Finite Group Databasebeta

← #125350

Subgroups — #125350

Order 512.

342 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 82 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#33211C211yesmaximal, characteristic
2C4×#349C211yesmaximal, characteristic
3#7862C211yesmaximal, characteristic
4#7127C212yesmaximal, complemented
5#7126C212yesmaximal, complemented
6C2×#499C411yescharacteristic
7#2920—21no—
8C2×C4×M16C2211yescharacteristic
9#2851—21no—
10C2×#349—21no—
11C4×M32—21no—
12#1991—21no—
13#1944C2211yescharacteristic
14#1944C2211yescharacteristic
15#1943—21no—
16#1846C411yescharacteristic
17#1809—21no—
18#1809C411yescharacteristic
19#1809C411yescharacteristic
20#1377C2212yes—
21#1275—21no—
22#1275C2212yes—
23C22×M16C2×C411yescharacteristic
24C22×M16D411yescharacteristic
25C2×#125—21no—
26C22×C42D411yescharacteristic
27C2×M32—22no—
28#499—41no—
29#499—41no—
30#432—41no—
31#432—41no—
32#431—22no—
33#431—22no—
34C2×#107—21no—
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37C2×#106—21no—
38C2×#106C2×C411yescharacteristic
39C2×#106C2×C411yescharacteristic
40#423—41no—
41#407—41no—
42C4×M16—22no—
43C4×M16—41no—
44C4×M16—41no—
45C2×#97D411yescharacteristic
46C2×#97C2311yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#389—21no—
50#389—21no—
51#387—41no—
52#387—41no—
53#385—41no—
54#385—41no—
55#382—21no—
56#380—21no—
57#380—21no—
58#349—24no—
59#346—41no—
60C4×C16—41no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340—21no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4SD1611yescharacteristic
68C2×M16—22no—
69C2×M16—22no—
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—42no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79#125—41no—
80#125—41no—
81#125—41no—
82#125—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42D811yescharacteristic
99C2×C42SD1611yescharacteristic
100C2×C42C2×D411yescharacteristic
101M32—24no—
102M32—42no—
103C2×C16—42no—
104#107—42no—
105#107—41no—
106#107—41no—
107#106—22no—
108#106—22no—
109#106—42no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114#97—41no—
115#97—41no—
116#97—42no—
117C4×C8—24no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—21no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128C24#10211yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—81no—
144C22×C4—81no—
145C22×C4#10211yescharacteristic
146C22×C4#10211yescharacteristic
147C22×C4#10211yescharacteristic
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C2×D811yescharacteristic
150C22×C4C2×SD1611yescharacteristic
151M16—44no—
152M16—44no—
153M16—84no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—44no—
163C2×C8#10112yes—
164C2×C8#10412yes—
165C2×C8#13112yes—
166C2×C8#13712yes—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C22⋊C4—82no—
182C22⋊C4—82no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42#9911yescharacteristic
190C42#9911yescharacteristic
191C42#10411yescharacteristic
192C42#10411yescharacteristic
193C16—44no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—41no—
198C23—41no—
199C23C2×#10211yescharacteristic
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4#32711yescharacteristic
230C2×C4#32711yescharacteristic
231C2×C4#32711yescharacteristic
232C2×C4#32711yescharacteristic
233C2×C4#40811yescharacteristic
234C2×C4C2×#9911yescharacteristic
235C2×C4C2×#10111yescharacteristic
236C2×C4#41911yescharacteristic
237C2×C4C2×#10411yescharacteristic
238C2×C4C2×#10411yescharacteristic
239C8—22no—
240C8—22no—
241C8—22no—
242C8—22no—
243C8—84no—
244C22—21no—
245C22—21no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22#149211yescharacteristic
252C22C2×#32711yescharacteristic
253C22#154611yescharacteristic
254C4—21no—
255C4—21no—
256C4—21no—
257C4—21no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—161no—
262C4—161no—
263C4—161no—
264C4—161no—
265C4#147211yescharacteristic
266C4#148511yescharacteristic
267C4#150811yescharacteristic
268C4#152611yescharacteristic
269C2—21no—
270C2—21no—
271C2—81no—
272C2#768511yescharacteristic
273C2#1168511yescharacteristic
274C2#1168811yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7127C2semidirect
5. #7126C2semidirect

Maximal subgroups

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

subgroupindex
#332112
C4×#3492
#78622
#71272
#71262