Finite Group Databasebeta

← #496911

Subgroups — #496911

Order 512.

526 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2225C211yesmaximal, characteristic
2C4×#500C211yesmaximal, characteristic
3#21276C211yesmaximal, characteristic
4#13275C211yesmaximal, characteristic
5#13176C211yesmaximal, characteristic
6#13167C212yesmaximal
7#13150C212yesmaximal
8#13133C212yesmaximal
9#13110C212yesmaximal
10#12812C211yesmaximal, characteristic
11#12653C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×C4×Q16C2211yescharacteristic
14C4×C4⋊Q8—21no—
15#2295—21no—
16#2225C418yescomplemented
17C2×#367C2212yes—
18C2×#366C2212yes—
19C2×C4⋊C16C2211yescharacteristic
20C2×C4⋊C16C2211yescharacteristic
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#346C2212yes—
24#1990—21no—
25#1969C2211yescharacteristic
26#1969C2211yescharacteristic
27#1942—21no—
28#1942C2211yescharacteristic
29#1942C2211yescharacteristic
30#1942C2212yes—
31#1933—21no—
32#1933—21no—
33#1845C2211yescharacteristic
34C4×#107—21no—
35C4×#107C2211yescharacteristic
36C4×#107C2212yes—
37C4×#103—21no—
38C42×C8C2211yescharacteristic
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1380C2212yes—
43#1374C2212yes—
44C22×Q16C2311yescharacteristic
45C2×C4⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—41no—
48C22×C16C2312yes—
49C22×C16C2312yes—
50#500—22no—
51#500C2×C414yes—
52C4×Q16—22no—
53C4×Q16C2×C414yes—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107C2312yes—
57C2×#107C2311yescharacteristic
58C2×#107C2311yescharacteristic
59C2×#107C2312yes—
60C2×#107C2311yescharacteristic
61C2×#103—21no—
62C2×#103—21no—
63C2×#103—21no—
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66C2×C4×C8D411yescharacteristic
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391—21no—
71#391—21no—
72#391—41no—
73#389—22no—
74#389—21no—
75#389—21no—
76#389—41no—
77#387—41no—
78#387—41no—
79#384—21no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—41no—
83C4×C4⋊C4—41no—
84#376—41no—
85C43D411yescharacteristic
86#367C2×C418yes—
87#366—42no—
88C4⋊C16—22no—
89C4⋊C16C2×C414yes—
90#358—42no—
91#358C2×C418yes—
92#346—42no—
93#340—21no—
94#340—21no—
95#340—21no—
96#340—22no—
97#340—22no—
98C22×Q8—21no—
99C22×Q8—41no—
100C2×Q16—21no—
101C2×Q16—21no—
102C2×Q16—21no—
103C2×Q16—21no—
104C2×Q16—21no—
105C2×Q16C22×C412yes—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2411yescharacteristic
110C4⋊Q8—24no—
111C4⋊Q8—42no—
112C4×Q8—24no—
113C4×Q8—42no—
114C4×Q8—84no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C42—41no—
128C2×C42—42no—
129C2×C42—41no—
130C2×C42D811yescharacteristic
131C2×C42D811yescharacteristic
132C2×C42D811yescharacteristic
133C2×C42D811yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C16—22no—
138C2×C16—22no—
139C2×C16—22no—
140C2×C16—22no—
141C2×C16—22no—
142C2×C16C22×C414yes—
143#107—24no—
144#107—21no—
145#107—22no—
146#107—22no—
147#107—21no—
148#107—21no—
149#107—21no—
150#107—22no—
151#107—21no—
152#107—42no—
153#107C22×C414yes—
154#107C22×C412yes—
155#103—24no—
156#103—42no—
157#103—42no—
158C4×C8—22no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8—22no—
164C4×C8—41no—
165C4×C8C22×C412yes—
166C4×C8#4112yes—
167#95—41no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172#95—42no—
173#95—41no—
174#95—41no—
175#95—42no—
176C2×Q8—22no—
177C2×Q8—41no—
178C2×Q8—44no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—82no—
184C2×Q8—82no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4C2×D811yescharacteristic
192C22×C4C2×D811yescharacteristic
193C22×C4C2×D811yescharacteristic
194C22×C4C2×D811yescharacteristic
195C22×C4C2×D811yescharacteristic
196C22×C4C2×D811yescharacteristic
197C22×C4C22×D411yescharacteristic
198Q16—22no—
199Q16—22no—
200Q16—41no—
201Q16—41no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8C4×D412yes—
210C2×C8#13712yes—
211C2×C8C23×C411yescharacteristic
212C2×C8C2×#4111yescharacteristic
213C2×C8H(5,𝔽2)11yescharacteristic
214C2×C82−1+411yescharacteristic
215C2×C82−1+411yescharacteristic
216C2×C82−1+411yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—41no—
242C42—44no—
243C42—84no—
244C42—82no—
245C42C4×D412yes—
246C42#13512yes—
247C16—24no—
248C16—42no—
249C23C22×D811yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—88no—
253Q8—81no—
254Q8—81no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4C4×D812yes—
287C2×C4C4×D812yes—
288C2×C4#50912yes—
289C2×C4#51012yes—
290C2×C4#51012yes—
291C2×C4C2×C4×D411yescharacteristic
292C2×C4#53611yescharacteristic
293C2×C4#53711yescharacteristic
294C2×C4C2×#13511yescharacteristic
295C2×C4C2×#13711yescharacteristic
296C2×C4#57511yescharacteristic
297C8—22no—
298C8—22no—
299C8—41no—
300C8#51911yescharacteristic
301C8#52011yescharacteristic
302C8#54111yescharacteristic
303C8#54211yescharacteristic
304C22C2×C4×D811yescharacteristic
305C22#322611yescharacteristic
306C22#324511yescharacteristic
307C22C2×#50911yescharacteristic
308C22C2×#51011yescharacteristic
309C22C2×#51011yescharacteristic
310C22#341411yescharacteristic
311C4—22no—
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—82no—
317C4—84no—
318C4—82no—
319C4—161no—
320C4—162no—
321C4—161no—
322C4#217514yes—
323C4#232311yescharacteristic
324C4C4×#13711yescharacteristic
325C4#323311yescharacteristic
326C4#324811yescharacteristic
327C2#2211011yescharacteristic
328C2C2×#217511yescharacteristic
329C2C4×#51011yescharacteristic
330C2#3380311yescharacteristic
331C2#3380511yescharacteristic
332C2#3381711yescharacteristic
333C2#3382111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2225C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22252
C4×#5002
#212762
#132752
#131762
#131672
#131502
#131332
#131102
#128122
#126532