Finite Group Databasebeta

← #110146

Subgroups — #110146

Order 512.

755 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 755 conjugacy classes of subgroups — too many to draw. Only its 146 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#21748C211yesmaximal, characteristic
2#20529C211yesmaximal, characteristic, complemented
3#17582C212yesmaximal
4#8102C211yesmaximal, characteristic, complemented
5#7943C212yesmaximal, complemented
6C2×#525C2211yescharacteristic
7C2×#407C2212yes—
8#2280C2212yes—
9#2280C2211yescharacteristic
10C2×#381C2211yescharacteristic
11#1899—22no—
12#1898—22no—
13#1820C412yescomplemented
14#1782C412yescomplemented
15#1753—24no—
16#1750—22no—
17#1516—24no—
18#1513—24no—
19#1508C414yescomplemented
20#1507C414yescomplemented
21#1304—22no—
22#1284—24no—
23C22×#41C2×C411yescharacteristic
24C24×C4C2311yescharacteristic
25C22×M16—22no—
26#525—22no—
27#525—22no—
28#525—22no—
29#525—22no—
30C2×#123—21no—
31C2×C22⋊Q8C2×C412yes—
32C2×#121C2×C412yes—
33C2×#120—21no—
34C2×C4×D4—22no—
35C2×#117—22no—
36C2×#117C2×C411yescharacteristic
37C2×#117D412yes—
38C22×C22⋊C4—21no—
39C2×#99—24no—
40#407—22no—
41#407—22no—
42#407C2×C414yes—
43#407D414yes—
44C2×C22⋊C8—22no—
45C2×C22⋊C8D414yes—
46C2×#97—22no—
47C2×C4×C8—22no—
48#381—22no—
49#381—22no—
50#381—21no—
51#381C2×C412yes—
52#381C2×C412yes—
53#380—22no—
54#380—22no—
55#380—22no—
56#379—22no—
57#379D412yes—
58C4×C22⋊C4—22no—
59C4×C22⋊C4—22no—
60C4×C22⋊C4C2×C414yes—
61C2×#95—22no—
62#342—24no—
63#336—24no—
64#328—28no—
65#327—28no—
66C25C2×D411yescharacteristic
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—82no—
70C22×Q8C4211yescharacteristic
71C22×D4—21no—
72C22×D4—22no—
73C22×D4C4211yescharacteristic
74C23×C4—21no—
75C23×C4—41no—
76C23×C4—82no—
77C23×C4C22×C411yescharacteristic
78C23×C4C22×C412yes—
79C23×C4C2×D412yes—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D412yes—
82C23×C4#4112yes—
83C23×C4#4112yes—
84C23×C4#4112yes—
85C2×M16—24no—
86C2×M16—24no—
87C2×M16—84no—
88C22×C8—24no—
89#123—41no—
90#123—42no—
91#123—41no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96#121—22no—
97#121—22no—
98#121—22no—
99#121—22no—
100#120—42no—
101#120—41no—
102#120—41no—
103C4×D4—42no—
104C4×D4—42no—
105C4×D4—42no—
106C4×D4—42no—
107#117—22no—
108#117—22no—
109#117—82no—
110#117—84no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4C4212yes—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—82no—
125C2×C22⋊C4C22⋊C414yes—
126C2×C42—21no—
127C2×C42—22no—
128C2×C42—24no—
129#99—84no—
130C22⋊C8—24no—
131C22⋊C8—24no—
132C22⋊C8—24no—
133C22⋊C8C22⋊C418yes—
134#97—82no—
135C4×C8—82no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—42no—
140#95—42no—
141C24—21no—
142C24—42no—
143C24—82no—
144C24C2×C22⋊C411yescharacteristic
145C24C4×D412yes—
146C24#12112yes—
147C24#12411yescharacteristic
148C24#12711yescharacteristic
149#41—84no—
150#41—84no—
151C2×Q8—22no—
152C2×Q8—81no—
153C2×D4—22no—
154C2×D4—44no—
155C2×D4—41no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—41no—
159C2×D4—81no—
160C2×D4—84no—
161C22×C4—21no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—84no—
178C22×C4C2×C4211yescharacteristic
179C22×C4C2×C22⋊C412yes—
180C22×C4#11712yes—
181C22×C4#11711yescharacteristic
182C22×C4C4×D412yes—
183C22×C4C4×D412yes—
184C22×C4C4×D412yes—
185C22×C4C4×D412yes—
186C22×C4C4×D412yes—
187C22×C4#12011yescharacteristic
188C22×C4#12112yes—
189C22×C4#12112yes—
190C22×C4#12311yescharacteristic
191C22×C4#12312yes—
192C22×C4#12412yes—
193C22×C4#12612yes—
194M16—84no—
195M16—84no—
196C2×C8—28no—
197C2×C8—44no—
198C2×C8—44no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—82no—
204C4⋊C4—84no—
205C22⋊C4—41no—
206C22⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—84no—
214C22⋊C4—84no—
215C22⋊C4—84no—
216C22⋊C4—82no—
217C22⋊C4—84no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—84no—
222C23—22no—
223C23—24no—
224C23—41no—
225C23—42no—
226C23—41no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—41no—
236C23—82no—
237C23—88no—
238C23C4×C22⋊C411yescharacteristic
239C23#38611yescharacteristic
240C23#38812yes—
241C23#39012yes—
242C23#39011yescharacteristic
243Q8—82no—
244D4—82no—
245D4—82no—
246D4—84no—
247D4—84no—
248D4—82no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—48no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—84no—
273C2×C4—82no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4—84no—
277C2×C4—164no—
278C2×C4C4×C22⋊C412yes—
279C2×C4#38612yes—
280C2×C4#38812yes—
281C2×C4#38812yes—
282C8—48no—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—44no—
295C22—82no—
296C22—84no—
297C22—82no—
298C22—84no—
299C22—81no—
300C22—82no—
301C22—162no—
302C22#143711yescharacteristic
303C22#191312yes—
304C4—22no—
305C4—22no—
306C4—22no—
307C4—22no—
308C4—84no—
309C4—81no—
310C4—82no—
311C4—164no—
312C4—162no—
313C2—22no—
314C2—41no—
315C2—42no—
316C2—42no—
317C2—41no—
318C2—161no—
319C2#1093811yescharacteristic
320C2#1095112yes—
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20529C2semidirect
4. #8102C2semidirect
5. #7943C2semidirect
13. #1820C4semidirect
14. #1782C4semidirect
19. #1508C4semidirect
20. #1507C4semidirect

Maximal subgroups

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

subgroupindex
#217482
#205292
#175822
#81022
#79432