Finite Group Databasebeta

← C2×#22824 #6380690

Subgroups — C2×#22824 #6380690

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2295C211yesmaximal, characteristic
2#22824C2116yesmaximal, direct product factor
3C2×#1985C212yesmaximal
4C2×#1917C212yesmaximal
5C2×#1916C212yesmaximal
6C2×#1842C211yesmaximal, characteristic
7C2×#1832C212yesmaximal
8C2×#1771C212yesmaximal
9C2×#1767C211yesmaximal, characteristic
10C2×#1768C212yesmaximal
11C22×C4×Q8C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×#97C2212yes—
19C22×#97C2212yes—
20C22×C4×C8C2212yes—
21#2295C418yes—
22#2295C2218yes—
23C2×#391C411yescharacteristic
24C2×#391C411yescharacteristic
25C2×#389C411yescharacteristic
26C2×#389C411yescharacteristic
27C2×#387C412yes—
28C2×#387C412yes—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×#376C2212yes—
32C2×C43C2211yescharacteristic
33#1985C22116yes—
34#1917C22116yes—
35#1916C22116yes—
36#1842C2218yes—
37#1832C22116yes—
38#1771C22116yes—
39#1768C22116yes—
40#1767C2218yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C23×Q8C2×C411yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×Q8—24no—
54C2×C4×Q8—24no—
55C2×C4×Q8C2×C414yes—
56C2×C4×Q8C2314yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—28no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—28no—
75C2×C4⋊C8—42no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8Q818yescomplemented
78C2×C4⋊C8C2314yes—
79C2×C4⋊C8C2318yes—
80C2×C4⋊C8C2318yes—
81C2×C4⋊C8C2314yes—
82C2×C4⋊C8C2314yes—
83C2×C4⋊C8C2314yes—
84C2×#97—28no—
85C2×#97—44no—
86C2×#97C2318yes—
87C2×#97C2318yes—
88C2×C4×C8—44no—
89C2×C4×C8C2318yes—
90#391C2×C418yes—
91#391C2×C418yes—
92#389C2×C418yes—
93#389C2×C418yes—
94#387C2×C4116yes—
95#387C2×C4116yes—
96C4×C4⋊C4C2×C414yes—
97C4×C4⋊C4C2×C418yes—
98C4×C4⋊C4C2318yes—
99C4×C4⋊C4C2314yes—
100#376C2×C418yes—
101#376C2318yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109C43C2×C414yes—
110C43C2314yes—
111#336C2318yes—
112#336C2318yes—
113#336C2318yes—
114#336C2318yes—
115#336C2318yes—
116#336C2318yes—
117#336C2318yes—
118C22×Q8—24no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8C22×C414yes—
122C23×C4C22×C412yes—
123C23×C4C22×C412yes—
124C23×C4C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—24no—
134C22×C8—24no—
135C22×C8—24no—
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—24no—
144C22×C8C2×Q818yes—
145C22×C8C2414yes—
146C22×C8C2414yes—
147C22×C8C2414yes—
148C22×C8C2414yes—
149C4×Q8—216no—
150C4×Q8—28no—
151C4×Q8—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4C22×C418yes—
159C2×C4⋊C4C22×C414yes—
160C2×C4⋊C4C22×C418yes—
161C2×C4⋊C4C22×C418yes—
162C2×C4⋊C4C22×C414yes—
163C2×C4⋊C4C22×C414yes—
164C2×C42—24no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—24no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42C22×C414yes—
175C2×C42C22×C412yes—
176C2×C42C22×C414yes—
177C2×C42C22×C414yes—
178C2×C42C22×C414yes—
179C2×C42C22×C414yes—
180C2×C42C22×C414yes—
181C2×C42C22×C412yes—
182C2×C42C22×C412yes—
183C2×C42C2×Q814yes—
184C2×C42#4114yes—
185C2×C42#4114yes—
186C2×C42#4114yes—
187C2×C42C2414yes—
188C2×C42C2412yes—
189C2×C42C2414yes—
190C2×C42C2412yes—
191C2×C42C2412yes—
192C4⋊C8—216no—
193C4⋊C8—28no—
194C4⋊C8—216no—
195C4⋊C8—44no—
196C4⋊C8—44no—
197C4⋊C8C2×Q8116yescomplemented
198#97—216no—
199#97—48no—
200C4×C8—48no—
201#95C22×C418yes—
202#95C22×C418yes—
203#95C22×C418yes—
204#95C22×C418yes—
205#95C22×C418yes—
206#95C22×C418yes—
207#95C22×C418yes—
208C24C23×C411yescharacteristic
209C2×Q8—28no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—416no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4#11714yes—
239C22×C4#11714yes—
240C22×C4C4×Q814yes—
241C22×C4C4×Q814yes—
242C22×C4#13114yes—
243C22×C4C23×C414yes—
244C22×C4C23×C414yes—
245C22×C4C23×C414yes—
246C22×C4C23×C412yes—
247C22×C4C23×C414yes—
248C22×C4C23×C412yes—
249C22×C4C23×C412yes—
250C22×C4C23×C412yes—
251C22×C4C23×C412yes—
252C22×C4C23×C412yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×Q812yes—
255C22×C4C2×#4112yes—
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C22×C4H(5,𝔽2)12yes—
259C22×C42−1+412yes—
260C22×C4C2511yescharacteristic
261C2×C8—216no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—48no—
274C2×C8—48no—
275C2×C8—48no—
276C2×C8C22×Q818yes—
277C4⋊C4—28no—
278C4⋊C4—216no—
279C4⋊C4—216no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—48no—
283C42—28no—
284C42—24no—
285C42—28no—
286C42—28no—
287C42—24no—
288C42—24no—
289C42—24no—
290C42—22no—
291C42—22no—
292C42—24no—
293C42—44no—
294C42—44no—
295C42—44no—
296C42#11718yes—
297C42C4×Q818yes—
298C42C22×Q814yes—
299C42C2×#4114yes—
300C42C2×#4114yes—
301C42C2×#4114yes—
302C23C2×#11712yes—
303C23C2×C4×Q812yes—
304C23#52012yes—
305C23C2×#13112yes—
306C23#56812yes—
307C23#56812yes—
308C23#56812yes—
309C23C24×C411yescharacteristic
310Q8—416no—
311C2×C4—22no—
312C2×C4—28no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4C2×#11714yes—
347C2×C4C2×#11714yes—
348C2×C4C2×#11712yes—
349C2×C4C2×C4×Q814yes—
350C2×C4C2×C4×Q814yes—
351C2×C4C2×C4×Q812yes—
352C2×C4#52012yes—
353C2×C4#55414yes—
354C2×C4#55614yes—
355C2×C4#55714yes—
356C2×C4#55914yes—
357C2×C4C2×#13114yes—
358C2×C4C23×Q811yescharacteristic
359C2×C4C22×#4111yescharacteristic
360C2×C4C22×#4111yescharacteristic
361C2×C4C22×#4111yescharacteristic
362C2×C4C2×H(5,𝔽2)11yescharacteristic
363C2×C4C2×2−1+411yescharacteristic
364C8—28no—
365C8—44no—
366C8—44no—
367C8—44no—
368C22#235514yes—
369C22#292114yes—
370C22#292514yes—
371C22#296114yes—
372C22#296214yes—
373C22#299214yes—
374C22#299514yes—
375C22C22×#11711yescharacteristic
376C22C22×C4×Q811yescharacteristic
377C22C2×#52011yescharacteristic
378C22C22×#13111yescharacteristic
379C22C2×#56811yescharacteristic
380C22C2×#56811yescharacteristic
381C22C2×#56811yescharacteristic
382C4—24no—
383C4—24no—
384C4—24no—
385C4—24no—
386C4—24no—
387C4—44no—
388C4—42no—
389C4—44no—
390C4—44no—
391C4—44no—
392C4#235514yes—
393C4C2×#55411yescharacteristic
394C4C2×#55611yescharacteristic
395C4C2×#55711yescharacteristic
396C4C2×#55911yescharacteristic
397C2#2282418yesdirect product factor
398C2C2×#235511yescharacteristic
399C2C2×#292111yescharacteristic
400C2C2×#292511yescharacteristic
401C2C2×#296111yescharacteristic
402C2C2×#296211yescharacteristic
403C2C2×#299211yescharacteristic
404C2C2×#299511yescharacteristic
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socle, prime radicalC24C23×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22824C2direct (N × H)
77. C2×C4⋊C8Q8semidirect
197. C4⋊C8C2×Q8semidirect
397. C2#22824direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#22952
#228242
C2×#19852
C2×#19172
C2×#19162
C2×#18422
C2×#18322
C2×#17712
C2×#17672
C2×#17682