Finite Group Databasebeta

← #99945

Subgroups — #99945

Order 512.

463 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 112 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#22307C211yesmaximal, characteristic, complemented
2#22268C211yesmaximal, characteristic
3#9642C211yesmaximal, characteristic
4#7590C211yesmaximal, characteristic, complemented
5#7587C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic, complemented
7#7014C211yesmaximal, characteristic, complemented
8C2×#492C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#1984—22no—
12#1957C411yescharacteristic
13#1956C411yescharacteristic
14#1939—21no—
15#1916—22no—
16#1880—22no—
17#1877—22no—
18#1834C411yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21C8×C4⋊C4—21no—
22#1744C2211yescharacteristic
23#1744C2211yescharacteristic
24#1744C2211yescharacteristic
25C2×D4⋊C8—21no—
26C2×D4⋊C8—21no—
27#1272—22no—
28#1270—22no—
29C22×SD16—22no—
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×#127C2×C411yescharacteristic
32C2×C4×Q8D412yes—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35C2×C4×D4—21no—
36#492—22no—
37#492C814yescomplemented
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—22no—
41C2×#103—22no—
42C2×#102—22no—
43C2×C4×C8—21no—
44C2×C4×C8—22no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#391D412yes—
50#390—21no—
51#390—21no—
52#390—21no—
53#389—21no—
54#389—21no—
55#384C2×C411yescharacteristic
56#384C2×C411yescharacteristic
57C4×C4⋊C4—21no—
58C4×C4⋊C4—21no—
59C43C2311yescharacteristic
60#340—22no—
61#336—22no—
62#336—22no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336D412yes—
73D4⋊C8—42no—
74D4⋊C8—42no—
75C22×Q8C22⋊C412yes—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C23×C4—41no—
80C2×SD16—28no—
81C2×SD16—82no—
82C2×SD16—82no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—41no—
89C22×C8—41no—
90C4⋊Q8—21no—
91C4⋊Q8—21no—
92C4⋊Q8C2×C812yes—
93C4⋊Q8M1612yes—
94#127—21no—
95#127—21no—
96#127—21no—
97#127C2×C812yes—
98C4×Q8—82no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—82no—
104C4×D4—82no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C22⋊C412yes—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C42—22no—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—82no—
130#103—82no—
131#102—82no—
132C4×C8—22no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—41no—
140C4×C8—82no—
141C4×C8C2×C812yes—
142C4×C8SD1612yescomplemented
143#95—22no—
144#95—22no—
145C24—41no—
146C2×Q8—22no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8C22⋊C814yes—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—81no—
163C2×D4—81no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—41no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4#12011yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12311yescharacteristic
178SD16—88no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—81no—
211C4⋊C4—82no—
212C4⋊C4—81no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C42—21no—
218C42—21no—
219C42—41no—
220C42—41no—
221C42—41no—
222C42—41no—
223C42—42no—
224C42—42no—
225C42C22×C811yescharacteristic
226C42C2×M1611yescharacteristic
227C42#13111yescharacteristic
228C42#13111yescharacteristic
229C42C2×SD1611yescharacteristic
230C42#13511yescharacteristic
231C42AGL(1,ℤ/8)11yescharacteristic
232C42#13711yescharacteristic
233C23—42no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23#38611yescharacteristic
241Q8—44no—
242Q8—82no—
243D4—44no—
244D4—42no—
245D4—42no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4—81no—
250D4—81no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—162no—
275C2×C4—162no—
276C2×C4C2×C22⋊C811yescharacteristic
277C2×C4#40811yescharacteristic
278C2×C4C8×D411yescharacteristic
279C2×C4C8×D411yescharacteristic
280C2×C4#43511yescharacteristic
281C2×C4#43511yescharacteristic
282C2×C4C4×SD1611yescharacteristic
283C2×C4#44011yescharacteristic
284C2×C4#44311yescharacteristic
285C2×C4#44411yescharacteristic
286C2×C4#44811yescharacteristic
287C2×C4#44911yescharacteristic
288C2×C4#46011yescharacteristic
289C2×C4#46411yescharacteristic
290C8—42no—
291C8—42no—
292C8—82no—
293C8—81no—
294C8—81no—
295C8—82no—
296C8—161no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—161no—
301C22—161no—
302C22—161no—
303C22—161no—
304C22#186911yescharacteristic
305C22#187611yescharacteristic
306C22#188011yescharacteristic
307C22#190311yescharacteristic
308C22#190511yescharacteristic
309C22#191011yescharacteristic
310C22#191111yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—44no—
322C4—82no—
323C4—161no—
324C4C8×SD1611yescharacteristic
325C4SD16⋊C811yescharacteristic
326C4#158311yescharacteristic
327C4#158811yescharacteristic
328C2—162no—
329C2#894811yescharacteristic
330C2#909011yescharacteristic
331C2#910111yescharacteristic
332C2#928211yescharacteristic
333C2#928311yescharacteristic
334C2#929411yescharacteristic
335C2#930111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22307C2semidirect
4. #7590C2semidirect
6. #7018C2semidirect
7. #7014C2semidirect
37. #492C8semidirect
142. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#223072
#222682
#96422
#75902
#75872
#70182
#70142