Finite Group Databasebeta

← C2×#9026 #194708

Subgroups — C2×#9026 #194708

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3C2×#1823C212yesmaximal
4C22×#324C211yesmaximal, characteristic
5#9026C218yesmaximal, direct product factor
6C2×#1294C212yesmaximal
7C23×C4⋊C4—21no—
8C23×C4⋊C4C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11C2×#398—24no—
12C2×#398—22no—
13C2×#398C418yescomplemented
14C2×#398C2214yes—
15#2313—24no—
16#2313—22no—
17#2313C2214yes—
18C2×#387—24no—
19C2×#381—21no—
20C2×#380C2212yes—
21C2×C4×C22⋊C4—21no—
22C22×#95—21no—
23C22×#95C2211yescharacteristic
24#1823—24no—
25#1823C2218yes—
26C2×#340—24no—
27C2×#324—24no—
28C2×#324C2214yes—
29#1294C2218yes—
30C24×C4—21no—
31C24×C4—21no—
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C24×C4C2311yescharacteristic
35C23×C8—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—24no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—28no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—48no—
47C22×C4⋊C4C2×C414yes—
48C22×C4⋊C4C2×C414yes—
49C22×C4⋊C4D412yes—
50C22×C4⋊C4D414yes—
51C22×C4⋊C4D414yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C22⋊C4—22no—
55C22×C42—22no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8C2314yes—
60#398—216no—
61#398—28no—
62#398C2×C4116yescomplemented
63#387—216no—
64#387—216no—
65#381—24no—
66#381—24no—
67#380—22no—
68#380—22no—
69#380D414yes—
70#380C2314yes—
71C4×C22⋊C4—24no—
72C4×C22⋊C4—24no—
73C2×#95—22no—
74C2×#95—24no—
75C2×#95—28no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—24no—
79C2×#95—22no—
80C2×#95C2×C414yes—
81C2×#95C2312yes—
82#340—216no—
83#324—28no—
84C25C2×D411yescharacteristic
85C23×C4—28no—
86C23×C4—24no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—22no—
99C23×C4—44no—
100C23×C4—44no—
101C23×C4—44no—
102C23×C4—44no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—42no—
106C23×C4—44no—
107C23×C4—44no—
108C23×C4C22⋊C414yes—
109C23×C4C22⋊C414yes—
110C23×C4C22×C412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4#4114yes—
118C23×C4C2411yescharacteristic
119C22×C8—24no—
120C22×C8—48no—
121C22×C8—44no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—216no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—416no—
139C2×C4⋊C4—432no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—48no—
145C2×C4⋊C4C22⋊C418yes—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C2×D414yes—
149C2×C4⋊C4C2×D418yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C42—28no—
155C2×C42—84no—
156C22⋊C8—24no—
157C22⋊C8—24no—
158C22⋊C8—44no—
159#95—28no—
160#95—216no—
161#95—216no—
162#95—24no—
163#95—24no—
164#95—22no—
165#95—24no—
166#95C22×C414yes—
167C24—24no—
168C24—24no—
169C24C2×C22⋊C412yes—
170C24#12011yescharacteristic
171C24#12011yescharacteristic
172C24#12312yes—
173C24#13514yes—
174C24AGL(1,ℤ/8)14yes—
175C24C22×D411yescharacteristic
176C22×C4—28no—
177C22×C4—216no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—28no—
189C22×C4—28no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—28no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—48no—
199C22×C4—42no—
200C22×C4—432no—
201C22×C4—48no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—48no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—48no—
209C22×C4—48no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—48no—
221C22×C4—48no—
222C22×C4—84no—
223C22×C4—82no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—88no—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C414yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C4×D414yes—
231C22×C4C4×D414yes—
232C22×C4#12012yes—
233C22×C4#12114yes—
234C22×C4#12114yes—
235C22×C4#12312yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D412yes—
238C22×C4C22×D411yescharacteristic
239C22×C4C2×#4112yes—
240C2×C8—48no—
241C2×C8—44no—
242C2×C8—88no—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4—216no—
246C4⋊C4—28no—
247C4⋊C4—48no—
248C4⋊C4—432no—
249C4⋊C4—44no—
250C4⋊C4—48no—
251C4⋊C4—84no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—48no—
255C42—88no—
256C23—216no—
257C23—216no—
258C23—24no—
259C23—24no—
260C23—24no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23#38612yes—
267C23#38612yes—
268C23#41614yes—
269C23#41814yes—
270C23#44914yes—
271C23#44914yes—
272C23#45614yes—
273C23#45814yes—
274C23#48218yes—
275C23C22×C22⋊C411yescharacteristic
276C23C2×#12011yescharacteristic
277C23C2×#12311yescharacteristic
278C23C2×#13512yes—
279C23C2×AGL(1,ℤ/8)12yes—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—432no—
297C2×C4—416no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—816no—
317C2×C4—88no—
318C2×C4—88no—
319C2×C4—84no—
320C2×C4#38614yes—
321C2×C4C2×C4×D412yes—
322C2×C4C2×#12112yes—
323C8—84no—
324C22—28no—
325C22—28no—
326C22—216no—
327C22—24no—
328C22—24no—
329C22—28no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22#1609116yes—
337C22#187514yes—
338C22#187514yes—
339C22#188614yes—
340C22#189014yes—
341C22C2×#38611yescharacteristic
342C22C2×#41611yescharacteristic
343C22C2×#41811yescharacteristic
344C22C2×#44912yes—
345C22C2×#45612yes—
346C22C2×#45812yes—
347C22C2×#48212yes—
348C4—22no—
349C4—22no—
350C4—24no—
351C4—48no—
352C4—44no—
353C4—44no—
354C4—84no—
355C4—88no—
356C2—21no—
357C2—22no—
358C2—21no—
359C2—24no—
360C2#902618yesdirect product factor
361C2C2×#160914yes—
362C2C2×#187511yescharacteristic
363C2C2×#188611yescharacteristic
364C2C2×#189011yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9026C2direct (N × H)
13. C2×#398C4semidirect
62. #398C2×C4semidirect
360. C2#9026direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3982
C2×#23132
C2×#18232
C22×#3242
#90262
C2×#12942