Finite Group Databasebeta

← C22×#1352 #481486

Subgroups — C22×#1352 #481486

Order 512.

2170 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2170 conjugacy classes of subgroups — too many to draw. Only its 322 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
1C22×#399C211yesmaximal, characteristic
2C22×#342C211yesmaximal, characteristic
3C22×#323C211yesmaximal, characteristic
4C2×#1352C2112yesmaximal, direct product factor
5C23×C4⋊C4—21no—
6C23×C4⋊C4C411yescharacteristic, complemented
7C23×C22⋊C4—21no—
8C23×C22⋊C4—21no—
9C23×C22⋊C4C2211yescharacteristic
10C22×C22⋊C8—21no—
11C2×#399—23no—
12C2×#399—23no—
13C2×#399C416yescomplemented
14C2×#399C2216yes—
15C22×#95C411yescharacteristic, complemented
16C2×#342—23no—
17C2×#342C2216yes—
18C2×#323—23no—
19C2×#323C2216yes—
20#1352C22116yesdirect product factor
21C26D411yescharacteristic
22C24×C4—21no—
23C24×C4—21no—
24C24×C4—41no—
25C24×C4C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C23×C8—41no—
28C22×C4⋊C4—23no—
29C22×C4⋊C4—26no—
30C22×C4⋊C4—26no—
31C22×C4⋊C4—43no—
32C22×C4⋊C4—41no—
33C22×C4⋊C4—48no—
34C22×C4⋊C4C2×C416yescomplemented
35C22×C22⋊C4—28no—
36C22×C22⋊C4—26no—
37C22×C22⋊C4—26no—
38C22×C22⋊C4—26no—
39C22×C22⋊C4—43no—
40C22×C22⋊C4—41no—
41C22×C22⋊C4—48no—
42C22×C22⋊C4—43no—
43C22×C22⋊C4—44no—
44C22×C22⋊C4C2×C413yes—
45C22×C22⋊C4D413yes—
46C22×C22⋊C4D413yes—
47C22×C22⋊C4C2313yes—
48C2×C22⋊C8—26no—
49C2×C22⋊C8—43no—
50#399—212no—
51#399—26no—
52#399—26no—
53#399C2×C4112yescomplemented
54#399C2314yes—
55C2×#95—23no—
56C2×#95C2×C416yescomplemented
57#342—26no—
58#342C2314yes—
59#323—26no—
60#323C2314yes—
61C25—21no—
62C25—23no—
63C25—24no—
64C25—44no—
65C25—44no—
66C25C22⋊C413yes—
67C25C22⋊C411yescharacteristic
68C25SD1614yes—
69C25Q1614yes—
70C25C2×D413yes—
71C23×C4—28no—
72C23×C4—23no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—26no—
76C23×C4—23no—
77C23×C4—26no—
78C23×C4—44no—
79C23×C4—44no—
80C23×C4—44no—
81C23×C4—44no—
82C23×C4—48no—
83C23×C4—46no—
84C23×C4—44no—
85C23×C4—44no—
86C23×C4—43no—
87C23×C4—41no—
88C23×C4—84no—
89C23×C4—84no—
90C23×C4C22⋊C413yes—
91C23×C4C22⋊C413yes—
92C23×C4C22×C413yes—
93C23×C4C2×D413yes—
94C22×C8—46no—
95C22×C8—84no—
96C2×C4⋊C4—212no—
97C2×C4⋊C4—26no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—46no—
100C2×C4⋊C4—46no—
101C2×C4⋊C4—424no—
102C2×C4⋊C4—424no—
103C2×C4⋊C4—46no—
104C2×C4⋊C4—46no—
105C2×C4⋊C4C22×C414yescomplemented
106C2×C22⋊C4—224no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—212no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—43no—
111C2×C22⋊C4—43no—
112C2×C22⋊C4—43no—
113C2×C22⋊C4—43no—
114C2×C22⋊C4—43no—
115C2×C22⋊C4—43no—
116C2×C22⋊C4—43no—
117C2×C22⋊C4—43no—
118C2×C22⋊C4—424no—
119C2×C22⋊C4—424no—
120C2×C22⋊C4—424no—
121C2×C22⋊C4—46no—
122C2×C22⋊C4—412no—
123C2×C22⋊C4C22⋊C416yes—
124C2×C22⋊C4C22×C413yes—
125C2×C22⋊C4C2×D413yes—
126C2×C22⋊C4C2×D413yes—
127C2×C22⋊C4C2411yescharacteristic
128C22⋊C8—24no—
129C22⋊C8—46no—
130#95—26no—
131#95C22×C414yescomplemented
132C24—24no—
133C24—24no—
134C24—24no—
135C24—24no—
136C24—23no—
137C24—23no—
138C24—212no—
139C24—212no—
140C24—23no—
141C24—23no—
142C24—26no—
143C24—26no—
144C24—46no—
145C24—46no—
146C24—46no—
147C24—416no—
148C24—416no—
149C24—46no—
150C24—44no—
151C24—44no—
152C24—86no—
153C24—816no—
154C24—86no—
155C24#9913yes—
156C24#9911yescharacteristic
157C24#10314yes—
158C24#103112yes—
159C24#10414yes—
160C24C2×C22⋊C413yes—
161C24C2×C22⋊C413yes—
162C24C2×SD1616yes—
163C24C2×Q1616yes—
164C24C22×D411yescharacteristic
165C22×C4—224no—
166C22×C4—24no—
167C22×C4—23no—
168C22×C4—23no—
169C22×C4—23no—
170C22×C4—23no—
171C22×C4—23no—
172C22×C4—23no—
173C22×C4—23no—
174C22×C4—23no—
175C22×C4—24no—
176C22×C4—412no—
177C22×C4—432no—
178C22×C4—412no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—412no—
182C22×C4—46no—
183C22×C4—46no—
184C22×C4—46no—
185C22×C4—46no—
186C22×C4—44no—
187C22×C4—424no—
188C22×C4—412no—
189C22×C4—412no—
190C22×C4—46no—
191C22×C4—46no—
192C22×C4—86no—
193C22×C4—812no—
194C22×C4—812no—
195C22×C4—832no—
196C22×C4—812no—
197C22×C4—84no—
198C22×C4C2×C22⋊C413yes—
199C22×C4C2×C22⋊C413yes—
200C22×C4C23×C411yescharacteristic
201C22×C4C22×D411yescharacteristic
202C2×C8—44no—
203C2×C8—812no—
204C4⋊C4—44no—
205C4⋊C4—424no—
206C4⋊C4—48no—
207C4⋊C4—412no—
208C22⋊C4—28no—
209C22⋊C4—46no—
210C22⋊C4—43no—
211C22⋊C4—43no—
212C22⋊C4—43no—
213C22⋊C4—41no—
214C22⋊C4—424no—
215C22⋊C4—48no—
216C22⋊C4—424no—
217C22⋊C4—44no—
218C23—26no—
219C23—26no—
220C23—216no—
221C23—216no—
222C23—26no—
223C23—26no—
224C23—21no—
225C23—23no—
226C23—26no—
227C23—212no—
228C23—26no—
229C23—21no—
230C23—23no—
231C23—26no—
232C23—212no—
233C23—26no—
234C23—43no—
235C23—41no—
236C23—424no—
237C23—43no—
238C23—412no—
239C23—412no—
240C23—41no—
241C23—46no—
242C23—46no—
243C23—416no—
244C23—416no—
245C23—46no—
246C23—46no—
247C23—83no—
248C23—83no—
249C23—83no—
250C23—83no—
251C23—812no—
252C23—812no—
253C23—824no—
254C23—824no—
255C23#328112yes—
256C23#32814yes—
257C23#35214yes—
258C23#35314yes—
259C23C2×#9913yes—
260C23C2×#9913yes—
261C23C2×#10316yes—
262C23C2×#10316yes—
263C23C2×#10416yes—
264C23C22×C22⋊C411yescharacteristic
265C23C22×SD1611yescharacteristic
266C23C22×Q1611yescharacteristic
267C2×C4—28no—
268C2×C4—23no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—23no—
272C2×C4—44no—
273C2×C4—448no—
274C2×C4—46no—
275C2×C4—46no—
276C2×C4—46no—
277C2×C4—46no—
278C2×C4—46no—
279C2×C4—46no—
280C2×C4—44no—
281C2×C4—43no—
282C2×C4—41no—
283C2×C4—43no—
284C2×C4—41no—
285C2×C4—48no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—83no—
290C2×C4—83no—
291C2×C4—812no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—848no—
295C2×C4—812no—
296C8—84no—
297C22—21no—
298C22—23no—
299C22—21no—
300C22—23no—
301C22—212no—
302C22—212no—
303C22—224no—
304C22—21no—
305C22—21no—
306C22—26no—
307C22—41no—
308C22—41no—
309C22—46no—
310C22—41no—
311C22—43no—
312C22—41no—
313C22—43no—
314C22—412no—
315C22—424no—
316C22—412no—
317C22—81no—
318C22—83no—
319C22—86no—
320C22—812no—
321C22—86no—
322C22#1352116yesdirect product factor
323C22C2×#32816yes—
324C22C2×#32816yes—
325C22C2×#35216yes—
326C22C2×#35316yes—
327C22C22×#9911yescharacteristic
328C22C22×#10311yescharacteristic
329C22C22×#10411yescharacteristic
330C4—48no—
331C4—43no—
332C4—41no—
333C4—41no—
334C4—43no—
335C4—88no—
336C4—84no—
337C2—21no—
338C2—21no—
339C2—26no—
340C2—41no—
341C2—41no—
342C2—46no—
343C2C2×#1352112yesdirect product factor
344C2C22×#32811yescharacteristic
345C2C22×#35211yescharacteristic
346C2C22×#35311yescharacteristic
subgroupG/H
center, socleC24#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1352C2direct (N × H)
6. C23×C4⋊C4C4semidirect
13. C2×#399C4semidirect
15. C22×#95C4semidirect
20. #1352C22direct (N × H)
34. C22×C4⋊C4C2×C4semidirect
53. #399C2×C4semidirect
56. C2×#95C2×C4semidirect
105. C2×C4⋊C4C22×C4semidirect
131. #95C22×C4semidirect
322. C22#1352direct (N × H)
343. C2C2×#1352direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3992
C22×#3422
C22×#3232
C2×#13522