Finite Group Databasebeta

← C2×#22708 #6380217

Subgroups — C2×#22708 #6380217

Order 512.

2674 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#2354C212yesmaximal
3#22708C2116yesmaximal, direct product factor
4C2×#1948C212yesmaximal, complemented
5C2×#1854C214yesmaximal, complemented
6C2×#1834C211yesmaximal, characteristic, complemented
7C2×#1807C212yesmaximal, complemented
8C2×C4×#107C211yesmaximal, characteristic, complemented
9C2×#1735C212yesmaximal, complemented
10C24×C8C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×#117C2214yes—
13C2×C8×D4C4116yescomplemented
14C2×C8×D4C2218yes—
15C22×#107C2212yes—
16C22×#106C2214yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2354—28no—
21#2354C22116yes—
22C2×#390—22no—
23C2×#388—24no—
24C2×#384C2212yes—
25C2×#380C2214yes—
26C2×C4×C4⋊C4—22no—
27C2×C4×C4⋊C4C2212yes—
28#1948C22116yescomplemented
29#1854C22132yescomplemented
30#1834C2218yescomplemented
31#1807C22116yescomplemented
32C4×#107C2218yescomplemented
33#1735C22116yescomplemented
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2214yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C23×D4D411yescharacteristic
40C24×C4C2312yes—
41C23×C8—48no—
42C23×C8C2×C4116yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2318yes—
47C2×C4×D4C2×C418yes—
48C2×C4×D4D414yes—
49C2×C4×D4Q818yes—
50C2×C4×D4C2314yes—
51C2×#117—28no—
52C2×#117—216no—
53C2×#117—48no—
54C2×#117C23116yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2314yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—24no—
62C22×C22⋊C4—24no—
63C22×C22⋊C4D412yes—
64C22×C42—24no—
65C22×C42—22no—
66C22×C42C2311yescharacteristic
67C8×D4C2×C4164yescomplemented
68C2×#107—28no—
69C2×#107C2318yes—
70C2×#106—48no—
71C2×#106C23116yes—
72C2×C4⋊C8C2×C418yes—
73C2×C4⋊C8C2314yes—
74C2×C22⋊C8C2×C4116yes—
75C2×C22⋊C8C2318yes—
76C2×C4×C8C2×C418yes—
77C2×C4×C8C2314yes—
78#390—216no—
79#388—232no—
80#384—24no—
81#384C2318yes—
82#380—28no—
83#380C23116yes—
84C4×C4⋊C4—216no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4C2318yes—
87C2×#95—24no—
88C2×#95—22no—
89#340C2314yes—
90#340C2318yes—
91#340C2318yes—
92#340C23116yes—
93#340C2314yes—
94C25C2×D412yes—
95C22×D4—24no—
96C22×D4—24no—
97C22×D4C4⋊C418yes—
98C22×D4C2×D414yes—
99C23×C4—28no—
100C23×C4—28no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—22no—
104C23×C4C22×C418yes—
105C23×C4C2×D414yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×Q818yes—
110C23×C4C2414yes—
111C23×C4C2411yescharacteristic
112C22×C8—24no—
113C22×C8—24no—
114C22×C8—28no—
115C22×C8—432no—
116C22×C8—416no—
117C22×C8C22×C414yes—
118C22×C8C22×C418yes—
119C22×C8C22×C414yes—
120C22×C8C22×C4132yes—
121C22×C8C2412yes—
122C22×C8C2414yes—
123C22×C8C2412yes—
124C4×D4C4⋊C4132yes—
125C4×D4C22×C4116yes—
126C4×D4C2×Q8116yes—
127#117—232no—
128#117—416no—
129#117—432no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—48no—
141C2×C4⋊C4—84no—
142C2×C4⋊C4C4⋊C418yes—
143C2×C4⋊C4C2×D414yes—
144C2×C4⋊C4#4118yes—
145C2×C4⋊C4C2418yes—
146C2×C4⋊C4C2414yes—
147C2×C4⋊C4C2414yes—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—216no—
150C2×C22⋊C4—416no—
151C2×C22⋊C4—88no—
152C2×C22⋊C4C4⋊C4116yes—
153C2×C22⋊C4C2×D418yes—
154C2×C42—216no—
155C2×C42—28no—
156C2×C42—416no—
157C2×C42—48no—
158C2×C42C22×C414yes—
159C2×C42C2×D412yes—
160C2×C42C2×Q814yes—
161C2×C42C2412yes—
162#107—216no—
163#106—416no—
164C4⋊C8C22×C4116yes—
165C22⋊C8C22×C4132yes—
166C4×C8C22×C4116yes—
167#95—216no—
168#95—28no—
169C24—28no—
170C24—24no—
171C24—24no—
172C24—28no—
173C24C2×C4⋊C418yes—
174C24C22×D414yes—
175C24C22×D411yescharacteristic
176C2×D4—28no—
177C2×D4—28no—
178C2×D4—232no—
179C2×D4C2×C4⋊C4116yes—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—216no—
183C22×C4—216no—
184C22×C4—216no—
185C22×C4—232no—
186C22×C4—28no—
187C22×C4—28no—
188C22×C4—28no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—28no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—48no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—48no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4C2×C4⋊C414yes—
201C22×C4C2×C4⋊C418yes—
202C22×C4C2×C4⋊C4116yes—
203C22×C4C2×C4⋊C414yes—
204C22×C4#13514yes—
205C22×C4#13514yes—
206C22×C4C23×C418yes—
207C22×C4C23×C412yes—
208C22×C4C22×D414yes—
209C22×C4C22×D412yes—
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D412yes—
212C22×C4C22×Q818yes—
213C22×C4C22×Q812yes—
214C22×C4C2×#4114yes—
215C22×C4H(5,𝔽2)14yes—
216C22×C4C2511yescharacteristic
217C2×C8—28no—
218C2×C8—28no—
219C2×C8—216no—
220C2×C8—24no—
221C2×C8—28no—
222C2×C8—24no—
223C2×C8—432no—
224C2×C8C23×C414yes—
225C2×C8C23×C414yes—
226C2×C8C23×C418yes—
227C4⋊C4—28no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—28no—
231C4⋊C4—24no—
232C4⋊C4—28no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—416no—
236C4⋊C4—88no—
237C4⋊C4C2×C4⋊C4116yes—
238C4⋊C4C2×#4118yes—
239C22⋊C4—432no—
240C22⋊C4—816no—
241C22⋊C4C2×C4⋊C4132yes—
242C42—432no—
243C42—416no—
244C42C2×C4⋊C418yes—
245C42C23×C414yes—
246C42C22×Q814yes—
247C23—216no—
248C23—216no—
249C23—216no—
250C23—28no—
251C23—24no—
252C23—24no—
253C23—28no—
254C23—416no—
255C23—416no—
256C23C22×C4⋊C418yes—
257C23C22×C4⋊C412yes—
258C23#54614yes—
259C23C2×#13512yes—
260C23C2×#13512yes—
261C23#57714yes—
262C23C23×D411yescharacteristic
263D4—232no—
264C2×C4—216no—
265C2×C4—28no—
266C2×C4—28no—
267C2×C4—24no—
268C2×C4—232no—
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—216no—
272C2×C4—28no—
273C2×C4—416no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—48no—
282C2×C4—88no—
283C2×C4—816no—
284C2×C4#427116yes—
285C2×C4C22×C4⋊C414yes—
286C2×C4C22×C4⋊C418yes—
287C2×C4C22×C4⋊C412yes—
288C2×C4C22×C4⋊C414yes—
289C2×C4C4×#4114yes—
290C2×C4#51814yes—
291C2×C4#54614yes—
292C2×C4#55418yes—
293C2×C4C2×#13514yes—
294C2×C4C2×#13514yes—
295C2×C4C24×C411yescharacteristic
296C2×C4C23×Q811yescharacteristic
297C2×C4C22×#4112yes—
298C2×C4C2×H(5,𝔽2)12yes—
299C8—24no—
300C8—24no—
301C8—28no—
302C22—28no—
303C22—28no—
304C22—28no—
305C22—432no—
306C22—48no—
307C22—48no—
308C22#234914yes—
309C22C2×#42714yes—
310C22#291214yes—
311C22#330318yes—
312C22#330418yes—
313C22C23×C4⋊C411yescharacteristic
314C22C2×#54612yes—
315C22C22×#13511yescharacteristic
316C22C22×#13511yescharacteristic
317C22C2×#57712yes—
318C4—28no—
319C4—24no—
320C4—48no—
321C4—84no—
322C4—88no—
323C4#234914yes—
324C4C2×#42718yes—
325C4C2×C4×#4111yescharacteristic
326C4C2×#51811yescharacteristic
327C4C2×#55412yes—
328C2—48no—
329C2#2270818yesdirect product factor
330C2C2×#234911yescharacteristic
331C2C22×#42711yescharacteristic
332C2C2×#291211yescharacteristic
333C2C2×#330312yes—
334C2C2×#330412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22708C2direct (N × H)
4. C2×#1948C2semidirect
5. C2×#1854C2semidirect
6. C2×#1834C2semidirect
7. C2×#1807C2semidirect
8. C2×C4×#107C2semidirect
9. C2×#1735C2semidirect
13. C2×C8×D4C4semidirect
28. #1948C22semidirect
29. #1854C22semidirect
30. #1834C22semidirect
31. #1807C22semidirect
32. C4×#107C22semidirect
33. #1735C22semidirect
67. C8×D4C2×C4semidirect
329. C2#22708direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#23542
#227082
C2×#19482
C2×#18542
C2×#18342
C2×#18072
C2×C4×#1072
C2×#17352