Finite Group Databasebeta

← #99940

Subgroups — #99940

Order 512.

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

normal subgroups only

This group has 415 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#22247C211yesmaximal, characteristic
2#22238C211yesmaximal, characteristic
3#9641C211yesmaximal, characteristic
4#7588C211yesmaximal, characteristic
5#7588C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic
7#7014C211yesmaximal, characteristic
8C2×#500C411yescharacteristic
9C2×C8×Q8—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12#1990C411yescharacteristic
13#1990C411yescharacteristic
14#1984—21no—
15#1918—22no—
16#1918—22no—
17#1916—22no—
18#1916—22no—
19#1834C411yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1744C2211yescharacteristic
23#1744C2211yescharacteristic
24#1744C2211yescharacteristic
25C2×Q8⋊C8—21no—
26C2×Q8⋊C8—21no—
27#1271—22no—
28#1271—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×C4×Q8—21no—
32C2×C4×Q8—21no—
33#500—22no—
34#500C814yescomplemented
35C8×Q8—42no—
36C8×Q8—42no—
37C2×#107—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8—22no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—21no—
48#391—21no—
49#389—21no—
50#389—21no—
51#389—21no—
52#389D412yes—
53#384C2×C411yescharacteristic
54#384C2×C411yescharacteristic
55C4×C4⋊C4—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—21no—
58C4×C4⋊C4D412yes—
59C43C2311yescharacteristic
60#340—22no—
61#340—22no—
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—
74Q8⋊C8—42no—
75Q8⋊C8—42no—
76C22×Q8—21no—
77C22×Q8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90C4⋊Q8C2×C812yes—
91C4⋊Q8C2×C812yes—
92C4⋊Q8M1612yes—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×Q8—41no—
100C4×Q8—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C22⋊C412yes—
112C2×C4⋊C4C22⋊C412yes—
113C2×C42—22no—
114C2×C42—41no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122#107—28no—
123C4⋊C8—44no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4×C8—22no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—41no—
136C4×C8—44no—
137C4×C8C2×C812yes—
138C4×C8D812yes—
139#95—22no—
140#95—22no—
141#95—41no—
142#95—41no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4C2×C22⋊C411yescharacteristic
160C22×C4C4×D411yescharacteristic
161C22×C4C4×D411yescharacteristic
162C22×C4#12011yescharacteristic
163C22×C4#12111yescharacteristic
164C22×C4#12111yescharacteristic
165C22×C4#12311yescharacteristic
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—42no—
181C2×C8—41no—
182C2×C8—81no—
183C2×C8—81no—
184C2×C8—81no—
185C2×C8—81no—
186C2×C8—81no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4C22⋊C814yes—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—41no—
211C42—41no—
212C42—41no—
213C42—42no—
214C42—42no—
215C42—81no—
216C42—81no—
217C42C22×C811yescharacteristic
218C42C2×M1611yescharacteristic
219C42#13111yescharacteristic
220C42#13111yescharacteristic
221C42C2×D811yescharacteristic
222C42#13511yescharacteristic
223C42AGL(1,ℤ/8)11yescharacteristic
224C42AGL(1,ℤ/8)11yescharacteristic
225C23#38611yescharacteristic
226Q8—42no—
227Q8—42no—
228Q8—42no—
229Q8—42no—
230Q8—42no—
231Q8—42no—
232Q8—81no—
233Q8—81no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4C2×C22⋊C811yescharacteristic
261C2×C4#40811yescharacteristic
262C2×C4C8×D411yescharacteristic
263C2×C4C8×D411yescharacteristic
264C2×C4#43511yescharacteristic
265C2×C4#43511yescharacteristic
266C2×C4C4×D811yescharacteristic
267C2×C4#44211yescharacteristic
268C2×C4#44311yescharacteristic
269C2×C4#44411yescharacteristic
270C2×C4#44711yescharacteristic
271C2×C4#44911yescharacteristic
272C2×C4#45911yescharacteristic
273C2×C4#46311yescharacteristic
274C8—42no—
275C8—42no—
276C8—82no—
277C8—81no—
278C8—81no—
279C8—82no—
280C8—161no—
281C22#186911yescharacteristic
282C22#187611yescharacteristic
283C22#187911yescharacteristic
284C22#190211yescharacteristic
285C22#190511yescharacteristic
286C22#190911yescharacteristic
287C22#191211yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—41no—
297C4—41no—
298C4—44no—
299C4—82no—
300C4—82no—
301C4—161no—
302C4C8×D811yescharacteristic
303C4#158011yescharacteristic
304C4#158111yescharacteristic
305C4#158611yescharacteristic
306C2#894711yescharacteristic
307C2#908811yescharacteristic
308C2#910011yescharacteristic
309C2#928011yescharacteristic
310C2#928611yescharacteristic
311C2#929111yescharacteristic
312C2#930011yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
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
34. #500C8semidirect

Maximal subgroups

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

subgroupindex
#222472
#222382
#96412
#75882
#70182
#70142