Finite Group Databasebeta

← #7625429

Subgroups — #7625429

Order 512.

8134 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Q8C211yesmaximal, characteristic
2#34196C218yesmaximal, complemented
3#34192C218yesmaximal, complemented
4#34187C216yesmaximal, complemented
5#34182C212yesmaximal, complemented
6#34178C216yesmaximal
7#34173C212yesmaximal
8#34128C218yesmaximal, complemented
9Q8×#117C218yesmaximal, complemented
10#34059C216yesmaximal, complemented
11#34012C216yesmaximal
12C4×#519C212yesmaximal, complemented
13C24×Q8C2211yescharacteristic
14C2×#519C2214yes—
15C22×C4×Q8C4132yescomplemented
16C22×C4×Q8C2213yes—
17C22×C4×Q8C2214yes—
18C22×C4×Q8C2214yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2213yes—
21C22×C4×Q8C2214yes—
22C22×C4×Q8C2214yes—
23C22×C4×Q8C2211yescharacteristic
24C23×C4⋊C4C2213yes—
25C23×C42C2213yes—
26#2356C22148yescomplemented
27#2355C22148yescomplemented
28#2351C22116yescomplemented
29Q8×C4⋊C4C22116yescomplemented
30#2345C22148yes—
31#2344C22148yes—
32Q8×C22⋊C4C22116yes—
33Q8×C22⋊C4C22116yes—
34#2320C22148yescomplemented
35#2316C22112yes—
36#2313C22124yes—
37#2310C22112yes—
38#2295C22148yescomplemented
39#2285C22112yes—
40#2285C22112yes—
41#2285C22112yes—
42#2283C22112yes—
43#2283C22112yes—
44#2283C22112yes—
45#2283C22112yes—
46#2283C22112yes—
47#2283C22112yes—
48#2280C22112yes—
49#2280C22112yes—
50#2280C2214yes—
51#2276C22116yescomplemented
52#2274C22112yes—
53C42×Q8C22116yescomplemented
54C4×#117C22112yes—
55C4×#117C22112yes—
56C4×#117C22112yes—
57C23×Q8—216no—
58C23×Q8C2×C4116yes—
59C23×Q8C2313yes—
60C23×Q8C2314yes—
61C23×Q8C2314yes—
62C23×Q8C2311yescharacteristic
63C24×C4C2313yes—
64C24×C4C2311yescharacteristic
65C24×C4C2313yes—
66#519—232no—
67C2×C4×Q8C2×C4148yes—
68C2×C4×Q8C2×C4164yes—
69C2×C4×Q8C2×C4164yes—
70C2×C4×Q8C2×C4148yes—
71C2×C4×Q8C2×C4164yes—
72C2×C4×Q8C2×C4164yes—
73C2×C4×Q8C2×C4116yes—
74C2×C4×Q8C2×C4116yes—
75C2×C4×Q8C23132yes—
76C2×C4×Q8C23112yes—
77C2×C4×Q8C2314yes—
78C2×C4×Q8C2314yes—
79C2×C4×Q8C23112yes—
80C2×C4×Q8C23112yes—
81C2×C4×Q8C2314yes—
82C2×C4×Q8C2314yes—
83C2×C4×Q8C23112yes—
84C2×#117C23124yes—
85C2×#117C2318yes—
86C2×#117C23124yes—
87C22×C4⋊C4C2×C4148yes—
88C22×C4⋊C4C2316yes—
89C22×C4⋊C4C2313yes—
90C22×C4⋊C4C23112yes—
91C22×C4⋊C4C23112yes—
92C22×C4⋊C4C2313yes—
93C22×C42C2×C4148yes—
94C22×C42C2313yes—
95C22×C42C2316yes—
96C22×C42C2316yes—
97C22×C42C2316yes—
98C22×C42C2313yes—
99C22×C42C2313yes—
100C22×C42C2316yes—
101C22×C42C2313yes—
102#391C23196yes—
103#389C23196yes—
104#387C231192yes—
105#384C23124yes—
106#383C23124yes—
107#383C23124yes—
108#382C23124yes—
109#382C23124yes—
110#382C23124yes—
111#381C23124yes—
112#381C23124yes—
113#381C2318yes—
114#380C23124yes—
115#380C23124yes—
116#380C23124yes—
117#380C23124yes—
118#380C2318yes—
119#380C2318yes—
120#379C2318yes—
121C4×C4⋊C4C23124yes—
122C4×C4⋊C4C23196yes—
123C4×C4⋊C4C23124yes—
124C4×C4⋊C4C23124yes—
125C4×C4⋊C4C23124yes—
126C4×C4⋊C4C23124yes—
127C4×C4⋊C4C23124yes—
128C4×C22⋊C4C23124yes—
129C4×C22⋊C4C23124yes—
130C4×C22⋊C4C23124yes—
131C4×C22⋊C4C23124yes—
132C4×C22⋊C4C2318yes—
133C4×C22⋊C4C2318yes—
134#376C23124yes—
135#376C23124yes—
136#376C23124yes—
137C43C23124yes—
138C25C2411yescharacteristic
139C22×Q8—2128no—
140C22×Q8—232no—
141C22×Q8—232no—
142C22×Q8—224no—
143C22×Q8C22×C4132yes—
144C22×Q8C22×C4132yes—
145C22×Q8C22×C4124yes—
146C22×Q8C22×C418yes—
147C22×Q8#41116yes—
148C22×Q8C24112yes—
149C22×Q8C2414yes—
150C23×C4—224no—
151C23×C4—28no—
152C23×C4—412no—
153C23×C4C22×C4124yes—
154C23×C4C22×C418yes—
155C23×C4C22×C4124yes—
156C23×C4C2416yes—
157C23×C4C2416yes—
158C23×C4C2416yes—
159C23×C4C2413yes—
160C23×C4C2411yescharacteristic
161C23×C4C2411yescharacteristic
162C23×C4C2411yescharacteristic
163C23×C4C2411yescharacteristic
164C23×C4C2411yescharacteristic
165C23×C4C2413yes—
166C23×C4C2416yes—
167C23×C4C2413yes—
168C23×C4C2413yes—
169C23×C4C2411yescharacteristic
170C4×Q8—2128no—
171C4×Q8C22×C4196yes—
172C4×Q8C22×C4196yes—
173C4×Q8C22×C4196yes—
174C4×Q8C22×C4196yes—
175C4×Q8C22×C4132yes—
176C4×Q8C22×C4132yes—
177C4×Q8C22×C4132yes—
178C4×Q8C22×C4132yes—
179#117—296no—
180#117—232no—
181#117—448no—
182C2×C4⋊C4C22×C4196yes—
183C2×C4⋊C4C22×C4148yes—
184C2×C4⋊C4C22×C4196yes—
185C2×C4⋊C4C22×C4124yes—
186C2×C4⋊C4C22×C4124yes—
187C2×C4⋊C4C24148yes—
188C2×C4⋊C4C24116yes—
189C2×C4⋊C4C24148yes—
190C2×C4⋊C4C24112yes—
191C2×C4⋊C4C24124yes—
192C2×C4⋊C4C24112yes—
193C2×C22⋊C4C24116yes—
194C2×C22⋊C4C24116yes—
195C2×C22⋊C4C24116yes—
196C2×C42C22×C4148yes—
197C2×C42C22×C4148yes—
198C2×C42C22×C4124yes—
199C2×C42C22×C4148yes—
200C2×C42C22×C4124yes—
201C2×C42C22×C4124yes—
202C2×C42C22×C4124yes—
203C2×C42C22×C4148yes—
204C2×C42C24148yes—
205C2×C42C24116yes—
206C2×C42C24148yes—
207C2×C42C2416yes—
208C2×C42C2416yes—
209C2×C42C2416yes—
210C2×C42C2416yes—
211C2×C42C2416yes—
212C2×C42C2416yes—
213C2×C42C2416yes—
214C2×C42C2416yes—
215#95C24116yes—
216#95C24116yes—
217#95C24148yes—
218#95C24148yes—
219#95C24148yes—
220#95C24148yes—
221C24—24no—
222C24—24no—
223C24—24no—
224C24C23×C414yes—
225C24C2511yescharacteristic
226C24C2511yescharacteristic
227C24C2511yescharacteristic
228C2×Q8—2128no—
229C2×Q8—2128no—
230C2×Q8—296no—
231C2×Q8—248no—
232C2×Q8#117164yescomplemented
233C2×Q8C23×C4148yes—
234C2×Q8C23×C4116yes—
235C2×Q8C2×#41132yes—
236C22×C4—212no—
237C22×C4—24no—
238C22×C4—224no—
239C22×C4—296no—
240C22×C4—232no—
241C22×C4—224no—
242C22×C4—224no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—448no—
248C22×C4—46no—
249C22×C4—412no—
250C22×C4—46no—
251C22×C4C23×C414yes—
252C22×C4C23×C414yes—
253C22×C4C23×C414yes—
254C22×C4C23×C414yes—
255C22×C4C23×C414yes—
256C22×C4C23×C4124yes—
257C22×C4C23×C4112yes—
258C22×C4C23×C4124yes—
259C22×C4C23×C4124yes—
260C22×C4C23×C4112yes—
261C22×C4C23×C4112yes—
262C22×C4C23×C4124yes—
263C22×C4C23×C4112yes—
264C22×C4C23×C414yes—
265C22×C4C2×#41124yes—
266C22×C42−1+418yes—
267C22×C4C25132yes—
268C22×C4C2511yescharacteristic
269C22×C4C2511yescharacteristic
270C22×C4C2516yes—
271C22×C4C2516yes—
272C22×C4C2516yes—
273C22×C4C2516yes—
274C22×C4C2511yescharacteristic
275C22×C4C2511yescharacteristic
276C4⋊C4—296no—
277C4⋊C4—232no—
278C4⋊C4—448no—
279C4⋊C4C23×C4196yes—
280C4⋊C4C23×C4148yes—
281C4⋊C4C23×C4148yes—
282C22⋊C4—232no—
283C22⋊C4—232no—
284C22⋊C4—232no—
285C42—296no—
286C42—232no—
287C42—448no—
288C42C23×C4124yes—
289C42C23×C4124yes—
290C42C23×C4124yes—
291C42C23×C4124yes—
292C42C23×C4124yes—
293C42C23×C4124yes—
294C42C23×C4124yes—
295C42C23×C4124yes—
296C23—232no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—28no—
304C23—28no—
305C23—28no—
306C23C24×C412yes—
307C23C24×C412yes—
308C23C24×C412yes—
309C23C22×#4114yes—
310C23C2×2−1+414yes—
311C23#58514yes—
312C23C2611yescharacteristic
313Q8—296no—
314Q8C2×#117164yes—
315C2×C4—212no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—212no—
319C2×C4—212no—
320C2×C4—224no—
321C2×C4—28no—
322C2×C4—248no—
323C2×C4—248no—
324C2×C4—248no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—28no—
329C2×C4—232no—
330C2×C4—232no—
331C2×C4—232no—
332C2×C4—412no—
333C2×C4—424no—
334C2×C4—412no—
335C2×C4C2×#117148yes—
336C2×C4#519116yescomplemented
337C2×C4C24×C412yes—
338C2×C4C24×C412yes—
339C2×C4C24×C4112yes—
340C2×C4C24×C4112yes—
341C2×C4C24×C4112yes—
342C2×C4C24×C4112yes—
343C2×C4C24×C412yes—
344C2×C4C24×C412yes—
345C2×C4C22×#41124yes—
346C2×C4C2×2−1+414yes—
347C2×C4C2×2−1+414yes—
348C22—232no—
349C22—28no—
350C22—28no—
351C22—28no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22C22×#11714yes—
359C22C2×#51914yes—
360C22#342814yes—
361C22C25×C411yescharacteristic
362C22C23×#4112yes—
363C22C22×2−1+412yes—
364C22C2×#58512yes—
365C4—212no—
366C4—22no—
367C4—22no—
368C4—212no—
369C4—212no—
370C4—432no—
371C4C22×#117124yes—
372C4C2×#51914yes—
373C4C2×#51914yes—
374C2—24no—
375C2—24no—
376C2—24no—
377C2C23×#11711yescharacteristic
378C2C22×#51911yescharacteristic
379C2C2×#342811yescharacteristic
380C2#6278814yes—
subgroupG/H
centerC22×C4C25
commutatorC22C25×C4
Frattini, socle, cosocleC23C26
prime radicalC25C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34196C2semidirect
3. #34192C2semidirect
4. #34187C2semidirect
5. #34182C2semidirect
8. #34128C2semidirect
9. Q8×#117C2semidirect
10. #34059C2semidirect
12. C4×#519C2semidirect
15. C22×C4×Q8C4semidirect
26. #2356C22semidirect
27. #2355C22semidirect
28. #2351C22semidirect
29. Q8×C4⋊C4C22semidirect
34. #2320C22semidirect
38. #2295C22semidirect
51. #2276C22semidirect
53. C42×Q8C22semidirect
232. C2×Q8#117semidirect
336. C2×C4#519semidirect

Maximal subgroups

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

subgroupindex
C23×C4×Q82
#341962
#341922
#341872
#341822
#341782
#341732
#341282
Q8×#1172
#340592
#340122
C4×#5192