Finite Group Databasebeta

← #331456

Subgroups — #331456

Order 512.

680 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 158 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#33294C211yesmaximal, characteristic
2#33255C211yesmaximal, characteristic
3#30280C211yesmaximal, characteristic
4#30262C211yesmaximal, characteristic
5#30229C211yesmaximal, characteristic
6#30229C211yesmaximal, characteristic
7#10319C212yesmaximal
8#10318C212yesmaximal
9#10318C212yesmaximal
10#10318C212yesmaximal
11#8134C211yesmaximal, characteristic
12C2×#498C2212yes—
13C2×#498C2211yescharacteristic
14C2×#494C2212yes—
15C2×#494C2211yescharacteristic
16C2×#487C2212yes—
17C2×#487C2212yes—
18C2×#487C2211yescharacteristic
19C2×#487C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2211yescharacteristic
22#2862C2211yescharacteristic
23#2859C2211yescharacteristic
24#2101—24no—
25#2101—24no—
26#2096—24no—
27#2016—24no—
28#1992—22no—
29#1990C2212yes—
30#1990C2212yes—
31#1990C2212yes—
32#1990C2212yes—
33#1990C2212yes—
34#1990C2212yes—
35#1971—22no—
36#1971—22no—
37#1836—22no—
38#1744C2211yescharacteristic
39#1744C2211yescharacteristic
40C4×#97C2211yescharacteristic
41#1273C2212yes—
42#1273C2212yes—
43C22×Q16—22no—
44C22×Q16—24no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2312yes—
51C2×#125—21no—
52#498—42no—
53#498—42no—
54#498—42no—
55#494—42no—
56#494—42no—
57#494—42no—
58#487—42no—
59#487—42no—
60#487—42no—
61#487—42no—
62#487—42no—
63#487—42no—
64C2×#106—24no—
65C2×#106—22no—
66C2×#103—24no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—24no—
70C2×#97D412yes—
71C2×#97D412yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2312yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—22no—
82#389—22no—
83#389—22no—
84#384—21no—
85#384—22no—
86#384—21no—
87#384C2312yes—
88#382—21no—
89#382—22no—
90#382—22no—
91C43C2311yescharacteristic
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#336D414yes—
99#336D414yes—
100C22×Q8—22no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C2×Q16—44no—
106C2×Q16—44no—
107C2×Q16—48no—
108C2×Q16—44no—
109C2×Q16—88no—
110C22×C8C2×D414yes—
111C22×C8C2×D414yes—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—84no—
127#125—44no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D411yescharacteristic
148C2×C42C2411yescharacteristic
149#106—44no—
150#106—48no—
151#103—44no—
152#103—44no—
153#103—48no—
154#103—48no—
155#97—24no—
156#97—42no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163#95—42no—
164#95—42no—
165#95—42no—
166#95—42no—
167#95—42no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—48no—
181C2×Q8—44no—
182C2×Q8—44no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4#12712yes—
190C22×C4#12712yes—
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194Q16—88no—
195Q16—816no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—28no—
204C2×C8—28no—
205C2×C8#13714yes—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—84no—
224C4⋊C4—84no—
225C4⋊C4—84no—
226C42—22no—
227C42—22no—
228C42—24no—
229C42—24no—
230C42H(5,𝔽2)12yes—
231C42H(5,𝔽2)12yes—
232C23C2×#12711yescharacteristic
233Q8—82no—
234Q8—82no—
235Q8—82no—
236Q8—82no—
237Q8—84no—
238Q8—84no—
239Q8—84no—
240Q8—84no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4#53511yescharacteristic
263C2×C4#53511yescharacteristic
264C2×C4#53512yes—
265C2×C4#53512yes—
266C2×C4C2×#13712yes—
267C2×C4#57512yes—
268C2×C4#57714yes—
269C2×C4#57814yes—
270C8—24no—
271C8—42no—
272C8—44no—
273C8—44no—
274C22#265711yescharacteristic
275C22#315211yescharacteristic
276C22#315311yescharacteristic
277C22#315412yes—
278C22#315612yes—
279C4—42no—
280C4—42no—
281C4—162no—
282C4—161no—
283C4—161no—
284C4—162no—
285C4—161no—
286C4—161no—
287C4#321512yes—
288C4#321612yes—
289C4#322214yes—
290C4#322314yes—
291C2#1971612yes—
292C2#1972012yes—
293C2#3026911yescharacteristic
294C2#3027111yescharacteristic
295C2#3027211yescharacteristic
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
#332942
#332552
#302802
#302622
#302292
#103192
#103182
#81342