Finite Group Databasebeta

← #99946

Subgroups — #99946

Order 512.

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

normal subgroups only

This group has 447 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#22219C211yesmaximal, characteristic
3#9642C211yesmaximal, characteristic
4#7591C211yesmaximal, characteristic, complemented
5#7587C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic, complemented
7#7014C211yesmaximal, characteristic, complemented
8C2×#486C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#1990C411yescharacteristic
12#1984—22no—
13#1957C411yescharacteristic
14#1939—21no—
15#1926—22no—
16#1916—22no—
17#1864—22no—
18#1844C411yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21C8×C4⋊C4—21no—
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
37C2×#106—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—22no—
41C2×#102—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#391—21no—
50#390—21no—
51#390—21no—
52#390—21no—
53#389—21no—
54#389D412yes—
55#384C2×C411yescharacteristic
56#384C2×C411yescharacteristic
57C4×C4⋊C4—21no—
58C4×C4⋊C4D412yes—
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×Q8—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C23×C4—41no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—22no—
85C22×C8—41no—
86C22×C8—41no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8C2×C812yes—
90C4⋊Q8M1612yes—
91#127—21no—
92#127—21no—
93#127—21no—
94#127C2×C812yes—
95C4×Q8—82no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—82no—
101C4×D4—82no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4C22⋊C412yes—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C42—22no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122#106—82no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—82no—
128#102—28no—
129#102—82no—
130C4×C8—22no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—41no—
138C4×C8—82no—
139C4×C8C2×C812yes—
140C4×C8SD1612yes—
141#95—22no—
142#95—22no—
143C24—41no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—81no—
147C2×Q8—81no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—81no—
161C2×D4—81no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—41no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C4×D411yescharacteristic
171C22×C4C4×D411yescharacteristic
172C22×C4#12011yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12311yescharacteristic
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—81no—
207C4⋊C4—82no—
208C4⋊C4—81no—
209C4⋊C4C22⋊C814yes—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—41no—
218C42—41no—
219C42—41no—
220C42—42no—
221C42—42no—
222C42C22×C811yescharacteristic
223C42C2×M1611yescharacteristic
224C42#13111yescharacteristic
225C42#13111yescharacteristic
226C42C2×SD1611yescharacteristic
227C42#13511yescharacteristic
228C42AGL(1,ℤ/8)11yescharacteristic
229C42#13711yescharacteristic
230C23—42no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23#38611yescharacteristic
238Q8—82no—
239Q8—82no—
240D4—44no—
241D4—42no—
242D4—42no—
243D4—82no—
244D4—82no—
245D4—82no—
246D4—81no—
247D4—81no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—162no—
272C2×C4—162no—
273C2×C4C2×C22⋊C811yescharacteristic
274C2×C4#40811yescharacteristic
275C2×C4C8×D411yescharacteristic
276C2×C4C8×D411yescharacteristic
277C2×C4#43511yescharacteristic
278C2×C4#43511yescharacteristic
279C2×C4C4×SD1611yescharacteristic
280C2×C4#44011yescharacteristic
281C2×C4#44311yescharacteristic
282C2×C4#44411yescharacteristic
283C2×C4#44811yescharacteristic
284C2×C4#44911yescharacteristic
285C2×C4#46011yescharacteristic
286C2×C4#46411yescharacteristic
287C8—42no—
288C8—42no—
289C8—82no—
290C8—81no—
291C8—81no—
292C8—82no—
293C8—161no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—161no—
298C22—161no—
299C22—161no—
300C22—161no—
301C22#186911yescharacteristic
302C22#187611yescharacteristic
303C22#188011yescharacteristic
304C22#190311yescharacteristic
305C22#190411yescharacteristic
306C22#191011yescharacteristic
307C22#191211yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—44no—
319C4—82no—
320C4—161no—
321C4C8×SD1611yescharacteristic
322C4SD16⋊C811yescharacteristic
323C4#158311yescharacteristic
324C4#158811yescharacteristic
325C2—162no—
326C2#894811yescharacteristic
327C2#908911yescharacteristic
328C2#910211yescharacteristic
329C2#928111yescharacteristic
330C2#928411yescharacteristic
331C2#929411yescharacteristic
332C2#930211yescharacteristic
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. #7591C2semidirect
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
#223072
#222192
#96422
#75912
#75872
#70182
#70142