Finite Group Databasebeta

← #373639

Subgroups — #373639

Order 512.

2196 conjugacy classes of subgroups; only the 344 normal subgroups are shown.

normal subgroups only

This group has 2196 conjugacy classes of subgroups — too many to draw. Only its 344 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60064C212yesmaximal
2#20892C214yesmaximal, complemented
3#20584C211yesmaximal, characteristic
4#11071C214yesmaximal, complemented
5#10757C214yesmaximal, complemented
6C23×#41C2211yescharacteristic
7C2×#568C2212yes—
8C2×#568C2212yes—
9#2905C2214yescomplemented
10C2×#410C2214yes—
11#2880—216no—
12#2879—24no—
13#2879C2214yes—
14C2×#407—24no—
15C2×#407C2214yes—
16#2280C2212yes—
17#1868—216no—
18#1782C414yes—
19#1781C414yes—
20C2×#344C2218yescomplemented
21#1742C2214yescomplemented
22C22×#41—28no—
23C22×#41—28no—
24C22×#41—42no—
25C22×#41—42no—
26C22×#41C2×C412yes—
27C22×#41D412yes—
28C22×#41Q812yes—
29C22×#41C2312yes—
30C23×Q8Q811yescharacteristic
31C23×D4C2×C412yes—
32C23×D4Q811yescharacteristic
33C24×C4D412yes—
34C24×C4C2311yescharacteristic
35#568—28no—
36#568—28no—
37#568—44no—
38#568—44no—
39C2×#131—28no—
40C2×#131—28no—
41C22×M16—24no—
42C22×M16—24no—
43C22×M16C2314yes—
44C22×M16C2314yes—
45C2×#117C2314yes—
46#424—216no—
47#423—44no—
48#410—28no—
49#410—28no—
50C2×#99C2×C4116yes—
51#408—216no—
52#408—216no—
53#408—28no—
54#408D4116yes—
55#407—216no—
56#407—24no—
57#407—28no—
58#407—24no—
59#407—28no—
60C2×C22⋊C8—216no—
61C2×C22⋊C8—28no—
62#381C2×C414yes—
63#380—42no—
64#379C2×C414yes—
65C4×C22⋊C4—42no—
66#344—216no—
67#343—44no—
68C25C4⋊C412yes—
69C25C22×C411yescharacteristic
70C2×#41—28no—
71C2×#41—216no—
72C2×#41—28no—
73C2×#41—28no—
74C2×#41—28no—
75C2×#41—24no—
76C2×#41—24no—
77C2×#41—24no—
78C2×#41—24no—
79C2×#41—42no—
80C2×#41—42no—
81C2×#41—44no—
82C2×#41—44no—
83C2×#41—416no—
84C2×#41—432no—
85C2×#41—416no—
86C2×#41C2×D418yes—
87C2×#41#4118yes—
88C22×Q8—28no—
89C22×Q8—42no—
90C22×Q8C4⋊C412yes—
91C22×Q8C2×Q812yes—
92C22×D4—216no—
93C22×D4—28no—
94C22×D4—42no—
95C22×D4—44no—
96C22×D4C4⋊C414yes—
97C22×D4C4⋊C412yes—
98C22×D4C22×C414yes—
99C22×D4C2×Q812yes—
100C23×C4—28no—
101C23×C4—28no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4C4⋊C412yes—
109C23×C4C4⋊C414yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D414yes—
114C23×C4C2×Q811yescharacteristic
115C23×C4C2×Q811yescharacteristic
116C23×C4C2411yescharacteristic
117#131—432no—
118#131—432no—
119C2×M16—28no—
120C2×M16—28no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—28no—
125C2×M16—28no—
126C2×M16—28no—
127C2×M16—28no—
128C2×M16—48no—
129C2×M16—48no—
130C2×M16—48no—
131C2×M16—48no—
132C2×M16C2×D418yes—
133C22×C8—28no—
134C22×C8—28no—
135C22×C8—24no—
136C22×C8C2×D418yes—
137#117—28no—
138#117—28no—
139C2×C4⋊C4—24no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4C22×C418yes—
142C2×C42—24no—
143C4⋊C8—416no—
144#99—232no—
145C22⋊C8—216no—
146C22⋊C8—216no—
147C22⋊C8—232no—
148#95—44no—
149C24—24no—
150C24—22no—
151C24—28no—
152C24—42no—
153C24—82no—
154C24—84no—
155C24C2×C4⋊C411yescharacteristic
156C24C2×C4⋊C411yescharacteristic
157C24C2×C4⋊C414yes—
158C24C23×C411yescharacteristic
159#41—216no—
160#41—216no—
161#41—216no—
162#41—216no—
163#41—48no—
164#41—416no—
165#41—48no—
166#41—432no—
167#41—432no—
168C2×Q8—24no—
169C2×Q8—24no—
170C2×Q8—28no—
171C2×Q8—42no—
172C2×Q8—416no—
173C2×Q8C22⋊Q818yes—
174C2×D4—28no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—28no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—28no—
181C2×D4—44no—
182C2×D4—42no—
183C2×D4—432no—
184C2×D4—416no—
185C2×D4C4×D4116yes—
186C2×D4C22⋊Q818yes—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—28no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—48no—
212C22×C4—416no—
213C22×C4—48no—
214C22×C4—44no—
215C22×C4—48no—
216C22×C4—44no—
217C22×C4C2×C4⋊C411yescharacteristic
218C22×C4C2×C4⋊C411yescharacteristic
219C22×C4C2×C4⋊C414yes—
220C22×C4#12018yes—
221C22×C4#12318yes—
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D414yes—
224C22×C4C22×Q811yescharacteristic
225C22×C4C2×#4114yes—
226C22×C4H(5,𝔽2)12yes—
227C22×C4H(5,𝔽2)12yes—
228M16—216no—
229M16—416no—
230M16—416no—
231M16—44no—
232M16—44no—
233M16—44no—
234M16—44no—
235M16—48no—
236C2×C8—28no—
237C2×C8—28no—
238C2×C8—216no—
239C2×C8—416no—
240C2×C8—416no—
241C2×C8—48no—
242C4⋊C4—48no—
243C22⋊C4—216no—
244C22⋊C4—88no—
245C42—44no—
246C42—44no—
247C23—21no—
248C23—22no—
249C23—24no—
250C23—24no—
251C23—21no—
252C23—22no—
253C23—42no—
254C23—48no—
255C23—44no—
256C23—48no—
257C23—416no—
258C23—44no—
259C23—84no—
260C23—81no—
261C23—81no—
262C23—84no—
263C23—82no—
264C23—84no—
265C23—88no—
266C23—84no—
267C23#38518yes—
268C23C22×C4⋊C411yescharacteristic
269C23C2×C4×D414yes—
270C23#51812yes—
271Q8—216no—
272Q8—48no—
273D4—232no—
274D4—216no—
275D4—416no—
276D4—48no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—28no—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4—416no—
289C2×C4—44no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—48no—
299C2×C4—88no—
300C2×C4#38518yes—
301C2×C4C2×#12012yes—
302C2×C4C2×C22⋊Q814yes—
303C2×C4C2×#12312yes—
304C2×C4#53411yescharacteristic
305C2×C4#53411yescharacteristic
306C2×C4#54312yes—
307C2×C4D4214yes—
308C2×C4#54618yes—
309C2×C4#55114yes—
310C8—48no—
311C8—48no—
312C22—24no—
313C22—22no—
314C22—28no—
315C22—42no—
316C22—41no—
317C22—44no—
318C22—48no—
319C22—48no—
320C22—42no—
321C22—41no—
322C22—44no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—84no—
327C22—88no—
328C22—816no—
329C22C2×#38512yes—
330C22#231511yescharacteristic
331C22#235814yes—
332C4—28no—
333C4—41no—
334C4—41no—
335C4—88no—
336C4#240311yescharacteristic
337C4#240512yes—
338C4#243714yes—
339C4#248311yescharacteristic
340C2—24no—
341C2—22no—
342C2—21no—
343C2—42no—
344C2—48no—
345C2—81no—
346C2#1412711yescharacteristic
subgroupG/H
centerC4#2403
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC2#14127
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20892C2semidirect
4. #11071C2semidirect
5. #10757C2semidirect
9. #2905C22semidirect
20. C2×#344C22semidirect
21. #1742C22semidirect

Maximal subgroups

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

subgroupindex
#600642
#208922
#205842
#110712
#107572