Finite Group Databasebeta

← #122274

Subgroups — #122274

Order 512.

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

normal subgroups only

This group has 426 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#21787C211yesmaximal, characteristic
2#11460C211yesmaximal, characteristic, complemented
3#11459C211yesmaximal, characteristic, complemented
4#7058C214yesmaximal, complemented
5C2×#532C411yescharacteristic
6C2×#431C411yescharacteristic
7C2×#431C2211yescharacteristic
8#2163—21no—
9#2162—21no—
10#1954—21no—
11#1954—21no—
12#1954—21no—
13#1954—21no—
14#1780C412yes—
15#1780C2212yes—
16#1322—21no—
17#1322—21no—
18#1314C2212yes—
19#1314C2212yes—
20C22×#41C2×C412yes—
21#532—41no—
22#532—41no—
23#532—41no—
24#532—41no—
25C2×C4⋊Q8C811yescharacteristic
26C2×#127C811yescharacteristic
27C2×#124C812yes—
28C2×#121—21no—
29C2×#121—21no—
30C2×C4×D4—21no—
31C2×C4×D4—21no—
32C22×C42C2×C411yescharacteristic
33#431—41no—
34#431—41no—
35#431—41no—
36#431—41no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8D411yescharacteristic
39C2×C4⋊C8D411yescharacteristic
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8C2×C412yes—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8C2311yescharacteristic
47C2×C4×C8—21no—
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C4⋊C16—41no—
51C4⋊C16—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54#346—41no—
55C4×C16—41no—
56C2×#41—42no—
57C2×#41—42no—
58C2×#41—82no—
59C2×#41—82no—
60C22×Q8C2×C812yes—
61C22×D4—22no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2×C812yes—
65C23×C4—22no—
66C23×C4M1611yescharacteristic
67C23×C4M1611yescharacteristic
68C23×C4C22×C411yescharacteristic
69C22×C8—22no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4⋊Q8—41no—
75C4⋊Q8—41no—
76#127—41no—
77#127—41no—
78#124—42no—
79#124—42no—
80#121—42no—
81#121—42no—
82#121—42no—
83#121—42no—
84C4×D4—42no—
85C4×D4—42no—
86C4×D4—41no—
87C4×D4—41no—
88C4×D4—41no—
89C4×D4—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C22⋊C411yescharacteristic
100C2×C42C2×C811yescharacteristic
101C2×C42C2×C811yescharacteristic
102C2×C16—42no—
103C2×C16—42no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C22⋊C8—41no—
109C22⋊C8—41no—
110C22⋊C8—42no—
111C22⋊C8—82no—
112C22⋊C8—82no—
113C22⋊C8—82no—
114C22⋊C8—82no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—42no—
120C24—22no—
121C24C2×M1611yescharacteristic
122#41—88no—
123#41—84no—
124#41—84no—
125C2×Q8—42no—
126C2×Q8—82no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—44no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—42no—
136C2×D4—82no—
137C2×D4—82no—
138C2×D4—82no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4C22⋊C811yescharacteristic
158C22×C4C22⋊C811yescharacteristic
159C22×C4C22⋊C811yescharacteristic
160C22×C4C22⋊C811yescharacteristic
161C22×C4C2×C22⋊C411yescharacteristic
162C22×C4C22×C811yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C22⋊C4—44no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42#9911yescharacteristic
187C42#9911yescharacteristic
188C42#10011yescharacteristic
189C42#10011yescharacteristic
190C16—82no—
191C16—82no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—41no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23C2×C22⋊C811yescharacteristic
205Q8—84no—
206D4—84no—
207D4—82no—
208D4—84no—
209D4—88no—
210D4—82no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4#32311yescharacteristic
237C2×C4#32311yescharacteristic
238C2×C4#32311yescharacteristic
239C2×C4#32311yescharacteristic
240C2×C4#35012yes—
241C2×C4C2×#9911yescharacteristic
242C2×C4C2×#10011yescharacteristic
243C8—41no—
244C8—41no—
245C8—41no—
246C8—41no—
247C8—82no—
248C22—21no—
249C22—21no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—81no—
254C22—81no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22C2×#32311yescharacteristic
263C22C2×#35011yescharacteristic
264C22#211711yescharacteristic
265C4—21no—
266C4—21no—
267C4—22no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4#132112yes—
274C4#212211yescharacteristic
275C4#212211yescharacteristic
276C2—21no—
277C2—21no—
278C2—81no—
279C2—162no—
280C2#1142211yescharacteristic
281C2#1142711yescharacteristic
282C2C2×#132111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C22⋊C8C23
socleC22C2×#323
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11460C2semidirect
3. #11459C2semidirect
4. #7058C2semidirect

Maximal subgroups

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

subgroupindex
#217872
#114602
#114592
#70582