Finite Group Databasebeta

← C2×#8044 #181587

Subgroups — C2×#8044 #181587

Order 512.

1994 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2C23×C4⋊C8C211yesmaximal, characteristic
3C2×#2285C211yesmaximal, characteristic
4#8044C218yesmaximal, direct product factor
5C2×#1296C214yesmaximal, complemented
6C24×C8—21no—
7C22×#117C412yes—
8C23×C42C2211yescharacteristic
9C2×#432—24no—
10C2×#432—22no—
11C2×#432C418yescomplemented
12C2×#432C2214yes—
13C22×C4⋊C8—24no—
14C22×C4⋊C8—21no—
15C22×C4⋊C8—22no—
16C22×C4⋊C8C418yescomplemented
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2212yes—
19C22×C4⋊C8C2214yes—
20C22×C22⋊C8—21no—
21C22×#97—21no—
22#2285—22no—
23#2285—22no—
24#2285C414yes—
25#2285C2214yes—
26C2×#384C2212yes—
27C2×#382C412yes—
28C2×#380—21no—
29C2×#380—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32#1296C22116yescomplemented
33C24×C4C2×C411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C23×C8—28no—
37C23×C8—22no—
38C23×C8—22no—
39C23×C8—24no—
40C23×C8—48no—
41C2×#117—24no—
42C2×#117—48no—
43C2×#117C2×C418yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2×C414yes—
47C22×C22⋊C4—22no—
48C22×C42—22no—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C412yes—
51C22×C42C2×C414yes—
52C22×C42C2×C414yes—
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D412yes—
56C22×C42D412yes—
57C22×C42D414yes—
58C22×C42Q814yes—
59C22×C42C2311yescharacteristic
60C22×C42C2312yes—
61#432—216no—
62#432—28no—
63#432C2×C4116yescomplemented
64C2×C4⋊C8—216no—
65C2×C4⋊C8—28no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—28no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8C2×C418yes—
73C2×C4⋊C8C2×C418yes—
74C2×C4⋊C8C2×C4116yescomplemented
75C2×C4⋊C8C2314yes—
76C2×C4⋊C8C2314yes—
77C2×C22⋊C8—28no—
78C2×C22⋊C8—24no—
79C2×#97—28no—
80C2×#97—24no—
81#384—22no—
82#384—22no—
83#384C2×C414yes—
84#384C2314yes—
85#382—24no—
86#382C2×C418yes—
87#380—24no—
88#380—24no—
89#380—24no—
90#380—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95C2×#95—22no—
96C2×#95—22no—
97C25C22⋊C411yescharacteristic
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—28no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4C4214yes—
105C23×C4C22⋊C412yes—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C414yes—
112C23×C4C22⋊C414yes—
113C23×C4C4⋊C414yes—
114C23×C4C4⋊C414yes—
115C23×C4C4⋊C414yes—
116C23×C4C22×C412yes—
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C22×C8—28no—
123C22×C8—216no—
124C22×C8—28no—
125C22×C8—24no—
126C22×C8—24no—
127C22×C8—416no—
128C22×C8—432no—
129C22×C8—48no—
130C22×C8—48no—
131#117—416no—
132#117—416no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—48no—
139C2×C4⋊C4—48no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4C22×C418yes—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—48no—
145C2×C42—28no—
146C2×C42—48no—
147C2×C42C4214yes—
148C2×C42C4218yes—
149C2×C42C22⋊C412yes—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C412yes—
152C2×C42C22⋊C414yes—
153C2×C42C22⋊C418yes—
154C2×C42C22⋊C414yes—
155C2×C42C22⋊C418yes—
156C2×C42C4⋊C414yes—
157C2×C42C4⋊C418yes—
158C2×C42C4⋊C418yes—
159C2×C42C4⋊C418yes—
160C2×C42C4⋊C414yes—
161C2×C42C4⋊C414yes—
162C2×C42C22×C412yes—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C412yes—
165C2×C42C22×C412yes—
166C2×C42C22×C414yes—
167C2×C42C22×C414yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D412yes—
173C2×C42C2×D414yes—
174C2×C42C2×Q814yes—
175C2×C42C2×Q812yes—
176C2×C42C2411yescharacteristic
177C4⋊C8—28no—
178C4⋊C8—216no—
179C4⋊C8—28no—
180C4⋊C8—24no—
181C4⋊C8—24no—
182C4⋊C8C22×C418yes—
183C4⋊C8C22×C418yes—
184C22⋊C8—216no—
185#97—216no—
186#95—28no—
187#95—28no—
188C24—24no—
189C24—24no—
190C24#9514yes—
191C24#10014yes—
192C24#10114yes—
193C24C2×C22⋊C412yes—
194C24C2×C22⋊C411yescharacteristic
195C22×C4—216no—
196C22×C4—216no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—216no—
212C22×C4—28no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—216no—
216C22×C4—216no—
217C22×C4—28no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—48no—
221C22×C4#9514yes—
222C22×C4#9514yes—
223C22×C4#9518yes—
224C22×C4#9518yes—
225C22×C4#9518yes—
226C22×C4#9514yes—
227C22×C4C2×C4214yes—
228C22×C4C2×C4212yes—
229C22×C4C2×C22⋊C414yes—
230C22×C4C2×C22⋊C414yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C4⋊C414yes—
240C22×C4C2×C4⋊C414yes—
241C22×C4C2×C4⋊C414yes—
242C22×C4C2×C4⋊C412yes—
243C22×C4C2×C4⋊C412yes—
244C22×C4C2×C4⋊C412yes—
245C22×C4#13518yes—
246C22×C4AGL(1,ℤ/8)14yes—
247C22×C4#13714yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C2×C8—28no—
252C2×C8—28no—
253C2×C8—432no—
254C2×C8—48no—
255C2×C8—48no—
256C2×C8—416no—
257C2×C8—416no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—48no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—48no—
264C4⋊C4—48no—
265C22⋊C4—416no—
266C42—416no—
267C42#9518yes—
268C42#9518yes—
269C42#9518yes—
270C42C2×C4214yes—
271C42C2×C4212yes—
272C42C2×C22⋊C414yes—
273C42C2×C22⋊C412yes—
274C42C2×C22⋊C414yes—
275C42C2×C22⋊C412yes—
276C42C2×C4⋊C414yes—
277C42C2×C4⋊C412yes—
278C42C2×C4⋊C414yes—
279C42C2×C4⋊C412yes—
280C42C2×C4⋊C414yes—
281C42C2×C4⋊C412yes—
282C42C23×C411yescharacteristic
283C42C23×C411yescharacteristic
284C42C22×D411yescharacteristic
285C42C22×Q811yescharacteristic
286C23—216no—
287C23—216no—
288C23—24no—
289C23—24no—
290C23—28no—
291C23—28no—
292C23#343116yes—
293C23C2×#9514yes—
294C23C2×#9512yes—
295C23C2×#10014yes—
296C23C2×#10012yes—
297C23C2×#10114yes—
298C23C2×#10112yes—
299C23#41614yes—
300C23#41714yes—
301C23C22×C22⋊C411yescharacteristic
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—28no—
311C2×C4—28no—
312C2×C4—216no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—216no—
324C2×C4—216no—
325C2×C4—48no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—816no—
329C2×C4C2×#9514yes—
330C2×C4C2×#9514yes—
331C2×C4C2×#9514yes—
332C2×C4C2×#9512yes—
333C2×C4C2×#9512yes—
334C2×C4C2×#9512yes—
335C2×C4#41618yes—
336C2×C4#41614yes—
337C2×C4#41714yes—
338C2×C4#41814yes—
339C2×C4#41914yes—
340C2×C4#42718yes—
341C2×C4#42818yes—
342C2×C4C22×C4211yescharacteristic
343C2×C4C22×C22⋊C411yescharacteristic
344C2×C4C22×C22⋊C411yescharacteristic
345C2×C4C22×C4⋊C411yescharacteristic
346C2×C4C22×C4⋊C411yescharacteristic
347C2×C4C22×C4⋊C411yescharacteristic
348C2×C4C2×#13514yes—
349C2×C4C2×AGL(1,ℤ/8)12yes—
350C2×C4C2×#13712yes—
351C8—48no—
352C8—48no—
353C22—28no—
354C22—28no—
355C22—232no—
356C22—24no—
357C22—24no—
358C22#155118yes—
359C22#155218yes—
360C22#173514yes—
361C22#173614yes—
362C22C2×#34318yes—
363C22C2×#34314yes—
364C22C22×#9511yescharacteristic
365C22C22×#10011yescharacteristic
366C22C22×#10111yescharacteristic
367C22C2×#41612yes—
368C22C2×#41712yes—
369C4—24no—
370C4—24no—
371C4—21no—
372C4—24no—
373C4—22no—
374C4—21no—
375C4—88no—
376C4#173514yes—
377C4#173614yes—
378C4C2×#41612yes—
379C4C2×#41811yescharacteristic
380C4C2×#41911yescharacteristic
381C4C2×#42712yes—
382C4C2×#42812yes—
383C2—28no—
384C2#804418yesdirect product factor
385C2C2×#155112yes—
386C2C2×#155212yes—
387C2C2×#173511yescharacteristic
388C2C2×#173611yescharacteristic
389C2C22×#34311yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8044C2direct (N × H)
5. C2×#1296C2semidirect
11. C2×#432C4semidirect
16. C22×C4⋊C8C4semidirect
32. #1296C22semidirect
63. #432C2×C4semidirect
74. C2×C4⋊C8C2×C4semidirect
384. C2#8044direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4322
C23×C4⋊C82
C2×#22852
#80442
C2×#12962