Finite Group Databasebeta

← #11785743

Subgroups — #11785743

Order 1152.

639 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 122 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×#63849C211yesmaximal, characteristic
2#10592825C211yesmaximal, characteristic
3#10591074C211yesmaximal, characteristic
4C3×#4554C212yesmaximal, complemented
5#10589167C212yesmaximal, complemented
6#68991—32nomaximal
7C22×C4×C3⋊D3C2211yescharacteristic
8C2×C32×M16C2211yescharacteristic
9C2×#3748C2211yescharacteristic
10C2×C3×C4⋊Dic3C2212yes—
11C2×C3×#208C2212yes—
12C2×#3722C412yes—
13C2×Dic32C412yes—
14#63849—22no—
15#63849—22no—
16#63647—22no—
17#63557—22no—
18C2×#892—32no—
19#5126—32no—
20#5126—32no—
21#5126—61no—
22C3×#343—32no—
23#4554—32no—
24#4554D312yescomplemented
25#4530—32no—
26C23×C3⋊D3C2×C411yescharacteristic
27C22×C32×C4C2311yescharacteristic
28C22×C3⋊Dic3C2×C411yescharacteristic
29C2×C4×C3⋊D3—22no—
30C2×C4×C3⋊D3—22no—
31C2×C4×C3⋊D3D412yes—
32C2×C4×C3⋊D3D412yes—
33C22×C3×Dic3C2×C414yes—
34C32×M16—22no—
35C2×C32×C8—21no—
36#3748—22no—
37C2×C3⋊(C3⋊C8)—21no—
38C3×C4⋊Dic3—22no—
39C3×C4⋊Dic3—22no—
40C3×#208—22no—
41C3×#208—22no—
42C2×C3×C3⋊C8—22no—
43#3722—24no—
44Dic32—24no—
45#1822—91noSylow
46C22×C4×D3—32no—
47C22×C4×D3—61no—
48C2×C3×M16—21no—
49C2×C3×M16—32no—
50C2×C3×M16C2×D312yes—
51C2×C3×C4⋊C4—32no—
52C2×(C2×D3)⋊C4—34no—
53C2×C4⋊Dic3C2×D312yes—
54C2×C4×Dic3—34no—
55C2×#208—32no—
56C2×#208—61no—
57C2×#208C2×D312yes—
58#892—62no—
59#892—62no—
60#892—62no—
61#892—62no—
62(C2×D3)⋊C8—62no—
63(C2×D3)⋊C8—62no—
64(C2×D3)⋊C8—62no—
65(C2×D3)⋊C8—62no—
66(C2×D3)⋊C8—64no—
67C23×C32C22×C411yescharacteristic
68C22×C3⋊D3—22no—
69C22×C3⋊D3—42no—
70C22×C3⋊D3C22⋊C412yes—
71C2×C32×C4C2×D411yescharacteristic
72C2×C32×C4C2×D411yescharacteristic
73C2×C32×C4#4112yes—
74C2×C32×C4#4112yes—
75C2×C3⋊Dic3—22no—
76C2×C3⋊Dic3C22⋊C412yes—
77C4×C3⋊D3—22no—
78C4×C3⋊D3—22no—
79C4×C3⋊D3—22no—
80C4×C3⋊D3—22no—
81C2×C3×Dic3—24no—
82C2×C3×Dic3—44no—
83C32×C8—41no—
84C3⋊(C3⋊C8)—41no—
85C3×C3⋊C8—42no—
86C2×#117—91no—
87#407—91no—
88#407—91no—
89#343—32no—
90#343—92no—
91C23×D3—32no—
92C23×D3—61no—
93C22×C3×C4—21no—
94C22×C3×C4—34no—
95C22×C3×C4C22×D312yes—
96C22×Dic3—32no—
97C22×Dic3—61no—
98C22×Dic3C4×D314yes—
99C2×C4×D3—32no—
100C2×C4×D3—32no—
101C2×C4×D3—32no—
102C2×C4×D3—32no—
103C2×C4×D3—62no—
104C2×C4×D3—62no—
105C2×C4×D3—62no—
106C2×C4×D3—62no—
107C2×C4×D3—62no—
108C2×C4×D3—62no—
109C2×C4×D3—122no—
110C2×C4×D3—122no—
111C3×M16—22no—
112C3×M16—22no—
113C3×M16—42no—
114C3×M16—62no—
115C3×M16—62no—
116C2×C3×C8—22no—
117C2×C3×C8—22no—
118C2×C3×C8—62no—
119C3×C4⋊C4—62no—
120C3×C4⋊C4—62no—
121(C2×D3)⋊C4—64no—
122(C2×D3)⋊C4—64no—
123C4⋊Dic3—22no—
124C4⋊Dic3—22no—
125C4×Dic3—64no—
126C4×Dic3—64no—
127#208—22no—
128#208—22no—
129#208—62no—
130#208—62no—
131#208—122no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—62no—
134C2×C3⋊C8—62no—
135C22×C32—22no—
136C22×C32C2×C22⋊C411yescharacteristic
137C22×C32#11712yes—
138C2×C3⋊D3—22no—
139C2×C3⋊D3—22no—
140C2×C3⋊D3—22no—
141C2×C3⋊D3—44no—
142C32×C4#12311yescharacteristic
143C32×C4#12312yes—
144C32×C4#12311yescharacteristic
145C3⋊Dic3—22no—
146C3×Dic3—44no—
147C23×C4—91no—
148C2×M16—32no—
149C2×M16—91no—
150C2×M16D3211yescharacteristic
151#117—182no—
152#117—182no—
153C2×C4⋊C4—32no—
154C2×C22⋊C4—92no—
155C2×C42—92no—
156C22⋊C8—182no—
157C22⋊C8—182no—
158C23×C3—21no—
159C23×C3C2×C4×D312yes—
160C22×D3—32no—
161C22×D3—32no—
162C22×D3—62no—
163C22×D3—62no—
164C22×D3—62no—
165C22×D3—122no—
166C22×D3—122no—
167C22×D3—122no—
168C22×D3—124no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—22no—
172C2×C3×C4—21no—
173C2×C3×C4—64no—
174C2×C3×C4—124no—
175C2×C3×C4#23512yes—
176C2×C3×C4#23512yes—
177C2×C3×C4D3×D412yes—
178C2×C3×C4D3×D412yes—
179C2×C3×C4#23712yes—
180C2×C3×C4#23912yes—
181C2×Dic3—24no—
182C2×Dic3—32no—
183C2×Dic3—32no—
184C2×Dic3—44no—
185C2×Dic3—62no—
186C2×Dic3—62no—
187C2×Dic3—62no—
188C2×Dic3—122no—
189C4×D3—62no—
190C4×D3—62no—
191C4×D3—62no—
192C4×D3—62no—
193C4×D3—62no—
194C4×D3—62no—
195C4×D3—62no—
196C4×D3—62no—
197C4×D3—122no—
198C4×D3—122no—
199C4×D3—122no—
200C4×D3—122no—
201C3×C8—42no—
202C3×C8—42no—
203C3×C8—122no—
204C3⋊C8—42no—
205C3⋊C8—122no—
206C3⋊C8—122no—
207C2×C32—22no—
208C2×C32#38111yescharacteristic
209C2×C32#41312yes—
210C3⋊D3—42no—
211C24—91no—
212C22×C4—34no—
213C22×C4—91no—
214C22×C4—92no—
215C22×C4—92no—
216C22×C4—182no—
217C22×C4—182no—
218C22×C4C2×D3211yescharacteristic
219M16—22no—
220M16—62no—
221M16—62no—
222M16—182no—
223C2×C8—21no—
224C2×C8—62no—
225C2×C8—181no—
226C4⋊C4—62no—
227C4⋊C4—62no—
228C22⋊C4—184no—
229C42—184no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—21no—
234C22×C3—42no—
235C22×C3#87212yes—
236C22×C3D3×C22⋊C412yes—
237C22×C3#89212yes—
238C2×D3—64no—
239C2×D3—62no—
240C2×D3—62no—
241C2×D3—62no—
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—62no—
245C2×D3—122no—
246C2×D3—122no—
247C2×D3—124no—
248C2×D3—124no—
249C2×D3—128no—
250C3×C4—22no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—124no—
254C3×C4#88312yes—
255C3×C4#88312yes—
256C3×C4#89412yes—
257C3×C4#89412yes—
258Dic3—44no—
259Dic3—62no—
260Dic3—62no—
261Dic3—122no—
262C32#182211yescharacteristic, complemented, Sylow
263C23—92no—
264C23—182no—
265C23—362no—
266C23C4×D3211yescharacteristic
267C2×C4—64no—
268C2×C4—92no—
269C2×C4—124no—
270C2×C4—182no—
271C2×C4—182no—
272C2×C4—182no—
273C2×C4—182no—
274C2×C4—182no—
275C2×C4#379912yes—
276C2×C4#380211yescharacteristic
277C2×C4#380412yes—
278C2×C4#381111yescharacteristic
279C8—41no—
280C8—122no—
281C8—361no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—21no—
286C2×C3—42no—
287C2×C3#466412yes—
288C2×C3#474512yes—
289C2×C3#475112yes—
290D3—124no—
291D3—122no—
292D3—122no—
293C22—22no—
294C22—182no—
295C22—182no—
296C22—182no—
297C22—364no—
298C22#6383612yes—
299C22#6392111yescharacteristic
300C4—124no—
301C4—182no—
302C4#6385412yes—
303C4#6389711yescharacteristic
304C4#6397511yescharacteristic
305C3—21no—
306C3#6899112yescomplemented
307C2—22no—
308C2—362no—
309C2#1059101012yes—
310C2#1059137111yescharacteristic
subgroupG/H
centerC22#63921
commutatorC23×C32C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32C2×C22⋊C4
prime radicalC23×C3⋊D3C2×C4
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1822
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. C3×#4554C2semidirect
5. #10589167C2semidirect
24. #4554D3semidirect
262. C32#1822semidirect
306. C3#68991semidirect

Maximal subgroups

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

subgroupindex
C2×#638492
#105928252
#105910742
C3×#45542
#105891672
#689913