Finite Group Databasebeta

← #497992

Subgroups — #497992

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q32C211yesmaximal, characteristic
2#22809C211yesmaximal, characteristic
3#22685C211yesmaximal, characteristic
4#13150C212yesmaximal
5#13112C212yesmaximal
6#13015C212yesmaximal
7#12889C211yesmaximal, characteristic
8#12814C211yesmaximal, characteristic
9#12814C211yesmaximal, characteristic
10#12662C211yesmaximal, characteristic
11#12638C212yesmaximal
12C22×Q32C2211yescharacteristic
13C2×C4×Q16C2212yes—
14C2×C4×Q16C2211yescharacteristic
15C2×C4×Q16C2211yescharacteristic
16#2356—21no—
17#2351—21no—
18C4×Q32C418yescomplemented
19C2×#366C2211yescharacteristic
20C2×C4⋊C16C2211yescharacteristic
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#346C2212yes—
26C2×C4×C16C2211yescharacteristic
27#1969C2212yes—
28#1942C2212yes—
29#1927—22no—
30#1865—22no—
31#1846C2211yescharacteristic
32#1846C2211yescharacteristic
33#1805—21no—
34#1805—21no—
35C4×#107C2212yes—
36C8×C4⋊C4C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2212yes—
42C22×Q16C2311yescharacteristic
43C22×Q16C2311yescharacteristic
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—22no—
47C2×Q32—22no—
48C2×Q32—22no—
49C2×Q32C2×C414yes—
50C22×C16C2312yes—
51C22×C16C2311yescharacteristic
52C22×C16C2311yescharacteristic
53C4×Q16—24no—
54C4×Q16—22no—
55C4×Q16—22no—
56C4×Q16—42no—
57C4×Q16C2×C414yes—
58C4×Q16C2×C414yes—
59C2×#107C2312yes—
60C2×#107C2312yes—
61C2×#107C2311yescharacteristic
62C2×#107C2311yescharacteristic
63C2×#106—21no—
64C2×#106—21no—
65C2×C4⋊C8D411yescharacteristic
66C2×#103—21no—
67C2×#103—21no—
68C2×#103—22no—
69C2×#103—22no—
70C2×C4×C8C2312yes—
71C2×C4×C8C2311yescharacteristic
72#391—22no—
73#389—22no—
74#389—42no—
75#387—42no—
76#383—41no—
77#382—21no—
78#382—21no—
79#382—41no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—41no—
82C4×C4⋊C4D411yescharacteristic
83#376—41no—
84#366C2×C414yes—
85C4⋊C16—22no—
86#358—24no—
87#358—42no—
88#358C2×C414yes—
89#358C2×C414yes—
90#346—42no—
91C4×C16C2×C414yes—
92#340—22no—
93#340—22no—
94#336D412yes—
95C22×Q8—21no—
96C22×Q8—21no—
97C2×Q16—21no—
98C2×Q16—21no—
99C2×Q16—21no—
100C2×Q16—21no—
101C2×Q16—21no—
102C2×Q16—21no—
103C2×Q16—21no—
104C2×Q16—21no—
105C2×Q16C22×C412yes—
106C2×Q16C22×C412yes—
107C2×Q16#4112yes—
108C2×Q16#4112yes—
109C22×C8C2×D412yes—
110C22×C8C2×D411yescharacteristic
111C22×C8C2411yescharacteristic
112C4×Q8—24no—
113C4×Q8—24no—
114C4×Q8—44no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8—82no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4D812yes—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—42no—
130C2×C42—42no—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133Q32—28no—
134C2×C16—21no—
135C2×C16—21no—
136C2×C16—21no—
137C2×C16—21no—
138C2×C16—22no—
139C2×C16—22no—
140C2×C16—22no—
141C2×C16C22×C412yes—
142C2×C16C22×C412yes—
143#107—21no—
144#107—21no—
145#107—22no—
146#107—22no—
147#107—22no—
148#107—22no—
149#107C22×C412yes—
150#107C22×C412yes—
151#106—81no—
152#106—81no—
153C4⋊C8—22no—
154#103—24no—
155#103—24no—
156#103—44no—
157#103—82no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8C22×C412yes—
161C4×C8#4112yes—
162#95—41no—
163#95—42no—
164#95—42no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—42no—
169#95—42no—
170#95D811yescharacteristic
171#95D811yescharacteristic
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C2×D812yes—
191C22×C4C2×D812yes—
192C22×C4C2×D811yescharacteristic
193C22×C4C2×D811yescharacteristic
194C22×C4C22×D411yescharacteristic
195Q16—22no—
196Q16—22no—
197Q16—22no—
198Q16—22no—
199Q16—22no—
200Q16—41no—
201Q16—41no—
202Q16#11714yes—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8C4×D412yes—
209C2×C8C23×C411yescharacteristic
210C2×C8C2×#4111yescharacteristic
211C2×C8C2×#4111yescharacteristic
212C2×C8C2×#4111yescharacteristic
213C2×C8H(5,𝔽2)11yescharacteristic
214C2×C82−1+411yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C42—22no—
229C42—22no—
230C42—44no—
231C42—44no—
232C42—84no—
233C42—84no—
234C42C4×D412yes—
235C42#13512yes—
236C16—22no—
237C16—22no—
238C16—42no—
239C23C22×D811yescharacteristic
240Q8—24no—
241Q8—42no—
242Q8—42no—
243Q8—42no—
244Q8—42no—
245Q8—42no—
246Q8—81no—
247Q8—81no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4C4×D812yes—
273C2×C4C4×D812yes—
274C2×C4C2×C4×D411yescharacteristic
275C2×C4#54611yescharacteristic
276C2×C4#54811yescharacteristic
277C2×C4C2×#13511yescharacteristic
278C2×C4#57711yescharacteristic
279C2×C4#57711yescharacteristic
280C8—22no—
281C8—42no—
282C8C2×#11711yescharacteristic
283C8#52011yescharacteristic
284C8#55211yescharacteristic
285C8#55311yescharacteristic
286C22C2×C4×D811yescharacteristic
287C22#329411yescharacteristic
288C22#335811yescharacteristic
289C22#341412yes—
290C22#341712yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—44no—
296C4—82no—
297C4—82no—
298C4—82no—
299C4—162no—
300C4—162no—
301C4#235211yescharacteristic
302C4#295211yescharacteristic
303C4#330311yescharacteristic
304C4#336011yescharacteristic
305C2#2276611yescharacteristic
306C2#3345111yescharacteristic
307C2#3345811yescharacteristic
308C2#3389412yes—
309C2#3394412yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C4×Q32C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q322
#228092
#226852
#131502
#131122
#130152
#128892
#128142
#126622
#126382