Finite Group Databasebeta

← #415316

Subgroups — #415316

Order 512.

1816 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1816 conjugacy classes of subgroups — too many to draw. Only its 314 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#35011C211yesmaximal, characteristic
2C2×#2502C212yesmaximal
3C2×#1996C212yesmaximal
4#13924C211yesmaximal, characteristic
5#11344C218yesmaximal
6C4×#342C211yesmaximal, characteristic
7C22×C22⋊Q8C2212yes—
8C23×C4⋊C4C2211yescharacteristic
9#2857—22no—
10#2659—22no—
11#2583—24no—
12#2502—28no—
13#2502—22no—
14#2502—28no—
15#2434—22no—
16C2×#398—24no—
17#2315C2214yes—
18C2×#391—22no—
19C2×#385C2218yes—
20C2×#384—22no—
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1996—28no—
25C2×#342C2218yes—
26C2×#342C2214yes—
27#1445—22no—
28C4⋊#95—22no—
29C22⋊#95C2214yes—
30C23×Q8D412yes—
31C24×C4C2312yes—
32C24×C4C2311yescharacteristic
33C2×C4⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—28no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D414yes—
51C22×C4⋊C4D414yes—
52C22×C4⋊C4D412yes—
53C22×C22⋊C4C2312yes—
54C22×C22⋊C4C2314yes—
55C22×C22⋊C4C2312yes—
56C22×C22⋊C4C2314yes—
57C22×C42—22no—
58C22×C42D412yes—
59#398—28no—
60#398—232no—
61#395—28no—
62#391—24no—
63#391—216no—
64#389—28no—
65#385—28no—
66#385—28no—
67#385—216no—
68#385—216no—
69#384—22no—
70#384—216no—
71#380—28no—
72#380—24no—
73C4×C22⋊C4—22no—
74C4×C22⋊C4—22no—
75C4×C22⋊C4—24no—
76C4×C22⋊C4D418yes—
77C2×#95—22no—
78C2×#95—28no—
79C2×#95—21no—
80C2×#95—28no—
81C2×#95D412yes—
82#342—216no—
83#342Q8116yescomplemented
84C25C2411yescharacteristic
85C22×Q8—22no—
86C22×Q8—24no—
87C22×Q8—22no—
88C22×Q8—24no—
89C22×Q8—48no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—44no—
97C23×C4—84no—
98C23×C4C2×D412yes—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D414yes—
101C23×C4C2×D414yes—
102C23×C4C2×D414yes—
103C23×C4C2×D414yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D414yes—
108C23×C4#4112yes—
109C4⋊Q8—84no—
110C22⋊Q8—44no—
111C22⋊Q8—44no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C22⋊Q8—44no—
116C22⋊Q8—44no—
117C22⋊Q8—44no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—232no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—816no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4C2×D418yes—
156C2×C22⋊C4C2×Q818yes—
157C2×C22⋊C4C2×Q8116yes—
158C2×C42—22no—
159C2×C42—28no—
160C2×C42—22no—
161C2×C42—48no—
162C2×C42—44no—
163#95—232no—
164#95—216no—
165#95—28no—
166#95—24no—
167#95—42no—
168#95—88no—
169C24—42no—
170C24—42no—
171C24C22×D412yes—
172C24C22×D411yescharacteristic
173C24C22×D411yescharacteristic
174C24C22×D412yes—
175C24C22×Q814yes—
176C24C2×#4111yescharacteristic
177C242−1+414yes—
178C2×Q8—416no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×Q8—48no—
182C2×Q8—84no—
183C2×Q8—88no—
184C22×C4—24no—
185C22×C4—28no—
186C22×C4—28no—
187C22×C4—24no—
188C22×C4—216no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—28no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—48no—
204C22×C4—48no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—48no—
210C22×C4—416no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—84no—
219C22×C4—88no—
220C22×C4—84no—
221C22×C4#12014yes—
222C22×C4#12012yes—
223C22×C4#12011yescharacteristic
224C22×C4#12014yes—
225C22×C4#12011yescharacteristic
226C22×C4#12112yes—
227C22×C4#12114yes—
228C22×C4#12114yes—
229C22×C4#12112yes—
230C22×C4#12712yes—
231C22×C4#12712yes—
232C22×C4C4⋊Q818yes—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—84no—
241C4⋊C4—88no—
242C4⋊C4—832no—
243C4⋊C4—88no—
244C22⋊C4—28no—
245C22⋊C4—216no—
246C22⋊C4—216no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—48no—
252C22⋊C4C4⋊Q8116yescomplemented
253C42—416no—
254C42—44no—
255C42—82no—
256C23—24no—
257C23—22no—
258C23—24no—
259C23—22no—
260C23—41no—
261C23—42no—
262C23—41no—
263C23—41no—
264C23—42no—
265C23—41no—
266C23—48no—
267C23—48no—
268C23C2×#12012yes—
269C23C2×#12011yescharacteristic
270C23C2×#12111yescharacteristic
271C23C2×#12112yes—
272C23C2×#12711yescharacteristic
273C23C2×C4⋊Q812yes—
274C23C2×C4⋊Q814yes—
275C23#53612yes—
276C23#53614yes—
277C23D4×Q818yes—
278C23D4×Q814yes—
279C23#55714yes—
280Q8—88no—
281Q8—88no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—216no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—432no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—48no—
300C2×C4—48no—
301C2×C4—416no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—84no—
310C2×C4—88no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—88no—
318C2×C4—82no—
319C2×C4#39212yes—
320C2×C4#39214yes—
321C2×C4#39212yes—
322C2×C4#45814yes—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—21no—
327C22—22no—
328C22—21no—
329C22—22no—
330C22—21no—
331C22—24no—
332C22—216no—
333C22—24no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22—44no—
338C22C2×#39211yescharacteristic
339C22#243414yes—
340C22#243412yes—
341C22#254612yes—
342C22#254614yes—
343C22#265811yescharacteristic
344C22#286811yescharacteristic
345C22C2×#45812yes—
346C22#302712yes—
347C4—22no—
348C4—24no—
349C4—416no—
350C4—48no—
351C4—84no—
352C4—84no—
353C4—84no—
354C4—162no—
355C4#202214yes—
356C2—24no—
357C2—21no—
358C2—21no—
359C2—24no—
360C2—22no—
361C2#1519011yescharacteristic
362C2C2×#202211yescharacteristic
363C2#2344511yescharacteristic
364C2#2585314yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
83. #342Q8semidirect
252. C22⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#350112
C2×#25022
C2×#19962
#139242
#113442
C4×#3422