Finite Group Databasebeta

← #100273

Subgroups — #100273

Order 512.

435 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 96 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#33073C211yesmaximal, characteristic
2#20572C211yesmaximal, characteristic
3#9686C211yesmaximal, characteristic
4#8173C211yesmaximal, characteristic
5#7683C211yesmaximal, characteristic
6#7042C211yesmaximal, characteristic
7#7030C211yesmaximal, characteristic
8C2×C8×Q8—21no—
9#2859C2211yescharacteristic
10C42×Q8C2211yescharacteristic
11#1948C411yescharacteristic
12#1947C411yescharacteristic
13#1920—21no—
14#1918—21no—
15#1865—21no—
16#1863—21no—
17#1831C2211yescharacteristic
18#1803C411yescharacteristic
19#1803C411yescharacteristic
20#1770—22no—
21#1770—21no—
22#1767—21no—
23#1767C2211yescharacteristic
24#1767C2211yescharacteristic
25#1744—21no—
26#1744C2211yescharacteristic
27C4×#97C2211yescharacteristic
28C2×#330—22no—
29C2×Q8⋊C8—21no—
30#1279—21no—
31#1271—21no—
32#1271—21no—
33C2×#125—21no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8D411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C8×Q8—24no—
38C8×Q8—24no—
39C2×#107—21no—
40C2×#106—21no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C4⋊C8C2×C411yescharacteristic
46C2×#103—21no—
47C2×#103—21no—
48C2×#97—21no—
49C2×#97—21no—
50C2×#97—22no—
51C2×#97—21no—
52C2×#97D412yes—
53C2×C4×C8—22no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#389—21no—
60#384C2×C411yescharacteristic
61#384C2×C411yescharacteristic
62#382—21no—
63C4×C4⋊C4—21no—
64C4×C4⋊C4—21no—
65C4×C4⋊C4—21no—
66C4×C4⋊C4C2×C411yescharacteristic
67C4×C4⋊C4D411yescharacteristic
68C4×C4⋊C4D411yescharacteristic
69C43—21no—
70C43C2311yescharacteristic
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—22no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336D411yescharacteristic
82#336D411yescharacteristic
83#330—44no—
84Q8⋊C8—24no—
85C22×Q8C22⋊C411yescharacteristic
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—41no—
92C22×C8—41no—
93#125—42no—
94#125—42no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—22no—
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—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4C22⋊C411yescharacteristic
113C2×C4⋊C4C22⋊C411yescharacteristic
114C2×C4⋊C4C22⋊C411yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42C22×C411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128#107—42no—
129#106—42no—
130C4⋊C8—28no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137#103—42no—
138#103—42no—
139#97—22no—
140#97—22no—
141#97—42no—
142#97—41no—
143#97—41no—
144#97—81no—
145#97—82no—
146C4×C8—28no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8—22no—
150C4×C8—41no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—41no—
155#95—41no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8#10112yes—
160C2×Q8#10412yes—
161C2×Q8#10412yes—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4C2×C22⋊C411yescharacteristic
169C22×C4C4×D411yescharacteristic
170C22×C4C4×D411yescharacteristic
171C22×C4#12011yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12311yescharacteristic
175C2×C8—22no—
176C2×C8—24no—
177C2×C8—22no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—81no—
195C2×C8—81no—
196C4⋊C4—24no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—81no—
213C4⋊C4—82no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4#10412yes—
217C42—24no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—41no—
231C42—42no—
232C42—42no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42—41no—
238C42—41no—
239C23#38611yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—22no—
243Q8#33014yescomplemented
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
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—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4C2×#10111yescharacteristic
278C2×C4#41311yescharacteristic
279C2×C4C2×#10411yescharacteristic
280C2×C4C2×#10411yescharacteristic
281C2×C4C2×#10411yescharacteristic
282C2×C4#42111yescharacteristic
283C2×C4#44311yescharacteristic
284C2×C4#44411yescharacteristic
285C2×C4#45311yescharacteristic
286C2×C4#45411yescharacteristic
287C2×C4#47111yescharacteristic
288C2×C4#47311yescharacteristic
289C8—44no—
290C8—42no—
291C8—42no—
292C8—44no—
293C8—82no—
294C8—81no—
295C8—81no—
296C22#187111yescharacteristic
297C22#187611yescharacteristic
298C22#188311yescharacteristic
299C22#189611yescharacteristic
300C22#190111yescharacteristic
301C22#190511yescharacteristic
302C22#190811yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—24no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—42no—
314C4—81no—
315C4—81no—
316C4—161no—
317C4—161no—
318C4C2×#33011yescharacteristic
319C4#152411yescharacteristic
320C4#153111yescharacteristic
321C4#153211yescharacteristic
322C2#897011yescharacteristic
323C2#912111yescharacteristic
324C2#916611yescharacteristic
325C2#921211yescharacteristic
326C2#924611yescharacteristic
327C2#932611yescharacteristic
328C2#935811yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
243. Q8#330semidirect

Maximal subgroups

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

subgroupindex
#330732
#205722
#96862
#81732
#76832
#70422
#70302