Finite Group Databasebeta

← #35228134

Subgroups — #35228134

Order 20864.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782398C213yesmaximal, complemented
2#23782391C211yesmaximal, characteristic, complemented
3#23782383C213yesmaximal, complemented
4#23782365C213yesmaximal, complemented
5#23782359C213yesmaximal, complemented
6#23782334C211yesmaximal, characteristic, complemented
7#23782327C211yesmaximal, characteristic, complemented
8#23782322C211yesmaximal, characteristic, complemented
9#23782212C213yesmaximal, complemented
10#23782199C213yesmaximal, complemented
11#23782197C213yesmaximal, complemented
12C4×#14579700C211yesmaximal, characteristic, complemented
13C4×#14579699C213yesmaximal, complemented
14C4×Q8×D163C211yesmaximal, characteristic, complemented
15C4×#41×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579699C2213yes—
18C2×Q8×D163C2211yescharacteristic
19#19104734C2213yes—
20#19104731C2213yes—
21#19104729C2216yes—
22#19104728C2216yes—
23#19104727C2213yes—
24#19104726C2213yes—
25#19104723C2213yes—
26#19104722C2213yes—
27#19104721C2216yes—
28#19104718C2216yes—
29#19104717C2216yes—
30#19104715C2213yes—
31#19104714C2213yes—
32#19104710C2213yes—
33#19104709C2213yes—
34#19104708C2216yes—
35#19104706C2213yes—
36#19104706C2213yes—
37Q8×Dic163C2211yescharacteristic
38Q8×Dic163C2211yescharacteristic
39C4×Dic326C2213yes—
40C4×Dic326C2213yes—
41C4×Dic326C2213yes—
42#19104700C2213yes—
43C4×D652C2213yes—
44C4×D652C2213yes—
45C4×D652C2213yes—
46#19104697C2213yes—
47C4×C22⋊D163C2216yes—
48C4×C22⋊D163C2213yes—
49C4×C22⋊D163C2213yes—
50C4×C22⋊D163C2213yes—
51C4×C22⋊D163C2213yes—
52#19104694C2213yes—
53#19104693C2213yes—
54#19104692C2213yes—
55#19104692C2211yescharacteristic
56C4⋊C4×D163C2213yes—
57C42×D163C2213yes—
58C42×D163C2213yes—
59C2×#41×C163C2211yescharacteristic
60C4×Q8×C163C2211yescharacteristic
61C4×Q8×C163C2211yescharacteristic
62C4×D4×C163C2213yes—
63C4×D4×C163C2213yes—
64#117×C163C2213yes—
65C2×C42×C163C2213yes—
66#14579700—24no—
67#14579699—212no—
68C2×Dic326C2313yes—
69C2×Dic326C2313yes—
70Q8×D163—24no—
71C2×D652C2313yes—
72C2×D652C2313yes—
73C2×C22⋊D163C2313yes—
74C2×C22⋊D163C2313yes—
75C2×C22⋊D163C2316yes—
76C2×C4×D163C2313yes—
77C2×C4×D163C2313yes—
78C2×C4×D163C2313yes—
79C2×C4×D163C2313yes—
80C2×C4×D163C2313yes—
81C2×C4×D163C2311yescharacteristic
82C4⋊Dic163C2313yes—
83C4⋊Dic163C2313yes—
84C4⋊Dic163C2313yes—
85C4⋊Dic163C2313yes—
86Dic163⋊C4C2316yes—
87Dic163⋊C4C2311yescharacteristic
88Dic163⋊C4C2311yescharacteristic
89Dic163⋊C4C2313yes—
90Dic163⋊C4C2313yes—
91Dic163⋊C4C2316yes—
92C22⋊Dic163C2313yes—
93C22⋊Dic163C2313yes—
94C22⋊Dic163C2316yes—
95(C2×D163)⋊C4C2311yescharacteristic
96(C2×D163)⋊C4C2311yescharacteristic
97(C2×D163)⋊C4C2313yes—
98(C2×D163)⋊C4C2313yes—
99(C2×D163)⋊C4C2316yes—
100(C2×D163)⋊C4C2316yes—
101C4×Dic163C2311yescharacteristic
102C4×Dic163C2313yes—
103C4×Dic163C2313yes—
104C4×Dic163C2313yes—
105C4×Dic163C2313yes—
106C4×Dic163C2313yes—
107#41×C163—24no—
108C2×Q8×C163C2311yescharacteristic
109C2×D4×C163C2313yes—
110C22×C4×C163C2313yes—
111C22×C4×C163C2313yes—
112C22×C4×C163C2313yes—
113C4⋊C4×C163C2313yes—
114C4⋊C4×C163C2313yes—
115C22⋊C4×C163C2316yes—
116C42×C163C2311yescharacteristic
117C42×C163C2313yes—
118C42×C163C2313yes—
119C42×C163C2313yes—
120C22×D163C2413yes—
121C22×D163C2411yescharacteristic
122Dic326—26no—
123Dic326—26no—
124D652—26no—
125D652—26no—
126C22⋊D163—26no—
127C22⋊D163—26no—
128C22⋊D163—46no—
129C2×Dic163C2411yescharacteristic
130C2×Dic163C2411yescharacteristic
131C2×Dic163C2413yes—
132C2×Dic163C2413yes—
133C2×Dic163C2413yes—
134C2×Dic163C2411yescharacteristic
135C4×D163—26no—
136C4×D163—26no—
137C4×D163—26no—
138C4×D163—43no—
139C4×D163—43no—
140C4×D163#4114yescomplemented
141C23×C163C2413yes—
142Q8×C163#4114yescomplemented
143D4×C163—26no—
144C2×C4×C163—26no—
145C2×C4×C163—26no—
146C2×C4×C163—43no—
147C2×C4×C163C2411yescharacteristic
148C2×C4×C163C2411yescharacteristic
149C2×C4×C163C2413yes—
150C2×C4×C163C2413yes—
151C2×C4×C163C2413yes—
152C2×C4×C163C2411yescharacteristic
153C22×C163—23no—
154C22×C163—23no—
155C22×C163—23no—
156C22×C163C2511yescharacteristic
157C2×D163—21no—
158C2×D163—21no—
159C2×D163—23no—
160C2×D163—23no—
161C2×D163—23no—
162C2×D163C2×#4112yes—
163C4×C163—21no—
164C4×C163—23no—
165C4×C163—21no—
166C4×C163—23no—
167C4×C163C2×#4112yes—
168C4×C163C2×#4116yes—
169Dic163—21no—
170Dic163—23no—
171Dic163—23no—
172Dic163—21no—
173Dic163—23no—
174Dic163C2×#4112yes—
175C2×C163—43no—
176C2×C163C22×#4111yescharacteristic
177C2×C163C22×#4111yescharacteristic
178C2×C163#58511yescharacteristic
179D163—22no—
180D163—43no—
181C163#348011yescharacteristic, complemented, Sylow
182#3480—1631nomaximal, Sylow
183#559—1633no—
184#557—1631no—
185#556—1633no—
186#553—1633no—
187#553—1633no—
188#550—1631no—
189#548—1631no—
190#548—1631no—
191#532—1633no—
192#529—1633no—
193#529—1633no—
194C4×#41—1631no—
195C4×#41—1633no—
196C4×#41D16311yescharacteristic, complemented
197C2×C4×Q8—1631no—
198C2×#41—1631no—
199C2×#41—1633no—
200C2×#41C2×D16311yescharacteristic
201C22×Q8—1631no—
202C4⋊Q8—1633no—
203#127—1633no—
204#126—1636no—
205#126—1636no—
206#125—1633no—
207#125—1633no—
208#124—1633no—
209#124—1636no—
210#124—1633no—
211#123—1636no—
212#123—1636no—
213C22⋊Q8—1633no—
214C22⋊Q8—1633no—
215#121—1633no—
216#121—1633no—
217#121—1633no—
218#121—1633no—
219#121—1636no—
220C4×Q8—1631no—
221C4×Q8—1631no—
222C4×Q8—1633no—
223C4×Q8—1633no—
224C4×Q8—1633no—
225C4×Q8—1633no—
226C4×Q8C2×D16311yescharacteristic
227C4×Q8C2×D16311yescharacteristic
228C4×D4—1633no—
229C4×D4—1633no—
230C4×D4—1633no—
231C4×D4—1633no—
232C4×D4—1636no—
233C4×D4—1633no—
234C4×D4—1633no—
235C4×D4—1633no—
236C4×D4—1633no—
237C4×D4C2×D16313yes—
238C4×D4C2×D16313yes—
239#117—1633no—
240#117—1633no—
241#117—1631no—
242#117—1633no—
243#117C2×D16313yes—
244C2×C4⋊C4—1633no—
245C2×C42—1633no—
246C2×C42—1633no—
247C2×C42C2×D16313yes—
248#41—24no—
249#41—32612no—
250#41—3264no—
251C2×Q8—3264no—
252C2×Q8—1633no—
253C2×Q8—1633no—
254C2×Q8C22×D16311yescharacteristic
255C2×D4—1633no—
256C2×D4—1633no—
257C2×D4—1636no—
258C2×D4—1633no—
259C2×D4—1633no—
260C2×D4C22×D16313yes—
261C22×C4—1633no—
262C22×C4—1633no—
263C22×C4—1633no—
264C22×C4—1633no—
265C22×C4—1633no—
266C22×C4—1631no—
267C22×C4C22×D16313yes—
268C22×C4C22×D16313yes—
269C22×C4C22×D16313yes—
270C4⋊C4—1633no—
271C4⋊C4—1636no—
272C4⋊C4—1631no—
273C4⋊C4—1631no—
274C4⋊C4—1633no—
275C4⋊C4—1633no—
276C4⋊C4—1633no—
277C4⋊C4—1633no—
278C4⋊C4—1633no—
279C4⋊C4—1636no—
280C4⋊C4C22×D16313yes—
281C4⋊C4C22×D16313yes—
282C22⋊C4—1633no—
283C22⋊C4—1633no—
284C22⋊C4—1636no—
285C22⋊C4—1631no—
286C22⋊C4—1631no—
287C22⋊C4—1633no—
288C22⋊C4—1636no—
289C22⋊C4—1633no—
290C22⋊C4—1636no—
291C22⋊C4C22×D16316yes—
292C42—1633no—
293C42—1633no—
294C42—1633no—
295C42—1633no—
296C42—1631no—
297C42—1633no—
298C42C22×D16313yes—
299C42C22×D16313yes—
300C42C22×D16313yes—
301C42C22×D16311yescharacteristic
302C23—1633no—
303C23—1631no—
304C23C23×D16313yes—
305Q8—3266no—
306Q8—3266no—
307Q8#1457969914yescomplemented
308D4—26no—
309D4—3266no—
310D4—3266no—
311D4—3266no—
312D4—3266no—
313D4—6526no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—43no—
317C2×C4—3266no—
318C2×C4—3266no—
319C2×C4—3266no—
320C2×C4—6523no—
321C2×C4—6523no—
322C2×C4—1631no—
323C2×C4—1631no—
324C2×C4—1633no—
325C2×C4—1633no—
326C2×C4—1633no—
327C2×C4—1631no—
328C2×C4—1634no—
329C2×C4C23×D16313yes—
330C2×C4C23×D16311yescharacteristic
331C2×C4C23×D16311yescharacteristic
332C2×C4C23×D16313yes—
333C2×C4C23×D16313yes—
334C2×C4C23×D16311yescharacteristic
335C22—23no—
336C22—23no—
337C22—23no—
338C22—3261no—
339C22—3261no—
340C22—3263no—
341C22—3263no—
342C22—3263no—
343C22—1632no—
344C22C24×D16311yescharacteristic
345C4—23no—
346C4—21no—
347C4—21no—
348C4—23no—
349C4—3263no—
350C4—3261no—
351C4—3263no—
352C4—3261no—
353C4—3263no—
354C4—1632no—
355C4#41×D16312yes—
356C4C2×#1457969916yes—
357C2—43no—
358C2—3262no—
359C2—6523no—
360C2C2×#41×D16311yescharacteristic
361C2C22×#1457969911yescharacteristic
362C2#2378250111yescharacteristic
subgroupG/H
centerC2×C4C23×D163
commutator, socle, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC4×#41×C163C2
hypercenterC4×#41D163
hypocenterC163#3480
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #23782398C2semidirect
2. #23782391C2semidirect
3. #23782383C2semidirect
4. #23782365C2semidirect
5. #23782359C2semidirect
6. #23782334C2semidirect
7. #23782327C2semidirect
8. #23782322C2semidirect
9. #23782212C2semidirect
10. #23782199C2semidirect
11. #23782197C2semidirect
12. C4×#14579700C2semidirect
13. C4×#14579699C2semidirect
14. C4×Q8×D163C2semidirect
15. C4×#41×C163C2semidirect
140. C4×D163#41semidirect
142. Q8×C163#41semidirect
181. C163#3480semidirect
196. C4×#41D163semidirect
307. Q8#14579699semidirect

Maximal subgroups

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

subgroupindex
#237823982
#237823912
#237823832
#237823652
#237823592
#237823342
#237823272
#237823222
#237822122
#237821992
#237821972
C4×#145797002
C4×#145796992
C4×Q8×D1632
C4×#41×C1632
#3480163