Finite Group Databasebeta

← #121135

Subgroups — #121135

Order 512.

351 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 106 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#8128C211yesmaximal, characteristic
2#8113C211yesmaximal, characteristic
3#8110C211yesmaximal, characteristic
4#8093C211yesmaximal, characteristic
5#7969C211yesmaximal, characteristic
6#7962C211yesmaximal, characteristic
7#7944C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1309—22no—
16#1306—22no—
17#1305—22no—
18#1303—22no—
19#1287—22no—
20#1286—22no—
21#1284C414yes—
22C22×M16—21no—
23C22×C42C2311yescharacteristic
24#432—21no—
25#432—21no—
26#432—21no—
27#432—21no—
28#432—21no—
29#432—21no—
30#432—21no—
31#432—21no—
32#431—21no—
33#431—21no—
34#431—21no—
35#431—21no—
36#431—21no—
37#431—21no—
38#431—21no—
39#431D412yes—
40#423—21no—
41#423—21no—
42#423—21no—
43#423—21no—
44#423—21no—
45#423—21no—
46#423—21no—
47#423D412yes—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50#407—21no—
51#407—21no—
52#407C2×C412yes—
53#407D412yes—
54C4×M16—21no—
55C4×M16—21no—
56C4×M16—21no—
57C4×M16Q812yes—
58#383—21no—
59#383—21no—
60#382—21no—
61#382—21no—
62#381—21no—
63#381C2×C412yes—
64#380—21no—
65#380—21no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4C2×C412yes—
70#376—21no—
71#376—21no—
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4#4111yescharacteristic
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—82no—
82C22×C8—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C42—22no—
88C2×C42—41no—
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D411yescharacteristic
91C2×C42C2×Q811yescharacteristic
92C2×C42#4111yescharacteristic
93C2×C42#4111yescharacteristic
94C2×C42#4111yescharacteristic
95C2×C42#4111yescharacteristic
96C2×C42#4111yescharacteristic
97C2×C42#4111yescharacteristic
98C2×C42#4111yescharacteristic
99C2×C42#4111yescharacteristic
100C2×C42#4111yescharacteristic
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—22no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C22⋊C8—22no—
116C22⋊C8—22no—
117C22⋊C8—22no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121#97—22no—
122#97—41no—
123#97—41no—
124#97—41no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—41no—
128C4×C8—41no—
129#95—22no—
130#95—22no—
131#95—22no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—41no—
137C24C4×D411yescharacteristic
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—81no—
143C22×C4—81no—
144C22×C4#11711yescharacteristic
145C22×C4#11711yescharacteristic
146C22×C4#11711yescharacteristic
147C22×C4C4×D411yescharacteristic
148C22×C4C4×D411yescharacteristic
149C22×C4C4×Q811yescharacteristic
150C22×C4#12111yescharacteristic
151C22×C4#12111yescharacteristic
152C22×C4C22⋊Q811yescharacteristic
153C22×C4#12311yescharacteristic
154C22×C4#12311yescharacteristic
155C22×C4#12311yescharacteristic
156C22×C4#12411yescharacteristic
157C22×C4#12411yescharacteristic
158C22×C4#12511yescharacteristic
159C22×C4#12611yescharacteristic
160C22×C4#12611yescharacteristic
161C22×C4#12611yescharacteristic
162M16—82no—
163M16—82no—
164M16—82no—
165M16—82no—
166M16—82no—
167M16—82no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—82no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—82no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C42—81no—
180C42—81no—
181C42—82no—
182C42#12111yescharacteristic
183C42C22⋊Q811yescharacteristic
184C42C22⋊Q811yescharacteristic
185C42#12311yescharacteristic
186C42#12311yescharacteristic
187C42#12311yescharacteristic
188C42#12411yescharacteristic
189C42#12511yescharacteristic
190C42#12511yescharacteristic
191C42#12611yescharacteristic
192C42#12611yescharacteristic
193C42#12611yescharacteristic
194C42#12611yescharacteristic
195C42#12611yescharacteristic
196C42#12611yescharacteristic
197C42#12611yescharacteristic
198C23—41no—
199C23—41no—
200C23—41no—
201C23#38711yescharacteristic
202C23#38811yescharacteristic
203C23#38811yescharacteristic
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4#38311yescharacteristic
232C2×C4#38711yescharacteristic
233C2×C4#38711yescharacteristic
234C2×C4#38811yescharacteristic
235C2×C4#39611yescharacteristic
236C2×C4#39711yescharacteristic
237C2×C4#39711yescharacteristic
238C2×C4#39711yescharacteristic
239C2×C4#40011yescharacteristic
240C2×C4#40011yescharacteristic
241C2×C4#40011yescharacteristic
242C2×C4#40111yescharacteristic
243C8—82no—
244C8—82no—
245C8—82no—
246C8—82no—
247C8—82no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—82no—
258C22#144311yescharacteristic
259C4—21no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—162no—
271C4—162no—
272C4#208111yescharacteristic
273C4#208211yescharacteristic
274C2—41no—
275C2—41no—
276C2—41no—
277C2#1130811yescharacteristic
278C2#1130911yescharacteristic
279C2#1131911yescharacteristic
subgroupG/H
center, socleC22#1443
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#81282
#81132
#81102
#80932
#79692
#79622
#79442