Finite Group Databasebeta

← C2×#23782416 #35228667

Subgroups — C2×#23782416 #35228667

Order 20864.

1024 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782416C2116yesmaximal, direct product factor
2C2×#19104734C211yesmaximal, characteristic, complemented
3C2×#19104733C211yesmaximal, characteristic, complemented
4C2×#19104730C211yesmaximal, characteristic, complemented
5C2×#19104730C211yesmaximal, characteristic, complemented
6C2×#19104728C211yesmaximal, characteristic, complemented
7C2×#19104728C211yesmaximal, characteristic, complemented
8C2×#19104726C211yesmaximal, characteristic, complemented
9C2×#19104725C211yesmaximal, characteristic, complemented
10C2×#19104721C211yesmaximal, characteristic
11C2×#19104718C211yesmaximal, characteristic
12C2×#19104717C211yesmaximal, characteristic
13C2×#19104714C211yesmaximal, characteristic, complemented
14C2×#19104713C211yesmaximal, characteristic, complemented
15C2×#19104711C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C163C211yesmaximal, characteristic, complemented
17C22×Dic326C2211yescharacteristic
18C22×Dic326C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21#19104734C2218yescomplemented
22#19104733C2218yescomplemented
23#19104730C2218yescomplemented
24#19104730C2218yescomplemented
25#19104728C2218yescomplemented
26#19104728C2218yescomplemented
27#19104726C2218yescomplemented
28#19104725C2218yescomplemented
29#19104721C2218yes—
30#19104718C2218yes—
31#19104717C2218yes—
32#19104714C2218yescomplemented
33#19104713C2218yescomplemented
34#19104711C2218yescomplemented
35C2×C4⋊Dic163C2211yescharacteristic
36C2×C4⋊Dic163C2211yescharacteristic
37C2×C4⋊Dic163C2211yescharacteristic
38C2×C4⋊Dic163C2211yescharacteristic
39C2×Dic163⋊C4C2211yescharacteristic
40C2×Dic163⋊C4C2211yescharacteristic
41C2×Dic163⋊C4C2211yescharacteristic
42C2×Dic163⋊C4C2211yescharacteristic
43C2×Dic163⋊C4C2211yescharacteristic
44C2×Dic163⋊C4C2211yescharacteristic
45C2×Dic163⋊C4C2211yescharacteristic
46C2×Dic163⋊C4C2211yescharacteristic
47C2×Dic163⋊C4C2211yescharacteristic
48C2×C22⋊Dic163C2211yescharacteristic
49C2×C22⋊Dic163C2211yescharacteristic
50C2×C22⋊Dic163C2211yescharacteristic
51C2×(C2×D163)⋊C4C2211yescharacteristic
52C2×(C2×D163)⋊C4C2211yescharacteristic
53C2×(C2×D163)⋊C4C2211yescharacteristic
54C2×(C2×D163)⋊C4C2211yescharacteristic
55C2×(C2×D163)⋊C4C2211yescharacteristic
56C2×C4×Dic163C2211yescharacteristic
57C2×C4×Dic163C2211yescharacteristic
58C2×C4×Dic163C2211yescharacteristic
59C22×Q8×C163C2211yescharacteristic
60C23×C4×C163C2211yescharacteristic
61C22⋊Q8×C163C2218yescomplemented
62C2×C4⋊C4×C163C2211yescharacteristic
63C2×C4⋊C4×C163C2211yescharacteristic
64C2×C4⋊C4×C163C2211yescharacteristic
65C2×C22⋊C4×C163C2211yescharacteristic
66C2×C22⋊C4×C163C2211yescharacteristic
67C23×D163C2311yescharacteristic
68C2×Dic326—42no—
69C2×Dic326—42no—
70C2×Dic326C2314yes—
71C2×Dic326C2314yes—
72C2×C22⋊D163—42no—
73C2×C22⋊D163C2314yes—
74C22×Dic163C2311yescharacteristic
75C22×Dic163C2311yescharacteristic
76C22×Dic163C2311yescharacteristic
77C22×Dic163C2311yescharacteristic
78C22×Dic163C2311yescharacteristic
79C22×Dic163C2311yescharacteristic
80C22×Dic163C2311yescharacteristic
81C2×C4×D163—42no—
82C2×C4×D163C2314yes—
83C4⋊Dic163C2314yes—
84C4⋊Dic163C2314yes—
85C4⋊Dic163C2314yes—
86C4⋊Dic163C2314yes—
87Dic163⋊C4C2314yes—
88Dic163⋊C4C2314yes—
89Dic163⋊C4C2314yes—
90Dic163⋊C4C2314yes—
91Dic163⋊C4C2314yes—
92Dic163⋊C4C2314yes—
93Dic163⋊C4C2314yes—
94Dic163⋊C4C2314yes—
95Dic163⋊C4C2314yes—
96C22⋊Dic163C2314yes—
97C22⋊Dic163C2314yes—
98C22⋊Dic163C2314yes—
99(C2×D163)⋊C4C2314yes—
100(C2×D163)⋊C4C2314yes—
101(C2×D163)⋊C4C2314yes—
102(C2×D163)⋊C4C2314yes—
103(C2×D163)⋊C4C2314yes—
104C4×Dic163C2314yes—
105C4×Dic163C2314yes—
106C4×Dic163C2314yes—
107C24×C163C2311yescharacteristic
108C2×Q8×C163—42no—
109C2×Q8×C163C2314yes—
110C22×C4×C163—42no—
111C22×C4×C163C2311yescharacteristic
112C22×C4×C163C2311yescharacteristic
113C22×C4×C163C2311yescharacteristic
114C22×C4×C163C2311yescharacteristic
115C22×C4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117C22×C4×C163C2314yes—
118C4⋊C4×C163C2314yes—
119C4⋊C4×C163C2314yes—
120C4⋊C4×C163C2314yes—
121C22⋊C4×C163C2314yes—
122C22⋊C4×C163C2314yes—
123C22×D163—22no—
124C22×D163—22no—
125C22×D163—22no—
126C22×D163C2412yes—
127Dic326—44no—
128Dic326—44no—
129C22⋊D163—44no—
130C2×Dic163—22no—
131C2×Dic163—22no—
132C2×Dic163—22no—
133C2×Dic163—22no—
134C2×Dic163—22no—
135C2×Dic163—22no—
136C2×Dic163—22no—
137C2×Dic163C2412yes—
138C2×Dic163C2412yes—
139C2×Dic163C2412yes—
140C2×Dic163C2412yes—
141C2×Dic163C2412yes—
142C2×Dic163C2412yes—
143C2×Dic163C2412yes—
144C4×D163—44no—
145C23×C163—22no—
146C23×C163—22no—
147C23×C163—22no—
148C23×C163C2412yes—
149C23×C163C2411yescharacteristic
150Q8×C163—44no—
151C2×C4×C163—22no—
152C2×C4×C163—22no—
153C2×C4×C163—22no—
154C2×C4×C163—22no—
155C2×C4×C163—22no—
156C2×C4×C163—22no—
157C2×C4×C163—44no—
158C2×C4×C163C2412yes—
159C2×C4×C163C2412yes—
160C2×C4×C163C2412yes—
161C2×C4×C163C2412yes—
162C2×C4×C163C2412yes—
163C2×C4×C163C2412yes—
164C22×C163—22no—
165C22×C163—22no—
166C22×C163—22no—
167C22×C163—44no—
168C22×C163H(5,𝔽2)12yes—
169C22×C1632−1+412yes—
170C22×C1632−1+412yes—
171C22×C163C2511yescharacteristic
172C2×D163—22no—
173C2×D163—22no—
174C2×D163—22no—
175C2×D163—44no—
176C4×C163—22no—
177C4×C163—22no—
178C4×C163—22no—
179C4×C163—22no—
180C4×C163—22no—
181C4×C163—22no—
182Dic163—22no—
183Dic163—22no—
184Dic163—22no—
185Dic163—22no—
186Dic163—22no—
187Dic163—22no—
188Dic163—22no—
189C2×C163—42no—
190C2×C163#56314yescomplemented
191C2×C163C2×H(5,𝔽2)11yescharacteristic
192C2×C163C2×2−1+411yescharacteristic
193C2×C163C2×2−1+411yescharacteristic
194D163—42no—
195C163C2×#56311yescharacteristic, complemented, Sylow
196C2×#563—1631nomaximal, Sylow
197#563—16316no—
198C2×C4⋊Q8—1631no—
199C2×C4⋊Q8—1631no—
200C2×#126—1631no—
201C2×#126—1631no—
202C2×#126—1631no—
203C2×#126—1631no—
204C2×#125—1631no—
205C2×#125—1631no—
206C2×#124—1631no—
207C2×#123—1631no—
208C2×#123—1631no—
209C2×C22⋊Q8—1631no—
210C2×C22⋊Q8—1631no—
211C2×C22⋊Q8—1631no—
212C2×C22⋊Q8D16311yescharacteristic, complemented
213C22×Q8—1631no—
214C22×Q8—1631no—
215C22×Q8C2×D16311yescharacteristic
216C22×D4—1631no—
217C23×C4—1631no—
218C23×C4C2×D16311yescharacteristic
219C4⋊Q8—1638no—
220C4⋊Q8—1638no—
221#126—1638no—
222#126—1638no—
223#126—1638no—
224#126—1638no—
225#125—1638no—
226#125—1638no—
227#124—1638no—
228#123—1638no—
229#123—1638no—
230C22⋊Q8—1638no—
231C22⋊Q8—1638no—
232C22⋊Q8—1638no—
233C22⋊Q8C2×D16318yescomplemented
234C2×C4⋊C4—1631no—
235C2×C4⋊C4—1631no—
236C2×C4⋊C4—1631no—
237C2×C4⋊C4—1631no—
238C2×C4⋊C4—1631no—
239C2×C4⋊C4—1631no—
240C2×C4⋊C4—1631no—
241C2×C4⋊C4—1631no—
242C2×C4⋊C4—1631no—
243C2×C4⋊C4—1631no—
244C2×C4⋊C4—1631no—
245C2×C4⋊C4—1631no—
246C2×C4⋊C4—1631no—
247C2×C4⋊C4C2×D16311yescharacteristic
248C2×C4⋊C4C2×D16311yescharacteristic
249C2×C4⋊C4C2×D16311yescharacteristic
250C2×C22⋊C4—1631no—
251C2×C22⋊C4—1631no—
252C2×C22⋊C4—1631no—
253C2×C22⋊C4—1631no—
254C2×C22⋊C4—1631no—
255C2×C22⋊C4—1631no—
256C2×C22⋊C4—1631no—
257C2×C22⋊C4—1631no—
258C2×C22⋊C4C2×D16311yescharacteristic
259C2×C22⋊C4C2×D16311yescharacteristic
260C2×C42—1631no—
261C2×C42—1631no—
262C2×C42—1631no—
263C24—1631no—
264C24C22×D16311yescharacteristic
265C2×Q8—42no—
266C2×Q8—6522no—
267C2×Q8—6522no—
268C2×Q8—1634no—
269C2×Q8—1634no—
270C2×Q8C22×D16314yes—
271C2×D4—6522no—
272C2×D4—1634no—
273C22×C4—42no—
274C22×C4—6522no—
275C22×C4—1631no—
276C22×C4—1631no—
277C22×C4—1631no—
278C22×C4—1631no—
279C22×C4—1631no—
280C22×C4—1631no—
281C22×C4—1631no—
282C22×C4—1634no—
283C22×C4C22×D16311yescharacteristic
284C22×C4C22×D16311yescharacteristic
285C22×C4C22×D16311yescharacteristic
286C22×C4C22×D16311yescharacteristic
287C22×C4C22×D16311yescharacteristic
288C22×C4C22×D16311yescharacteristic
289C22×C4C22×D16314yes—
290C4⋊C4—1634no—
291C4⋊C4—1634no—
292C4⋊C4—1634no—
293C4⋊C4—1634no—
294C4⋊C4—1634no—
295C4⋊C4—1634no—
296C4⋊C4—1634no—
297C4⋊C4—1634no—
298C4⋊C4—1634no—
299C4⋊C4—1634no—
300C4⋊C4—1634no—
301C4⋊C4—1634no—
302C4⋊C4—1634no—
303C4⋊C4C22×D16314yes—
304C4⋊C4C22×D16314yes—
305C4⋊C4C22×D16314yes—
306C22⋊C4—1634no—
307C22⋊C4—1634no—
308C22⋊C4—1634no—
309C22⋊C4—1634no—
310C22⋊C4—1634no—
311C22⋊C4—1634no—
312C22⋊C4—1634no—
313C22⋊C4—1634no—
314C22⋊C4C22×D16314yes—
315C22⋊C4C22×D16314yes—
316C42—1634no—
317C42—1634no—
318C42—1634no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—3262no—
323C23—3262no—
324C23—3262no—
325C23—1632no—
326C23C23×D16312yes—
327C23C23×D16311yescharacteristic
328Q8—44no—
329Q8—6524no—
330Q8—6524no—
331D4—6524no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—44no—
339C2×C4—3262no—
340C2×C4—3262no—
341C2×C4—3262no—
342C2×C4—3262no—
343C2×C4—3262no—
344C2×C4—3262no—
345C2×C4—3262no—
346C2×C4—6524no—
347C2×C4—1632no—
348C2×C4—1632no—
349C2×C4—1632no—
350C2×C4—1632no—
351C2×C4—1632no—
352C2×C4—1632no—
353C2×C4—1632no—
354C2×C4C23×D16312yes—
355C2×C4C23×D16312yes—
356C2×C4C23×D16312yes—
357C2×C4C23×D16312yes—
358C2×C4C23×D16312yes—
359C2×C4C23×D16312yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—3262no—
365C22—3262no—
366C22—3262no—
367C22—6524no—
368C22#1910478012yes—
369C22#1910478212yes—
370C22#1910478312yes—
371C22C24×D16311yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—3262no—
379C4—3262no—
380C4—3262no—
381C4—3262no—
382C4—3262no—
383C4—3262no—
384C4—3262no—
385C2—42no—
386C2—6522no—
387C2#2378241614yesdirect product factor
388C2C2×#1910478011yescharacteristic
389C2C2×#1910478211yescharacteristic
390C2C2×#1910478311yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×C22⋊Q8×C163C2
socleC23×C163C24
prime radicalC22×C22⋊D163C22
hypercenterC2×C22⋊Q8D163
hypocenterC163C2×#563
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #23782416C2direct (N × H)
2. C2×#19104734C2semidirect
3. C2×#19104733C2semidirect
4. C2×#19104730C2semidirect
5. C2×#19104730C2semidirect
6. C2×#19104728C2semidirect
7. C2×#19104728C2semidirect
8. C2×#19104726C2semidirect
9. C2×#19104725C2semidirect
13. C2×#19104714C2semidirect
14. C2×#19104713C2semidirect
15. C2×#19104711C2semidirect
16. C2×C22⋊Q8×C163C2semidirect
21. #19104734C22semidirect
22. #19104733C22semidirect
23. #19104730C22semidirect
24. #19104730C22semidirect
25. #19104728C22semidirect
26. #19104728C22semidirect
27. #19104726C22semidirect
28. #19104725C22semidirect
32. #19104714C22semidirect
33. #19104713C22semidirect
34. #19104711C22semidirect
61. C22⋊Q8×C163C22semidirect
190. C2×C163#563semidirect
195. C163C2×#563semidirect
212. C2×C22⋊Q8D163semidirect
233. C22⋊Q8C2×D163semidirect
387. C2#23782416direct (N × H)

Maximal subgroups

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

subgroupindex
#237824162
C2×#191047342
C2×#191047332
C2×#191047302
C2×#191047282
C2×#191047262
C2×#191047252
C2×#191047212
C2×#191047182
C2×#191047172
C2×#191047142
C2×#191047132
C2×#191047112
C2×C22⋊Q8×C1632
C2×#563163