Finite Group Databasebeta

← #315137

Subgroups — #315137

Order 512.

539 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28829C211yesmaximal, characteristic
2#28744C211yesmaximal, characteristic
3#27816C212yesmaximal
4#10423C212yesmaximal
5#10357C212yesmaximal
6#9839C211yesmaximal, characteristic
7#9733C211yesmaximal, characteristic
8#9565C212yesmaximal
9#8651C211yesmaximal, characteristic
10#8285C211yesmaximal, characteristic
11#8211C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#446C2211yescharacteristic
14#2568C2212yes—
15#2552C2212yes—
16#2101—22no—
17#2084—22no—
18#2056—22no—
19#2026—22no—
20#1991—22no—
21#1990C2212yes—
22#1990C2212yes—
23#1988C2212yes—
24#1981C2212yes—
25#1971—21no—
26#1971—21no—
27#1948C2212yes—
28#1941—21no—
29#1933—22no—
30#1844—21no—
31#1844C2211yescharacteristic
32#1803C2212yes—
33C4×#106C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#332C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×#322C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C22×Q16—21no—
46C22×Q16—21no—
47C2×C4⋊Q8C2312yes—
48C2×#125—22no—
49C2×C4×Q8C2311yescharacteristic
50#494—22no—
51#494—22no—
52#446—22no—
53#446—22no—
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—22no—
65C2×#103—22no—
66C2×#103D412yes—
67C2×#97—21no—
68C2×C4×C8—21no—
69C2×C4×C8—22no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#400—42no—
74#398—42no—
75#391—22no—
76#391—22no—
77#391—22no—
78#389—22no—
79#389—22no—
80#384—22no—
81#384C2312yes—
82#382—22no—
83C4×C4⋊C4C2312yes—
84C4×C4⋊C4C2311yescharacteristic
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—42no—
92#340D412yes—
93#340D412yes—
94#336—21no—
95#336—22no—
96#335—22no—
97#332—24no—
98Q8⋊C8—22no—
99#322—22no—
100C22×Q8—22no—
101C22×Q8C2×D411yescharacteristic
102C2×Q16—42no—
103C2×Q16—42no—
104C2×Q16—42no—
105C2×Q16—42no—
106C2×Q16—81no—
107C2×Q16—81no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8C2×D412yes—
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119#125—44no—
120#125—82no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C4⋊C4C2×D412yes—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—21no—
141C2×C42C2411yescharacteristic
142#106—22no—
143#106—22no—
144#106—42no—
145#106—82no—
146#106—81no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—42no—
153C4⋊C8#4112yes—
154#103—24no—
155#103—82no—
156#103—82no—
157#103—82no—
158#97—42no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8—42no—
164C4×C8—82no—
165C4×C8#4112yes—
166C4×C8#4112yes—
167#95—21no—
168#95—22no—
169#95—42no—
170#95—42no—
171#95—42no—
172#95—42no—
173#95—42no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4H(5,𝔽2)11yescharacteristic
194Q16—84no—
195Q16—84no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8#12412yes—
213C2×C8#12412yes—
214C2×C8#13512yes—
215C2×C8#13512yes—
216C4⋊C4—22no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—82no—
236C4⋊C4—162no—
237C4⋊C4#13714yes—
238C42—41no—
239C42—41no—
240C42—41no—
241C42—42no—
242C42—42no—
243C42—41no—
244C42C2×#4111yescharacteristic
245C42C2×#4111yescharacteristic
246C42C2×#4111yescharacteristic
247C422−1+411yescharacteristic
248C23D4211yescharacteristic
249Q8—42no—
250Q8—41no—
251Q8—41no—
252Q8—82no—
253Q8—82no—
254Q8—82no—
255Q8—82no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4C2×#12411yescharacteristic
277C2×C4#53211yescharacteristic
278C2×C4#54211yescharacteristic
279C2×C4#54711yescharacteristic
280C2×C4#54811yescharacteristic
281C2×C4#55511yescharacteristic
282C2×C4C2×#13511yescharacteristic
283C2×C4C2×#13511yescharacteristic
284C2×C4C2×#13712yes—
285C2×C4#57611yescharacteristic
286C2×C4#57712yes—
287C2×C4#57711yescharacteristic
288C8—22no—
289C8—42no—
290C8—42no—
291C8—81no—
292C8—81no—
293C8—82no—
294C8#48814yes—
295C22#249111yescharacteristic
296C22#326612yes—
297C22#328411yescharacteristic
298C22#328711yescharacteristic
299C22#328912yes—
300C4—21no—
301C4—21no—
302C4—44no—
303C4—82no—
304C4—81no—
305C4—82no—
306C4—81no—
307C4—162no—
308C4—162no—
309C4C2×#48811yescharacteristic
310C4#313611yescharacteristic
311C4#317311yescharacteristic
312C4#318511yescharacteristic
313C4#335612yes—
314C4#336111yescharacteristic
315C4#336511yescharacteristic
316C2#1985312yes—
317C2#1986912yes—
318C2#2777011yescharacteristic
319C2#2780311yescharacteristic
320C2#2781011yescharacteristic
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
#287442
#278162
#104232
#103572
#98392
#97332
#95652
#86512
#82852
#82112