Finite Group Databasebeta

← C2×#30092 #6434780

Subgroups — C2×#30092 #6434780

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#30092C2116yesmaximal, direct product factor
4C2×#2100C212yesmaximal
5C2×#2099C212yesmaximal
6C2×#2084C212yesmaximal
7C2×#2058C212yesmaximal
8C2×#1920C214yesmaximal
9C2×#1839C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#125—21no—
12C22×C4×Q8—21no—
13C22×#106C2214yes—
14#2859—24no—
15#2859C2218yes—
16C4×C4⋊Q8—24no—
17C4×C4⋊Q8C2218yes—
18C2×#391—21no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389—21no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2212yes—
31C2×C43C2211yescharacteristic
32#2100C22116yes—
33#2099C22116yes—
34#2084C22116yes—
35#2058C22116yes—
36#1920C22132yes—
37#1839C2218yes—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C23×Q8—21no—
45C23×C8C2314yes—
46C2×C4⋊Q8—28no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2314yes—
49C2×#125—28no—
50C2×#125—48no—
51C2×C4×Q8—28no—
52C2×C4×Q8—48no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C42—22no—
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#106—216no—
76C2×#106C23116yes—
77#391—28no—
78#389—24no—
79#389—28no—
80#389—24no—
81#389—24no—
82#389—28no—
83#389—28no—
84#389C2318yes—
85#389C2318yes—
86#389C2318yes—
87#384—22no—
88#384—22no—
89#384—22no—
90#384C2314yes—
91#384C2314yes—
92#384C2314yes—
93#382—28no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—28no—
96C4×C4⋊C4C2318yes—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C43D414yes—
102C43C2314yes—
103#340C23116yes—
104#340C23116yes—
105#340C23116yes—
106#336C2318yes—
107#336C2318yes—
108#336C2318yes—
109C22×Q8—28no—
110C22×Q8—24no—
111C22×Q8—44no—
112C22×Q8—44no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—28no—
125C22×C8—28no—
126C22×C8—28no—
127C22×C8C2418yes—
128C4⋊Q8—216no—
129C4⋊Q8—216no—
130#125—432no—
131C4×Q8—432no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4#4118yes—
161C2×C4⋊C4C2414yes—
162C2×C4⋊C4C2414yes—
163C2×C4⋊C4C2414yes—
164C2×C4⋊C4C2414yes—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—28no—
171C2×C42—42no—
172C2×C42—48no—
173C2×C42SD1618yes—
174C2×C42C2×D414yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D414yes—
179C2×C42C2×D414yes—
180C2×C42C2×D414yes—
181C2×C42C2412yes—
182C2×C42C2412yes—
183C2×C42C2412yes—
184#106—232no—
185#95—28no—
186#95—28no—
187#95—28no—
188#95—28no—
189C24C22×D411yescharacteristic
190C2×Q8—216no—
191C2×Q8—432no—
192C2×Q8—48no—
193C2×Q8—48no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—28no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4C2×SD1614yes—
224C22×C4C2×SD1614yes—
225C22×C4C2×SD1614yes—
226C22×C4AGL(1,ℤ/8)14yes—
227C22×C4#13714yes—
228C22×C4C22×D412yes—
229C22×C4C22×D412yes—
230C22×C4C22×D412yes—
231C22×C4C22×D412yes—
232C22×C4C22×D412yes—
233C22×C4C22×D412yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4114yes—
236C22×C42−1+414yes—
237C22×C42−1+414yes—
238C22×C4C2511yescharacteristic
239C2×C8—28no—
240C2×C8—28no—
241C2×C8—28no—
242C2×C8—416no—
243C4⋊C4—216no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C4⋊C4—48no—
252C4⋊C4—416no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—416no—
257C4⋊C4—48no—
258C4⋊C4—48no—
259C4⋊C4—416no—
260C4⋊C4—48no—
261C4⋊C4C2×#4118yes—
262C42—24no—
263C42—24no—
264C42—24no—
265C42—28no—
266C42—28no—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—44no—
271C42—416no—
272C42C2×SD16116yes—
273C23#52512yes—
274C23#53612yes—
275C23#53612yes—
276C23C22×SD1612yes—
277C23C2×AGL(1,ℤ/8)12yes—
278C23C2×#13712yes—
279C23#57512yes—
280C23C23×D411yescharacteristic
281Q8—432no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—28no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—416no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—88no—
315C2×C4—88no—
316C2×C4—88no—
317C2×C4#52512yes—
318C2×C4#53612yes—
319C2×C4#53612yes—
320C2×C4#541116yes—
321C2×C4C22×SD1614yes—
322C2×C4C22×SD1614yes—
323C2×C4C22×SD1614yes—
324C2×C4C2×AGL(1,ℤ/8)14yes—
325C2×C4C2×#13714yes—
326C2×C4C22×#4112yes—
327C2×C4C2×2−1+412yes—
328C2×C4C2×2−1+412yes—
329C8—48no—
330C22#263414yes—
331C22#310114yes—
332C22#311214yes—
333C22#322714yes—
334C22#322914yes—
335C22#323514yes—
336C22#323614yes—
337C22C2×#52511yescharacteristic
338C22C2×#53611yescharacteristic
339C22C2×#53611yescharacteristic
340C22C23×SD1611yescharacteristic
341C22C22×AGL(1,ℤ/8)11yescharacteristic
342C22C22×#13711yescharacteristic
343C22C2×#57511yescharacteristic
344C4—24no—
345C4—24no—
346C4—24no—
347C4—24no—
348C4—24no—
349C4—48no—
350C4—42no—
351C4—84no—
352C4—84no—
353C4—84no—
354C4#263414yes—
355C4C2×#54114yes—
356C2#3009218yesdirect product factor
357C2C2×#263411yescharacteristic
358C2C2×#310111yescharacteristic
359C2C2×#311211yescharacteristic
360C2C2×#322911yescharacteristic
361C2C2×#322711yescharacteristic
362C2C2×#323611yescharacteristic
363C2C2×#323511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30092C2direct (N × H)
356. C2#30092direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28592
C2×C4×C4⋊Q82
#300922
C2×#21002
C2×#20992
C2×#20842
C2×#20582
C2×#19202
C2×#18392