Finite Group Databasebeta

← #122673

Subgroups — #122673

Order 512.

316 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 72 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#29879C211yesmaximal, characteristic
2#11460C211yesmaximal, characteristic
3#8001C211yesmaximal, characteristic
4#7091C212yesmaximal, complemented
5#7089C212yesmaximal, complemented
6C2×#499C411yescharacteristic
7C2×#431C2211yescharacteristic
8C2×#423—21no—
9#2851—21no—
10#2163—21no—
11#2088—21no—
12#2087C2211yescharacteristic
13#2086C2211yescharacteristic
14#2077—21no—
15#2076C411yescharacteristic
16#2075C411yescharacteristic
17#1991—21no—
18#1835C411yescharacteristic
19#1322C2212yes—
20#1314—21no—
21#1292—21no—
22#1292C2212yes—
23C22×M16—41no—
24C2×#125—21no—
25C22×C42D411yescharacteristic
26#499—41no—
27#499—41no—
28#431—41no—
29#431—41no—
30C2×#107C2×C411yescharacteristic
31C2×#107C2×C411yescharacteristic
32C2×#106—21no—
33#423—22no—
34#423—22no—
35#423—41no—
36#423—41no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—41no—
39C2×C4⋊C8D411yescharacteristic
40C2×C4⋊C8C2311yescharacteristic
41C2×C22⋊C8C2×C411yescharacteristic
42C2×C22⋊C8D411yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#389—21no—
46#389—21no—
47#387—41no—
48#387—41no—
49#385—41no—
50#385—41no—
51#382—21no—
52#380—21no—
53#380—21no—
54C4⋊C16—41no—
55C22⋊C16—22no—
56#346—41no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4D811yescharacteristic
66C23×C4SD1611yescharacteristic
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—82no—
70C2×M16—82no—
71C22×C8—42no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76#125—41no—
77#125—41no—
78#125—41no—
79#125—41no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42D811yescharacteristic
96C2×C42SD1611yescharacteristic
97C2×C42C2×D411yescharacteristic
98C2×C16—42no—
99#107—41no—
100#107—41no—
101#106—42no—
102C4⋊C8—24no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C4⋊C8—42no—
108C22⋊C8—41no—
109C22⋊C8—41no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120C24#10211yescharacteristic
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—81no—
137C22×C4#10211yescharacteristic
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143M16—84no—
144M16—82no—
145M16—82no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—44no—
155C2×C8—84no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—82no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C22⋊C4—82no—
172C22⋊C4—82no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42#9911yescharacteristic
180C42#9911yescharacteristic
181C42#10411yescharacteristic
182C42#10411yescharacteristic
183C16—82no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—41no—
188C23#36012yes—
189C23C2×#10211yescharacteristic
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4#32711yescharacteristic
223C2×C4#32711yescharacteristic
224C2×C4#32711yescharacteristic
225C2×C4#32711yescharacteristic
226C2×C4C2×#9911yescharacteristic
227C2×C4C2×#10411yescharacteristic
228C2×C4#50911yescharacteristic
229C2×C4#51011yescharacteristic
230C8—41no—
231C8—41no—
232C8—41no—
233C8—41no—
234C8—84no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—81no—
242C22C2×#32711yescharacteristic
243C22#213911yescharacteristic
244C22C2×#36011yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—41no—
253C4—41no—
254C4—161no—
255C4—161no—
256C4—161no—
257C4—161no—
258C4#212311yescharacteristic
259C4#213511yescharacteristic
260C2—21no—
261C2—21no—
262C2—42no—
263C2#1168811yescharacteristic
264C2#1170011yescharacteristic
265C2#1170411yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7091C2semidirect
5. #7089C2semidirect

Maximal subgroups

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

subgroupindex
#298792
#114602
#80012
#70912
#70892