Finite Group Databasebeta

← #11858327

Subgroups — #11858327

Order 1152.

782 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 189 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×#64286C211yesmaximal, characteristic
2C2×#63818C212yesmaximal
3#10593328C214yesmaximal, complemented
4#10593327C218yesmaximal, complemented
5C22×C4×C3⋊D3C2211yescharacteristic
6C2×C22⋊#157C412yes—
7#64286C414yes—
8#64286C2214yes—
9C2×C4⋊#157C412yes—
10C2×C4×#157C2212yes—
11C2×C3×#208—22no—
12C2×#3712—22no—
13#63818—22no—
14#63818—22no—
15#63818C2214yes—
16#63818C2214yes—
17C2×#3710C2214yes—
18#63732C22116yescomplemented
19C2×D3×M16—62no—
20C23×C3⋊D3C2×C411yescharacteristic
21C22×#157C2×C414yes—
22C22×C32×C4D411yescharacteristic
23C22×C3⋊Dic3Q811yescharacteristic
24C2×C4×C3⋊D3C2×C411yescharacteristic
25C2×C4×C3⋊D3C2×C412yes—
26C2×C4×C3⋊D3D411yescharacteristic
27C2×C4×C3⋊D3D412yes—
28C2×C4×C3⋊D3Q812yes—
29C2×C4×C3⋊D3Q811yescharacteristic
30C2×C4×C3⋊D3C2311yescharacteristic
31C2×C4×C3⋊D3C2312yes—
32C22⋊#157C2×C418yes—
33C4⋊#157C2×C418yes—
34C4×#157C2×C414yes—
35C4×#157C2314yes—
36C3×#208—24no—
37C3×#208—24no—
38C2×C3×C3⋊C8—24no—
39#3712—24no—
40#3712—24no—
41#3710—24no—
42#3710C2318yes—
43#2905—91nomaximal, Sylow
44C22×C4×D3—62no—
45C2×C3×M16—62no—
46C2×#208—22no—
47D3×M16—64no—
48D3×M16—64no—
49D3×M16—64no—
50D3×M16—64no—
51C2×ΓL(1,𝔽25)—64no—
52C2×D3×C8—64no—
53C23×C32C4⋊C411yescharacteristic
54C22×C3⋊D3—42no—
55C22×C3⋊D3C4⋊C411yescharacteristic
56C22×C3⋊D3C4⋊C412yes—
57C22×C3⋊D3C22×C411yescharacteristic
58C22×C3⋊D3C22×C412yes—
59C2×#157—44no—
60C2×#157C22×C418yes—
61C2×C32×C4C4⋊C411yescharacteristic
62C2×C32×C4C4⋊C412yes—
63C2×C32×C4C2×D412yes—
64C2×C32×C4C2×D411yescharacteristic
65C2×C3⋊Dic3C4⋊C411yescharacteristic
66C2×C3⋊Dic3C4⋊C412yes—
67C2×C3⋊Dic3C2×Q811yescharacteristic
68C2×C3⋊Dic3C2×Q812yes—
69C4×C3⋊D3C4⋊C412yes—
70C4×C3⋊D3C4⋊C412yes—
71C4×C3⋊D3C4⋊C412yes—
72C4×C3⋊D3C22×C411yescharacteristic
73C4×C3⋊D3C22×C412yes—
74C4×C3⋊D3C2×D412yes—
75C4×C3⋊D3C2×D411yescharacteristic
76C4×C3⋊D3C2×Q811yescharacteristic
77C4×C3⋊D3C2×Q812yes—
78C4×C3⋊D3C2411yescharacteristic
79C3×C3⋊C8—28no—
80C22×M16—92no—
81C2×#117—91no—
82#424—98no—
83#423—94no—
84C23×D3—62no—
85C22×C3×C4—22no—
86C22×Dic3—62no—
87C2×C4×D3—62no—
88C2×C4×D3—62no—
89C2×C4×D3—62no—
90C2×C4×D3—62no—
91C2×C4×D3—62no—
92C2×C4×D3—62no—
93C2×C4×D3—62no—
94C2×C4×D3—62no—
95C2×C4×D3—62no—
96C2×C4×D3—62no—
97C2×C4×D3—62no—
98C2×C4×D3—62no—
99C3×M16—64no—
100C3×M16—64no—
101C2×C3×C8—64no—
102#208—24no—
103#208—24no—
104C2×C3⋊C8—24no—
105ΓL(1,𝔽25)—68no—
106ΓL(1,𝔽25)—68no—
107D3×C8—68no—
108D3×C8—68no—
109C22×C32—41no—
110C22×C32C2×C4⋊C412yes—
111C22×C32C2×C4⋊C411yescharacteristic
112C2×C3⋊D3—24no—
113C2×C3⋊D3—22no—
114C2×C3⋊D3—41no—
115C2×C3⋊D3—42no—
116C2×C3⋊D3C2×C4⋊C412yes—
117C2×C3⋊D3C2×C4⋊C411yescharacteristic
118C2×C3⋊D3C23×C411yescharacteristic
119#157—28no—
120C32×C4C2×C4⋊C411yescharacteristic
121C32×C4C2×C4⋊C412yes—
122C32×C4C22×D411yescharacteristic
123C3⋊Dic3C2×C4⋊C411yescharacteristic
124C3⋊Dic3C2×C4⋊C412yes—
125C3⋊Dic3C22×Q811yescharacteristic
126C23×C4—91no—
127C2×M16—62no—
128C2×M16—94no—
129C2×M16—94no—
130C2×M16—182no—
131C2×M16—182no—
132C2×M16—182no—
133C22×C8—94no—
134#117—94no—
135#117—94no—
136C2×C4⋊C4—92no—
137C2×C22⋊C4—92no—
138C2×C42—92no—
139C4⋊C8—916no—
140C23×C3—22no—
141C22×D3—62no—
142C22×D3—62no—
143C22×D3—62no—
144C22×D3—62no—
145C22×D3—62no—
146C22×D3—62no—
147C22×D3—128no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C4×D3—62no—
161C4×D3—62no—
162C4×D3—62no—
163C4×D3—62no—
164C4×D3—62no—
165C4×D3—62no—
166C4×D3—62no—
167C4×D3—62no—
168C4×D3—62no—
169C4×D3—62no—
170C4×D3—62no—
171C4×D3—62no—
172C4×D3—62no—
173C4×D3—62no—
174C4×D3—62no—
175C4×D3—62no—
176C3×C8—68no—
177C3⋊C8—28no—
178C2×C32—22no—
179C2×C32—21no—
180C2×C32C22×C4⋊C411yescharacteristic
181C3⋊D3—22no—
182C3⋊D3—21no—
183C3⋊D3#56812yes—
184C24—91no—
185C22×C4—91no—
186C22×C4—91no—
187C22×C4—92no—
188C22×C4—91no—
189C22×C4—92no—
190C22×C4—91no—
191C22×C4—92no—
192C22×C4—91no—
193C22×C4—92no—
194C22×C4—94no—
195C22×C4GO+(4,𝔽2)11yescharacteristic
196M16—64no—
197M16—64no—
198M16—184no—
199M16—184no—
200C2×C8—64no—
201C2×C8—98no—
202C2×C8—184no—
203C4⋊C4—98no—
204C22⋊C4—98no—
205C42—94no—
206C42—94no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—44no—
211C2×D3—62no—
212C2×D3—62no—
213C2×D3—62no—
214C2×D3—62no—
215C2×D3—68no—
216C2×D3—124no—
217C2×D3—124no—
218C2×D3—124no—
219C2×D3—124no—
220C2×D3—124no—
221C3×C4—22no—
222C3×C4—22no—
223C3×C4—22no—
224C3×C4—22no—
225Dic3—62no—
226Dic3—62no—
227Dic3—62no—
228Dic3—62no—
229C32#290511yescharacteristic, complemented, Sylow
230C23—92no—
231C23—91no—
232C23—91no—
233C23—92no—
234C23—362no—
235C23#377311yescharacteristic
236C2×C4—91no—
237C2×C4—92no—
238C2×C4—91no—
239C2×C4—92no—
240C2×C4—91no—
241C2×C4—92no—
242C2×C4—91no—
243C2×C4—91no—
244C2×C4—92no—
245C2×C4—91no—
246C2×C4—92no—
247C2×C4—92no—
248C2×C4—92no—
249C2×C4—92no—
250C2×C4—98no—
251C2×C4—364no—
252C2×C4#377311yescharacteristic
253C2×C4#377312yes—
254C2×C4C2×GO+(4,𝔽2)12yes—
255C2×C4C2×GO+(4,𝔽2)11yescharacteristic
256C8—68no—
257C2×C3—24no—
258C2×C3—22no—
259C2×C3—42no—
260C2×C3—42no—
261D3—64no—
262D3—64no—
263D3—122no—
264D3—122no—
265C22—41no—
266C22—91no—
267C22—92no—
268C22—91no—
269C22—182no—
270C22—184no—
271C22—362no—
272C22—361no—
273C22C2×#377312yes—
274C22C2×#377311yescharacteristic
275C4—91no—
276C4—92no—
277C4—91no—
278C4—188no—
279C4C2×#377311yescharacteristic
280C4C2×#377312yes—
281C4C22×GO+(4,𝔽2)11yescharacteristic
282C3—22no—
283C2—22no—
284C2—21no—
285C2—92no—
286C2—182no—
287C2—181no—
288C2C22×#377311yescharacteristic
subgroupG/H
center, FrattiniC4C22×GO+(4,𝔽2)
commutatorC2×C3⋊D3C23×C4
FittingC22×C32×C4D4
socleC2×C32C22×C4⋊C4
prime radicalC23×C3⋊D3C2×C4
cosocleC4×C3⋊D3C24
hypercenterC22×C4GO+(4,𝔽2)
hypocenterC32#2905
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #10593328C2semidirect
4. #10593327C2semidirect
18. #63732C22semidirect
229. C32#2905semidirect

Maximal subgroups

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

subgroupindex
C2×#642862
C2×#638182
#105933282
#105933272
#29059