Finite Group Databasebeta

← #11785119

Subgroups — #11785119

Order 1152.

762 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×#64090C211yesmaximal, characteristic
2C2×#63788C212yesmaximal
3#10593805C212yesmaximal, complemented
4#10590208C212yesmaximal, complemented
5C22×#638C2211yescharacteristic
6C23×C3⋊Dic3C411yescharacteristic
7C22×C4×C3⋊D3C411yescharacteristic
8C2×#3792C2214yes—
9C2×C32×C22⋊C4C411yescharacteristic
10C2×#3752C411yescharacteristic
11C2×#3750C2211yescharacteristic
12C2×C4×C3⋊Dic3C2211yescharacteristic
13#64090—22no—
14#64090—22no—
15#63788—24no—
16#63779—22no—
17C4⋊#622—22no—
18C2×#67⋊C4—62no—
19C24×C32C2×C411yescharacteristic
20C23×C3⋊D3C2×C411yescharacteristic
21C22×C32×C4C2×C411yescharacteristic
22C22×C32×C4C2×C411yescharacteristic
23C2×#638—22no—
24C2×#638—22no—
25C2×#638D412yes—
26C2×#638Q812yes—
27C22×C3⋊Dic3—21no—
28C22×C3⋊Dic3—21no—
29C22×C3⋊Dic3—42no—
30C22×C3⋊Dic3C2×C411yescharacteristic
31C22×C3⋊Dic3C2×C411yescharacteristic
32C22×C3⋊Dic3C2311yescharacteristic
33C2×C4×C3⋊D3—21no—
34C2×C4×C3⋊D3—21no—
35C2×C4×C3⋊D3—42no—
36#3792—24no—
37#3792—24no—
38C2×#622—24no—
39C32×C22⋊C4—22no—
40#3752—22no—
41#3750—22no—
42C4×C3⋊Dic3D412yes—
43C4×C3⋊Dic3D412yes—
44#1800—91nomaximal, Sylow
45C22×#67—62no—
46C23×Dic3—62no—
47C22×C4×D3—62no—
48C2×C3×C22⋊C4—22no—
49C2×(C2×D3)⋊C4—62no—
50C2×Dic3⋊C4—62no—
51C2×C4×Dic3—62no—
52#67⋊C4—68no—
53#67⋊C4—68no—
54C23×C32—21no—
55C23×C32—21no—
56C23×C32—41no—
57C23×C32—41no—
58C23×C32C4⋊C412yes—
59C23×C32C22×C411yescharacteristic
60C22×C3⋊D3—21no—
61C22×C3⋊D3—21no—
62C22×C3⋊D3—41no—
63C22×C3⋊D3—41no—
64C22×C3⋊D3C4⋊C412yes—
65C2×C32×C4—21no—
66C2×C32×C4—41no—
67C2×C32×C4—41no—
68C2×C32×C4C22⋊C412yes—
69#638—24no—
70#638—24no—
71C2×C3⋊Dic3—22no—
72C2×C3⋊Dic3—21no—
73C2×C3⋊Dic3—42no—
74C2×C3⋊Dic3—42no—
75C2×C3⋊Dic3—41no—
76C2×C3⋊Dic3C22⋊C411yescharacteristic
77C2×C3⋊Dic3C22⋊C411yescharacteristic
78C2×C3⋊Dic3C2×D411yescharacteristic
79C2×C3⋊Dic3C2×D412yes—
80C2×C3⋊Dic3C2×Q811yescharacteristic
81C2×C3⋊Dic3#4112yes—
82C4×C3⋊D3—42no—
83C4×C3⋊D3—42no—
84#622—44no—
85C2×C4×D4—91no—
86#423—92no—
87C2×#100—92no—
88#343—92no—
89C24×C3—22no—
90C23×D3—62no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C2×#67—64no—
94C2×#67—64no—
95C2×#67—68no—
96C2×#67—68no—
97C22×Dic3—62no—
98C22×Dic3—62no—
99C22×Dic3—62no—
100C22×Dic3—64no—
101C22×Dic3—64no—
102C22×Dic3—128no—
103C2×C4×D3—64no—
104C2×C4×D3—64no—
105C2×C4×D3—128no—
106C3×C22⋊C4—28no—
107(C2×D3)⋊C4—68no—
108Dic3⋊C4—68no—
109C4×Dic3—64no—
110C4×Dic3—64no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—22no—
114C22×C32—42no—
115C22×C32—42no—
116C22×C32—41no—
117C22×C32—41no—
118C22×C32C2×C22⋊C411yescharacteristic
119C22×C32C2×C4⋊C411yescharacteristic
120C22×C32#11711yescharacteristic
121C2×C3⋊D3—22no—
122C2×C3⋊D3—42no—
123C2×C3⋊D3—41no—
124C2×C3⋊D3—41no—
125C2×C3⋊D3—42no—
126C32×C4—22no—
127C32×C4—41no—
128C3⋊Dic3—21no—
129C3⋊Dic3—41no—
130C3⋊Dic3#10012yescomplemented
131C3⋊Dic3#12112yes—
132C3⋊Dic3C22⋊Q812yes—
133C22×D4—91no—
134C23×C4—91no—
135C23×C4—91no—
136C2×M16—94no—
137C4×D4—182no—
138C4×D4—182no—
139C2×C4⋊C4—91no—
140C2×C22⋊C4—91no—
141C2×C22⋊C4#15711yescharacteristic
142C2×C42—91no—
143C4⋊C8—182no—
144C4⋊C8—182no—
145#100—184no—
146C23×C3—24no—
147C23×C3—24no—
148C23×C3—22no—
149C23×C3—24no—
150C23×C3—44no—
151C23×C3—44no—
152C22×D3—64no—
153C22×D3—64no—
154C22×D3—64no—
155C22×D3—124no—
156C22×D3—124no—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4—44no—
160C2×C3×C4—44no—
161#67—616no—
162#67—616no—
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—64no—
166C2×Dic3—62no—
167C2×Dic3—62no—
168C2×Dic3—64no—
169C2×Dic3—68no—
170C2×Dic3—64no—
171C2×Dic3—128no—
172C2×Dic3—128no—
173C2×Dic3—124no—
174C4×D3—128no—
175C4×D3—128no—
176C2×C32—21no—
177C2×C32—21no—
178C2×C32—42no—
179C2×C32#38011yescharacteristic
180C2×C32C2×#10011yescharacteristic
181C2×C32#41311yescharacteristic
182C3⋊D3—42no—
183C24—91no—
184C24C2×#15711yescharacteristic
185C2×D4—92no—
186C2×D4—92no—
187C2×D4—182no—
188C2×D4—182no—
189C22×C4—91no—
190C22×C4—91no—
191C22×C4—91no—
192C22×C4—181no—
193C22×C4—181no—
194C22×C4—181no—
195C22×C4—181no—
196C22×C4—362no—
197C22×C4—362no—
198C22×C4C2×#15711yescharacteristic
199C22×C4C2×#15711yescharacteristic
200M16—184no—
201M16—184no—
202C2×C8—184no—
203C4⋊C4—182no—
204C22⋊C4—22no—
205C22⋊C4—182no—
206C42—92no—
207C42—92no—
208C22×C3—24no—
209C22×C3—24no—
210C22×C3—28no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—44no—
215C22×C3—48no—
216C22×C3—48no—
217C22×C3—44no—
218C2×D3—68no—
219C2×D3—124no—
220C2×D3—128no—
221C2×D3—128no—
222C2×D3—124no—
223C3×C4—44no—
224C3×C4—44no—
225Dic3—64no—
226Dic3—64no—
227Dic3—64no—
228Dic3—64no—
229Dic3—124no—
230C32#180011yescharacteristic, complemented, Sylow
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23—92no—
236C23—181no—
237C23—181no—
238C23—361no—
239C23—361no—
240C23C4⋊#15712yes—
241C23C22×#15711yescharacteristic
242D4—184no—
243D4—184no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—92no—
250C2×C4—92no—
251C2×C4—91no—
252C2×C4—91no—
253C2×C4—182no—
254C2×C4—181no—
255C2×C4—362no—
256C2×C4—362no—
257C2×C4—362no—
258C2×C4—362no—
259C2×C4—361no—
260C2×C4C22⋊#15712yes—
261C8—364no—
262C2×C3—24no—
263C2×C3—24no—
264C2×C3—22no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—48no—
268D3—128no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—42no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22—182no—
277C22—362no—
278C22—362no—
279C22—361no—
280C22—361no—
281C22C2×C4⋊#15711yescharacteristic
282C22#6428611yescharacteristic
283C22C2×C22⋊#15711yescharacteristic
284C4—22no—
285C4—41no—
286C4—92no—
287C4—92no—
288C4—92no—
289C4—181no—
290C4—361no—
291C3—22no—
292C2—21no—
293C2—21no—
294C2—42no—
295C2—362no—
296C2#1059381211yescharacteristic
297C2#1059382411yescharacteristic
298C2#1059382511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#157
commutatorC23×C32C22×C4
FrattiniC23C22×#157
FittingC2×C32×C22⋊C4C4
socleC22×C32C2×C22⋊C4
prime radicalC22×#638C22
cosocleC22×C3⋊Dic3C23
hypercenterC2×C22⋊C4#157
hypocenterC32#1800
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #10593805C2semidirect
4. #10590208C2semidirect
130. C3⋊Dic3#100semidirect
230. C32#1800semidirect

Maximal subgroups

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

subgroupindex
C2×#640902
C2×#637882
#105938052
#105902082
#18009