Finite Group Databasebeta

← C22×#1528 #6343745

Subgroups — C22×#1528 #6343745

Order 512.

3458 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#125C211yesmaximal, characteristic
2C22×#432C211yesmaximal, characteristic
3C22×C4×M16C211yesmaximal, characteristic
4C2×#1528C2124yesmaximal, direct product factor
5C22×#330C214yesmaximal, complemented
6C23×M16—21no—
7C22×#125—21no—
8C22×#125—26no—
9C22×#125—21no—
10C22×#125—26no—
11C22×#125C4112yes—
12C22×#125C412yes—
13C22×#125C2212yes—
14C22×#125C22112yes—
15C23×C4⋊C4—21no—
16C23×C4⋊C4—21no—
17C23×C4⋊C4C411yescharacteristic
18C23×C4⋊C4C411yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#432—26no—
21C2×#432C22112yes—
22C22×C4⋊C8—21no—
23C22×C22⋊C8—21no—
24C2×C4×M16—26no—
25C2×C4×M16C22112yes—
26C22×#97C2212yes—
27C22×#97C2212yes—
28C22×C4×C8—21no—
29#1528C22164yesdirect product factor
30C2×#330C22148yescomplemented
31C24×C4—21no—
32C24×C4—21no—
33C24×C4C2×C411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C22×M16—216no—
37C22×M16—212no—
38C23×C8—22no—
39C23×C8—22no—
40C2×#125—212no—
41C2×#125—212no—
42C2×#125—212no—
43C2×#125—248no—
44C2×#125—212no—
45C2×#125—26no—
46C2×#125—26no—
47C2×#125—224no—
48C2×#125—212no—
49C2×#125—224no—
50C2×#125C2×C4112yes—
51C2×#125C2×C4124yes—
52C2×#125C2×C4148yes—
53C2×#125C23112yes—
54C2×#125C23116yes—
55C22×C4⋊C4—212no—
56C22×C4⋊C4—212no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—26no—
60C22×C4⋊C4—26no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—212no—
64C22×C4⋊C4—212no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2×C4112yes—
68C22×C4⋊C4C2×C4112yes—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C412yes—
71C22×C42C2×C416yes—
72C22×C42C2×C411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D416yes—
75C22×C42D411yescharacteristic
76C22×C42D416yes—
77C22×C42C2311yescharacteristic
78C22×C42C2316yes—
79#432—224no—
80#432C23116yes—
81C2×C4⋊C8—212no—
82C2×C22⋊C8—212no—
83C4×M16—224no—
84C4×M16C23116yes—
85C2×#97—26no—
86C2×#97—26no—
87C2×#97C23112yes—
88C2×#97C23112yes—
89C2×C4×C8—212no—
90#330C23164yescomplemented
91C25C22⋊C411yescharacteristic
92C23×C4—28no—
93C23×C4—28no—
94C23×C4—212no—
95C23×C4—212no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—44no—
99C23×C4—48no—
100C23×C4—48no—
101C23×C4C22⋊C416yes—
102C23×C4C22⋊C416yes—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C416yes—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22×C416yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D416yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D416yes—
113C2×M16—296no—
114C2×M16—216no—
115C22×C8—216no—
116C22×C8—212no—
117C22×C8—212no—
118C22×C8—48no—
119#125—224no—
120#125—248no—
121#125—248no—
122#125—248no—
123#125—216no—
124#125—212no—
125#125—212no—
126#125—216no—
127#125C22×C4124yes—
128#125C22×C4132yes—
129#125C2418yes—
130C2×C4⋊C4—248no—
131C2×C4⋊C4—212no—
132C2×C4⋊C4—224no—
133C2×C4⋊C4—224no—
134C2×C4⋊C4—212no—
135C2×C4⋊C4—248no—
136C2×C4⋊C4—26no—
137C2×C4⋊C4—26no—
138C2×C4⋊C4—224no—
139C2×C4⋊C4—224no—
140C2×C4⋊C4—212no—
141C2×C4⋊C4—212no—
142C2×C4⋊C4—212no—
143C2×C4⋊C4—212no—
144C2×C4⋊C4—216no—
145C2×C4⋊C4—216no—
146C2×C4⋊C4C22×C4112yes—
147C2×C4⋊C4C22×C4112yes—
148C2×C4⋊C4C22×C4116yes—
149C2×C4⋊C4C22×C4116yes—
150C2×C42C22⋊C416yes—
151C2×C42C22⋊C416yes—
152C2×C42C22⋊C416yes—
153C2×C42C22⋊C4124yes—
154C2×C42C22×C413yes—
155C2×C42C22×C416yes—
156C2×C42C22×C4112yes—
157C2×C42C2×D416yes—
158C2×C42C2×D413yes—
159C2×C42C2×D413yes—
160C2×C42C2×D4112yes—
161C2×C42C2×D416yes—
162C2×C42C2×D4112yes—
163C2×C42C2413yes—
164C2×C42C2414yes—
165C4⋊C8—216no—
166C22⋊C8—216no—
167#97—212no—
168#97—212no—
169#97C2418yes—
170#97C2418yes—
171C4×C8—216no—
172C24—24no—
173C24—24no—
174C24#10418yes—
175C24C2×C22⋊C416yes—
176C24C2×C22⋊C411yescharacteristic
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—224no—
180C22×C4—224no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—224no—
185C22×C4—224no—
186C22×C4—216no—
187C22×C4—216no—
188C22×C4—212no—
189C22×C4—212no—
190C22×C4—412no—
191C22×C4—412no—
192C22×C4—448no—
193C22×C4—448no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4#10118yes—
197C22×C4C2×C22⋊C4112yes—
198C22×C4C2×C22⋊C413yes—
199C22×C4C2×C22⋊C416yes—
200C22×C4C2×C22⋊C416yes—
201C22×C4C2×C22⋊C413yes—
202C22×C4C2×C22⋊C413yes—
203C22×C4C2×C22⋊C4112yes—
204C22×C4C2×C22⋊C4112yes—
205C22×C4C2×C22⋊C416yes—
206C22×C4C23×C413yes—
207C22×C4C23×C414yes—
208C22×C4C22×D413yes—
209C22×C4C22×D413yes—
210C22×C4C22×D414yes—
211C22×C4C22×D414yes—
212M16—264no—
213C2×C8—248no—
214C2×C8—28no—
215C2×C8—28no—
216C2×C8—424no—
217C4⋊C4—224no—
218C4⋊C4—232no—
219C4⋊C4—232no—
220C4⋊C4—224no—
221C4⋊C4—212no—
222C4⋊C4—212no—
223C4⋊C4—216no—
224C4⋊C4—216no—
225C4⋊C4—28no—
226C4⋊C4—28no—
227C4⋊C4C23×C418yes—
228C4⋊C4C23×C418yes—
229C42C2×C22⋊C416yes—
230C42C2×C22⋊C4112yes—
231C42C2×C22⋊C4112yes—
232C42C2×C22⋊C4112yes—
233C42C23×C413yes—
234C42C23×C414yes—
235C42C22×D414yes—
236C42C22×D413yes—
237C42C22×D413yes—
238C42C22×D414yes—
239C42C2511yescharacteristic
240C23—212no—
241C23—212no—
242C23—232no—
243C23C2×#10114yes—
244C23C2×#104124yes—
245C23C2×#10414yes—
246C23#42114yes—
247C23C22×C22⋊C414yes—
248C23C22×C22⋊C413yes—
249C2×C4—26no—
250C2×C4—26no—
251C2×C4—26no—
252C2×C4—28no—
253C2×C4—224no—
254C2×C4—26no—
255C2×C4—212no—
256C2×C4—26no—
257C2×C4—26no—
258C2×C4—212no—
259C2×C4—224no—
260C2×C4—28no—
261C2×C4—28no—
262C2×C4—28no—
263C2×C4—412no—
264C2×C4—44no—
265C2×C4—432no—
266C2×C4—432no—
267C2×C4—424no—
268C2×C4—424no—
269C2×C4C2×#101124yes—
270C2×C4C22×C22⋊C413yes—
271C2×C4C22×C22⋊C414yes—
272C2×C4C22×C22⋊C414yes—
273C2×C4C22×C22⋊C413yes—
274C2×C4C22×C22⋊C413yes—
275C2×C4C22×C22⋊C414yes—
276C2×C4C24×C411yescharacteristic
277C2×C4C23×D411yescharacteristic
278C2×C4C23×D411yescharacteristic
279C8—216no—
280C8—48no—
281C22—24no—
282C22—24no—
283C22—248no—
284C22#1528116yesdirect product factor
285C22C22×#10116yes—
286C22C22×#10418yes—
287C22C22×#10416yes—
288C22C2×#42116yes—
289C22C23×C22⋊C411yescharacteristic
290C4—23no—
291C4—21no—
292C4—21no—
293C4—23no—
294C4—24no—
295C4—21no—
296C4—23no—
297C4—24no—
298C4—48no—
299C4—48no—
300C4C22×#10118yes—
301C2—28no—
302C2C2×#1528112yesdirect product factor
303C2C23×#10111yescharacteristic
304C2C23×#10411yescharacteristic
305C2C22×#42111yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1528C2direct (N × H)
5. C22×#330C2semidirect
29. #1528C22direct (N × H)
30. C2×#330C22semidirect
90. #330C23semidirect
284. C22#1528direct (N × H)
302. C2C2×#1528direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1252
C22×#4322
C22×C4×M162
C2×#15282
C22×#3302