Finite Group Databasebeta

← #11778364

Subgroups — #11778364

Order 1152.

567 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 567 conjugacy classes of subgroups — too many to draw. Only its 80 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
1#10594813C211yesmaximal, characteristic
2#10592929C211yesmaximal, characteristic
3C3×#5228C212yesmaximal
4#10589152C212yesmaximal
5#10589148C211yesmaximal, characteristic
6#70983—32nomaximal
7C22×C32×Q8C2211yescharacteristic
8C2×#3750—21no—
9C2×#3750C2211yescharacteristic
10C2×C3×#216—22no—
11C3×#924C2212yes—
12#63982—21no—
13#63957—21no—
14#63847—22no—
15#63839—22no—
16#63658—21no—
17#63658C2211yescharacteristic
18C3×#835—22no—
19C3×#830C2212yes—
20#5623—32no—
21C3×#451—32no—
22#5230—32no—
23#5230—61no—
24#5228D312yes—
25#4538—32no—
26#4469—32no—
27#4467—32no—
28C2×C32×Q8—22no—
29C2×C32×Q8—42no—
30C22×C32×C4—21no—
31C22×C32×C4C2311yescharacteristic
32C22×C3⋊Dic3—21no—
33C22×C3⋊Dic3D411yescharacteristic
34C22×C3⋊Dic3D411yescharacteristic
35C2×C3×Dic6D412yes—
36C2×#627—21no—
37#3750—22no—
38#3750—42no—
39C3×#217—22no—
40C3×#216—44no—
41C3×C4⋊Dic3D412yes—
42C3×Dic3⋊C4—22no—
43C2×C3×C3⋊C8—22no—
44#3724—21no—
45#3724—21no—
46#3724—21no—
47Dic3⋊Dic3—22no—
48Dic32—22no—
49#1605—91noSylow
50C22×C3×Q8—21no—
51C22×C3×Q8C2×D312yes—
52C2×C3×Q16—62no—
53C3×C22⋊Q8—32no—
54#944—62no—
55C2×#216—22no—
56#924C2×D312yes—
57C2×Dic3⋊C4—32no—
58C2×Dic3⋊C4—61no—
59C2×Dic3⋊C4—62no—
60C2×Dic3⋊C4—62no—
61#889—62no—
62#889—62no—
63#888—62no—
64#878—62no—
65#877—62no—
66C3×#103—62no—
67C3×C22⋊C8—32no—
68#835—22no—
69C3⋊#95—32no—
70C3⋊#95—61no—
71C3⋊#95—62no—
72C3⋊#95—62no—
73#830C2×D312yes—
74C23×C32C2×D411yescharacteristic
75C32×Q8—24no—
76C32×Q8—42no—
77C2×C32×C4—22no—
78C2×C32×C4—41no—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C32×C4C2×D412yes—
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3—21no—
84C2×C3⋊Dic3—21no—
85C2×C3⋊Dic3—21no—
86C2×C3⋊Dic3—41no—
87C2×C3⋊Dic3—41no—
88C2×C3⋊Dic3—41no—
89C2×C3⋊Dic3Q1612yescomplemented
90C2×C3×Dic3—22no—
91C2×C3×Dic3—22no—
92C3×Dic6—42no—
93#627—81no—
94C3×C3⋊C8—42no—
95#544—91no—
96#451—32no—
97#395—91no—
98#328—92no—
99#324—91no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—32no—
103C2×C3×Q8—42no—
104C2×C3×Q8—44no—
105C2×C3×Q8—44no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4C22×D312yes—
110C22×Dic3—32no—
111C22×Dic3—32no—
112C22×Dic3—61no—
113C22×Dic3—61no—
114C22×Dic3—62no—
115C22×Dic3—62no—
116C2×Dic6—62no—
117C2×Dic6#6712yes—
118C3×Q16—124no—
119C2×C3×C8—62no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—62no—
122C3×C22⋊C4—62no—
123#217—22no—
124#216—44no—
125C4⋊Dic3—62no—
126C4⋊Dic3#6712yes—
127Dic3⋊C4—22no—
128Dic3⋊C4—62no—
129Dic3⋊C4—62no—
130Dic3⋊C4—62no—
131Dic3⋊C4—62no—
132Dic3⋊C4—62no—
133Dic3⋊C4—62no—
134Dic3⋊C4—122no—
135Dic3⋊C4—124no—
136Dic3⋊C4—124no—
137C4×Dic3—62no—
138C4×Dic3—62no—
139C2×C3⋊C8—22no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32#12011yescharacteristic
143C22×C32C2×Q1611yescharacteristic
144C22×C32#13711yescharacteristic
145C32×C4—22no—
146C32×C4—42no—
147C3⋊Dic3—22no—
148C3⋊Dic3—42no—
149C3⋊Dic3—41no—
150C3×Dic3—42no—
151C3×Dic3—42no—
152C22×Q8D3211yescharacteristic
153C2×Q16—62no—
154C4⋊Q8—182no—
155#125—182no—
156C22⋊Q8—32no—
157C22⋊Q8—181no—
158#117—181no—
159C2×C4⋊C4—91no—
160C2×C4⋊C4—181no—
161#103—62no—
162C22⋊C8—32no—
163#95—91no—
164#95—181no—
165C23×C3—21no—
166C23×C3D3×D412yes—
167C3×Q8—24no—
168C3×Q8—24no—
169C3×Q8—44no—
170C3×Q8—42no—
171C3×Q8—42no—
172C3×Q8—44no—
173C3×Q8—122no—
174C2×C3×C4—24no—
175C2×C3×C4—24no—
176C2×C3×C4—22no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4D3×D412yes—
186C2×C3×C4C2×#6712yes—
187C2×Dic3—22no—
188C2×Dic3—22no—
189C2×Dic3—34no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—64no—
194C2×Dic3—64no—
195C2×Dic3—62no—
196C2×Dic3—121no—
197C2×Dic3—121no—
198C2×Dic3—121no—
199C2×Dic3—122no—
200C2×Dic3—122no—
201C2×Dic3—122no—
202C2×Dic3—122no—
203C2×Dic3—122no—
204C2×Dic3—122no—
205Dic6—42no—
206Dic6—242no—
207C3×C8—122no—
208C3⋊C8—42no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32#45111yescharacteristic
212C2×C32#45611yescharacteristic
213C2×C32B(4,𝔽2)11yescharacteristic
214C2×Q8—22no—
215C2×Q8—32no—
216C2×Q8—42no—
217C2×Q8—181no—
218C22×C4—21no—
219C22×C4—91no—
220C22×C4—91no—
221C22×C4—181no—
222C22×C4C2×D3211yescharacteristic
223Q16—124no—
224C2×C8—62no—
225C4⋊C4—32no—
226C4⋊C4—62no—
227C4⋊C4—181no—
228C4⋊C4—181no—
229C4⋊C4—181no—
230C4⋊C4—182no—
231C4⋊C4—182no—
232C4⋊C4—362no—
233C22⋊C4—62no—
234C42—182no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3D3×Q1612yes—
243C22×C3#91812yes—
244C22×C3#93712yes—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—42no—
248C3×C4—44no—
249C3×C4—44no—
250C3×C4—122no—
251C3×C4—122no—
252Dic3—42no—
253Dic3—42no—
254Dic3—64no—
255Dic3—122no—
256Dic3—124no—
257Dic3—124no—
258Dic3—122no—
259Dic3—241no—
260C32#160511yescharacteristic, complemented, Sylow
261C23#381111yescharacteristic
262Q8—24no—
263Q8—42no—
264Q8—122no—
265Q8—721no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—92no—
273C2×C4—181no—
274C2×C4—181no—
275C2×C4—182no—
276C2×C4—181no—
277C2×C4—361no—
278C2×C4—361no—
279C2×C4—361no—
280C2×C4D3×#6712yes—
281C8—122no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3#503212yes—
290C2×C3#518612yes—
291C2×C3#520512yes—
292C22—21no—
293C22—21no—
294C22#6394411yescharacteristic
295C22#6394511yescharacteristic
296C22#6398311yescharacteristic
297C4—22no—
298C4—42no—
299C4—122no—
300C4—122no—
301C4—182no—
302C4—362no—
303C4—361no—
304C3—21no—
305C3#7098312yescomplemented
306C2—21no—
307C2—21no—
308C2#1059165411yescharacteristic
309C2#1059166011yescharacteristic
310C2#1059172511yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenterC22×Q8D32
hypocenterC32#1605
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
89. C2×C3⋊Dic3Q16semidirect
260. C32#1605semidirect
305. C3#70983semidirect

Maximal subgroups

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

subgroupindex
#105948132
#105929292
C3×#52282
#105891522
#105891482
#709833