Finite Group Databasebeta

← #215169

Subgroups — #215169

Order 512.

1044 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2592C211yesmaximal, characteristic
2#21914C212yesmaximal, complemented
3#9652C214yesmaximal, complemented
4#9651C214yesmaximal, complemented
5#8305C212yesmaximal
6#8303C212yesmaximal, complemented
7C22×#124C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2592—24no—
10#2592—24no—
11C2×#397C412yes—
12C2×#393C412yes—
13C2×#391C412yes—
14C2×#390C412yes—
15C2×#388C2214yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17#1917—24no—
18#1916—24no—
19#1817C2214yes—
20#1797C2214yes—
21#1749C2212yes—
22C4×C22⋊C8C2212yes—
23C2×#324C2212yes—
24C2×#324C2214yes—
25C2×#324C2212yes—
26C2×#323C2212yes—
27C2×#323C2214yes—
28C2×#323C2212yes—
29C23×Q8C2×C411yescharacteristic
30C23×Q8C2×C411yescharacteristic
31C23×D4C2×C411yescharacteristic
32C2×#124—24no—
33C2×#124—42no—
34C2×#124—42no—
35C2×C4×Q8—24no—
36C2×C4×Q8—42no—
37C2×C4×Q8—42no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2312yes—
41C22×C22⋊C4C2×C414yes—
42C22×C22⋊C4C2×C412yes—
43C22×C22⋊C4C2×C414yes—
44C22×C22⋊C4C2312yes—
45C22×C42C2×C412yes—
46C22×C42C2311yescharacteristic
47C22×C42C2312yes—
48C2×C4⋊C8—24no—
49C2×C22⋊C8C2314yes—
50C2×C22⋊C8C2314yes—
51C2×#97—22no—
52C2×C4×C8—22no—
53#397—22no—
54#397—24no—
55#397—22no—
56#393—22no—
57#393—24no—
58#393—22no—
59#391—24no—
60#391—22no—
61#391—22no—
62#390—24no—
63#390—22no—
64#390—22no—
65#388—24no—
66#388—24no—
67#388—24no—
68#388—24no—
69C4×C4⋊C4D414yes—
70C4×C4⋊C4D414yes—
71C2×#95C2×C414yes—
72C2×#95C2×C414yes—
73C2×#95C2×C412yes—
74#336—24no—
75#336—28no—
76#336—22no—
77#336—22no—
78#324—42no—
79#324—44no—
80#324—42no—
81#323—42no—
82#323—44no—
83#323—42no—
84C25C22×C412yes—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—24no—
88C22×Q8—41no—
89C22×Q8—41no—
90C22×Q8—42no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8—44no—
94C22×D4—22no—
95C22×D4—41no—
96C22×D4—41no—
97C22×D4—44no—
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C412yes—
101C23×C4C22×C412yes—
102C23×C4C22×C412yes—
103C23×C4C22×C414yes—
104C23×C4C2411yescharacteristic
105C22×C8—24no—
106C22×C8—24no—
107#124—416no—
108C4×Q8—416no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C22⋊C414yes—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—44no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—84no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42—82no—
142C2×C42C2×D414yes—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42C2×D414yes—
146C4⋊C8—84no—
147C22⋊C8—24no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C22⋊C8—24no—
151#97—82no—
152C4×C8—82no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95C22⋊C412yes—
158#95C22⋊C414yes—
159#95C22⋊C412yes—
160C24—22no—
161C24—22no—
162C24—22no—
163C24—24no—
164C24—24no—
165C24—42no—
166C24—44no—
167C24—42no—
168C24C23×C411yescharacteristic
169C2×Q8—48no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—48no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—84no—
178C2×Q8—84no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—84no—
183C2×D4—84no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4C2×C22⋊C414yes—
213C22×C4C2×C22⋊C414yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4#13114yes—
217C22×C4C22×D412yes—
218C22×C4H(5,𝔽2)12yes—
219C22×C4H(5,𝔽2)12yes—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—44no—
226C4⋊C4—28no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C22⋊C4—48no—
233C22⋊C4—48no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C22⋊C4—84no—
239C22⋊C4—84no—
240C42—28no—
241C42—42no—
242C42—42no—
243C42—44no—
244C42—44no—
245C42—44no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—42no—
250C23—44no—
251C23—42no—
252C23—42no—
253C23—44no—
254C23—44no—
255C23—42no—
256C23—42no—
257C23—82no—
258C23—82no—
259C23—84no—
260C23—84no—
261C23—84no—
262C23—84no—
263C23—82no—
264C23—82no—
265C23C22×C22⋊C411yescharacteristic
266C23#51811yescharacteristic
267C23#51811yescharacteristic
268C23C2×#13112yes—
269C23#56812yes—
270Q8—416no—
271Q8—88no—
272D4—88no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—84no—
305C2×C4—84no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4#41314yes—
311C2×C4#53514yes—
312C8—84no—
313C8—84no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—84no—
323C22—84no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—164no—
328C22—162no—
329C22—162no—
330C22#230911yescharacteristic
331C22#288012yes—
332C22#288311yescharacteristic
333C22#288411yescharacteristic
334C22C2×#41312yes—
335C22#297612yes—
336C22#297711yescharacteristic
337C22#297811yescharacteristic
338C4—24no—
339C4—24no—
340C4—44no—
341C4—44no—
342C4—42no—
343C4—44no—
344C4—82no—
345C4—82no—
346C2—22no—
347C2—21no—
348C2—21no—
349C2—162no—
350C2#1742812yes—
351C2#1742912yes—
352C2#2179211yescharacteristic
353C2#2179611yescharacteristic
354C2#2179811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21914C2semidirect
3. #9652C2semidirect
4. #9651C2semidirect
6. #8303C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25922
#219142
#96522
#96512
#83052
#83032