Finite Group Databasebeta

← #322491

Subgroups — #322491

Order 512.

762 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 186 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#33236C211yesmaximal, characteristic
2#28687C212yesmaximal
3#28616C212yesmaximal
4#21727C211yesmaximal, characteristic
5C2×#1716C212yesmaximal
6#10318C212yesmaximal
7#10317C212yesmaximal
8#9833C212yesmaximal
9#8263C211yesmaximal, characteristic
10C2×#498C2212yes—
11C2×#498C2212yes—
12C2×#498C2211yescharacteristic
13C2×#494C2212yes—
14C2×#487C2212yes—
15C2×#487C2212yes—
16C2×#487C2212yes—
17#2868C2211yescharacteristic
18#2862C2211yescharacteristic
19C4×C4⋊Q8C2212yes—
20#2084—24no—
21#2056—24no—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1960—22no—
27#1960—22no—
28#1956C2212yes—
29#1948—24no—
30#1944—22no—
31#1878—24no—
32#1845C2211yescharacteristic
33#1776—22no—
34C4×C4⋊C8C2212yes—
35C4×#97C2211yescharacteristic
36#1716—28no—
37C2×#322C2212yes—
38#1273C2211yescharacteristic
39#1273C2211yescharacteristic
40C22×Q16—24no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2312yes—
44C2×C4⋊Q8C2312yes—
45C2×C4⋊Q8C2312yes—
46C2×C4×Q8—22no—
47#498—24no—
48#498—24no—
49#498—22no—
50#498—42no—
51#498—42no—
52#498D414yes—
53#494—24no—
54#494—42no—
55#487—24no—
56#487—24no—
57#487—24no—
58#487—42no—
59#487—42no—
60#487—42no—
61C2×#107—22no—
62C2×#106—22no—
63C2×#106—24no—
64C2×#106—24no—
65C2×C4⋊C8D414yes—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#97D412yes—
71C2×#97D412yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2312yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#391—22no—
78#391—22no—
79#389—22no—
80#389—22no—
81#384—22no—
82#384—22no—
83#384C2312yes—
84#382—21no—
85#382—21no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C43C2311yescharacteristic
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#336D414yes—
94#322—24no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—22no—
98C2×Q16—48no—
99C2×Q16—48no—
100C2×Q16—84no—
101C22×C8C2×D414yes—
102C22×C8C2×D412yes—
103C22×C8C2×D412yes—
104C22×C8C2×D414yes—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—44no—
117C4⋊Q8—44no—
118C4⋊Q8C2×D414yes—
119C4×Q8—84no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C42—42no—
134C2×C42C2×D412yes—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D412yes—
137C2×C42C2×D411yescharacteristic
138C2×C42C2411yescharacteristic
139#107—44no—
140#106—28no—
141#106—48no—
142#106—48no—
143C4⋊C8—28no—
144#103—48no—
145#103—48no—
146#103—48no—
147#103—84no—
148#97—24no—
149#97—42no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8C2×D412yes—
156C4×C8#4112yes—
157#95—42no—
158#95—42no—
159#95—41no—
160#95—41no—
161#95—42no—
162C2×Q8—24no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—44no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4#12711yescharacteristic
179C22×C4#12711yescharacteristic
180C22×C4#12712yes—
181C22×C4C22×D411yescharacteristic
182C22×C4C22×D412yes—
183Q16—816no—
184C2×C8—24no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—24no—
188C2×C8—22no—
189C2×C8—24no—
190C2×C8—22no—
191C2×C8—24no—
192C2×C8—22no—
193C2×C8—24no—
194C2×C8#12114yes—
195C2×C8#13718yes—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—84no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—84no—
215C42—24no—
216C42—22no—
217C42—24no—
218C42—84no—
219C42#13714yes—
220C42C22×D411yescharacteristic
221C42C2×#4111yescharacteristic
222C42H(5,𝔽2)12yes—
223C23C2×#12711yescharacteristic
224Q8—48no—
225Q8—84no—
226Q8—84no—
227Q8—84no—
228Q8—84no—
229Q8—84no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—44no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4#49714yes—
249C2×C4C2×#12111yescharacteristic
250C2×C4#53511yescharacteristic
251C2×C4D4212yes—
252C2×C4#55012yes—
253C2×C4C2×#13714yes—
254C2×C4C2×#13712yes—
255C2×C4#57512yes—
256C2×C4#57714yes—
257C8—28no—
258C8—24no—
259C8—42no—
260C8—44no—
261C22#254111yescharacteristic
262C22C2×#49711yescharacteristic
263C22#315212yes—
264C22#315311yescharacteristic
265C22#315412yes—
266C4—24no—
267C4—41no—
268C4—41no—
269C4—84no—
270C4—162no—
271C4—162no—
272C4—162no—
273C4#303712yes—
274C4#321612yes—
275C4#326614yes—
276C4#340014yes—
277C2#1968012yes—
278C2#1972411yescharacteristic
279C2#1975111yescharacteristic
280C2#2845111yescharacteristic
281C2#2847812yes—
subgroupG/H
center, socle, prime radicalC23C2×#127
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#332362
#286872
#286162
#217272
C2×#17162
#103182
#103172
#98332
#82632