Finite Group Databasebeta

← #213630

Subgroups — #213630

Order 512.

1302 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1302 conjugacy classes of subgroups — too many to draw. Only its 214 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#36624C211yesmaximal, characteristic
2#20715C212yesmaximal, complemented
3#9124C218yesmaximal, complemented
4#8722C212yesmaximal, complemented
5#8337C212yesmaximal, complemented
6C2×#532C2211yescharacteristic
7C2×#525C412yes—
8C23×C42C2211yescharacteristic
9C2×#432C2214yes—
10#2857C412yes—
11#2856C412yes—
12#2854—22no—
13#2850C2214yes—
14#2847—22no—
15C4×#121—22no—
16C4×#121—22no—
17#2297C412yes—
18#2285C2211yescharacteristic
19#1817—22no—
20#1797—22no—
21#1797—22no—
22#1779—22no—
23#1749C2214yes—
24#1749C2214yes—
25C2×#329C2218yescomplemented
26#1295C2218yescomplemented
27C22×#41C2×C412yes—
28C24×C4C2×C411yescharacteristic
29C24×C4D412yes—
30#532—44no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×#127C2×C411yescharacteristic
33C2×#124C2312yes—
34#525—48no—
35C2×#123—22no—
36C2×C22⋊Q8C2×C414yes—
37C2×#121—22no—
38C2×#121—22no—
39C2×#121—22no—
40C2×#121C2×C414yes—
41C2×#120—22no—
42C2×C4×D4—24no—
43C2×C4×D4—22no—
44C2×#117C2×C412yes—
45C22×C42—44no—
46C22×C42C2×C412yes—
47C22×C42C2×C411yescharacteristic
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42D412yes—
51C22×C42C2312yes—
52C22×C42C2311yescharacteristic
53#432—48no—
54C2×C4⋊C8—24no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58C2×#97C2318yes—
59#391C2×C414yes—
60#390—22no—
61#390—22no—
62#390C2×C414yes—
63#389—22no—
64#389—22no—
65#388—24no—
66#388—24no—
67#386—24no—
68#386—24no—
69#386—24no—
70#384C2×C412yes—
71#382C2312yes—
72#380—22no—
73#380—22no—
74#380—22no—
75C4×C4⋊C4—22no—
76C4×C22⋊C4—22no—
77C4×C22⋊C4—22no—
78#336—24no—
79#336—24no—
80#329—48no—
81C25C22⋊C411yescharacteristic
82C2×#41—44no—
83C2×#41—44no—
84C2×#41—44no—
85C22×Q8C22×C412yes—
86C22×D4—24no—
87C22×D4—22no—
88C22×D4—22no—
89C22×D4C22×C412yes—
90C23×C4—24no—
91C23×C4—22no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—44no—
95C23×C4—44no—
96C23×C4—44no—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C22×C8—28no—
108C4⋊Q8—42no—
109#127—42no—
110#124—28no—
111#123—48no—
112C22⋊Q8—48no—
113#121—48no—
114#121—48no—
115#121—48no—
116#121—48no—
117#120—48no—
118C4×D4—48no—
119C4×D4—416no—
120#117—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4C22×C414yes—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C42—24no—
133C2×C42—28no—
134C2×C42—22no—
135C2×C42—42no—
136C2×C42—44no—
137C2×C42—44no—
138C2×C42—44no—
139C2×C42—44no—
140C2×C42C22⋊C411yescharacteristic
141C2×C42C22⋊C411yescharacteristic
142C2×C42C22⋊C412yes—
143C2×C42C22⋊C412yes—
144C2×C42C22×C411yescharacteristic
145C2×C42C22×C412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2411yescharacteristic
150C4⋊C8—48no—
151C22⋊C8—48no—
152C22⋊C8—48no—
153C22⋊C8—48no—
154#97—216no—
155#95—24no—
156#95—24no—
157#95—24no—
158C24—22no—
159C24—41no—
160C24—41no—
161C24—42no—
162C24—42no—
163C24#10014yes—
164C24C2×C22⋊C411yescharacteristic
165C24C2×C22⋊C412yes—
166#41—816no—
167C2×Q8—24no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×D4—24no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—44no—
174C2×D4—44no—
175C2×D4—44no—
176C2×D4—48no—
177C2×D4—48no—
178C2×D4—44no—
179C2×D4—48no—
180C2×D4—44no—
181C2×D4—44no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—416no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—432no—
218C22×C4—44no—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D412yes—
226C2×C8—48no—
227C2×C8—48no—
228C2×C8—48no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—48no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C22⋊C4—48no—
235C22⋊C4—48no—
236C22⋊C4—44no—
237C22⋊C4—48no—
238C22⋊C4—48no—
239C22⋊C4—48no—
240C42—24no—
241C42—22no—
242C42—24no—
243C42—24no—
244C42—24no—
245C42—42no—
246C42—42no—
247C42—41no—
248C42—41no—
249C42—44no—
250C42H(5,𝔽2)14yes—
251C23—21no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—44no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—44no—
270C23—44no—
271C23C2×#10014yes—
272C23C2×#10012yes—
273C23C22×C22⋊C411yescharacteristic
274Q8—84no—
275D4—84no—
276D4—88no—
277D4—816no—
278D4—88no—
279C2×C4—28no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—88no—
312C2×C4#41312yes—
313C2×C4#42114yes—
314C2×C4#42114yes—
315C2×C4#51812yes—
316C2×C4#53414yes—
317C8—88no—
318C22—24no—
319C22—24no—
320C22—24no—
321C22—22no—
322C22—24no—
323C22—416no—
324C22—41no—
325C22—42no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—82no—
332C22—82no—
333C22—84no—
334C22—84no—
335C22—82no—
336C22#230811yescharacteristic
337C22C22×#10011yescharacteristic
338C22C2×#41311yescharacteristic
339C22C2×#42112yes—
340C22C2×#42112yes—
341C4—24no—
342C4—24no—
343C4—24no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—42no—
348C4—84no—
349C4—82no—
350C2—44no—
351C2—42no—
352C2—162no—
353C2#1766511yescharacteristic
354C2#1767311yescharacteristic
355C2#1767512yes—
356C2#2175211yescharacteristic
357C2#2177412yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #9124C2semidirect
4. #8722C2semidirect
5. #8337C2semidirect
25. C2×#329C22semidirect
26. #1295C22semidirect

Maximal subgroups

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

subgroupindex
#366242
#207152
#91242
#87222
#83372