Finite Group Databasebeta

← #11868721

Subgroups — #11868721

Order 1152.

1128 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic
2#10595941C211yesmaximal, characteristic
3#10592924C212yesmaximal
4#10592721C214yesmaximal, complemented
5#10592720C212yesmaximal, complemented
6#10592719C212yesmaximal, complemented
7#10592663C212yesmaximal
8#10592623C211yesmaximal, characteristic
9#82431—34nomaximal
10C22×C32×Q8C2211yescharacteristic
11C32×C4×Q8—21no—
12C32×C4×Q8—21no—
13C32×C4×Q8C2212yes—
14C32×C4×Q8C2212yes—
15C2×C32×C4⋊C4—21no—
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C42—21no—
18C2×C32×C42C2211yescharacteristic
19#64135C2212yes—
20#64133—21no—
21#64133—21no—
22C2×#3748C2212yes—
23#64079C2212yes—
24#64078C2211yescharacteristic
25#64078C2211yescharacteristic
26C4×#634—21no—
27C4×#634—21no—
28#63662C2214yes—
29#63662C2214yes—
30#63635C2214yes—
31#63634C2214yes—
32C4⋊(C3⋊(C3⋊C8))C2212yes—
33C4⋊(C3⋊(C3⋊C8))C2212yes—
34C2×C3×C4×Q8D314yes—
35#5470—34no—
36#5225—38no—
37#5022—316no—
38#5021—38no—
39#5020—38no—
40#4964—38no—
41#4924—34no—
42C2×C32×Q8—21no—
43C2×C32×Q8—22no—
44C2×C32×Q8Q814yescomplemented
45C2×C32×Q8C2312yes—
46C22×C32×C4—21no—
47C22×C32×C4—21no—
48C22×C32×C4D411yescharacteristic
49C22×C32×C4D411yescharacteristic
50C22×C32×C4C2311yescharacteristic
51C2×#634—21no—
52C2×#634—21no—
53C32×C4⋊C4—22no—
54C32×C4⋊C4—22no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4C2312yes—
57C32×C42—22no—
58C32×C42—22no—
59C32×C42D411yescharacteristic
60C32×C42D411yescharacteristic
61C32×C42C2311yescharacteristic
62C32×C42C2311yescharacteristic
63#3757—22no—
64C4⋊(C3⋊Dic3)—21no—
65C4⋊(C3⋊Dic3)—21no—
66C4⋊(C3⋊Dic3)C2312yes—
67C4⋊(C3⋊Dic3)C2312yes—
68#3750—22no—
69#3750—22no—
70C4×C3⋊Dic3—22no—
71#3748—24no—
72C2×C3⋊(C3⋊C8)C2314yes—
73#3078—91noSylow
74C22×C3×Q8C2×D314yes—
75C3×C4×Q8—24no—
76C3×C4×Q8—24no—
77C3×C4×Q8C2×D318yes—
78C3×C4×Q8C2×D318yes—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4C2×D314yes—
81C2×C3×C42—24no—
82C2×C3×C42C2×D314yes—
83#926—38no—
84#924—64no—
85#924—64no—
86C2×#208—38no—
87#870—38no—
88#869—34no—
89#869—34no—
90C4×Dic6—64no—
91C4×Dic6—64no—
92#835—316no—
93#835—316no—
94#808—316no—
95#807—316no—
96C4⋊(C3⋊C8)—38no—
97C4⋊(C3⋊C8)—38no—
98C23×C32C2×D411yescharacteristic
99C32×Q8—24no—
100C32×Q8—22no—
101C32×Q8C2×Q814yes—
102C2×C32×C4—24no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—41no—
107C2×C32×C4—42no—
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×Q812yes—
114C2×C32×C4#4112yes—
115C2×C32×C4C2411yescharacteristic
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118#634—42no—
119#634—42no—
120C3⋊(C3⋊C8)—24no—
121#533—91no—
122C2×C4×Q8C3⋊D311yescharacteristic
123#479—94no—
124#477—92no—
125#475—92no—
126#428—92no—
127#423—91no—
128#419—92no—
129C2×C3×Q8—28no—
130C2×C3×Q8—24no—
131C2×C3×Q8Dic6116yescomplemented
132C2×C3×Q8C22×D318yes—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4#6714yes—
136C22×C3×C4#6714yes—
137C22×C3×C4C22×D314yes—
138C2×Dic6—64no—
139C2×Dic6—64no—
140C3×C4⋊C4—28no—
141C3×C4⋊C4—28no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4C22×D318yes—
144C3×C42—28no—
145C3×C42—28no—
146C3×C42#6714yes—
147C3×C42#6714yes—
148C3×C42C22×D314yes—
149C3×C42C22×D314yes—
150#217—68no—
151C4⋊Dic3—38no—
152C4⋊Dic3—38no—
153C4⋊Dic3—64no—
154C4⋊Dic3—64no—
155Dic3⋊C4—68no—
156Dic3⋊C4—68no—
157C4×Dic3—68no—
158#208—616no—
159C2×C3⋊C8—316no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32C22⋊Q812yes—
163C22×C32C22×D411yescharacteristic
164C32×C4—24no—
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4—42no—
169C32×C4C22⋊Q812yes—
170C32×C4#13712yes—
171C32×C4C22×Q811yescharacteristic
172C32×C4C2×#4111yescharacteristic
173C3⋊Dic3—42no—
174C3⋊Dic3—42no—
175C22×Q8C2×C3⋊D311yescharacteristic
176C2×M16—92no—
177C4⋊Q8—91no—
178C4⋊Q8—91no—
179#125—92no—
180C22⋊Q8—181no—
181C22⋊Q8—181no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—181no—
185C4×Q8—181no—
186C4×Q8C2×C3⋊D312yes—
187C4×Q8C2×C3⋊D312yes—
188#117—92no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C2×C3⋊D311yescharacteristic
191C2×C42—21no—
192C2×C42C2×C3⋊D311yescharacteristic
193#107—94no—
194#106—94no—
195C4⋊C8—92no—
196C4⋊C8—92no—
197#103—94no—
198#103—94no—
199C23×C3C2×#6714yes—
200C3×Q8—216no—
201C3×Q8—28no—
202C3×Q8C2×Dic6116yes—
203C2×C3×C4—216no—
204C2×C3×C4—28no—
205C2×C3×C4—28no—
206C2×C3×C4—28no—
207C2×C3×C4—44no—
208C2×C3×C4—48no—
209C2×C3×C4C2×Dic618yes—
210C2×C3×C4#23518yes—
211C2×C3×C4C2×#6714yes—
212C2×C3×C4C2×#6714yes—
213C2×C3×C4C2×#6714yes—
214C2×C3×C4C2×#6714yes—
215C2×C3×C4C2×#6714yes—
216C2×C3×C4C23×D314yes—
217C2×Dic3—68no—
218C2×Dic3—68no—
219Dic6—128no—
220Dic6—128no—
221C3⋊C8—616no—
222C2×C32—22no—
223C2×C32C2×C22⋊Q811yescharacteristic
224C2×C32C2×#13711yescharacteristic
225C2×C32#57511yescharacteristic
226C2×Q8—22no—
227C2×Q8—21no—
228C2×Q8—181no—
229C2×Q8—181no—
230C2×Q8#63414yescomplemented
231C2×Q8C22×C3⋊D312yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4#63811yescharacteristic
235C22×C4#63811yescharacteristic
236C22×C4C22×C3⋊D311yescharacteristic
237M16—184no—
238C2×C8—94no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—21no—
242C4⋊C4—92no—
243C4⋊C4—92no—
244C4⋊C4—182no—
245C4⋊C4—182no—
246C4⋊C4—181no—
247C4⋊C4—181no—
248C4⋊C4C22×C3⋊D312yes—
249C22⋊C4—182no—
250C42—22no—
251C42—22no—
252C42—182no—
253C42#63811yescharacteristic
254C42#63811yescharacteristic
255C42C22×C3⋊D311yescharacteristic
256C42C22×C3⋊D311yescharacteristic
257C22×C3—24no—
258C22×C3—24no—
259C22×C3#92418yes—
260C22×C3C22×#6714yes—
261C3×C4—216no—
262C3×C4—24no—
263C3×C4—24no—
264C3×C4—24no—
265C3×C4—48no—
266C3×C4#92418yes—
267C3×C4#94218yes—
268C3×C4C22×Dic614yes—
269C3×C4C2×#23514yes—
270Dic3—128no—
271Dic3—128no—
272C32#307811yescharacteristic, complemented, Sylow
273C23C2×#63811yescharacteristic
274Q8—22no—
275Q8—24no—
276Q8—362no—
277Q8—362no—
278Q8C2×#63414yes—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—182no—
286C2×C4—182no—
287C2×C4C2×#63412yes—
288C2×C4#382812yes—
289C2×C4C2×#63811yescharacteristic
290C2×C4C2×#63811yescharacteristic
291C2×C4C2×#63811yescharacteristic
292C2×C4C2×#63811yescharacteristic
293C2×C4C2×#63811yescharacteristic
294C2×C4C23×C3⋊D311yescharacteristic
295C8—184no—
296C2×C3—28no—
297C2×C3C2×#92414yes—
298C2×C3C2×#94214yes—
299C2×C3#582214yes—
300C22—21no—
301C22—21no—
302C22#6413312yes—
303C22C22×#63811yescharacteristic
304C4—24no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—42no—
309C4—362no—
310C4—362no—
311C4#6413312yes—
312C4#6415112yes—
313C4C22×#63411yescharacteristic
314C4C2×#382811yescharacteristic
315C3#8243114yescomplemented
316C2—22no—
317C2C2×#6413311yescharacteristic
318C2C2×#6415111yescharacteristic
319C2#1059629311yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×C4×Q8C2
socleC22×C32C22×D4
prime radicalC23×C32C2×D4
hypercenterC2×C4×Q8C3⋊D3
hypocenterC32#3078
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
4. #10592721C2semidirect
5. #10592720C2semidirect
6. #10592719C2semidirect
44. C2×C32×Q8Q8semidirect
131. C2×C3×Q8Dic6semidirect
230. C2×Q8#634semidirect
272. C32#3078semidirect
315. C3#82431semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
#105959412
#105929242
#105927212
#105927202
#105927192
#105926632
#105926232
#824313