Finite Group Databasebeta

← #506584

Subgroups — #506584

Order 512.

716 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2241C211yesmaximal, characteristic, complemented
2C2×#2225C211yesmaximal, characteristic, complemented
3C2×#2208C211yesmaximal, characteristic
4C2×C4×SD32C211yesmaximal, characteristic
5#28527C211yesmaximal, characteristic
6#28462C211yesmaximal, characteristic, complemented
7#13531C212yesmaximal, complemented
8#13327C212yesmaximal
9#13237C212yesmaximal
10#13203C212yesmaximal, complemented
11C16×C4⋊C4C211yesmaximal, characteristic, complemented
12C22×SD32C2211yescharacteristic
13C22×SD32C2212yes—
14C2×#500C2211yescharacteristic
15C2×#494C2211yescharacteristic
16C2×#494C2212yes—
17C2×#474—21no—
18C2×#462—21no—
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2546—21no—
22#2542—21no—
23#2241—24no—
24#2225—24no—
25#2208—24no—
26C4×SD32—24no—
27C2×#367C2212yes—
28C2×#367C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#358C2212yes—
31C2×#358C2212yes—
32C2×#358C2212yes—
33C2×#358C2211yescharacteristic
34C2×#358C2212yes—
35C2×#357C2211yescharacteristic
36C2×C4×C16C2212yes—
37C2×C4×C16C2211yescharacteristic
38#2093—22no—
39#2016—22no—
40#1970C2212yes—
41#1969C2212yes—
42#1845C2212yes—
43C8×C4⋊C4C2211yescharacteristic
44#1374C2212yes—
45C22×Q16—22no—
46C22×Q16—21no—
47C22×Q16C2312yes—
48C22×Q16C2311yescharacteristic
49C22×D8C2311yescharacteristic
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C22⋊Q8—42no—
54C2×C4×Q8—21no—
55C2×C4×D4—21no—
56C2×SD32—22no—
57C2×SD32—22no—
58C2×SD32—24no—
59C2×SD32—22no—
60C2×SD32—24no—
61C2×SD32—42no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2312yes—
65#500—22no—
66#500—22no—
67#494—24no—
68#494—22no—
69#494—22no—
70#494—42no—
71#474—44no—
72#462—44no—
73C4×Q16—22no—
74C4×Q16—22no—
75C4×D8—22no—
76C4×D8—22no—
77C2×#107—21no—
78C2×#107—22no—
79C2×#107C2311yescharacteristic
80C2×#107C2312yes—
81C2×#107C2311yescharacteristic
82C2×C4⋊C8D411yescharacteristic
83C2×#103—21no—
84C2×#103—22no—
85C2×#102—21no—
86C2×#102—22no—
87C2×C4×C8C2312yes—
88C2×C4×C8C2311yescharacteristic
89#399—42no—
90#398—42no—
91#391—22no—
92#390—22no—
93#389—42no—
94#384—22no—
95C4×C4⋊C4D411yescharacteristic
96#367—22no—
97#367—24no—
98C4⋊C16D414yes—
99#358—22no—
100#358—24no—
101#358—24no—
102#358—24no—
103#358—24no—
104#357—22no—
105C4×C16—42no—
106C4×C16D414yes—
107#336D412yes—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—41no—
111C22×D4—21no—
112C23×C4—41no—
113C2×Q16—21no—
114C2×Q16—21no—
115C2×Q16—22no—
116C2×Q16—22no—
117C2×Q16—21no—
118C2×Q16—22no—
119C2×Q16—22no—
120C2×Q16—21no—
121C2×Q16—22no—
122C2×Q16—21no—
123C2×Q16—22no—
124C2×Q16—44no—
125C2×Q16—42no—
126C2×Q16—44no—
127C2×Q16—82no—
128C2×Q16—81no—
129C2×Q16—81no—
130C2×Q16#4112yes—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8#4112yes—
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D412yes—
139C22×C8C2411yescharacteristic
140C4⋊Q8—44no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—82no—
146C22⋊Q8—88no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×D4—42no—
150C4×D4—42no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4D812yes—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—44no—
163C2×C42—41no—
164C2×C42C2×D412yes—
165C2×C42C2×D411yescharacteristic
166SD32—28no—
167SD32—48no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—22no—
171C2×C16—21no—
172C2×C16—22no—
173C2×C16—21no—
174C2×C16C2×D412yes—
175C2×C16C2×D412yes—
176C2×C16C2×D414yes—
177#107—21no—
178#107—21no—
179#107—22no—
180#107—21no—
181#107—22no—
182#107—21no—
183#107—42no—
184#107—44no—
185C4⋊C8D814yes—
186#103—42no—
187#103—44no—
188#102—42no—
189#102—44no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8C2×D412yes—
193C4×C8C2×D412yes—
194#95—44no—
195#95D812yes—
196C24—41no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—41no—
203C2×Q8—42no—
204C2×Q8—41no—
205C2×Q8—42no—
206C2×Q8—41no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—82no—
210C2×Q8—84no—
211C2×D4—22no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—82no—
223C22×C4—84no—
224C22×C4C2×D811yescharacteristic
225C22×C4C2×D814yes—
226C22×C4C2×D811yescharacteristic
227C22×C4C22×D411yescharacteristic
228Q16—22no—
229Q16—22no—
230Q16—42no—
231Q16—41no—
232Q16—42no—
233Q16—41no—
234Q16—42no—
235Q16—42no—
236Q16—88no—
237Q16—84no—
238D8—22no—
239D8—22no—
240D8—41no—
241D8—41no—
242C2×C8—22no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—22no—
246C2×C8#12712yes—
247C2×C8C2×D814yes—
248C2×C8C22×D411yescharacteristic
249C2×C8C22×D411yescharacteristic
250C2×C8C2×#4111yescharacteristic
251C2×C8C2×#4111yescharacteristic
252C2×C8H(5,𝔽2)11yescharacteristic
253C2×C82−1+411yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—88no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—88no—
266C4⋊C4SD3218yescomplemented
267C22⋊C4—82no—
268C22⋊C4—88no—
269C42—22no—
270C42—22no—
271C42—82no—
272C42#12712yes—
273C42C2×D812yes—
274C16—22no—
275C16—22no—
276C16—24no—
277C23—42no—
278C23—81no—
279C23—82no—
280C23—82no—
281C23—81no—
282C23C22×D811yescharacteristic
283Q8—42no—
284Q8—42no—
285Q8—81no—
286Q8—81no—
287Q8—82no—
288Q8—82no—
289Q8—82no—
290Q8—82no—
291Q8—164no—
292D4—42no—
293D4—42no—
294D4—81no—
295D4—81no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—164no—
312C2×C4#49312yes—
313C2×C4#49312yes—
314C2×C4C2×SD3218yes—
315C2×C4C2×SD3214yes—
316C2×C4C2×#12711yescharacteristic
317C2×C4#54611yescharacteristic
318C2×C4#54811yescharacteristic
319C2×C4C22×D811yescharacteristic
320C2×C4#57711yescharacteristic
321C2×C4#57711yescharacteristic
322C8—22no—
323C8—24no—
324C8#52511yescharacteristic
325C8#53711yescharacteristic
326C8#54611yescharacteristic
327C8#54811yescharacteristic
328C22—84no—
329C22—82no—
330C22—162no—
331C22—161no—
332C22—161no—
333C22C2×#49311yescharacteristic
334C22#329411yescharacteristic
335C22#335811yescharacteristic
336C22C22×SD3212yes—
337C22#341712yes—
338C4—28no—
339C4—22no—
340C4—82no—
341C4—161no—
342C4—162no—
343C4—161no—
344C4—162no—
345C4#224814yes—
346C4#254011yescharacteristic
347C4#315411yescharacteristic
348C4#329411yescharacteristic
349C4#335811yescharacteristic
350C2—162no—
351C2#2839711yescharacteristic
352C2C2×#224811yescharacteristic
353C2#3372111yescharacteristic
354C2#3386512yes—
355C2#3392612yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×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
1. C2×#2241C2semidirect
2. C2×#2225C2semidirect
6. #28462C2semidirect
7. #13531C2semidirect
10. #13203C2semidirect
11. C16×C4⋊C4C2semidirect
266. C4⋊C4SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#22412
C2×#22252
C2×#22082
C2×C4×SD322
#285272
#284622
#135312
#133272
#132372
#132032
C16×C4⋊C42