Finite Group Databasebeta

← #280931

Subgroups — #280931

Order 512.

427 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 126 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#28297C211yesmaximal, characteristic
2#22815C211yesmaximal, characteristic
3#10639C212yesmaximal
4#10471C212yesmaximal
5#9669C211yesmaximal, characteristic
6#9630C211yesmaximal, characteristic
7#9178C212yesmaximal
8#8793C211yesmaximal, characteristic
9#8769C212yesmaximal
10#8675C211yesmaximal, characteristic
11#8256C211yesmaximal, characteristic
12#3078C2212yes—
13#2960C2211yescharacteristic
14C2×C4×Q16C2211yescharacteristic
15#2356C2212yes—
16#2064—22no—
17#1979—21no—
18#1971—21no—
19#1952—21no—
20#1928—22no—
21#1848C2211yescharacteristic
22#1846C2211yescharacteristic
23#1805C2212yes—
24#1801C2212yes—
25#1776—21no—
26#1770C2211yescharacteristic
27#1571C2212yes—
28#1564C2211yescharacteristic
29#1542C2211yescharacteristic
30#1518C2212yes—
31#1503C2211yescharacteristic
32#1479C2212yes—
33#1461C2212yes—
34#1450C2211yescharacteristic
35#1303C2211yescharacteristic
36#1303C2212yes—
37#1301C2211yescharacteristic
38#1286C2212yes—
39#1286C2212yes—
40#1286C2212yes—
41C22×Q16D411yescharacteristic
42#533C2311yescharacteristic
43C2×C4×Q8C2312yes—
44#479—42no—
45#477—22no—
46#475—22no—
47#446—21no—
48#446—21no—
49C8×Q8—21no—
50C8×Q8—21no—
51C4×Q16—21no—
52C4×Q16—21no—
53C4×Q16—21no—
54C4×Q16—21no—
55#431C2311yescharacteristic
56#431C2311yescharacteristic
57#431C2312yes—
58C2×#108—21no—
59C2×#108—21no—
60#428—22no—
61C2×#107—21no—
62C2×#107D411yescharacteristic
63C2×#106—21no—
64#424—21no—
65#423C2312yes—
66#423C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68#419—22no—
69C2×#103—22no—
70C2×#103D412yes—
71C2×#101—22no—
72#405—21no—
73C2×#97—21no—
74C2×C4×C8C2311yescharacteristic
75#391—22no—
76#387—42no—
77#383—22no—
78#382C2312yes—
79C4×C4⋊C4—22no—
80C4×C4⋊C4C2311yescharacteristic
81#376—22no—
82#343—22no—
83#340—22no—
84#336—21no—
85#335—21no—
86#335—42no—
87#334—21no—
88#333—21no—
89#332—21no—
90#328—22no—
91#328—42no—
92Q8⋊C8—42no—
93#324—42no—
94#322—21no—
95C82—21no—
96C22×Q8C2×D412yes—
97C2×Q16—21no—
98C2×Q16—21no—
99C2×Q16—22no—
100C2×Q16—22no—
101C2×M16—21no—
102C2×M16—22no—
103C22×C8—21no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C4⋊Q8#4111yescharacteristic
107C4⋊Q8#4111yescharacteristic
108#125—21no—
109C22⋊Q8—21no—
110C22⋊Q8—21no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×Q8—21no—
115C4×Q8—21no—
116C4×Q8—22no—
117#117—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C2×D412yes—
122C2×C42—21no—
123C2×C42—22no—
124C2×C42—42no—
125C2×C42C2411yescharacteristic
126#108—42no—
127#108—81no—
128#107—21no—
129#107—21no—
130#107—42no—
131#107—81no—
132#106—42no—
133#106—42no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136C4⋊C8—22no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—22no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—22no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145#103—22no—
146#103—22no—
147#103—42no—
148#103—42no—
149#103—82no—
150#101—82no—
151C22⋊C8—21no—
152C22⋊C8—21no—
153C22⋊C8—22no—
154#97—41no—
155#97—42no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—22no—
159C4×C8—41no—
160C4×C8#4111yescharacteristic
161C4×C8#4111yescharacteristic
162C4×C8#4111yescharacteristic
163C4×C8#4111yescharacteristic
164#95—22no—
165#95—22no—
166#95—42no—
167#95—42no—
168#95—42no—
169#95#4112yes—
170C2×Q8—22no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4C22×D411yescharacteristic
183C22×C4C2×#4111yescharacteristic
184C22×C4H(5,𝔽2)11yescharacteristic
185Q16—42no—
186Q16—42no—
187M16—42no—
188M16—41no—
189M16—42no—
190M16—41no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8#12311yescharacteristic
202C2×C8#12311yescharacteristic
203C2×C8#12311yescharacteristic
204C2×C8#12311yescharacteristic
205C2×C8#13512yes—
206C2×C8#13712yes—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C22⋊C4—41no—
221C42—41no—
222C42—42no—
223C42—42no—
224C42—41no—
225C42—41no—
226C42—84no—
227C42C2×#4111yescharacteristic
228C42C2×#4111yescharacteristic
229C42C2×#4111yescharacteristic
230C42H(5,𝔽2)11yescharacteristic
231C23#54611yescharacteristic
232Q8—42no—
233Q8—42no—
234Q8—42no—
235Q8—42no—
236Q8—81no—
237Q8—81no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4C2×#12311yescharacteristic
254C2×C4#52911yescharacteristic
255C2×C4#54011yescharacteristic
256C2×C4#54611yescharacteristic
257C2×C4#55311yescharacteristic
258C2×C4#55511yescharacteristic
259C2×C4C2×#13511yescharacteristic
260C2×C4C2×#13711yescharacteristic
261C2×C4#57611yescharacteristic
262C2×C4#57811yescharacteristic
263C8—21no—
264C8—21no—
265C8—82no—
266C8—81no—
267C8—81no—
268C8—81no—
269C8—81no—
270C8—82no—
271C8#48512yes—
272C8#48512yes—
273C22—21no—
274C22—21no—
275C22#242011yescharacteristic
276C22#330211yescharacteristic
277C22#333311yescharacteristic
278C4—21no—
279C4—21no—
280C4—21no—
281C4—21no—
282C4—82no—
283C4—82no—
284C4—82no—
285C4—81no—
286C4—81no—
287C4—162no—
288C4C2×#48511yescharacteristic
289C4#309811yescharacteristic
290C4#330511yescharacteristic
291C4#332811yescharacteristic
292C2—21no—
293C2—21no—
294C2#2497011yescharacteristic
295C2#2498611yescharacteristic
296C2#2498911yescharacteristic
subgroupG/H
center, socleC22#2420
commutator, Frattini, cosocleC2×C42C24
prime radicalC23#546
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ 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
#282972
#228152
#106392
#104712
#96692
#96302
#91782
#87932
#87692
#86752
#82562