Finite Group Databasebeta

← C2×#11385 #358771

Subgroups — C2×#11385 #358771

Order 512.

3338 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#125C211yesmaximal, characteristic
2C22×#340C214yesmaximal
3C22×#336C211yesmaximal, characteristic
4C2×C4⋊#95C211yesmaximal, characteristic
5#11385C218yesmaximal, direct product factor
6C24×C8C2212yes—
7C22×#125—24no—
8C22×#125—22no—
9C22×#125—216no—
10C22×#125C418yescomplemented
11C22×#125C2214yes—
12C23×C4⋊C4—22no—
13C23×C4⋊C4C2212yes—
14C23×C4⋊C4C2212yes—
15C23×C42C2211yescharacteristic
16C22×#95—22no—
17C2×#340—216no—
18C2×#340C22116yes—
19C2×#336—24no—
20C2×#336C2214yes—
21C4⋊#95—22no—
22C4⋊#95C2214yes—
23C24×C4—22no—
24C24×C4—22no—
25C24×C4D412yes—
26C24×C4C2311yescharacteristic
27C23×C8—24no—
28C23×C8—28no—
29C23×C8—24no—
30C23×C8—28no—
31C23×C8Q818yes—
32C23×C8C2314yes—
33C2×#125—264no—
34C2×#125—232no—
35C2×#125—264no—
36C2×#125—232no—
37C2×#125—216no—
38C2×#125—28no—
39C2×#125C2×C4116yescomplemented
40C22×C4⋊C4—232no—
41C22×C4⋊C4—216no—
42C22×C4⋊C4—28no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—28no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—28no—
47C22×C4⋊C4—28no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4C2×C418yes—
51C22×C4⋊C4D4116yes—
52C22×C4⋊C4Q818yes—
53C22×C4⋊C4C2314yes—
54C22×C4⋊C4C2314yes—
55C22×C42—28no—
56C22×C42C2×C414yes—
57C22×C42D412yes—
58C22×C42D414yes—
59C22×C42C2312yes—
60C2×#95—28no—
61C2×#95—48no—
62#340—232no—
63#336—28no—
64C25C2×D411yescharacteristic
65C23×C4—28no—
66C23×C4—216no—
67C23×C4—28no—
68C23×C4—28no—
69C23×C4—24no—
70C23×C4—24no—
71C23×C4—24no—
72C23×C4—28no—
73C23×C4—24no—
74C23×C4—48no—
75C23×C4—44no—
76C23×C4—44no—
77C23×C4—44no—
78C23×C4—44no—
79C23×C4—48no—
80C23×C4—48no—
81C23×C4C22⋊C418yes—
82C23×C4C22×C412yes—
83C23×C4C2×D414yes—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D418yes—
86C23×C4C2×D412yes—
87C23×C4C2×Q814yes—
88C23×C4C2×Q814yes—
89C23×C4#4114yes—
90C23×C4#4114yes—
91C23×C4C2411yescharacteristic
92C22×C8—232no—
93C22×C8—216no—
94C22×C8—24no—
95C22×C8—28no—
96C22×C8—24no—
97C22×C8—28no—
98C22×C8—48no—
99C22×C8—416no—
100C22×C8—48no—
101C22×C8C2×Q818yes—
102#125—264no—
103#125—2128no—
104#125—264no—
105C2×C4⋊C4—264no—
106C2×C4⋊C4—2128no—
107C2×C4⋊C4—232no—
108C2×C4⋊C4—216no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—232no—
111C2×C4⋊C4—216no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—232no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—416no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4C22⋊C4132yes—
123C2×C4⋊C4C22×C418yes—
124C2×C4⋊C4C2×D4116yes—
125C2×C4⋊C4C2×Q818yes—
126C2×C42—216no—
127C2×C42—216no—
128C2×C42—28no—
129C2×C42—28no—
130C2×C42C22⋊C418yes—
131C2×C42C22×C414yes—
132C2×C42C2×D414yes—
133#95—416no—
134C24C2×C22⋊C412yes—
135C24#12014yes—
136C24C22⋊Q814yes—
137C24#12412yes—
138C24C4⋊Q812yes—
139C24#13518yes—
140C24AGL(1,ℤ/8)14yes—
141C24#13714yes—
142C24C22×D411yescharacteristic
143C22×C4—28no—
144C22×C4—28no—
145C22×C4—216no—
146C22×C4—28no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—216no—
150C22×C4—232no—
151C22×C4—216no—
152C22×C4—216no—
153C22×C4—28no—
154C22×C4—48no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—416no—
158C22×C4—416no—
159C22×C4—416no—
160C22×C4—432no—
161C22×C4—416no—
162C22×C4—432no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—48no—
166C22×C4—48no—
167C22×C4—416no—
168C22×C4—432no—
169C22×C4—816no—
170C22×C4—88no—
171C22×C4—88no—
172C22×C4C2×C22⋊C418yes—
173C22×C4C2×C22⋊C418yes—
174C22×C4C2×C22⋊C412yes—
175C22×C4C4×Q818yes—
176C22×C4C4×Q818yes—
177C22×C4#12018yes—
178C22×C4#12014yes—
179C22×C4C22⋊Q8116yes—
180C22×C4C22⋊Q8116yes—
181C22×C4C22⋊Q818yes—
182C22×C4C22⋊Q814yes—
183C22×C4#124116yes—
184C22×C4#12414yes—
185C22×C4#12412yes—
186C22×C4C4⋊Q8116yes—
187C22×C4C4⋊Q814yes—
188C22×C4C4⋊Q812yes—
189C22×C4C23×C411yescharacteristic
190C22×C4C22×D414yes—
191C22×C4C22×D411yescharacteristic
192C22×C4C22×Q812yes—
193C22×C4C22×Q812yes—
194C22×C4C2×#4112yes—
195C22×C4C2×#4112yes—
196C2×C8—28no—
197C2×C8—216no—
198C2×C8—44no—
199C2×C8—44no—
200C2×C8—48no—
201C2×C8—432no—
202C4⋊C4—2128no—
203C4⋊C4—28no—
204C4⋊C4—216no—
205C4⋊C4—232no—
206C4⋊C4—216no—
207C4⋊C4—48no—
208C4⋊C4—416no—
209C4⋊C4C2×C22⋊C4116yes—
210C42—28no—
211C42—28no—
212C42—216no—
213C23#37914yes—
214C23#39114yes—
215C23#39114yes—
216C23#39318yes—
217C23#39518yes—
218C23#41618yes—
219C23#41718yes—
220C23#449116yes—
221C23#452116yes—
222C23#478116yes—
223C23#479116yes—
224C23#48818yes—
225C23#49014yes—
226C23#49114yes—
227C23#49918yes—
228C23#50118yes—
229C23C22×C22⋊C411yescharacteristic
230C23C2×#12012yes—
231C23C2×C22⋊Q812yes—
232C23C2×#12411yescharacteristic
233C23C2×C4⋊Q811yescharacteristic
234C23C2×#13514yes—
235C23C2×AGL(1,ℤ/8)12yes—
236C23C2×#13712yes—
237C2×C4—28no—
238C2×C4—28no—
239C2×C4—28no—
240C2×C4—216no—
241C2×C4—48no—
242C2×C4—48no—
243C2×C4—48no—
244C2×C4—416no—
245C2×C4—48no—
246C2×C4—416no—
247C2×C4—432no—
248C2×C4—464no—
249C2×C4—416no—
250C2×C4—48no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—88no—
254C2×C4—832no—
255C2×C4#37914yes—
256C2×C4#39114yes—
257C2×C4#39114yes—
258C2×C4#391132yes—
259C2×C4#393116yes—
260C2×C4#39318yes—
261C2×C4#395116yes—
262C2×C4#39518yes—
263C2×C4C22×C22⋊C412yes—
264C2×C4C2×C4×Q812yes—
265C2×C4C2×C4×Q812yes—
266C2×C4C2×#12012yes—
267C2×C4C2×C22⋊Q812yes—
268C2×C4C2×C22⋊Q814yes—
269C2×C4C2×C22⋊Q814yes—
270C2×C4C2×#12411yescharacteristic
271C2×C4C2×#12414yes—
272C2×C4C2×C4⋊Q811yescharacteristic
273C2×C4C2×C4⋊Q814yes—
274C8—48no—
275C22#144618yes—
276C22#1788116yes—
277C22#1989116yes—
278C22#199118yes—
279C22#199218yes—
280C22#2025116yes—
281C22#2026116yes—
282C22#2059132yes—
283C22C2×#37911yescharacteristic
284C22C2×#39111yescharacteristic
285C22C2×#39111yescharacteristic
286C22C2×#39312yes—
287C22C2×#39512yes—
288C22C2×#41612yes—
289C22C2×#41712yes—
290C22C2×#44914yes—
291C22C2×#45214yes—
292C22C2×#47814yes—
293C22C2×#47914yes—
294C22C2×#48812yes—
295C22C2×#49011yescharacteristic
296C22C2×#49111yescharacteristic
297C22C2×#49912yes—
298C22C2×#50112yes—
299C4—48no—
300C4—416no—
301C4—88no—
302C4#144618yes—
303C4C2×#39114yes—
304C4C2×#39312yes—
305C4C2×#39512yes—
306C2#11385116yesdirect product factor
307C2C2×#144611yescharacteristic
308C2C2×#178812yes—
309C2C2×#198912yes—
310C2C2×#199111yescharacteristic
311C2C2×#199211yescharacteristic
312C2C2×#202512yes—
313C2C2×#202612yes—
314C2C2×#205914yes—
subgroupG/H
center, socle, prime radicalC25C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C25 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11385C2direct (N × H)
10. C22×#125C4semidirect
39. C2×#125C2×C4semidirect
306. C2#11385direct (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×#3402
C22×#3362
C2×C4⋊#952
#113852