Finite Group Databasebeta

← #11861507

Subgroups — #11861507

Order 1152.

686 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64096C211yesmaximal, characteristic
2#10593805C211yesmaximal, characteristic
3#10593805C212yesmaximal
4C4×#3792C212yesmaximal
5C4×#3792C211yesmaximal, characteristic
6#10593756C212yesmaximal, complemented
7#10593754C214yesmaximal, complemented
8#10593750C212yesmaximal, complemented
9C22×#634C411yescharacteristic
10C2×#3792C2212yes—
11C2×#3792C2211yescharacteristic
12C2×#3792C2211yescharacteristic
13C2×C32×C4⋊C4C411yescharacteristic
14C2×#3750C412yes—
15C2×C4×C3⋊Dic3C2211yescharacteristic
16C2×C4×C3⋊Dic3C2212yes—
17#64096C412yes—
18#64096C411yescharacteristic
19#64096C411yescharacteristic
20#64096C2211yescharacteristic
21#64096C2212yes—
22#64096C2211yescharacteristic
23#63779C2214yes—
24#63779C2212yes—
25#63777C2212yes—
26#63777C2214yes—
27C4⋊#622C2212yes—
28C4⋊#622C2214yes—
29C4×#622C2214yes—
30C4×#622C2212yes—
31C2×#887—62no—
32C22×C32×C4C2×C411yescharacteristic
33C22×C32×C4C2×C412yes—
34C22×C3⋊Dic3C2×C412yes—
35C22×C3⋊Dic3C2×C411yescharacteristic
36C22×C3⋊Dic3C2311yescharacteristic
37C2×#634—24no—
38C2×#634C2×C411yescharacteristic
39C2×#634C2×C412yes—
40C2×#634C2×C411yescharacteristic
41#3792—24no—
42#3792—22no—
43#3792Q814yescomplemented
44C2×#622C2314yes—
45C2×#622C2312yes—
46C2×#622C2312yes—
47C32×C4⋊C4C2×C412yes—
48C32×C4⋊C4C2×C411yescharacteristic
49C32×C4⋊C4C2×C411yescharacteristic
50#3750C2×C412yes—
51#3750C2×C412yes—
52#3750C2×C412yes—
53#3750C2×C412yes—
54C4×C3⋊Dic3C2×C411yescharacteristic
55C4×C3⋊Dic3C2×C411yescharacteristic
56C4×C3⋊Dic3C2×C412yes—
57C4×C3⋊Dic3C2×C412yes—
58C4×C3⋊Dic3C2311yescharacteristic
59C4×C3⋊Dic3C2312yes—
60C4×C3⋊Dic3C2312yes—
61C4×C3⋊Dic3C2311yescharacteristic
62Q8×M16—91nomaximal, Sylow
63C22×Dic6—62no—
64C2×C3×C4⋊C4—22no—
65C2×Dic3⋊C4—64no—
66C2×C4×Dic3—64no—
67C2×C4×Dic3—62no—
68#887—62no—
69#887—64no—
70#887—62no—
71#887—62no—
72#887—64no—
73#887—62no—
74C23×C32C22×C411yescharacteristic
75C2×C32×C4—22no—
76C2×C32×C4—42no—
77C2×C32×C4C22×C412yes—
78C2×C32×C4C22×C412yes—
79C2×C32×C4C22×C411yescharacteristic
80C2×C32×C4C22×C411yescharacteristic
81C2×C3⋊Dic3—24no—
82C2×C3⋊Dic3—41no—
83C2×C3⋊Dic3C22×C411yescharacteristic
84C2×C3⋊Dic3C22×C411yescharacteristic
85C2×C3⋊Dic3C22×C412yes—
86C2×C3⋊Dic3C22×C411yescharacteristic
87C2×C3⋊Dic3C22×C412yes—
88C2×C3⋊Dic3C2×Q812yes—
89C2×C3⋊Dic3#4112yes—
90C2×C3⋊Dic3C2411yescharacteristic
91#634—22no—
92#634—24no—
93#634M1614yescomplemented
94#622—24no—
95#622—22no—
96#622C2×Q814yescomplemented
97C2×C4×Q8—91no—
98#446—94no—
99#446—92no—
100C8×Q8—92no—
101#423—91no—
102#423—92no—
103C4×M16—92no—
104C4×M16—91no—
105C22×C3×C4—22no—
106C22×C3×C4—24no—
107C22×Dic3—62no—
108C22×Dic3—64no—
109C22×Dic3—62no—
110C2×Dic6—62no—
111C2×Dic6—64no—
112C2×Dic6—62no—
113C2×Dic6—616no—
114C3×C4⋊C4—24no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117Dic3⋊C4—64no—
118Dic3⋊C4—64no—
119Dic3⋊C4—64no—
120Dic3⋊C4—64no—
121C4×Dic3—64no—
122C4×Dic3—64no—
123C4×Dic3—62no—
124C4×Dic3—62no—
125C4×Dic3—64no—
126C4×Dic3—62no—
127C4×Dic3—64no—
128C4×Dic3—62no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32C4×Q812yes—
132C22×C32C23×C411yescharacteristic
133C32×C4—21no—
134C32×C4—22no—
135C32×C4—22no—
136C32×C4C2×M1612yes—
137C3⋊Dic3—22no—
138C3⋊Dic3—21no—
139C3⋊Dic3—21no—
140C3⋊Dic3C4×Q812yes—
141C3⋊Dic3C2×M1614yes—
142C3⋊Dic3C22×Q811yescharacteristic
143C3⋊Dic3C2×#4111yescharacteristic
144C22×Q8—91no—
145C2×M16—91no—
146C2×M16—92no—
147C2×M16—91no—
148C4×Q8—91no—
149C4×Q8—92no—
150C4×Q8—91no—
151C4×Q8—91no—
152C4×Q8—92no—
153C4×Q8—91no—
154C2×C4⋊C4—92no—
155C2×C4⋊C4#15711yescharacteristic
156C2×C42—92no—
157C2×C42—91no—
158C4⋊C8—92no—
159C4⋊C8—94no—
160C4⋊C8—94no—
161C4⋊C8—92no—
162#97—92no—
163#97—94no—
164C4×C8—94no—
165C4×C8—92no—
166C23×C3—22no—
167C2×C3×C4—28no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—24no—
171C2×C3×C4—24no—
172C2×C3×C4—48no—
173C2×Dic3—62no—
174C2×Dic3—64no—
175C2×Dic3—64no—
176C2×Dic3—64no—
177C2×Dic3—64no—
178C2×Dic3—62no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—616no—
182C2×Dic3—124no—
183Dic6—68no—
184Dic6—616no—
185Dic6—68no—
186C2×C32—22no—
187C2×C32C2×C4×Q811yescharacteristic
188C2×C32C22×M1611yescharacteristic
189C2×C32#56811yescharacteristic
190C2×Q8—91no—
191C2×Q8—92no—
192C2×Q8—91no—
193C2×Q8—184no—
194C22×C4—91no—
195C22×C4—92no—
196C22×C4—91no—
197C22×C4C2×#15711yescharacteristic
198C22×C4C2×#15712yes—
199M16—94no—
200M16—184no—
201M16—182no—
202C2×C8—92no—
203C2×C8—94no—
204C2×C8—92no—
205C4⋊C4—92no—
206C4⋊C4—92no—
207C4⋊C4—92no—
208C4⋊C4—92no—
209C4⋊C4C2×#15712yes—
210C4⋊C4C2×#15711yescharacteristic
211C4⋊C4C2×#15711yescharacteristic
212C42—92no—
213C42—91no—
214C42—91no—
215C42—92no—
216C42—92no—
217C42—91no—
218C42—91no—
219C42—92no—
220C22×C3—24no—
221C22×C3—24no—
222C22×C3—24no—
223C22×C3—22no—
224C3×C4—24no—
225C3×C4—24no—
226C3×C4—44no—
227C3×C4—44no—
228Dic3—68no—
229Dic3—64no—
230Dic3—62no—
231Dic3—62no—
232Dic3—68no—
233Dic3—122no—
234Dic3—122no—
235C32Q8×M1611yescharacteristic, complemented, Sylow
236C23C22×#15711yescharacteristic
237Q8—94no—
238Q8—184no—
239Q8—182no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—92no—
243C2×C4—92no—
244C2×C4—91no—
245C2×C4—91no—
246C2×C4—92no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—92no—
250C2×C4—184no—
251C2×C4—361no—
252C2×C4C22×#15712yes—
253C2×C4C22×#15712yes—
254C2×C4C22×#15711yescharacteristic
255C2×C4C22×#15711yescharacteristic
256C8—94no—
257C8—184no—
258C8—182no—
259C2×C3—28no—
260C2×C3—22no—
261C2×C3—22no—
262C2×C3—22no—
263C22—21no—
264C22—21no—
265C22Q8×#15712yes—
266C22C23×#15711yescharacteristic
267C4—21no—
268C4—22no—
269C4—22no—
270C4—94no—
271C4—91no—
272C4—91no—
273C4—92no—
274C4—182no—
275C4—181no—
276C4—181no—
277C4#6428312yes—
278C3—22no—
279C2—22no—
280C2C2×#6428311yescharacteristic
281C2#1059664011yescharacteristic
282C2C2×Q8×#15711yescharacteristic
subgroupG/H
center, FrattiniC22C23×#157
commutator, socleC22×C32C23×C4
FittingC2×C32×C4⋊C4C4
prime radicalC23×C32C22×C4
cosocleC2×C3⋊Dic3C24
hypercenterC2×C4⋊C4#157
hypocenterC32Q8×M16
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #10593756C2semidirect
7. #10593754C2semidirect
8. #10593750C2semidirect
43. #3792Q8semidirect
93. #634M16semidirect
96. #622C2×Q8semidirect
235. C32Q8×M16semidirect

Maximal subgroups

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

subgroupindex
C2×#640962
#105938052
C4×#37922
#105937562
#105937542
#105937502
Q8×M169