Finite Group Databasebeta

← #308922

Subgroups — #308922

Order 512.

566 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28813C211yesmaximal, characteristic
2#26442C212yesmaximal
3#23158C211yesmaximal, characteristic
4#10589C212yesmaximal
5#10349C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9615C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#8927C212yesmaximal
11C8×#106C211yesmaximal, characteristic
12C2×#500—21no—
13C2×#500C2211yescharacteristic
14#2551C2212yes—
15#2369C2212yes—
16#2088—22no—
17#2058—22no—
18#2057—22no—
19#1990C2212yes—
20#1988—22no—
21#1987C2212yes—
22#1987C2212yes—
23#1956C2212yes—
24#1948C2212yes—
25#1947C2212yes—
26#1942—21no—
27#1942—21no—
28#1939C2212yes—
29#1921—22no—
30#1919—22no—
31#1845C2211yescharacteristic
32C4×#107—21no—
33C4×#107C2211yescharacteristic
34C4×#106C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#334C2211yescharacteristic
39C2×#333C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2211yescharacteristic
44#1279C2212yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2312yes—
48#500—22no—
49#500—44no—
50#500Q814yescomplemented
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—22no—
55C2×#107—21no—
56C2×#107D411yescharacteristic
57C2×#107D411yescharacteristic
58C2×#106—22no—
59C2×#106D411yescharacteristic
60C2×#106D411yescharacteristic
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—42no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#395—42no—
72#391—42no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#384C2312yes—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4C2311yescharacteristic
81C4×C4⋊C4C2312yes—
82C4×C4⋊C4C2311yescharacteristic
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340D412yes—
92#340D411yescharacteristic
93#340D411yescharacteristic
94#336—21no—
95#336—22no—
96#336—21no—
97#335—22no—
98#334—22no—
99#333—24no—
100C82—22no—
101C22×Q8—22no—
102C22×Q8—42no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—44no—
115C4⋊Q8—44no—
116C4⋊Q8—82no—
117C4⋊Q8—82no—
118C4⋊Q8C2×Q814yes—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C42—21no—
134C2×C42—22no—
135C2×C42—21no—
136C2×C42—42no—
137C2×C42C2411yescharacteristic
138#107—24no—
139#107—24no—
140#107—22no—
141#107—42no—
142#107—44no—
143#107—81no—
144#107SD1614yes—
145#106—22no—
146#106—82no—
147#106Q1614yes—
148C4⋊C8—22no—
149C4⋊C8—21no—
150C4⋊C8—22no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—82no—
154C4⋊C8#4112yes—
155#103—84no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—42no—
161C4×C8—42no—
162C4×C8C2×Q812yes—
163C4×C8#4112yes—
164#95—21no—
165#95—22no—
166#95—21no—
167#95—42no—
168#95—42no—
169#95—42no—
170C2×Q8—24no—
171C2×Q8—42no—
172C2×Q8—84no—
173C2×Q8—84no—
174C2×Q8—82no—
175C2×Q8—82no—
176C2×Q8—162no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4C22×D411yescharacteristic
186C22×C4C22×D411yescharacteristic
187C22×C4H(5,𝔽2)11yescharacteristic
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—22no—
193C2×C8—21no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8C22⋊Q812yes—
208C2×C8C22⋊Q812yes—
209C2×C8C2×SD1612yes—
210C2×C8C2×Q1612yes—
211C2×C8#13712yes—
212C4⋊C4—21no—
213C4⋊C4—22no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—21no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C4⋊C4—82no—
231C4⋊C4—162no—
232C4⋊C4C2×SD1614yes—
233C4⋊C4C2×Q1612yes—
234C4⋊C4C2×Q1612yes—
235C4⋊C4#13512yes—
236C42—41no—
237C42—42no—
238C42—41no—
239C42—42no—
240C42—41no—
241C42—41no—
242C42—162no—
243C42C22×Q811yescharacteristic
244C42C2×#4111yescharacteristic
245C42C2×#4111yescharacteristic
246C422−1+411yescharacteristic
247C23D4211yescharacteristic
248Q8—84no—
249Q8—162no—
250Q8—168no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4C2×C22⋊Q811yescharacteristic
278C2×C4#53211yescharacteristic
279C2×C4#54211yescharacteristic
280C2×C4#54811yescharacteristic
281C2×C4D4×Q811yescharacteristic
282C2×C4#55411yescharacteristic
283C2×C4C22×SD1611yescharacteristic
284C2×C4C22×Q1611yescharacteristic
285C2×C4C2×#13511yescharacteristic
286C2×C4C2×#13711yescharacteristic
287C2×C4#57611yescharacteristic
288C2×C4#57711yescharacteristic
289C2×C4#57811yescharacteristic
290C2×C4#57811yescharacteristic
291C8—22no—
292C8—22no—
293C8—42no—
294C8—42no—
295C8—41no—
296C8—81no—
297C8—82no—
298C8#47514yes—
299C22#245211yescharacteristic
300C22#327611yescharacteristic
301C22#328011yescharacteristic
302C22D4×SD1611yescharacteristic
303C22D4×Q1611yescharacteristic
304C22#329011yescharacteristic
305C22#329211yescharacteristic
306C4—21no—
307C4—21no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4—84no—
313C4—81no—
314C4—82no—
315C4—162no—
316C4C2×#47511yescharacteristic
317C4#308211yescharacteristic
318C4#317011yescharacteristic
319C4#317511yescharacteristic
320C4#335911yescharacteristic
321C4#336911yescharacteristic
322C4Q8×Q1611yescharacteristic
323C4#338311yescharacteristic
324C2#1990511yescharacteristic
325C2#1993011yescharacteristic
326C2#1997611yescharacteristic
327C2SD16×Q1611yescharacteristic
328C2#2646211yescharacteristic
329C2#2648511yescharacteristic
330C2#2650111yescharacteristic
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

Complements

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

NH ≅ G/Ntype
50. #500Q8semidirect

Maximal subgroups

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

subgroupindex
#288132
#264422
#231582
#105892
#103492
#98232
#97302
#96152
#95942
#89272
C8×#1062