Finite Group Databasebeta

← #122683

Subgroups — #122683

Order 512.

357 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 74 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#23699C211yesmaximal, characteristic
2#11460C211yesmaximal, characteristic
3#8001C211yesmaximal, characteristic
4#7091C212yesmaximal, complemented
5#7090C212yesmaximal, complemented
6C2×C4×Q16C411yescharacteristic
7C2×#431C2211yescharacteristic
8C2×#423—21no—
9C4×C22⋊Q8—21no—
10#2163—21no—
11#2086C2211yescharacteristic
12#2075C411yescharacteristic
13#2066—21no—
14#2056C2211yescharacteristic
15#2035—21no—
16#2001C411yescharacteristic
17C4×#106C411yescharacteristic
18C4×#103—21no—
19#1322C2212yes—
20#1314—21no—
21#1292—21no—
22#1292C2212yes—
23C22×Q16C2×C411yescharacteristic
24C22×M16—41no—
25C2×C22⋊Q8—21no—
26C2×C4×Q8—21no—
27C22×C42D411yescharacteristic
28C4×Q16—41no—
29C4×Q16—41no—
30#431—41no—
31#431—41no—
32C2×#107C2×C411yescharacteristic
33C2×#106—21no—
34#423—22no—
35#423—22no—
36#423—41no—
37#423—41no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—41no—
40C2×C4⋊C8D411yescharacteristic
41C2×C4⋊C8C2311yescharacteristic
42C2×#103—21no—
43C2×#103—21no—
44C2×#103—21no—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8D411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#391—21no—
50#389—21no—
51#387—41no—
52#385—41no—
53#380—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—41no—
56C4×C22⋊C4—41no—
57C4⋊C16—41no—
58C22⋊C16—22no—
59#346—41no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C22×Q8—21no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4SD1611yescharacteristic
68C2×Q16—41no—
69C2×Q16—42no—
70C2×M16—42no—
71C2×M16—42no—
72C2×M16—82no—
73C2×M16—82no—
74C22×C8—42no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C22⋊Q8—42no—
80C22⋊Q8—42no—
81C4×Q8—41no—
82C4×Q8—41no—
83C4×Q8—41no—
84C4×Q8—41no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
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×C42—41no—
98C2×C42—41no—
99C2×C42C22⋊C411yescharacteristic
100C2×C42D811yescharacteristic
101C2×C42SD1611yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C16—42no—
104#107—41no—
105#106—42no—
106C4⋊C8—24no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—42no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112#103—42no—
113#103—42no—
114#103—42no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—21no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—41no—
126C24#10211yescharacteristic
127C2×Q8—22no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4#10211yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4#10211yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×D811yescharacteristic
154C22×C4C2×SD1611yescharacteristic
155Q16—42no—
156Q16—42no—
157M16—84no—
158M16—82no—
159M16—82no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—44no—
169C2×C8—84no—
170C4⋊C4—22no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—44no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C22⋊C4—44no—
182C22⋊C4—82no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—82no—
190C42—81no—
191C42—81no—
192C42#9911yescharacteristic
193C42#9911yescharacteristic
194C42#10411yescharacteristic
195C42#10411yescharacteristic
196C16—82no—
197C23—21no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23#36012yes—
202C23C2×#10211yescharacteristic
203Q8—42no—
204Q8—42no—
205Q8—81no—
206Q8—81no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
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—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4#32711yescharacteristic
240C2×C4#32711yescharacteristic
241C2×C4#32711yescharacteristic
242C2×C4#32711yescharacteristic
243C2×C4C2×#9911yescharacteristic
244C2×C4C2×#10411yescharacteristic
245C2×C4#50911yescharacteristic
246C2×C4#51011yescharacteristic
247C8—41no—
248C8—41no—
249C8—41no—
250C8—41no—
251C8—84no—
252C22—21no—
253C22—21no—
254C22—22no—
255C22—42no—
256C22—41no—
257C22—41no—
258C22—81no—
259C22C2×#32711yescharacteristic
260C22#213911yescharacteristic
261C22C2×#36011yescharacteristic
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—82no—
271C4—82no—
272C4—161no—
273C4—161no—
274C4#134212yes—
275C4#212011yescharacteristic
276C4#213211yescharacteristic
277C2—21no—
278C2—21no—
279C2—42no—
280C2C2×#134211yescharacteristic
281C2#1169711yescharacteristic
282C2#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. #7090C2semidirect

Maximal subgroups

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

subgroupindex
#236992
#114602
#80012
#70912
#70902