Finite Group Databasebeta

← #152246

Subgroups — #152246

Order 512.

332 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 70 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#21648C211yesmaximal, characteristic
2C2×#1414C212yesmaximal
3#12801C211yesmaximal, characteristic
4#12798C211yesmaximal, characteristic
5#7502C212yesmaximal
6C2×#488C2211yescharacteristic
7C4×#125C411yescharacteristic, complemented
8C4×#124C2211yescharacteristic
9C2×#355C2212yes—
10C2×#354C2212yes—
11#1991C411yescharacteristic, complemented
12#1958C2211yescharacteristic
13#1936—21no—
14#1935—21no—
15#1835C411yescharacteristic, complemented
16#1801—21no—
17#1799—21no—
18#1763—21no—
19#1762—21no—
20C4×#97C411yescharacteristic, complemented
21#1414—24no—
22#1394—21no—
23#1393—21no—
24C2×#125C2×C411yescharacteristic
25C2×#124C2311yescharacteristic
26C2×C4×Q8D411yescharacteristic
27C2×C4×D4D411yescharacteristic
28#488—21no—
29#488—21no—
30#488D412yes—
31#488Q812yes—
32#423—41no—
33C2×#103—21no—
34C2×#103—21no—
35C2×#102—21no—
36C2×#102—21no—
37C2×#101—22no—
38C2×#99—22no—
39C2×#97—21no—
40C2×#97—21no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43#391D411yescharacteristic
44#390D411yescharacteristic
45#389—21no—
46#388—41no—
47#387—41no—
48#382C2×C411yescharacteristic
49#382C2×C411yescharacteristic
50#380—41no—
51C4×C4⋊C4—21no—
52C4×C4⋊C4—41no—
53C4×C22⋊C4—41no—
54C43C2×C411yescharacteristic
55#355—22no—
56#355—22no—
57#354—22no—
58#354—22no—
59#343—41no—
60#342—41no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340—21no—
65C22×Q8C2×D411yescharacteristic
66C22×D4C2×D411yescharacteristic
67C23×C4—21no—
68C2×M16—42no—
69C22×C8—21no—
70C22×C8—21no—
71#125—21no—
72#125—41no—
73#125C4⋊C412yescomplemented
74#124—41no—
75#124C2×D411yescharacteristic
76#124C2×Q811yescharacteristic
77#124#4111yescharacteristic
78#124#4111yescharacteristic
79C4×Q8—41no—
80C4×Q8—41no—
81C4×D4—41no—
82C4×D4—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4C22⋊C411yescharacteristic
90C2×C4⋊C4C22⋊C411yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—42no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—41no—
99C2×C42C22⋊C411yescharacteristic
100C2×C42C22⋊C411yescharacteristic
101C2×C42C22×C411yescharacteristic
102C4⋊C8—81no—
103C4⋊C8—81no—
104#103—22no—
105#103—22no—
106#103—42no—
107#102—22no—
108#102—22no—
109#102—42no—
110#101—42no—
111#101—42no—
112#99—42no—
113#99—42no—
114#97—41no—
115#97—41no—
116#97—42no—
117C4×C8—21no—
118C4×C8—41no—
119C4×C8C4⋊C412yescomplemented
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—81no—
126C24—21no—
127C2×Q8—21no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8#12111yescharacteristic
131C2×Q8C22⋊Q811yescharacteristic
132C2×D4—21no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4#12111yescharacteristic
136C2×D4C22⋊Q811yescharacteristic
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4#9911yescharacteristic
149C22×C4#9911yescharacteristic
150C22×C4C2×C22⋊C411yescharacteristic
151M16—82no—
152M16—82no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—82no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—81no—
169C4⋊C4—82no—
170C4⋊C4—162no—
171C22⋊C4—41no—
172C22⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—81no—
176C22⋊C4—82no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—41no—
183C42—41no—
184C42—41no—
185C42—81no—
186C42—81no—
187C42—82no—
188C42C2×C4⋊C411yescharacteristic
189C42#11711yescharacteristic
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23—81no—
195C23—81no—
196C23C2×#9911yescharacteristic
197Q8—41no—
198Q8—41no—
199Q8—81no—
200D4—41no—
201D4—41no—
202D4—81no—
203C2×C4—21no—
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—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—161no—
231C2×C4—162no—
232C2×C4—162no—
233C2×C4—162no—
234C2×C4#35411yescharacteristic
235C2×C4#35411yescharacteristic
236C2×C4#38011yescharacteristic
237C2×C4#41011yescharacteristic
238C8—42no—
239C8—42no—
240C8—162no—
241C22—21no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—161no—
250C22#179911yescharacteristic
251C22C2×#35411yescharacteristic
252C22#212611yescharacteristic
253C4—21no—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—41no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—81no—
263C4—82no—
264C4—81no—
265C4—81no—
266C4—161no—
267C4—322no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—81no—
272C2#1279811yescharacteristic
273C2#1360311yescharacteristic
274C2#1360411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocleC2×#124C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C4×#125C4semidirect
11. #1991C4semidirect
15. #1835C4semidirect
20. C4×#97C4semidirect
73. #125C4⋊C4semidirect
119. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#216482
C2×#14142
#128012
#127982
#75022