Finite Group Databasebeta

← #129647

Subgroups — #129647

Order 512.

746 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C8×H(5,𝔽2)C211yesmaximal, characteristic
2#13189C211yesmaximal, characteristic, complemented
3#11621C212yesmaximal, complemented
4#11613C212yesmaximal, complemented
5#11487C211yesmaximal, characteristic, complemented
6C4×H(5,𝔽2)C411yescharacteristic
7#2988—22no—
8#2988C412yes—
9#2972—22no—
10#2972—22no—
11#2972C411yescharacteristic
12C8×#41—22no—
13C8×#41C2212yes—
14C2×C8×D4—22no—
15C2×C8×D4—22no—
16C2×C8×D4C2211yescharacteristic
17#2162C2212yescomplemented
18C2×C4⋊C16C2211yescharacteristic, complemented
19C2×C22⋊C16—22no—
20C2×C4×C16C2211yescharacteristic, complemented
21Q8⋊C16—22no—
22Q8⋊C16—22no—
23D4⋊C16—22no—
24D4⋊C16—22no—
25#1314—22no—
26C2×H(5,𝔽2)C811yescharacteristic
27C23×C8—22no—
28C23×C8D412yes—
29#520—22no—
30#520C812yes—
31#518—22no—
32#518—22no—
33#518C811yescharacteristic
34C4×#41—22no—
35C4×#41C2×C412yes—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4C2×C411yescharacteristic
39C22×C16—21no—
40C22×C16—21no—
41C8×Q8—22no—
42C8×Q8D412yes—
43C8×Q8D412yes—
44C8×D4—24no—
45C8×D4—24no—
46C8×D4—24no—
47C8×D4—24no—
48C8×D4—24no—
49C8×D4—22no—
50C8×D4—22no—
51C8×D4—22no—
52C8×D4—22no—
53C8×D4—22no—
54C8×D4—22no—
55C8×D4—22no—
56C8×D4D412yes—
57C8×D4D412yes—
58#431—24no—
59#431—22no—
60#431—22no—
61#431C2×C412yes—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2×C411yescharacteristic
65C2×C22⋊C8—22no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8D412yes—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8C2311yescharacteristic
72C4⋊C16—22no—
73C4⋊C16—22no—
74C22⋊C16—22no—
75C22⋊C16—44no—
76C4×C16—22no—
77C4×C16—22no—
78H(5,𝔽2)—24no—
79H(5,𝔽2)—22no—
80H(5,𝔽2)C1614yescomplemented
81C2×#41—22no—
82C2×#41C2×C812yes—
83C22×D4—22no—
84C22×D4—22no—
85C22×D4C2×C811yescharacteristic
86C23×C4—22no—
87C23×C4C22⋊C412yes—
88C22×C8—24no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—42no—
96C22×C8—42no—
97C22×C8—82no—
98C22×C8—82no—
99C22×C8C22⋊C412yes—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C4×Q8—22no—
103C4×Q8C22⋊C412yes—
104C4×Q8M1612yes—
105C4×D4—24no—
106C4×D4—24no—
107C4×D4—24no—
108C4×D4—24no—
109C4×D4—22no—
110C4×D4—24no—
111C4×D4—22no—
112C4×D4—22no—
113C4×D4—22no—
114C4×D4—22no—
115C4×D4—22no—
116C4×D4—22no—
117C4×D4C22⋊C412yes—
118C4×D4M1612yes—
119#117—24no—
120#117—22no—
121#117—22no—
122#117C2×C812yes—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4C2×C811yescharacteristic
126C2×C22⋊C4—24no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4M1612yes—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42C22×C411yescharacteristic
133C2×C16—22no—
134C2×C16—41no—
135C2×C16—41no—
136C2×C16—42no—
137C2×C16—42no—
138C2×C16—41no—
139C4⋊C8—24no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8C22⋊C412yes—
144C4⋊C8C22⋊C412yes—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—24no—
151C4×C8—24no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8C2×D412yes—
156C4×C8C2×D412yes—
157C24—22no—
158C24C22⋊C812yes—
159#41—28no—
160#41—28no—
161#41—22no—
162#41—22no—
163#41C2×C1614yes—
164#41M3214yes—
165C2×Q8C22⋊C812yes—
166C2×D4—24no—
167C2×D4—28no—
168C2×D4—28no—
169C2×D4—28no—
170C2×D4—22no—
171C2×D4—28no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4C22⋊C812yes—
175C2×D4C2×C1612yes—
176C2×D4M3212yes—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4C22⋊C812yes—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C22×C811yescharacteristic
193C22×C4C2×M1611yescharacteristic
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—84no—
209C2×C8—82no—
210C2×C8#12012yes—
211C2×C8#12011yescharacteristic
212C2×C8#12011yescharacteristic
213C4⋊C4—24no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4C22⋊C812yes—
218C4⋊C4C22⋊C812yes—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C42—24no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42C2×C22⋊C412yes—
230C42C2×M1612yes—
231C16—42no—
232C16—82no—
233C16—81no—
234C23—28no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—82no—
241C23—82no—
242C23C22⋊C1614yes—
243C23C2×C22⋊C811yescharacteristic
244Q8—22no—
245Q8C22⋊C1614yescomplemented
246D4—216no—
247D4—24no—
248D4—24no—
249D4—24no—
250D4—22no—
251D4—24no—
252D4C22⋊C1614yescomplemented
253C2×C4—28no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4#37911yescharacteristic
274C2×C4C2×C22⋊C812yes—
275C2×C4#40712yes—
276C2×C4#40711yescharacteristic
277C2×C4C22×C1611yescharacteristic
278C2×C4C2×M3212yes—
279C2×C4C2×M3211yescharacteristic
280C8—21no—
281C8—22no—
282C8—42no—
283C8—42no—
284C8—42no—
285C8#45311yescharacteristic
286C8#45511yescharacteristic
287C22—21no—
288C22—21no—
289C22—24no—
290C22—24no—
291C22—44no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—82no—
297C22—88no—
298C22#177911yescharacteristic
299C22C2×C22⋊C1611yescharacteristic
300C22#211211yescharacteristic
301C4—21no—
302C4—24no—
303C4—22no—
304C4—42no—
305C4—42no—
306C4—42no—
307C4#179211yescharacteristic
308C4#179411yescharacteristic
309C4C2×C22⋊C1612yes—
310C4#211212yes—
311C2—21no—
312C2—21no—
313C2—44no—
314C2—44no—
315C2#831811yescharacteristic
316C2#1267911yescharacteristic
317C2#1268511yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC2×H(5,𝔽2)C8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13189C2semidirect
3. #11621C2semidirect
4. #11613C2semidirect
5. #11487C2semidirect
17. #2162C22semidirect
18. C2×C4⋊C16C22semidirect
20. C2×C4×C16C22semidirect
80. H(5,𝔽2)C16semidirect
245. Q8C22⋊C16semidirect
252. D4C22⋊C16semidirect

Maximal subgroups

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

subgroupindex
C8×H(5,𝔽2)2
#131892
#116212
#116132
#114872