Finite Group Databasebeta

← #315291

Subgroups — #315291

Order 512.

523 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28829C211yesmaximal, characteristic
2#28682C211yesmaximal, characteristic
3#27824C212yesmaximal
4#10415C212yesmaximal
5#10355C212yesmaximal
6#9836C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9575C212yesmaximal
9#8651C211yesmaximal, characteristic
10#8286C211yesmaximal, characteristic
11#8211C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#446C2211yescharacteristic
14#2567C2212yes—
15#2552C2212yes—
16#2099—22no—
17#2086—22no—
18#2084—22no—
19#2026—22no—
20#1992—22no—
21#1990C2212yes—
22#1987C2212yes—
23#1987C2212yes—
24#1956C2212yes—
25#1948C2212yes—
26#1944—21no—
27#1944—21no—
28#1941—21no—
29#1935—22no—
30#1844—21no—
31#1844C2211yescharacteristic
32#1803C2212yes—
33C4×#106C2211yescharacteristic
34#1770C2212yes—
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#333C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×#322C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×C4⋊Q8C2312yes—
46C2×#125—22no—
47C2×C4×Q8C2311yescharacteristic
48#500—22no—
49#500—22no—
50#446—22no—
51#446—22no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#106—21no—
55C2×#106—21no—
56C2×#106—22no—
57C2×#106D411yescharacteristic
58C2×#106D411yescharacteristic
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×#103—22no—
64C2×#103—42no—
65C2×#103D412yes—
66C2×#97—21no—
67C2×#97—22no—
68C2×C4×C8—21no—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#398—42no—
73#395—42no—
74#391—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#384—22no—
80#384—22no—
81#384C2312yes—
82C4×C4⋊C4C2312yes—
83C4×C4⋊C4C2311yescharacteristic
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340D412yes—
93#340D412yes—
94#336—21no—
95#336—22no—
96#335—22no—
97#333—24no—
98Q8⋊C8—22no—
99#322—22no—
100C22×Q8—22no—
101C22×Q8C2×D411yescharacteristic
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8C2×D412yes—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113#125—44no—
114#125—82no—
115C4×Q8—21no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C42—21no—
135C2×C42—22no—
136C2×C42—21no—
137C2×C42C2411yescharacteristic
138#107—42no—
139#107—42no—
140#106—22no—
141#106—22no—
142#106—42no—
143#106—82no—
144#106—81no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—42no—
151C4⋊C8#4112yes—
152#103—24no—
153#103—82no—
154#103—84no—
155#97—42no—
156#97—82no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—42no—
162C4×C8#4112yes—
163C4×C8#4112yes—
164#95—22no—
165#95—21no—
166#95—42no—
167#95—42no—
168#95—42no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—82no—
176C2×Q8—82no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C22×D411yescharacteristic
185C22×C4C22×D411yescharacteristic
186C22×C4H(5,𝔽2)11yescharacteristic
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8#12412yes—
204C2×C8#12412yes—
205C2×C8#13712yes—
206C2×C8#13712yes—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—84no—
225C4⋊C4—84no—
226C4⋊C4—82no—
227C4⋊C4—162no—
228C4⋊C4—162no—
229C4⋊C4—162no—
230C4⋊C4#13714yes—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—42no—
235C42—42no—
236C42—41no—
237C42C2×#4111yescharacteristic
238C42C2×#4111yescharacteristic
239C42C2×#4111yescharacteristic
240C422−1+411yescharacteristic
241C23D4211yescharacteristic
242Q8—42no—
243Q8—41no—
244Q8—41no—
245Q8—84no—
246Q8—162no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4C2×#12411yescharacteristic
268C2×C4#53211yescharacteristic
269C2×C4#54211yescharacteristic
270C2×C4#54711yescharacteristic
271C2×C4#54811yescharacteristic
272C2×C4#55511yescharacteristic
273C2×C4C2×#13712yes—
274C2×C4C2×#13711yescharacteristic
275C2×C4C2×#13711yescharacteristic
276C2×C4#57712yes—
277C2×C4#57711yescharacteristic
278C2×C4#57811yescharacteristic
279C8—22no—
280C8—42no—
281C8—42no—
282C8—81no—
283C8—81no—
284C8—82no—
285C8#49114yes—
286C22#249111yescharacteristic
287C22#326612yes—
288C22#327611yescharacteristic
289C22#327811yescharacteristic
290C22#328812yes—
291C4—21no—
292C4—21no—
293C4—44no—
294C4—81no—
295C4—81no—
296C4—82no—
297C4—82no—
298C4—162no—
299C4—162no—
300C4C2×#49111yescharacteristic
301C4#313711yescharacteristic
302C4#317311yescharacteristic
303C4#318511yescharacteristic
304C4#335612yes—
305C4#337211yescharacteristic
306C4#337411yescharacteristic
307C2#2018112yes—
308C2#2018812yes—
309C2#2777711yescharacteristic
310C2#2780311yescharacteristic
311C2#2781111yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ 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
#288292
#286822
#278242
#104152
#103552
#98362
#97302
#95752
#86512
#82862
#82112