Finite Group Databasebeta

← #379728

Subgroups — #379728

Order 512.

2253 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 2253 conjugacy classes of subgroups — too many to draw. Only its 298 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
1C2×#2614C211yesmaximal, characteristic
2C2×#2310C211yesmaximal, characteristic, complemented
3C2×#1899C212yesmaximal
4#14336C211yesmaximal, characteristic, complemented
5#11197C214yesmaximal, complemented
6#10966C214yesmaximal
7#10736C212yesmaximal, complemented
8C24×Q8C2211yescharacteristic
9C22×#123C2211yescharacteristic
10C22×#123C2212yes—
11C22×#117C2212yes—
12C22×#99C2214yes—
13#2614—24no—
14#2614—24no—
15C2×#395C414yescomplemented
16C2×#393C414yescomplemented
17#2310—28no—
18C2×#391—24no—
19C2×#381C2212yes—
20C2×#381C2214yes—
21C2×#381C2214yes—
22C2×#381C2212yes—
23C2×#381C2211yescharacteristic
24#1899—28no—
25C2×#342C2214yes—
26C2×#342C2218yes—
27#1446—24no—
28C23×Q8—24no—
29C23×Q8—24no—
30C23×Q8—48no—
31C23×Q8C2×C414yes—
32C23×Q8D414yes—
33C23×Q8D414yes—
34C23×D4C2311yescharacteristic
35C24×C4C2312yes—
36C24×C4C2311yescharacteristic
37C2×#123—24no—
38C2×#123—24no—
39C2×#123—22no—
40C2×#123—22no—
41C2×#123—24no—
42C2×#123—24no—
43C2×#123—48no—
44C2×#117—24no—
45C2×#117—48no—
46C2×#117D418yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C414yes—
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C414yes—
52C22×C22⋊C4C2314yes—
53C22×C22⋊C4C2311yescharacteristic
54C22×C22⋊C4C2312yes—
55C22×C22⋊C4C2314yes—
56C22×C42—22no—
57C2×#99—28no—
58C2×#99—48no—
59#395—216no—
60#393—216no—
61#391—232no—
62#381—24no—
63#381—24no—
64#381—22no—
65#381—22no—
66#381—216no—
67#381—24no—
68#381—24no—
69#381—48no—
70C2×#95—28no—
71C2×#95—28no—
72C2×#95—24no—
73C2×#95C2×C418yes—
74C2×#95C2×C414yes—
75#342—28no—
76#342—48no—
77C25C22×C411yescharacteristic
78C25C2411yescharacteristic
79C22×Q8—216no—
80C22×Q8—28no—
81C22×Q8—216no—
82C22×Q8—28no—
83C22×Q8—416no—
84C22×Q8—48no—
85C22×Q8—48no—
86C22×Q8—416no—
87C22×Q8—416no—
88C22×Q8—416no—
89C22×Q8—432no—
90C22×Q8C22⋊C4116yes—
91C22×D4—21no—
92C22×D4—21no—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4—28no—
96C23×C4—28no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—28no—
100C23×C4—28no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C414yes—
105C23×C4C22×C412yes—
106C23×C4C22×C412yes—
107C23×C4C22×C414yes—
108C23×C4C2×D414yes—
109C23×C4C2×D414yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D414yes—
113C23×C4#4114yes—
114#123—416no—
115#123—416no—
116#123—44no—
117#123—48no—
118#123—44no—
119#117—416no—
120#117—416no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—48no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—48no—
145C2×C22⋊C4C2×D418yes—
146C2×C42—28no—
147C2×C42—48no—
148#99—416no—
149#95—232no—
150#95—216no—
151#95—28no—
152#95—416no—
153#95—48no—
154C24—21no—
155C24—22no—
156C24—24no—
157C24—44no—
158C24—44no—
159C24C23×C411yescharacteristic
160C24C22×D411yescharacteristic
161C24C22×D411yescharacteristic
162C24C22×D412yes—
163C24C2×#4112yes—
164C24H(5,𝔽2)18yes—
165C2×Q8—232no—
166C2×Q8—432no—
167C2×Q8—464no—
168C2×Q8—416no—
169C2×Q8—416no—
170C2×Q8—464no—
171C2×Q8—432no—
172C2×D4—41no—
173C2×D4—42no—
174C2×D4—41no—
175C2×D4—44no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—44no—
179C2×D4—48no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—28no—
184C22×C4—28no—
185C22×C4—28no—
186C22×C4—216no—
187C22×C4—28no—
188C22×C4—28no—
189C22×C4—28no—
190C22×C4—24no—
191C22×C4—28no—
192C22×C4—24no—
193C22×C4—28no—
194C22×C4—28no—
195C22×C4—28no—
196C22×C4—28no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—28no—
203C22×C4—24no—
204C22×C4—28no—
205C22×C4—48no—
206C22×C4—48no—
207C22×C4—416no—
208C22×C4—416no—
209C22×C4—48no—
210C22×C4—48no—
211C22×C4—44no—
212C22×C4—416no—
213C22×C4—44no—
214C22×C4—48no—
215C22×C4C2×C22⋊C418yes—
216C22×C4C2×C22⋊C414yes—
217C22×C4C4×D418yes—
218C22×C4#12014yes—
219C22×C4#12118yes—
220C22×C4#12314yes—
221C4⋊C4—416no—
222C4⋊C4—416no—
223C4⋊C4—48no—
224C22⋊C4—28no—
225C22⋊C4—28no—
226C22⋊C4—416no—
227C22⋊C4—41no—
228C22⋊C4—42no—
229C22⋊C4—41no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—48no—
233C22⋊C4—48no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—48no—
237C42—416no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—44no—
242C23—44no—
243C23—416no—
244C23—42no—
245C23—44no—
246C23—41no—
247C23—44no—
248C23—88no—
249C23—84no—
250C23C22×C22⋊C411yescharacteristic
251C23C2×C4×D412yes—
252C23#51814yes—
253C23C2×#12011yescharacteristic
254C23C2×#12112yes—
255C23C2×#12311yescharacteristic
256C23#53414yes—
257C23#53414yes—
258C23#53414yes—
259C23#53514yes—
260C23#53818yes—
261Q8—464no—
262D4—82no—
263D4—84no—
264D4—82no—
265C2×C4—28no—
266C2×C4—216no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—216no—
270C2×C4—28no—
271C2×C4—28no—
272C2×C4—28no—
273C2×C4—28no—
274C2×C4—416no—
275C2×C4—48no—
276C2×C4—416no—
277C2×C4—48no—
278C2×C4—416no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—48no—
283C2×C4—48no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—816no—
288C2×C4—88no—
289C2×C4—84no—
290C2×C4—816no—
291C2×C4—88no—
292C2×C4—84no—
293C2×C4#38618yes—
294C22—28no—
295C22—28no—
296C22—28no—
297C22—22no—
298C22—24no—
299C22—24no—
300C22—22no—
301C22—44no—
302C22—44no—
303C22—48no—
304C22—81no—
305C22—84no—
306C22—82no—
307C22—164no—
308C22C2×#38611yescharacteristic
309C22#230812yes—
310C22#232114yes—
311C22#261312yes—
312C22#261714yes—
313C22#262212yes—
314C22#288414yes—
315C4—28no—
316C4—48no—
317C4—88no—
318C4—81no—
319C4—84no—
320C4—82no—
321C4—164no—
322C2—22no—
323C2—24no—
324C2—24no—
325C2—22no—
326C2—161no—
327C2#1425511yescharacteristic
328C2#2098612yes—
329C2#2175114yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×Q8 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2310C2semidirect
4. #14336C2semidirect
5. #11197C2semidirect
7. #10736C2semidirect
15. C2×#395C4semidirect
16. C2×#393C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26142
C2×#23102
C2×#18992
#143362
#111972
#109662
#107362