Finite Group Databasebeta

← #392213

Subgroups — #392213

Order 512.

5510 conjugacy classes of subgroups; only the 506 normal subgroups are shown.

normal subgroups only

This group has 5510 conjugacy classes of subgroups — too many to draw. Only its 506 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
1C24×C4⋊C4C211yesmaximal, characteristic
2C2×#1821C214yesmaximal, complemented
3#20579C212yesmaximal
4#10727C218yesmaximal, complemented
5C25×C4C412yes—
6C25×C4C2211yescharacteristic
7C23×C4⋊C4—24no—
8C23×C4⋊C4—24no—
9C23×C4⋊C4—216no—
10C23×C4⋊C4C412yes—
11C23×C4⋊C4C414yes—
12C23×C4⋊C4C2214yes—
13C23×C4⋊C4C2212yes—
14C2×#407C2214yes—
15C2×#407C2218yes—
16#1821—216no—
17#1779—24no—
18#1779—24no—
19C2×#343C22116yescomplemented
20C26C2×C411yescharacteristic
21C24×C4—28no—
22C24×C4—216no—
23C24×C4—24no—
24C24×C4—24no—
25C24×C4—216no—
26C24×C4—416no—
27C24×C4C2×C411yescharacteristic
28C24×C4C2×C412yes—
29C24×C4C2×C418yes—
30C24×C4C2×C414yes—
31C24×C4D414yes—
32C24×C4D414yes—
33C24×C4C2311yescharacteristic
34C24×C4C2312yes—
35C22×M16C2318yes—
36C22×C4⋊C4—264no—
37C22×C4⋊C4—264no—
38C22×C4⋊C4—216no—
39C22×C4⋊C4—24no—
40C22×C4⋊C4—24no—
41C22×C4⋊C4—216no—
42C22×C4⋊C4—216no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—28no—
46C22×C4⋊C4—232no—
47C22×C4⋊C4—232no—
48C22×C4⋊C4C2×C418yes—
49C22×C4⋊C4C2×C414yes—
50C22×C4⋊C4D4116yes—
51C22×C4⋊C4C2314yes—
52#407—216no—
53#407—232no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—216no—
57#343—232no—
58C25—28no—
59C25—28no—
60C25—24no—
61C25—28no—
62C25C22⋊C412yes—
63C25C22⋊C412yes—
64C25C22×C411yescharacteristic
65C25C22×C412yes—
66C23×C4—2128no—
67C23×C4—24no—
68C23×C4—28no—
69C23×C4—28no—
70C23×C4—28no—
71C23×C4—28no—
72C23×C4—216no—
73C23×C4—216no—
74C23×C4—232no—
75C23×C4—232no—
76C23×C4—28no—
77C23×C4—28no—
78C23×C4—28no—
79C23×C4—232no—
80C23×C4—232no—
81C23×C4—216no—
82C23×C4—28no—
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—232no—
86C23×C4—4128no—
87C23×C4—432no—
88C23×C4—416no—
89C23×C4—416no—
90C23×C4—432no—
91C23×C4—416no—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C418yes—
95C23×C4C22⋊C4116yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C412yes—
98C23×C4C22×C418yes—
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×D418yes—
102C23×C4C2×D418yes—
103C23×C4#41116yes—
104C23×C4C2411yescharacteristic
105C2×M16—216no—
106C2×M16—232no—
107C22×C8—28no—
108C2×C4⋊C4—232no—
109C2×C4⋊C4—232no—
110C2×C4⋊C4—2128no—
111C2×C4⋊C4—2128no—
112C2×C4⋊C4—232no—
113C2×C4⋊C4—216no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—232no—
117C2×C4⋊C4—232no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4C22⋊C4116yes—
121C2×C4⋊C4C2×D4116yes—
122C22⋊C8—48no—
123C22⋊C8—48no—
124C22⋊C8—432no—
125C24—232no—
126C24—232no—
127C24—2128no—
128C24—232no—
129C24—24no—
130C24—24no—
131C24—24no—
132C24—24no—
133C24—24no—
134C24—28no—
135C24—28no—
136C24—28no—
137C24—24no—
138C24—28no—
139C24—24no—
140C24—28no—
141C24—24no—
142C24—28no—
143C24—24no—
144C24#10018yes—
145C24#10118yes—
146C24C2×C22⋊C414yes—
147C24C2×C22⋊C411yescharacteristic
148C24C2×C22⋊C411yescharacteristic
149C24C2×C22⋊C414yes—
150C24#11718yes—
151C24C23×C411yescharacteristic
152C22×C4—264no—
153C22×C4—232no—
154C22×C4—2128no—
155C22×C4—2128no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—216no—
162C22×C4—216no—
163C22×C4—28no—
164C22×C4—28no—
165C22×C4—216no—
166C22×C4—232no—
167C22×C4—264no—
168C22×C4—216no—
169C22×C4—216no—
170C22×C4—28no—
171C22×C4—28no—
172C22×C4—264no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—216no—
176C22×C4—216no—
177C22×C4—4128no—
178C22×C4—4128no—
179C22×C4—416no—
180C22×C4—432no—
181C22×C4—4128no—
182C22×C4—432no—
183C22×C4—432no—
184C22×C4—416no—
185C22×C4—432no—
186C22×C4—416no—
187C22×C4C2×C22⋊C414yes—
188C22×C4C2×C22⋊C412yes—
189C22×C4C2×C22⋊C414yes—
190C22×C4C2×C22⋊C4116yes—
191C22×C4#11718yes—
192C22×C4#12018yes—
193C22×C4#12018yes—
194C22×C4#123132yes—
195C22×C4C22×D412yes—
196C22×C4C22×D414yes—
197C22×C4C2×#4118yes—
198M16—432no—
199C2×C8—48no—
200C2×C8—416no—
201C4⋊C4—264no—
202C4⋊C4—232no—
203C4⋊C4—232no—
204C23—216no—
205C23—216no—
206C23—216no—
207C23—216no—
208C23—2256no—
209C23—216no—
210C23—216no—
211C23—232no—
212C23—232no—
213C23—232no—
214C23—232no—
215C23—232no—
216C23—232no—
217C23—232no—
218C23—22no—
219C23—24no—
220C23—22no—
221C23—22no—
222C23—24no—
223C23—22no—
224C23—24no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—28no—
229C23—216no—
230C23—216no—
231C23—28no—
232C23—216no—
233C23—28no—
234C23#37912yes—
235C23#37912yes—
236C23#38118yes—
237C23C2×#10018yes—
238C23C2×#10014yes—
239C23C2×#10118yes—
240C23C2×#10114yes—
241C23C22×C22⋊C412yes—
242C23C22×C22⋊C411yescharacteristic
243C23C2×#11714yes—
244C2×C4—264no—
245C2×C4—216no—
246C2×C4—216no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—28no—
251C2×C4—216no—
252C2×C4—216no—
253C2×C4—216no—
254C2×C4—264no—
255C2×C4—216no—
256C2×C4—416no—
257C2×C4—432no—
258C2×C4—4128no—
259C2×C4—416no—
260C2×C4—432no—
261C2×C4—416no—
262C2×C4#37914yes—
263C2×C4#38118yes—
264C2×C4C2×#12014yes—
265C2×C4C2×#12318yes—
266C2×C4#525116yes—
267C8—88no—
268C22—264no—
269C22—264no—
270C22—264no—
271C22—232no—
272C22—22no—
273C22—22no—
274C22—28no—
275C22—22no—
276C22—22no—
277C22—28no—
278C22—22no—
279C22—22no—
280C22—28no—
281C22—28no—
282C22—28no—
283C22—28no—
284C22—28no—
285C22—28no—
286C22—28no—
287C22#178314yes—
288C22#178414yes—
289C22#1821116yescomplemented
290C22C2×#37911yescharacteristic
291C22C2×#38112yes—
292C22#228014yes—
293C22C22×#10012yes—
294C22C22×#10112yes—
295C4—24no—
296C4—28no—
297C4—416no—
298C4#284918yes—
299C2—216no—
300C2—24no—
301C2—24no—
302C2—24no—
303C2#1468111yescharacteristic
304C2C2×#178311yescharacteristic
305C2C2×#178411yescharacteristic
306C2C2×#182114yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1821C2semidirect
4. #10727C2semidirect
19. C2×#343C22semidirect
289. C22#1821semidirect

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C2×#18212
#205792
#107272