Finite Group Databasebeta

← #11870710

Subgroups — #11870710

Order 1152.

1284 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
2#10595948C211yesmaximal, characteristic
3#10592926C212yesmaximal, complemented
4#10592726C212yesmaximal, complemented
5C4×#3755C214yesmaximal, complemented
6#10592719C212yesmaximal, complemented
7#10592667C212yesmaximal, complemented
8#10592624C211yesmaximal, characteristic, complemented
9#82442—34nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×#3828—21no—
12C32×C4×Q8—21no—
13C32×C4×Q8—21no—
14C32×C4×Q8C2212yes—
15C32×C4×Q8C2212yes—
16C2×C32×C4⋊C4—21no—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C42—21no—
19C2×C32×C42C2211yescharacteristic
20C2×#3755C2214yes—
21#64139—21no—
22#64139C2212yes—
23#64084—21no—
24#64083C2211yescharacteristic
25C4×C4⋊(C3⋊D3)—21no—
26C4×C4⋊(C3⋊D3)C2212yes—
27#64078C2211yescharacteristic
28#63662C2214yes—
29#63657C2212yes—
30#63636C2214yes—
31#63635C2214yes—
32C4⋊(C3⋊(C3⋊C8))C2212yes—
33C4×C3⋊(C3⋊C8)C2212yes—
34C2×C3×C4×Q8D314yescomplemented
35#5477—34no—
36#5227—38no—
37#5027—38no—
38C4×#215—316no—
39#5020—38no—
40#4968—38no—
41#4925—34no—
42C2×C32×Q8—24no—
43C2×C32×Q8—21no—
44C2×C32×Q8C2312yes—
45C22×C32×C4—21no—
46C22×C32×C4—21no—
47C22×C32×C4D411yescharacteristic
48C22×C32×C4D411yescharacteristic
49C22×C32×C4C2311yescharacteristic
50C2×#638—22no—
51#3828—44no—
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)C2312yes—
55C2×C4×C3⋊D3—22no—
56C2×#634—21no—
57C32×C4⋊C4—22no—
58C32×C4⋊C4—22no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2312yes—
61C32×C42—22no—
62C32×C42—22no—
63C32×C42D411yescharacteristic
64C32×C42D411yescharacteristic
65C32×C42C2311yescharacteristic
66C32×C42C2311yescharacteristic
67#3755—28no—
68#3752—22no—
69#3752—22no—
70C4⋊(C3⋊Dic3)—21no—
71C4⋊(C3⋊Dic3)C2312yes—
72C2×C3⋊(C3⋊C8)C2314yes—
73#3103—91noSylow
74C22×C3×Q8C2×D314yes—
75C2×#235—64no—
76C3×C4×Q8—24no—
77C3×C4×Q8—24no—
78C3×C4×Q8C2×D318yes—
79C3×C4×Q8C2×D318yes—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4C2×D314yes—
82C2×C3×C42—24no—
83C2×C3×C42C2×D314yes—
84C2×#215—316no—
85#930—38no—
86#930—64no—
87#875—64no—
88#874—34no—
89C4×D12—38no—
90C4×D12—64no—
91#869—34no—
92#835—316no—
93#830—38no—
94#809—316no—
95#808—316no—
96C4⋊(C3⋊C8)—38no—
97C4×C3⋊C8—38no—
98C23×C32C2×D411yescharacteristic
99C22×C3⋊D3—22no—
100C32×Q8—24no—
101C32×Q8—22no—
102C32×Q8#4114yescomplemented
103C2×C32×C4—24no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—41no—
108C2×C32×C4—42no—
109C2×C32×C4SD1614yescomplemented
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C32×C4C2411yescharacteristic
116#638—44no—
117C2×C3⋊Dic3—22no—
118C4⋊(C3⋊D3)—24no—
119C4⋊(C3⋊D3)—42no—
120C4⋊(C3⋊D3)—42no—
121C4×C3⋊D3—44no—
122#634—42no—
123C3⋊(C3⋊C8)—24no—
124#532—91no—
125C2×C4×Q8C3⋊D311yescharacteristic, complemented
126#481—92no—
127#475—92no—
128#460—92no—
129#448—92no—
130C4×SD16—94no—
131#431—91no—
132C2×C3×Q8—216no—
133C2×C3×Q8—24no—
134C2×C3×Q8C22×D318yes—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4#6714yes—
138C22×C3×C4#6714yes—
139C22×C3×C4C22×D314yes—
140C2×#67—68no—
141#235—1216no—
142C2×D12—38no—
143C2×D12—64no—
144C2×D12—64no—
145C2×C4×D3—68no—
146C2×Dic6—64no—
147C3×C4⋊C4—28no—
148C3×C4⋊C4—28no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4C22×D318yes—
151C3×C42—28no—
152C3×C42—28no—
153C3×C42#6714yes—
154C3×C42#6714yes—
155C3×C42C22×D314yes—
156C3×C42C22×D314yes—
157#215—632no—
158(C2×D3)⋊C4—68no—
159(C2×D3)⋊C4—68no—
160C4⋊Dic3—38no—
161C4⋊Dic3—64no—
162C2×C3⋊C8—316no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32C2×SD1612yes—
166C22×C32C22×D411yescharacteristic
167C2×C3⋊D3—42no—
168C2×C3⋊D3—42no—
169C2×C3⋊D3—42no—
170C32×C4—24no—
171C32×C4—21no—
172C32×C4—21no—
173C32×C4—21no—
174C32×C4—42no—
175C32×C4C2×SD1612yes—
176C32×C4C2×SD1612yes—
177C32×C4C2×#4112yes—
178C3⋊Dic3—42no—
179C2×#41—181no—
180C22×Q8C2×C3⋊D311yescharacteristic
181C2×SD16—94no—
182C4⋊Q8—91no—
183#127—91no—
184#124—181no—
185#121—92no—
186#121—181no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8C2×C3⋊D312yes—
190C4×Q8C2×C3⋊D312yes—
191C4×D4—92no—
192C4×D4—181no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×C3⋊D311yescharacteristic
195C2×C42—21no—
196C2×C42C2×C3⋊D311yescharacteristic
197#106—94no—
198C4⋊C8—92no—
199#103—94no—
200#102—94no—
201C22⋊C8—92no—
202C4×C8—92no—
203C23×C3C2×#6714yes—
204C22×D3—68no—
205C3×Q8—216no—
206C3×Q8—28no—
207C3×Q8#235116yescomplemented
208C2×C3×C4—216no—
209C2×C3×C4—28no—
210C2×C3×C4—28no—
211C2×C3×C4—28no—
212C2×C3×C4—44no—
213C2×C3×C4—48no—
214C2×C3×C4#215116yescomplemented
215C2×C3×C4C2×#6714yes—
216C2×C3×C4C2×#6714yes—
217C2×C3×C4C2×#6714yes—
218C2×C3×C4C2×#6714yes—
219C2×C3×C4C2×#6714yes—
220C2×C3×C4C23×D314yes—
221#67—1216no—
222C2×Dic3—68no—
223D12—616no—
224D12—128no—
225D12—128no—
226C4×D3—1216no—
227Dic6—128no—
228C3⋊C8—616no—
229C2×C32—22no—
230C2×C32#52511yescharacteristic
231C2×C32C22×SD1611yescharacteristic
232C2×C32#57511yescharacteristic
233C3⋊D3—82no—
234#41—364no—
235C2×Q8—24no—
236C2×Q8—21no—
237C2×Q8—181no—
238C2×Q8C22×C3⋊D312yes—
239C2×D4—92no—
240C2×D4—182no—
241C2×D4—181no—
242C2×D4—181no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—182no—
246C22×C4#63811yescharacteristic
247C22×C4#63811yescharacteristic
248C22×C4C22×C3⋊D311yescharacteristic
249SD16—188no—
250C2×C8—94no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—92no—
255C4⋊C4—181no—
256C4⋊C4C22×C3⋊D312yes—
257C22⋊C4—182no—
258C22⋊C4—182no—
259C42—22no—
260C42—22no—
261C42#63811yescharacteristic
262C42#63811yescharacteristic
263C42C22×C3⋊D311yescharacteristic
264C42C22×C3⋊D311yescharacteristic
265C22×C3—24no—
266C22×C3—24no—
267C22×C3C2×#21518yes—
268C22×C3C22×#6714yes—
269C2×D3—128no—
270C2×D3—128no—
271C2×D3—128no—
272C3×C4—216no—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—24no—
276C3×C4—48no—
277C3×C4C2×#21518yes—
278C3×C4C2×#21518yes—
279C3×C4C2×#23518yes—
280Dic3—128no—
281C32#310311yescharacteristic, complemented, Sylow
282C23—182no—
283C23C2×#63811yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—362no—
287Q8#382814yescomplemented
288D4—184no—
289D4—362no—
290D4—364no—
291D4—362no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—182no—
299C2×C4—364no—
300C2×C4#375514yescomplemented
301C2×C4C2×#63811yescharacteristic
302C2×C4C2×#63811yescharacteristic
303C2×C4C2×#63811yescharacteristic
304C2×C4C2×#63811yescharacteristic
305C2×C4C2×#63811yescharacteristic
306C2×C4C23×C3⋊D311yescharacteristic
307C8—184no—
308C2×C3—28no—
309C2×C3#579114yes—
310C2×C3C22×#21514yes—
311C2×C3#582214yes—
312D3—248no—
313C22—21no—
314C22—21no—
315C22—362no—
316C22—362no—
317C22—362no—
318C22C2×#375512yes—
319C22C22×#63811yescharacteristic
320C4—24no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—362no—
326C4C2×#375512yes—
327C4C2×#375512yes—
328C4C2×#382812yes—
329C3#8244214yescomplemented
330C2—22no—
331C2—722no—
332C2#1059626211yescharacteristic
333C2C22×#375511yescharacteristic
334C2#1059629311yescharacteristic
subgroupG/H
centerC2×C4C2×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×C4×Q8C2
socleC22×C32C22×D4
prime radical#10595948C2
hypercenterC2×C4×Q8C3⋊D3
hypocenterC32#3103
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×Q8C2semidirect
3. #10592926C2semidirect
4. #10592726C2semidirect
5. C4×#3755C2semidirect
6. #10592719C2semidirect
7. #10592667C2semidirect
8. #10592624C2semidirect
34. C2×C3×C4×Q8D3semidirect
102. C32×Q8#41semidirect
109. C2×C32×C4SD16semidirect
125. C2×C4×Q8C3⋊D3semidirect
207. C3×Q8#235semidirect
214. C2×C3×C4#215semidirect
281. C32#3103semidirect
287. Q8#3828semidirect
300. C2×C4#3755semidirect
329. C3#82442semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
#105959482
#105929262
#105927262
C4×#37552
#105927192
#105926672
#105926242
#824423