Finite Group Databasebeta

← #108724

Subgroups — #108724

Order 512.

796 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 158 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#13807C211yesmaximal, characteristic
2C4×#342C211yesmaximal, characteristic
3C2×#1297C212yesmaximal
4C2×#1284C212yesmaximal
5#7911C211yesmaximal, characteristic
6C2×#407C2212yes—
7C2×#381C2211yescharacteristic
8C2×C4×C22⋊C4C2212yes—
9C2×C4×C22⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11C2×#342—22no—
12#1437—22no—
13#1434—21no—
14#1434C412yescomplemented
15C4×#95—21no—
16C4×#95C412yescomplemented
17#1297C418yescomplemented
18#1284—24no—
19#1280—21no—
20#1280—21no—
21C24×C4C2311yescharacteristic
22C22×M16—22no—
23C22×C22⋊C4—22no—
24C22×C22⋊C4—21no—
25C22×C22⋊C4—21no—
26C22×C22⋊C4D412yes—
27C22×C42—21no—
28C22×C42—22no—
29C22×C42—22no—
30C22×C42C2×C412yes—
31#407—22no—
32#407—22no—
33#407—22no—
34#407C2×C414yes—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37#381—21no—
38#381—21no—
39#381D412yes—
40#381Q812yes—
41C4×C22⋊C4—22no—
42C4×C22⋊C4—22no—
43C4×C22⋊C4—22no—
44C4×C22⋊C4—22no—
45C4×C22⋊C4—22no—
46C4×C22⋊C4—22no—
47C4×C22⋊C4C2×C414yes—
48C4×C22⋊C4C2×C414yes—
49C2×#95—21no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C412yes—
57C2×#95D411yescharacteristic
58C2×#95D411yescharacteristic
59#342—28no—
60C25C2×D411yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—22no—
64C23×C4—42no—
65C23×C4—82no—
66C23×C4C22⋊C412yes—
67C23×C4C22⋊C412yes—
68C23×C4C22×C411yescharacteristic
69C23×C4C22×C412yes—
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×Q811yescharacteristic
73C23×C4#4112yes—
74C23×C4#4112yes—
75C23×C4#4111yescharacteristic
76C23×C4#4111yescharacteristic
77C23×C4#4112yes—
78C2×M16—24no—
79C2×M16—42no—
80C2×M16—84no—
81C22×C8—42no—
82C2×C22⋊C4—28no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—82no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—24no—
98C2×C42—24no—
99C2×C42—44no—
100C2×C42—44no—
101C2×C42—42no—
102C2×C42—82no—
103C2×C42—82no—
104C2×C42—84no—
105C2×C42—82no—
106C22⋊C8—24no—
107C22⋊C8—24no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110#95—21no—
111#95—21no—
112#95—22no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95—22no—
117#95—24no—
118#95—44no—
119#95—44no—
120#95—42no—
121#95—82no—
122#95C4212yes—
123#95C4212yes—
124#95C4⋊C414yes—
125C24—22no—
126C24—22no—
127C24—41no—
128C24—41no—
129C24—42no—
130C24C2×C22⋊C411yescharacteristic
131C24C4×D412yes—
132C24#12111yescharacteristic
133C24C22⋊Q811yescharacteristic
134C24#12312yes—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—48no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—82no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—82no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—84no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4#10414yes—
177C22×C4C2×C4211yescharacteristic
178C22×C4C2×C4⋊C411yescharacteristic
179C22×C4#11711yescharacteristic
180C22×C4#11711yescharacteristic
181C22×C4#11711yescharacteristic
182C22×C4#11712yes—
183C22×C4#11712yes—
184C22×C4#11711yescharacteristic
185C22×C4C4×D412yes—
186C22×C4C4×D412yes—
187C22×C4C4×Q812yes—
188C22×C4#12111yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4#12312yes—
191C22×C4#12412yes—
192C22×C4#12512yes—
193C22×C4#12612yes—
194C22×C4#12612yes—
195M16—84no—
196M16—84no—
197C2×C8—44no—
198C2×C8—84no—
199C22⋊C4—28no—
200C22⋊C4—48no—
201C22⋊C4—48no—
202C22⋊C4—48no—
203C22⋊C4—44no—
204C22⋊C4—82no—
205C22⋊C4—82no—
206C42—82no—
207C42—82no—
208C42—84no—
209C42—88no—
210C42—84no—
211C42—84no—
212C23—21no—
213C23—21no—
214C23—22no—
215C23—24no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—42no—
224C23—41no—
225C23—41no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—82no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23C4×C22⋊C411yescharacteristic
234C23#38011yescharacteristic
235C23#38111yescharacteristic
236C23#38712yes—
237C23#38812yes—
238C23C2×#10412yes—
239C23#42112yes—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—48no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—84no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—84no—
286C2×C4—88no—
287C2×C4—88no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4#37611yescharacteristic
291C2×C4C4×C4⋊C411yescharacteristic
292C2×C4#38211yescharacteristic
293C2×C4#38311yescharacteristic
294C2×C4#38712yes—
295C2×C4#38812yes—
296C8—84no—
297C22—22no—
298C22—22no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—44no—
312C22—44no—
313C22—82no—
314C22—82no—
315C22—82no—
316C22—82no—
317C22#143411yescharacteristic
318C22C4×#10411yescharacteristic
319C22#175911yescharacteristic
320C22#180612yes—
321C22#182612yes—
322C22#182711yescharacteristic
323C22#182711yescharacteristic
324C22#182812yes—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—22no—
331C4—44no—
332C4—44no—
333C4—84no—
334C4—82no—
335C4—88no—
336C4—82no—
337C2—21no—
338C2—22no—
339C2—21no—
340C2—42no—
341C2—42no—
342C2#877612yes—
343C2#877812yes—
344C2#1080811yescharacteristic
345C2#1082411yescharacteristic
346C2#1082911yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #1434C4semidirect
16. C4×#95C4semidirect
17. #1297C4semidirect

Maximal subgroups

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

subgroupindex
#138072
C4×#3422
C2×#12972
C2×#12842
#79112