Finite Group Databasebeta

← #99944

Subgroups — #99944

Order 512.

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

normal subgroups only

This group has 465 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#22238C211yesmaximal, characteristic, complemented
2#22220C211yesmaximal, characteristic
3#9642C211yesmaximal, characteristic
4#7588C211yesmaximal, characteristic, complemented
5#7587C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic, complemented
7#7014C211yesmaximal, characteristic, complemented
8C2×#486C411yescharacteristic
9C2×C8×Q8—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#1990C411yescharacteristic
13#1984—21no—
14#1957C411yescharacteristic
15#1925—22no—
16#1916—22no—
17#1916—22no—
18#1862—22no—
19#1845C411yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1744C2211yescharacteristic
23#1744C2211yescharacteristic
24#1744C2211yescharacteristic
25C2×D4⋊C8—21no—
26C2×D4⋊C8—21no—
27#1271—22no—
28#1270—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×#127C2×C411yescharacteristic
31C2×C4×Q8—21no—
32C2×C4×D4—21no—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35#486—22no—
36#486C814yescomplemented
37C8×Q8—42no—
38C8×Q8—42no—
39C2×#107—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×#102—22no—
44C2×#102—22no—
45C2×C4×C8—21no—
46C2×C4×C8—22no—
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51#391—21no—
52#390—21no—
53#390—21no—
54#390—21no—
55#389—21no—
56#389D412yes—
57#384C2×C411yescharacteristic
58#384C2×C411yescharacteristic
59C4×C4⋊C4—21no—
60C4×C4⋊C4D412yes—
61C43C2311yescharacteristic
62#340—22no—
63#336—24no—
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#336—21no—
73#336D412yes—
74D4⋊C8—42no—
75D4⋊C8—42no—
76C22×Q8—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C23×C4—41no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90C4⋊Q8C2×C812yes—
91C4⋊Q8M1612yes—
92#127—21no—
93#127—21no—
94#127—21no—
95#127C2×C812yes—
96C4×Q8—41no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×Q8—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—41no—
104C4×D4—82no—
105C4×D4—82no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C22⋊C412yes—
114C2×C4⋊C4C22⋊C412yes—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C42—22no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126#107—82no—
127C4⋊C8—44no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132#102—28no—
133#102—82no—
134C4×C8—22no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—41no—
142C4×C8—44no—
143C4×C8C2×C812yes—
144C4×C8D812yes—
145#95—22no—
146#95—22no—
147C24—41no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—81no—
167C2×D4—81no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4C2×C22⋊C411yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4C4×D411yescharacteristic
178C22×C4#12011yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12311yescharacteristic
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—81no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4C22⋊C814yes—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C42—21no—
222C42—21no—
223C42—41no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—42no—
228C42—42no—
229C42C22×C811yescharacteristic
230C42C2×M1611yescharacteristic
231C42#13111yescharacteristic
232C42#13111yescharacteristic
233C42C2×D811yescharacteristic
234C42#13511yescharacteristic
235C42AGL(1,ℤ/8)11yescharacteristic
236C42AGL(1,ℤ/8)11yescharacteristic
237C23—42no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23#38611yescharacteristic
245Q8—42no—
246Q8—42no—
247Q8—42no—
248Q8—42no—
249D4—44no—
250D4—42no—
251D4—42no—
252D4—82no—
253D4—82no—
254D4—82no—
255D4—81no—
256D4—81no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—162no—
281C2×C4—162no—
282C2×C4C2×C22⋊C811yescharacteristic
283C2×C4#40811yescharacteristic
284C2×C4C8×D411yescharacteristic
285C2×C4C8×D411yescharacteristic
286C2×C4#43511yescharacteristic
287C2×C4#43511yescharacteristic
288C2×C4C4×D811yescharacteristic
289C2×C4#44211yescharacteristic
290C2×C4#44311yescharacteristic
291C2×C4#44411yescharacteristic
292C2×C4#44711yescharacteristic
293C2×C4#44911yescharacteristic
294C2×C4#45911yescharacteristic
295C2×C4#46311yescharacteristic
296C8—42no—
297C8—42no—
298C8—82no—
299C8—81no—
300C8—81no—
301C8—82no—
302C8—161no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—161no—
307C22—161no—
308C22—161no—
309C22—161no—
310C22#186911yescharacteristic
311C22#187611yescharacteristic
312C22#187911yescharacteristic
313C22#190311yescharacteristic
314C22#190511yescharacteristic
315C22#191011yescharacteristic
316C22#191211yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—44no—
328C4—82no—
329C4—161no—
330C4C8×D811yescharacteristic
331C4#158011yescharacteristic
332C4#158111yescharacteristic
333C4#158611yescharacteristic
334C2—162no—
335C2#894711yescharacteristic
336C2#909011yescharacteristic
337C2#910211yescharacteristic
338C2#928211yescharacteristic
339C2#928411yescharacteristic
340C2#929311yescharacteristic
341C2#930011yescharacteristic
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. #22238C2semidirect
4. #7588C2semidirect
6. #7018C2semidirect
7. #7014C2semidirect
36. #486C8semidirect

Maximal subgroups

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

subgroupindex
#222382
#222202
#96422
#75882
#75872
#70182
#70142