Finite Group Databasebeta

← #11783423

Subgroups — #11783423

Order 1152.

941 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 941 conjugacy classes of subgroups — too many to draw. Only its 188 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×#64101C211yesmaximal, characteristic
2C2×#63818C212yesmaximal
3C3×#4530C212yesmaximal, complemented
4#10589168C212yesmaximal, complemented
5#68989—32nomaximal
6C22×C4×C3⋊D3C2211yescharacteristic
7C2×C32×C4⋊C4C2211yescharacteristic
8C2×#3752C412yes—
9C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C4×C3⋊Dic3C412yes—
11#64101—22no—
12#64101—22no—
13C2×C3×#208C2214yes—
14C2×#3712—22no—
15#63818—24no—
16#63818C418yescomplemented
17C2×#3710—22no—
18C2×D3×M16—34no—
19C2×#892—32no—
20C2×#892—61no—
21C3×#343—32no—
22#4554—34no—
23#4530D312yescomplemented
24C23×C3⋊D3C2×C411yescharacteristic
25C22×C32×C4C2×C412yes—
26C22×C32×C4C2311yescharacteristic
27C22×C3⋊Dic3C2×C412yes—
28C22×C3⋊Dic3C2×C411yescharacteristic
29C2×C4×C3⋊D3C2×C414yes—
30C2×C4×C3⋊D3D412yes—
31C2×C4×C3⋊D3D412yes—
32C2×C4×C3⋊D3D412yes—
33C2×C4×C3⋊D3Q812yes—
34C32×C4⋊C4—22no—
35#3752—24no—
36C4⋊(C3⋊Dic3)—22no—
37C4×C3⋊Dic3—22no—
38C4×C3⋊Dic3—22no—
39C3×#208—24no—
40C3×#208C2×C418yes—
41C2×C3×C3⋊C8—24no—
42#3712—28no—
43#3710—28no—
44#1742—91noSylow
45C22×C4×D3—32no—
46C22×C4×D3—61no—
47C2×C3×M16—34no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×D312yes—
50C2×(C2×D3)⋊C4—34no—
51C2×(C2×D3)⋊C4—62no—
52C2×C4⋊Dic3—32no—
53C2×C4⋊Dic3—61no—
54C2×C4×Dic3—34no—
55C2×C4×Dic3—62no—
56C2×#208C2×D314yes—
57D3×M16—316no—
58D3×M16—68no—
59C2×ΓL(1,𝔽25)—64no—
60C2×D3×C8—64no—
61#892—64no—
62#892—64no—
63#892—64no—
64#892—64no—
65C23×C32C22×C411yescharacteristic
66C22×C3⋊D3—41no—
67C22×C3⋊D3—41no—
68C22×C3⋊D3C4212yes—
69C22×C3⋊D3C22⋊C412yes—
70C22×C3⋊D3C4⋊C412yes—
71C2×C32×C4—22no—
72C2×C32×C4—42no—
73C2×C32×C4C22×C412yes—
74C2×C32×C4C2×D411yescharacteristic
75C2×C32×C4C2×D412yes—
76C2×C32×C4C2×Q811yescharacteristic
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3—42no—
79C2×C3⋊Dic3C4212yes—
80C2×C3⋊Dic3C22⋊C411yescharacteristic
81C2×C3⋊Dic3C22⋊C411yescharacteristic
82C2×C3⋊Dic3C4⋊C412yes—
83C4×C3⋊D3C22⋊C414yes—
84C4×C3⋊D3C22⋊C414yes—
85C4×C3⋊D3C4⋊C414yes—
86C4×C3⋊D3C4⋊C414yes—
87C3×C3⋊C8—28no—
88C22×M16—92no—
89C2×#117—91no—
90#343—32no—
91#343—92no—
92C23×D3—32no—
93C23×D3—61no—
94C22×C3×C4—22no—
95C22×C3×C4—21no—
96C22×C3×C4C4×D314yes—
97C22×C3×C4C22×D312yes—
98C22×Dic3—32no—
99C22×Dic3—34no—
100C22×Dic3—61no—
101C22×Dic3—62no—
102C2×C4×D3—38no—
103C2×C4×D3—34no—
104C2×C4×D3—34no—
105C2×C4×D3—34no—
106C2×C4×D3—34no—
107C2×C4×D3—62no—
108C2×C4×D3—62no—
109C2×C4×D3—64no—
110C2×C4×D3—62no—
111C2×C4×D3—62no—
112C3×M16—38no—
113C3×M16—64no—
114C2×C3×C8—64no—
115C3×C4⋊C4—24no—
116C3×C4⋊C4—24no—
117(C2×D3)⋊C4—68no—
118(C2×D3)⋊C4—68no—
119C4⋊Dic3—64no—
120C4⋊Dic3—64no—
121C4×Dic3—64no—
122C4×Dic3—64no—
123C4×Dic3—64no—
124C4×Dic3—64no—
125#208—24no—
126#208C4×D318yes—
127C2×C3⋊C8—24no—
128ΓL(1,𝔽25)—616no—
129D3×C8—616no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32C2×C4211yescharacteristic
133C22×C32C2×C22⋊C411yescharacteristic
134C22×C32C2×C4⋊C411yescharacteristic
135C2×C3⋊D3—22no—
136C2×C3⋊D3—22no—
137C2×C3⋊D3—42no—
138C2×C3⋊D3—42no—
139C2×C3⋊D3#9514yes—
140C32×C4—42no—
141C32×C4C2×C22⋊C412yes—
142C32×C4C2×C4⋊C412yes—
143C3⋊Dic3—42no—
144C3⋊Dic3#9512yes—
145C3⋊Dic3#9512yes—
146C23×C4—91no—
147C2×M16—34no—
148C2×M16—98no—
149C2×M16—182no—
150C2×M16—184no—
151C22×C8—182no—
152#117—182no—
153#117—182no—
154C2×C4⋊C4—91no—
155C2×C4⋊C4D3211yescharacteristic
156C2×C22⋊C4—92no—
157C2×C42—92no—
158C23×C3—21no—
159C23×C3C2×C4×D312yes—
160C22×D3—34no—
161C22×D3—34no—
162C22×D3—34no—
163C22×D3—62no—
164C22×D3—62no—
165C22×D3—62no—
166C22×D3—122no—
167C22×D3—122no—
168C22×D3—122no—
169C22×D3—122no—
170C2×C3×C4—24no—
171C2×C3×C4—24no—
172C2×C3×C4—21no—
173C2×C3×C4—22no—
174C2×C3×C4—21no—
175C2×C3×C4—22no—
176C2×C3×C4—44no—
177C2×C3×C4—44no—
178C2×C3×C4C2×C4×D314yes—
179C2×C3×C4D3×D412yes—
180C2×C3×C4D3×D414yes—
181C2×C3×C4D3×Q812yes—
182C2×Dic3—34no—
183C2×Dic3—32no—
184C2×Dic3—32no—
185C2×Dic3—34no—
186C2×Dic3—62no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—62no—
190C2×Dic3—64no—
191C2×Dic3—64no—
192C2×Dic3—124no—
193C2×Dic3—124no—
194C4×D3—38no—
195C4×D3—38no—
196C4×D3—38no—
197C4×D3—38no—
198C4×D3—64no—
199C4×D3—64no—
200C4×D3—64no—
201C4×D3—64no—
202C3×C8—68no—
203C3⋊C8—28no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32C2×#9511yescharacteristic
207C2×C32#41311yescharacteristic
208C2×C32#41311yescharacteristic
209C3⋊D3—24no—
210C24—91no—
211C22×C4—91no—
212C22×C4—92no—
213C22×C4—94no—
214C22×C4—92no—
215C22×C4—92no—
216C22×C4—92no—
217C22×C4—92no—
218C22×C4#62412yes—
219C22×C4C2×D3211yescharacteristic
220M16—38no—
221M16—64no—
222M16—188no—
223C2×C8—64no—
224C2×C8—188no—
225C4⋊C4—22no—
226C4⋊C4—182no—
227C22⋊C4—184no—
228C42—182no—
229C42—182no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3C42×D312yes—
238C22×C3D3×C22⋊C412yes—
239C22×C3D3×C4⋊C412yes—
240C2×D3—38no—
241C2×D3—64no—
242C2×D3—64no—
243C2×D3—64no—
244C2×D3—122no—
245C2×D3—122no—
246C2×D3—124no—
247C2×D3—124no—
248C2×D3—124no—
249C2×D3—124no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—44no—
253C3×C4—44no—
254C3×C4D3×C22⋊C414yes—
255C3×C4D3×C4⋊C414yes—
256Dic3—34no—
257Dic3—34no—
258Dic3—62no—
259Dic3—62no—
260Dic3—124no—
261Dic3—124no—
262C32#174211yescharacteristic, complemented, Sylow
263C23—92no—
264C23—92no—
265C23—92no—
266C23—361no—
267C23—361no—
268C23C2×#62411yescharacteristic
269C2×C4—22no—
270C2×C4—42no—
271C2×C4—92no—
272C2×C4—92no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—94no—
276C2×C4—94no—
277C2×C4—94no—
278C2×C4—94no—
279C2×C4—182no—
280C2×C4—362no—
281C2×C4#379711yescharacteristic
282C2×C4C4×D3212yes—
283C2×C4#380211yescharacteristic
284C2×C4#381112yes—
285C8—68no—
286C2×C3—22no—
287C2×C3—22no—
288C2×C3—22no—
289C2×C3—22no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3D3×#9512yes—
294C2×C3#474512yes—
295C2×C3#475112yes—
296D3—68no—
297D3—124no—
298C22—21no—
299C22—21no—
300C22—94no—
301C22—182no—
302C22—182no—
303C22—362no—
304C22—362no—
305C22C4×#62411yescharacteristic
306C22#6390011yescharacteristic
307C22#6397411yescharacteristic
308C4—42no—
309C4—92no—
310C4—92no—
311C4—362no—
312C4#6388312yes—
313C4#6392112yes—
314C3—21no—
315C3#6898912yescomplemented
316C2—21no—
317C2—21no—
318C2—184no—
319C2#1059137011yescharacteristic
320C2#1059150411yescharacteristic
321C2#1059164011yescharacteristic
subgroupG/H
centerC22#63974
commutatorC22×C32C2×C42
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32C2×C22⋊C4
prime radicalC23×C3⋊D3C2×C4
cosocleC22×C32×C4C23
hypercenterC2×C4⋊C4D32
hypocenterC32#1742
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#4530C2semidirect
4. #10589168C2semidirect
16. #63818C4semidirect
23. #4530D3semidirect
262. C32#1742semidirect
315. C3#68989semidirect

Maximal subgroups

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

subgroupindex
C2×#641012
C2×#638182
C3×#45302
#105891682
#689893