Finite Group Databasebeta

← #4308739

Subgroups — #4308739

Order 512.

2018 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C212yesmaximal
2#15256C211yesmaximal, characteristic
3#15253C211yesmaximal, characteristic
4#15251C214yesmaximal
5#15173C212yesmaximal
6#14903C214yesmaximal
7#14901C214yesmaximal
8#14688C214yesmaximal
9#14328C214yesmaximal
10#14327C214yesmaximal
11C4×#376C211yesmaximal, characteristic
12#2863—216no—
13C2×#398C2214yes—
14C2×#395C2214yes—
15C2×#391C2212yes—
16C2×#391C2212yes—
17C2×#391C2212yes—
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#387C2214yes—
22C2×#387C2214yes—
23C2×#387C2214yes—
24C2×#383C2214yes—
25C2×#376C2214yes—
26C2×C43C2211yescharacteristic
27C2×C43C2212yes—
28#1446C2214yes—
29#1445C2214yes—
30#1444C2214yes—
31#1444C2214yes—
32#1444C2218yes—
33#1442C2218yes—
34#1441C2218yes—
35#1441C2214yes—
36#1441C2218yes—
37#1441C2214yes—
38C4⋊#95C2218yes—
39C4⋊#95C2218yes—
40C4⋊#95C2214yes—
41C4⋊#95C2214yes—
42#1434C2214yes—
43#1434C2214yes—
44#1434C2214yes—
45#1434C2214yes—
46#1434C2218yes—
47C4×#95C2211yescharacteristic
48C4×#95C2212yes—
49C4×#95C2211yescharacteristic
50C4×#95C2214yes—
51C23×Q8C2312yes—
52C22×C4⋊C4C2314yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2318yes—
55C22×C4⋊C4C2318yes—
56C22×C4⋊C4C2314yes—
57C22×C4⋊C4C2314yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42C2312yes—
61C22×C42C2314yes—
62C22×C42C2311yescharacteristic
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C22×C42C2312yes—
66C22×C42C2314yes—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#398—216no—
70#395—216no—
71#391—28no—
72#391—28no—
73#391—28no—
74#389—28no—
75#389—28no—
76#389—28no—
77#387—216no—
78#387—216no—
79#387—216no—
80#383—216no—
81#376—216no—
82C2×#95C2318yes—
83C2×#95C2318yes—
84C2×#95C2318yes—
85C2×#95C2314yes—
86C2×#95C2314yes—
87C2×#95C2314yes—
88C2×#95C2314yes—
89C2×#95C2314yes—
90C2×#95C2314yes—
91C2×#95C2314yes—
92C2×#95C2314yes—
93C2×#95C2314yes—
94C2×#95C2314yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2318yes—
98C2×#95C2318yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C43D418yes—
103C43Q8116yescomplemented
104C22×Q8—24no—
105C22×Q8—28no—
106C22×Q8—44no—
107C22×Q8—44no—
108C23×C4C2414yes—
109C23×C4C2414yes—
110C23×C4C2414yes—
111C23×C4C2414yes—
112C23×C4C2412yes—
113C23×C4C2414yes—
114C23×C4C2411yescharacteristic
115C23×C4C2411yescharacteristic
116C23×C4C2411yescharacteristic
117C23×C4C2412yes—
118C23×C4C2414yes—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—216no—
129C2×C4⋊C4—216no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—48no—
137C2×C4⋊C4—48no—
138C2×C4⋊C4—48no—
139C2×C42—28no—
140C2×C42—216no—
141C2×C42—42no—
142C2×C42—44no—
143C2×C42—41no—
144C2×C42—41no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—48no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42C2×D418yes—
151C2×C42C2×D414yes—
152C2×C42C2×D418yes—
153C2×C42C2×D414yes—
154C2×C42C2×D414yes—
155C2×C42C2×Q8116yes—
156C2×C42C2×Q818yes—
157C2×C42C2×Q8116yes—
158C2×C42C2×Q818yes—
159C2×C42C2×Q818yes—
160#95—28no—
161#95—28no—
162#95—28no—
163#95—28no—
164#95—28no—
165#95—28no—
166#95—28no—
167#95—28no—
168#95—28no—
169#95—28no—
170#95—216no—
171#95—216no—
172#95—216no—
173#95—44no—
174#95—44no—
175#95—48no—
176#95—48no—
177#95—44no—
178#95—44no—
179#95#41116yes—
180C24C2511yescharacteristic
181C2×Q8—48no—
182C2×Q8—48no—
183C2×Q8—48no—
184C2×Q8—816no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—28no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—28no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—28no—
220C22×C4—28no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4C22×D414yes—
226C22×C4C22×D412yes—
227C22×C4C22×D412yes—
228C22×C4C22×D412yes—
229C22×C4C22×D414yes—
230C22×C4C22×Q818yes—
231C22×C4C22×Q814yes—
232C22×C4C22×Q814yes—
233C22×C4C22×Q814yes—
234C22×C4C22×Q818yes—
235C22×C4C2×#4118yes—
236C22×C4C2×#41116yes—
237C4⋊C4—48no—
238C4⋊C4—48no—
239C4⋊C4—416no—
240C4⋊C4—416no—
241C4⋊C4—416no—
242C4⋊C4—48no—
243C4⋊C4—816no—
244C4⋊C4—816no—
245C42—216no—
246C42—232no—
247C42—44no—
248C42—48no—
249C42—44no—
250C42—48no—
251C42—416no—
252C42—48no—
253C42C4⋊Q8116yes—
254C23C23×D411yescharacteristic
255C23C23×Q812yes—
256C23C22×#4114yes—
257C23#58512yes—
258C23#58511yescharacteristic
259C23#58514yes—
260C23#58511yescharacteristic
261Q8—816no—
262C2×C4—24no—
263C2×C4—28no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—24no—
272C2×C4—28no—
273C2×C4—24no—
274C2×C4—216no—
275C2×C4—216no—
276C2×C4—216no—
277C2×C4—24no—
278C2×C4—28no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—48no—
306C2×C4—44no—
307C2×C4—48no—
308C2×C4—44no—
309C2×C4C2×C4⋊Q814yes—
310C2×C4C2×C4⋊Q814yes—
311C2×C4C2×C4⋊Q814yes—
312C2×C4#53218yes—
313C2×C4#533116yes—
314C22C22×C4⋊Q811yescharacteristic
315C22C2×#53212yes—
316C22C2×#53314yes—
317C22#344911yescharacteristic
318C22#344912yes—
319C22#344911yescharacteristic
320C22#345812yes—
321C22#345814yes—
322C22#345812yes—
323C22#352118yes—
324C22#352114yes—
325C22#352414yes—
326C4—24no—
327C4—28no—
328C4—216no—
329C4—24no—
330C4—24no—
331C4—44no—
332C4—48no—
333C4—84no—
334C4—84no—
335C4—84no—
336C4—84no—
337C2#3464211yescharacteristic
338C2#3464211yescharacteristic
339C2#3514512yes—
340C2#3514614yes—
341C2#3521212yes—
342C2#3521314yes—
343C2#3664511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
103. C43Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28632
#152562
#152532
#152512
#151732
#149032
#149012
#146882
#143282
#143272
C4×#3762