Finite Group Databasebeta

← #94087

Subgroups — #94087

Order 512.

410 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 66 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#11431C211yesmaximal, characteristic
2#10751C211yesmaximal, characteristic
3#7563C211yesmaximal, characteristic
4C2×#568C411yescharacteristic
5C2×#410C411yescharacteristic
6#2297C2211yescharacteristic
7C4×C22⋊C8—21no—
8#1733—21no—
9C2×#339—21no—
10#1458—21no—
11#1319—21no—
12#1318C412yescomplemented
13#1272—21no—
14#1270C412yescomplemented
15C22×#41C2×C411yescharacteristic
16#568—21no—
17#568—42no—
18#568—41no—
19#568C812yescomplemented
20C2×#131—42no—
21C22×M16—21no—
22C22×M16—41no—
23C2×#117—21no—
24C2×#117D411yescharacteristic
25C22×C42D411yescharacteristic
26#431—41no—
27#410—21no—
28#410—41no—
29#410C812yescomplemented
30C2×#99—41no—
31C2×C22⋊C8—41no—
32C2×C4×C8—41no—
33#391D411yescharacteristic
34#391Q811yescharacteristic
35#390C2×C411yescharacteristic
36#390C2×C411yescharacteristic
37#386—41no—
38C4×C22⋊C4—41no—
39#339—42no—
40#339—42no—
41#336—22no—
42#324—22no—
43#323—41no—
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—41no—
47C2×#41—41no—
48C2×#41—41no—
49C2×#41—41no—
50C2×#41C2×C811yescharacteristic
51C2×#41C2×C811yescharacteristic
52C2×#41M1611yescharacteristic
53C2×#41M1611yescharacteristic
54C22×Q8C4⋊C411yescharacteristic
55C22×D4—21no—
56C22×D4C4211yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C22⋊C411yescharacteristic
60#131—44no—
61#131—48no—
62#131—44no—
63C2×M16—22no—
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—41no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—42no—
73C2×M16—42no—
74C22×C8—41no—
75C22×C8—42no—
76C22×C8—42no—
77#117—22no—
78#117—41no—
79#117—41no—
80#117—41no—
81#117—41no—
82#117—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4C22⋊C411yescharacteristic
85C2×C4⋊C4C4⋊C411yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C42—41no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C4⋊C411yescharacteristic
97C4⋊C8—81no—
98#99—42no—
99#99—42no—
100C22⋊C8—44no—
101C22⋊C8—42no—
102C4×C8—81no—
103C4×C8—81no—
104C4×C8—81no—
105#95—41no—
106#95—41no—
107C24—21no—
108C24#9911yescharacteristic
109#41—42no—
110#41—42no—
111#41—42no—
112#41—42no—
113#41—41no—
114#41—41no—
115#41—41no—
116#41—41no—
117#41—41no—
118#41—41no—
119#41—41no—
120#41—41no—
121C2×Q8—21no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8C4⋊C811yescharacteristic
125C2×Q8C4⋊C811yescharacteristic
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4C4×C811yescharacteristic
136C2×D4#9711yescharacteristic
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4#9511yescharacteristic
159C22×C4C22⋊C811yescharacteristic
160C22×C4C22⋊C811yescharacteristic
161C22×C4#9911yescharacteristic
162C22×C4#10011yescharacteristic
163C22×C4#10111yescharacteristic
164M16—24no—
165M16—42no—
166M16—44no—
167M16—44no—
168M16—42no—
169M16—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—44no—
173C2×C8—42no—
174C2×C8—44no—
175C2×C8—44no—
176C2×C8—82no—
177C2×C8—82no—
178C2×C8—82no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—81no—
183C22⋊C4—42no—
184C22⋊C4—82no—
185C22⋊C4—82no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C42—41no—
189C42—41no—
190C42—42no—
191C42—41no—
192C42—41no—
193C42—41no—
194C42—41no—
195C42—81no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—81no—
205C23—81no—
206C23#32311yescharacteristic
207C23#32411yescharacteristic
208C23#34211yescharacteristic
209Q8—41no—
210Q8—41no—
211Q8—42no—
212D4—41no—
213D4—41no—
214D4—42no—
215D4—42no—
216D4—42no—
217D4—42no—
218D4—42no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—84no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—161no—
254C2×C4—161no—
255C2×C4#32311yescharacteristic
256C2×C4#32411yescharacteristic
257C2×C4#33611yescharacteristic
258C2×C4#34311yescharacteristic
259C2×C4AΣL(1,𝔽16)11yescharacteristic
260C2×C4AΣL(1,𝔽16)11yescharacteristic
261C2×C4#35211yescharacteristic
262C2×C4#35211yescharacteristic
263C8—44no—
264C8—84no—
265C8—162no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—161no—
282C22#128011yescharacteristic
283C22#139011yescharacteristic
284C22#139011yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—162no—
294C4#131611yescharacteristic
295C4#131711yescharacteristic
296C4#132611yescharacteristic
297C4#132611yescharacteristic
298C2—21no—
299C2—21no—
300C2—41no—
301C2—41no—
302C2—81no—
303C2—81no—
304C2#718711yescharacteristic
305C2#748611yescharacteristic
306C2#748811yescharacteristic
subgroupG/H
centerC2×C4AΣL(1,𝔽16)
commutatorC2×D4C4×C8
Frattini, cosocle#2297C22
socleC22#1390
prime radicalC22×#41C2×C4
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G
LECSG ▷ #2297 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1318C4semidirect
14. #1270C4semidirect
19. #568C8semidirect
29. #410C8semidirect

Maximal subgroups

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

subgroupindex
#114312
#107512
#75632