Finite Group Databasebeta

← #7632387

Subgroups — #7632387

Order 512.

7936 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3497C211yesmaximal, characteristic
2C22×#557C211yesmaximal, characteristic, complemented
3#36583C218yesmaximal, complemented
4#36446C218yesmaximal, complemented
5#35854C2116yesmaximal, complemented
6#35649C2112yesmaximal, complemented
7#35289C216yesmaximal, complemented
8#34955C216yesmaximal, complemented
9#34257C214yesmaximal, complemented
10C4×#519C211yesmaximal, characteristic, complemented
11C22×2−1+4C2212yes—
12#3497—216no—
13C2×#557—216no—
14C2×#557C2218yes—
15C2×#557C2218yes—
16C2×#550C2214yes—
17C2×D4×Q8C2214yes—
18C2×#548C2218yes—
19C2×#537C2216yes—
20C2×#532C2216yes—
21C22×C4⋊Q8C2213yes—
22C22×#125C2216yes—
23C2×#519C2212yes—
24C22×C4×Q8C2213yes—
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27#2862C22112yes—
28#2858C22112yes—
29#2640C22124yes—
30#2592C22124yes—
31#2591C22124yes—
32#2568C22112yes—
33#2563C22124yes—
34#2548C22112yes—
35#2547C22124yes—
36#2504C22124yes—
37#2504C2214yes—
38#2490C22196yes—
39#2489C22196yes—
40#2436C22112yes—
41#2398C22124yes—
42#2341C22132yes—
43Q8×C22⋊C4C22132yes—
44#2311C22124yes—
45C4×#124C22124yes—
46#2297C22124yes—
47#2276C2218yes—
48#2274C2216yes—
49C42×Q8C2218yes—
50C4×#117C2216yes—
51C2×2−1+4—216no—
52C2×2−1+4—232no—
53C22×#41C23112yes—
54C22×#41C2318yes—
55C23×Q8C2318yes—
56C23×Q8C2313yes—
57C23×Q8C2311yescharacteristic
58#557—264no—
59#557—264no—
60#550—232no—
61D4×Q8—232no—
62#548—264no—
63#537—248no—
64#532—248no—
65C2×C4⋊Q8—224no—
66C2×C4⋊Q8C23112yes—
67C2×C4⋊Q8C2316yes—
68C2×C4⋊Q8C23112yes—
69C2×#127C2316yes—
70C2×#125—248no—
71C2×#125C23148yes—
72C2×#124C23112yes—
73C2×#124C23124yes—
74C2×C22⋊Q8C23124yes—
75C2×#121C23112yes—
76C2×#121C2314yes—
77C2×#121C23124yes—
78#519—216no—
79C2×C4×Q8—216no—
80C2×C4×Q8—224no—
81C2×C4×Q8D4116yes—
82C2×C4×Q8C23112yes—
83C2×C4×Q8C23112yes—
84C2×C4×Q8C23116yes—
85C2×C4×Q8C23116yes—
86C2×C4×Q8C2314yes—
87C2×C4×Q8C2314yes—
88C2×C4×D4C23112yes—
89C2×C4×D4C23112yes—
90C2×#117C23112yes—
91C22×C4⋊C4C2316yes—
92C22×C4⋊C4C2316yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2316yes—
95C22×C4⋊C4C2313yes—
96C22×C42C2316yes—
97C22×C42C2313yes—
98#400C23148yes—
99#397C23116yes—
100#397C23196yes—
101#394C23148yes—
102#393C23148yes—
103#391C23148yes—
104#391C23148yes—
105#391C23124yes—
106#390C23148yes—
107#390C23148yes—
108#390C23148yes—
109#390C2318yes—
110#388C23196yes—
111#386C23148yes—
112#386C23116yes—
113#382C23148yes—
114#382C2318yes—
115#380C23196yes—
116C4×C4⋊C4C23112yes—
117C4×C4⋊C4C23112yes—
118C4×C4⋊C4C23148yes—
119C4×C22⋊C4C23112yes—
120C4×C22⋊C4C23196yes—
121C4×C22⋊C4C2314yes—
122#376C23112yes—
123C43C23112yes—
1242−1+4—4128no—
125C2×#41—232no—
126C2×#41—264no—
127C2×#41—248no—
128C2×#41—296no—
129C22×Q8—232no—
130C22×Q8—264no—
131C22×Q8—212no—
132C22×Q8—412no—
133C22×Q8C2×D418yes—
134C22×Q8#41116yes—
135C22×Q8C2416yes—
136C22×Q8C24112yes—
137C22×Q8C2416yes—
138C22×Q8C2414yes—
139C22×D4C2414yes—
140C22×D4C2418yes—
141C22×D4C24112yes—
142C22×D4C24112yes—
143C23×C4C24124yes—
144C23×C4C2411yescharacteristic
145C23×C4C2416yes—
146C23×C4C2412yes—
147C23×C4C2413yes—
148C23×C4C2413yes—
149C4⋊Q8—248no—
150C4⋊Q8—248no—
151C4⋊Q8—224no—
152#127—224no—
153#125—2192no—
154#124—248no—
155#124—448no—
156C22⋊Q8—296no—
157#121—216no—
158#121—248no—
159#121—296no—
160C4×Q8—264no—
161C4×Q8—264no—
162C4×Q8—248no—
163C4×Q8—248no—
164C4×Q8C2×D4132yes—
165C4×Q8C2×D4132yes—
166C4×D4—248no—
167C4×D4—248no—
168#117—248no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—224no—
171C2×C4⋊C4—224no—
172C2×C4⋊C4—224no—
173C2×C4⋊C4C2×D4124yes—
174C2×C4⋊C4C2412yes—
175C2×C4⋊C4C2412yes—
176C2×C4⋊C4C24124yes—
177C2×C4⋊C4C24124yes—
178C2×C4⋊C4C24124yes—
179C2×C4⋊C4C24124yes—
180C2×C4⋊C4C2414yes—
181C2×C4⋊C4C24112yes—
182C2×C4⋊C4C24112yes—
183C2×C22⋊C4C2418yes—
184C2×C22⋊C4C2418yes—
185C2×C22⋊C4C24124yes—
186C2×C22⋊C4C24196yes—
187C2×C22⋊C4C24132yes—
188C2×C42—224no—
189C2×C42C2×D4124yes—
190C2×C42C24124yes—
191C2×C42C24124yes—
192C2×C42C2418yes—
193C2×C42C24124yes—
194C2×C42C2416yes—
195C2×C42C2416yes—
196#95C2418yes—
197#95C24124yes—
198#95C24132yes—
199#95C24196yes—
200C24C2518yes—
201C24C2511yescharacteristic
202#41—4128no—
203#41—4192no—
204C2×Q8—212no—
205C2×Q8—212no—
206C2×Q8—224no—
207C2×Q8—248no—
208C2×Q8—224no—
209C2×Q8—412no—
210C2×Q8—448no—
211C2×Q8—4128no—
212C2×Q8#124164yescomplemented
213C2×Q8C22×D4116yes—
214C2×Q8C2×#41132yes—
215C2×D4—216no—
216C2×D4—232no—
217C2×D4—28no—
218C2×D4—224no—
219C2×D4—224no—
220C2×D4—248no—
221C2×D4—216no—
222C2×D4—424no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—212no—
226C22×C4—248no—
227C22×C4—296no—
228C22×C4—46no—
229C22×C4—44no—
230C22×C4—412no—
231C22×C4C22×D414yes—
232C22×C4C22×D4112yes—
233C22×C4C22×D4112yes—
234C22×C4C2×#41124yes—
235C22×C4C2518yes—
236C22×C4C25116yes—
237C22×C4C2514yes—
238C22×C4C25112yes—
239C22×C4C2511yescharacteristic
240C22×C4C2516yes—
241C22×C4C2516yes—
242C22×C4C2511yescharacteristic
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—248no—
246C4⋊C4—248no—
247C4⋊C4—248no—
248C4⋊C4—28no—
249C4⋊C4—224no—
250C4⋊C4—248no—
251C4⋊C4C22×D4148yes—
252C22⋊C4—216no—
253C22⋊C4—216no—
254C22⋊C4—248no—
255C22⋊C4—432no—
256C22⋊C4—496no—
257C42—248no—
258C42—248no—
259C42—216no—
260C42—424no—
261C42C22×D4124yes—
262C42C22×D4124yes—
263C23—22no—
264C23—22no—
265C23—28no—
266C23—216no—
267C23—216no—
268C23—216no—
269C23C23×D412yes—
270C23C22×#4114yes—
271C23C2611yescharacteristic
272Q8—424no—
273Q8—448no—
274Q8—424no—
275Q8C2×#124164yes—
276D4—432no—
277D4—448no—
278D4—416no—
279D4—824no—
280C2×C4—22no—
281C2×C4—212no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—212no—
285C2×C4—28no—
286C2×C4—216no—
287C2×C4—216no—
288C2×C4—216no—
289C2×C4—224no—
290C2×C4—216no—
291C2×C4—28no—
292C2×C4—412no—
293C2×C4—48no—
294C2×C4—424no—
295C2×C4—496no—
296C2×C4C2×#124148yes—
297C2×C4C23×D412yes—
298C2×C4C23×D4112yes—
299C2×C4C23×D4112yes—
300C2×C4C23×D412yes—
301C2×C4C22×#41124yes—
302C2×C4#58514yes—
303C22—28no—
304C22—22no—
305C22—22no—
306C22—432no—
307C22—48no—
308C22—48no—
309C22—48no—
310C22C22×#12414yes—
311C22C24×D411yescharacteristic
312C22C23×#4112yes—
313C22C2×#58512yes—
314C222−1+612yes—
315C4—22no—
316C4—212no—
317C4—416no—
318C4—412no—
319C4—44no—
320C4—48no—
321C4C22×#124124yes—
322C4#348714yes—
323C2—24no—
324C2—88no—
325C2C23×#12411yescharacteristic
326C2C2×#348711yescharacteristic
327C2#6285911yescharacteristic
328C2#6300714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#3497C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#557C2semidirect
3. #36583C2semidirect
4. #36446C2semidirect
5. #35854C2semidirect
6. #35649C2semidirect
7. #35289C2semidirect
8. #34955C2semidirect
9. #34257C2semidirect
10. C4×#519C2semidirect
212. C2×Q8#124semidirect

Maximal subgroups

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

subgroupindex
C2×#34972
C22×#5572
#365832
#364462
#358542
#356492
#352892
#349552
#342572
C4×#5192