Finite Group Databasebeta

← #308908

Subgroups — #308908

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28814C211yesmaximal, characteristic
2#26506C212yesmaximal
3#23184C211yesmaximal, characteristic
4#10602C212yesmaximal
5#10356C212yesmaximal
6#9825C211yesmaximal, characteristic
7#9733C211yesmaximal, characteristic
8#9614C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#8923C212yesmaximal
11C8×#107C211yesmaximal, characteristic
12C2×#499—21no—
13C2×#494C2211yescharacteristic
14C2×C4×Q16—21no—
15#2552C2212yes—
16#2365C2212yes—
17#2099—22no—
18#2086—22no—
19#2059—22no—
20#1990C2212yes—
21#1990C2212yes—
22#1988C2212yes—
23#1981C2212yes—
24#1969—21no—
25#1948C2212yes—
26#1947—22no—
27#1947C2212yes—
28#1942—21no—
29#1939C2212yes—
30#1918—22no—
31#1878—22no—
32#1845C2211yescharacteristic
33C4×#107C2211yescharacteristic
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#335C2211yescharacteristic
39C2×#332C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2211yescharacteristic
44#1279C2212yes—
45C22×Q16—21no—
46C22×Q16—21no—
47C2×C4⋊Q8C2312yes—
48C2×#125—22no—
49C2×C4×Q8—22no—
50#499—44no—
51#494—22no—
52#494Q814yes—
53C4×Q16—44no—
54C2×#107—21no—
55C2×#107—22no—
56C2×#107—21no—
57C2×#107D411yescharacteristic
58C2×#107D411yescharacteristic
59C2×#107D411yescharacteristic
60C2×#107D411yescharacteristic
61C2×#106—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×#103—22no—
67C2×#103—22no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#398—42no—
74#391—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#387—42no—
79#384—22no—
80#384C2312yes—
81C4×C4⋊C4—22no—
82C4×C4⋊C4C2311yescharacteristic
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#340D411yescharacteristic
94#340D411yescharacteristic
95#336—21no—
96#336—22no—
97#336—21no—
98#335—22no—
99#335—22no—
100#332—24no—
101C82—22no—
102C22×Q8—22no—
103C2×Q16—24no—
104C2×Q16—24no—
105C2×Q16—42no—
106C2×Q16—42no—
107C2×Q16—81no—
108C2×Q16—81no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—21no—
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8C2×Q814yes—
121#125—44no—
122#125—82no—
123C4×Q8—44no—
124C4×Q8—82no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D412yes—
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—42no—
144C2×C42C2411yescharacteristic
145#107—22no—
146#107—22no—
147#107—22no—
148#107—42no—
149#107—44no—
150#107—81no—
151#107D814yes—
152#106—44no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—44no—
159C4⋊C8#4112yes—
160#103—44no—
161#103—82no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—42no—
167C4×C8—42no—
168C4×C8C2×Q812yes—
169C4×C8#4112yes—
170#95—21no—
171#95—22no—
172#95—21no—
173#95—42no—
174#95—42no—
175#95—42no—
176C2×Q8—24no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4H(5,𝔽2)11yescharacteristic
193Q16—48no—
194Q16—84no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—22no—
200C2×C8—21no—
201C2×C8—22no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8C22⋊Q812yes—
215C2×C8C22⋊Q812yes—
216C2×C8C2×D812yes—
217C2×C8#13512yes—
218C2×C8#13712yes—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—84no—
238C4⋊C4—84no—
239C4⋊C4—84no—
240C4⋊C4—82no—
241C4⋊C4—162no—
242C4⋊C4C2×D814yes—
243C4⋊C4#13512yes—
244C42—41no—
245C42—41no—
246C42—42no—
247C42—41no—
248C42—41no—
249C42—42no—
250C42—84no—
251C42C22×Q811yescharacteristic
252C42C2×#4111yescharacteristic
253C42C2×#4111yescharacteristic
254C422−1+411yescharacteristic
255C23D4211yescharacteristic
256Q8—44no—
257Q8—44no—
258Q8—82no—
259Q8—82no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4C2×C22⋊Q811yescharacteristic
287C2×C4#53211yescharacteristic
288C2×C4#54211yescharacteristic
289C2×C4#54811yescharacteristic
290C2×C4D4×Q811yescharacteristic
291C2×C4#55411yescharacteristic
292C2×C4C22×D811yescharacteristic
293C2×C4C2×#13511yescharacteristic
294C2×C4C2×#13511yescharacteristic
295C2×C4C2×#13711yescharacteristic
296C2×C4#57611yescharacteristic
297C2×C4#57711yescharacteristic
298C2×C4#57811yescharacteristic
299C2×C4#57811yescharacteristic
300C8—22no—
301C8—22no—
302C8—42no—
303C8—42no—
304C8—41no—
305C8—81no—
306C8—82no—
307C8#47414yes—
308C22#245211yescharacteristic
309C22#327811yescharacteristic
310C22D4×D811yescharacteristic
311C22#328311yescharacteristic
312C22#328511yescharacteristic
313C22#329011yescharacteristic
314C22#329311yescharacteristic
315C4—21no—
316C4—21no—
317C4—44no—
318C4—42no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—84no—
323C4—82no—
324C4—162no—
325C4C2×#47411yescharacteristic
326C4#308111yescharacteristic
327C4#316811yescharacteristic
328C4#317711yescharacteristic
329C4#335811yescharacteristic
330C4#337511yescharacteristic
331C4#338311yescharacteristic
332C4#338311yescharacteristic
333C2#1994411yescharacteristic
334C2#1995911yescharacteristic
335C2#1997111yescharacteristic
336C2#1997311yescharacteristic
337C2#2646011yescharacteristic
338C2#2648311yescharacteristic
339C2#2650211yescharacteristic
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
#288142
#265062
#231842
#106022
#103562
#98252
#97332
#96142
#95942
#89232
C8×#1072