Finite Group Databasebeta

← #128777

Subgroups — #128777

Order 512.

546 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C4×#467C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic
3C2×#1348C212yesmaximal
4C4×#340C211yesmaximal, characteristic
5#7258C212yesmaximal
6C2×#467C2211yescharacteristic
7C2×C4×Q16C411yescharacteristic, complemented
8C22×C4×C8C2211yescharacteristic
9C4×C22⋊Q8—21no—
10C2×C4×C4⋊C4—21no—
11C2×C22⋊C16C2212yes—
12C2×#346—21no—
13#1969C411yescharacteristic, complemented
14#1942C411yescharacteristic, complemented
15#1933—21no—
16#1892—21no—
17#1808C2211yescharacteristic
18C4×#107C411yescharacteristic, complemented
19C4×#103—21no—
20C2×#340—21no—
21C2×#340C2212yes—
22#1374—21no—
23#1348C418yescomplemented
24C22×Q16C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C23×C8C2311yescharacteristic
27C2×C22⋊Q8—21no—
28C2×C4×Q8—21no—
29C22×C4⋊C4—22no—
30C22×C42—41no—
31C22×C42D411yescharacteristic
32C22×C16—22no—
33#467—22no—
34#467C2×C414yes—
35C4×Q16—42no—
36C2×#107C2×C411yescharacteristic
37C2×#107C2×C411yescharacteristic
38C2×#107C2×C411yescharacteristic
39C2×#103—21no—
40C2×#103—21no—
41C2×#103—21no—
42C2×C4×C8—42no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—21no—
48#389—21no—
49#387—41no—
50#385—41no—
51#380—21no—
52C4×C4⋊C4—24no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—42no—
55C4×C4⋊C4—41no—
56C4×C22⋊C4—41no—
57C2×#95—41no—
58C22⋊C16—22no—
59C22⋊C16C2×C414yes—
60#346—42no—
61#340—24no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—22no—
66#340C2×C414yes—
67C22×Q8—21no—
68C23×C4—42no—
69C23×C4D811yescharacteristic
70C23×C4D811yescharacteristic
71C23×C4C2×D411yescharacteristic
72C2×Q16—21no—
73C2×Q16—41no—
74C2×Q16—41no—
75C2×Q16C4212yescomplemented
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C22×C412yes—
85C22×C8C2×D411yescharacteristic
86C22×C8#4112yes—
87C22⋊Q8—24no—
88C22⋊Q8—42no—
89C4×Q8—42no—
90C4×Q8—42no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C42—22no—
106C2×C42—22no—
107C2×C42—44no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—82no—
111C2×C42—82no—
112C2×C42C22⋊C411yescharacteristic
113C2×C42C22⋊C411yescharacteristic
114C2×C42SD1611yescharacteristic
115C2×C42SD1611yescharacteristic
116C2×C16—24no—
117C2×C16—42no—
118C2×C16—42no—
119#107—21no—
120#107—41no—
121#107—41no—
122#107C4212yescomplemented
123#103—24no—
124#103—42no—
125#103—42no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—41no—
129C4×C8—41no—
130#95—44no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136C24C2×D811yescharacteristic
137C2×Q8—22no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—41no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—44no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—82no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4#10211yescharacteristic
166C22×C4#10211yescharacteristic
167C22×C4#10211yescharacteristic
168C22×C4#10211yescharacteristic
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C4×D412yes—
171C22×C4C2×SD1611yescharacteristic
172C22×C4#13512yes—
173Q16—42no—
174Q16—42no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8#10412yes—
188C2×C8C2×C4211yescharacteristic
189C2×C8#11711yescharacteristic
190C2×C8C4×D412yes—
191C4⋊C4—28no—
192C4⋊C4—22no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—41no—
197C4⋊C4—44no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C22⋊C4—44no—
205C22⋊C4—81no—
206C22⋊C4—81no—
207C42—22no—
208C42—22no—
209C42—21no—
210C42—21no—
211C42—82no—
212C42—88no—
213C42—82no—
214C42#9912yes—
215C42#10412yes—
216C16—44no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23C2×#10211yescharacteristic
222C23C4×D812yes—
223C23#50911yescharacteristic
224C23#51011yescharacteristic
225C23#51012yes—
226Q8—42no—
227Q8—42no—
228Q8—81no—
229Q8—81no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—44no—
248C2×C4—48no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—84no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4#32712yes—
268C2×C4#32712yes—
269C2×C4#36012yes—
270C2×C4C4×C22⋊C411yescharacteristic
271C2×C4C2×#9911yescharacteristic
272C2×C4#41011yescharacteristic
273C2×C4#41611yescharacteristic
274C2×C4C2×#10411yescharacteristic
275C2×C4C2×#10411yescharacteristic
276C2×C4C4×SD1612yes—
277C8—22no—
278C8—21no—
279C8—21no—
280C8—22no—
281C8—41no—
282C22—22no—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—21no—
289C22—21no—
290C22C2×#32711yescharacteristic
291C22#150211yescharacteristic
292C22C4×#10211yescharacteristic
293C22#213911yescharacteristic
294C22#214111yescharacteristic
295C22C2×#36011yescharacteristic
296C22#214511yescharacteristic
297C22#217512yes—
298C22#217612yes—
299C4—22no—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—88no—
305C4—82no—
306C4—82no—
307C4—161no—
308C4—161no—
309C4#150011yescharacteristic
310C4#150111yescharacteristic
311C4C4×#9911yescharacteristic
312C4C4×#10411yescharacteristic
313C2—22no—
314C2—22no—
315C2C4×#32711yescharacteristic
316C2#1170411yescharacteristic
317C2#1170611yescharacteristic
318C2#1174511yescharacteristic
319C2#1174611yescharacteristic
320C2#1263811yescharacteristic
321C2C4×#36011yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC2×C8C2×C42
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q16C4semidirect
13. #1969C4semidirect
14. #1942C4semidirect
18. C4×#107C4semidirect
23. #1348C4semidirect
75. C2×Q16C42semidirect
122. #107C42semidirect

Maximal subgroups

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

subgroupindex
C4×#4672
#125992
C2×#13482
C4×#3402
#72582