Finite Group Databasebeta

← #373638

Subgroups — #373638

Order 512.

2378 conjugacy classes of subgroups; only the 346 normal subgroups are shown.

normal subgroups only

This group has 2378 conjugacy classes of subgroups — too many to draw. Only its 346 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#34172C212yesmaximal
2#20889C214yesmaximal, complemented
3C2×#1782C211yesmaximal, characteristic
4#11070C214yesmaximal, complemented
5#10754C214yesmaximal, complemented
6C23×#41C2211yescharacteristic
7C2×#518C2212yes—
8C2×#518C2212yes—
9C22×#99C2214yes—
10#2339—216no—
11#2315C2214yes—
12#2310—22no—
13#2308—22no—
14#2308C2214yes—
15#2280—22no—
16C2×#381—22no—
17C2×#381C2214yes—
18C2×#381C2212yes—
19#1867—216no—
20#1782C418yescomplemented
21C2×#342C2218yes—
22C2×#342C2214yes—
23C22×#41—28no—
24C22×#41—28no—
25C22×#41—42no—
26C22×#41—42no—
27C22×#41C2×C414yes—
28C22×#41Q814yes—
29C23×Q8D411yescharacteristic
30C23×D4D411yescharacteristic
31C23×D4C2312yes—
32C24×C4D412yes—
33C24×C4C2311yescharacteristic
34#518—28no—
35#518—28no—
36#518—44no—
37#518—44no—
38C2×C4×D4—28no—
39C2×C4×D4—28no—
40C2×#117—22no—
41C2×#117—22no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4C2314yes—
45C22×C22⋊C4C2314yes—
46C22×C22⋊C4C2314yes—
47C2×#99—28no—
48C2×#99—28no—
49C2×#99C2×C4116yescomplemented
50#391—28no—
51#391—28no—
52#386—28no—
53#386—28no—
54#386—28no—
55#386D4116yes—
56#385—216no—
57#381—28no—
58#381—24no—
59#381—24no—
60#381—24no—
61#381—28no—
62#381—42no—
63#381C2×C418yescomplemented
64#380—28no—
65#380—44no—
66#379—24no—
67C4×C22⋊C4—28no—
68C2×#95—28no—
69C2×#95—28no—
70C2×#95—42no—
71#342—216no—
72#342—44no—
73C25C2×D412yes—
74C25C2411yescharacteristic
75C2×#41—232no—
76C2×#41—232no—
77C2×#41—48no—
78C2×#41—48no—
79C2×#41—416no—
80C2×#41—432no—
81C2×#41—416no—
82C22×Q8—24no—
83C22×Q8—24no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8C4⋊C414yescomplemented
87C22×D4—24no—
88C22×D4—24no—
89C22×D4—24no—
90C22×D4—24no—
91C22×D4—24no—
92C22×D4—24no—
93C22×D4—41no—
94C22×D4—41no—
95C22×D4C4⋊C414yescomplemented
96C22×D4C22×C414yes—
97C22×D4C2×D418yes—
98C22×D4C2×Q814yes—
99C22×D4#4118yes—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4C4⋊C418yescomplemented
115C23×C4C22×C412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D418yes—
120C23×C4C2×Q812yes—
121C4×D4—432no—
122C4×D4—432no—
123#117—44no—
124#117—44no—
125#117—44no—
126#117—44no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—48no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—216no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4C22×C418yes—
148C2×C22⋊C4C2×D418yes—
149C2×C42—24no—
150C2×C42—24no—
151#99—232no—
152#95—216no—
153#95—216no—
154#95—216no—
155#95—48no—
156#95—48no—
157C24—22no—
158C24—22no—
159C24—22no—
160C24—22no—
161C24—82no—
162C24—84no—
163C24C2×C4⋊C414yes—
164C24#12014yes—
165C24#12314yes—
166C24C23×C411yescharacteristic
167C24C22×D411yescharacteristic
168C24C22×D414yes—
169C24C22×Q811yescharacteristic
170C24C2×#4114yes—
171C24H(5,𝔽2)12yes—
172C24H(5,𝔽2)12yes—
173#41—464no—
174#41—464no—
175C2×Q8—216no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—48no—
179C2×Q8—44no—
180C2×D4—28no—
181C2×D4—28no—
182C2×D4—28no—
183C2×D4—28no—
184C2×D4—216no—
185C2×D4—216no—
186C2×D4—216no—
187C2×D4—216no—
188C2×D4—216no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—48no—
192C2×D4—44no—
193C2×D4C4×D4116yes—
194C2×D4C22⋊Q8116yes—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—28no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—28no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—216no—
212C22×C4—216no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—216no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—48no—
221C22×C4—48no—
222C22×C4—44no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—48no—
228C22×C4—48no—
229C22×C4—44no—
230C22×C4—88no—
231C22×C4—88no—
232C22×C4—88no—
233C22×C4—88no—
234C22×C4—88no—
235C22×C4C2×C4⋊C412yes—
236C22×C4C2×C4⋊C412yes—
237C22×C4C2×C4⋊C414yes—
238C22×C4#12014yes—
239C22×C4#12314yes—
240C4⋊C4—48no—
241C4⋊C4—48no—
242C4⋊C4—168no—
243C22⋊C4—216no—
244C22⋊C4—216no—
245C22⋊C4—48no—
246C22⋊C4—48no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—48no—
252C22⋊C4—48no—
253C22⋊C4—816no—
254C42—48no—
255C42—48no—
256C23—24no—
257C23—24no—
258C23—24no—
259C23—24no—
260C23—24no—
261C23—24no—
262C23—48no—
263C23—44no—
264C23—42no—
265C23—48no—
266C23—48no—
267C23—84no—
268C23—81no—
269C23—81no—
270C23—84no—
271C23—82no—
272C23—84no—
273C23—84no—
274C23—84no—
275C23—84no—
276C23#38518yes—
277C23C22×C4⋊C411yescharacteristic
278C23C2×C4×D414yes—
279C23#51812yes—
280C23C2×#12012yes—
281C23C2×C22⋊Q814yes—
282C23C2×#12312yes—
283C23#53411yescharacteristic
284C23#53411yescharacteristic
285C23#54312yes—
286C23D4214yes—
287C23#54618yes—
288C23#55114yes—
289Q8—416no—
290D4—232no—
291D4—232no—
292D4—416no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—432no—
304C2×C4—48no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—48no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—416no—
313C2×C4—416no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—88no—
318C2×C4—816no—
319C2×C4—88no—
320C2×C4—168no—
321C2×C4—168no—
322C2×C4—164no—
323C2×C4—168no—
324C2×C4#38518yes—
325C22—24no—
326C22—28no—
327C22—22no—
328C22—28no—
329C22—42no—
330C22—48no—
331C22—44no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—48no—
339C22—48no—
340C22—48no—
341C22—48no—
342C22—84no—
343C22—84no—
344C22—84no—
345C22C2×#38512yes—
346C22#231511yescharacteristic
347C22#235814yes—
348C22#240311yescharacteristic
349C22#240512yes—
350C22#243714yes—
351C22#248311yescharacteristic
352C4—28no—
353C4—42no—
354C4—42no—
355C4—88no—
356C4—88no—
357C4—88no—
358C2—22no—
359C2—24no—
360C2—22no—
361C2—21no—
362C2—21no—
363C2—24no—
364C2—48no—
365C2#1412711yescharacteristic
366C2#2389612yes—
subgroupG/H
center, socleC22#2403
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20889C2semidirect
4. #11070C2semidirect
5. #10754C2semidirect
20. #1782C4semidirect
49. C2×#99C2×C4semidirect
63. #381C2×C4semidirect
86. C22×Q8C4⋊C4semidirect
95. C22×D4C4⋊C4semidirect
114. C23×C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341722
#208892
C2×#17822
#110702
#107542