32
hypocenterC32#3392
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#5570C2semidirect
3. C3×Q8×C3⋊C8C2semidirect
4. #10590379C2semidirect
5. #10590373C2semidirect
6. #10590367C2semidirect
7. C4×#3717C2semidirect
30. #5570D3semidirect
36. Q8×C3⋊C8D3semidirect
100. C32×Q8SD16semidirect
115. C3×C3⋊C8#41semidirect
206. C3×Q8#204semidirect
207. C3×Q8#215semidirect
244. C3⋊C8#239semidirect
300. C32#3392semidirect
304. Q8#3717semidirect
342. C3#79911semidirect
343. C3#83685semidirect

Maximal subgroups

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

subgroupindex
#105960482
C3×#55702
C3×Q8×C3⋊C82
#105903792
#105903732
#105903672
C4×#37172
#836853
#799113