Finite Group Databasebeta

← #35228152

Subgroups — #35228152

Order 20864.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782391C211yesmaximal, characteristic, complemented
2#23782384C213yesmaximal, complemented
3#23782355C213yesmaximal, complemented
4#23782349C213yesmaximal, complemented
5Q8×C22⋊D163C211yesmaximal, characteristic, complemented
6#23782322C211yesmaximal, characteristic, complemented
7#23782315C213yesmaximal, complemented
8#23782217C213yesmaximal, complemented
9#23782196C213yesmaximal, complemented
10#23782194C213yesmaximal, complemented
11#41×Dic163C211yesmaximal, characteristic, complemented
12C4×#14579698C213yesmaximal, complemented
13C2×Q8×Dic163C211yesmaximal, characteristic, complemented
14C4×Q8×D163C211yesmaximal, characteristic, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2213yes—
17C2×Q8×D163C2211yescharacteristic
18#19104734C2213yes—
19#19104733C2213yes—
20#19104728C2216yes—
21#19104728C2216yes—
22#19104727C2213yes—
23#19104726C2213yes—
24#19104725C2216yes—
25#19104722C2213yes—
26#19104717C2216yes—
27#19104716C2216yes—
28#19104714C2213yes—
29#19104714C2213yes—
30#19104712C2216yes—
31#19104711C2213yes—
32#19104706C2213yes—
33Q8×Dic163C2213yes—
34Q8×Dic163C2211yescharacteristic
35Q8×Dic163C2211yescharacteristic
36Q8×Dic163C2213yes—
37Q8×Dic163C2211yescharacteristic
38C4×Dic326C2213yes—
39C4×Dic326C2213yes—
40#19104700C2213yes—
41#19104700C2213yes—
42D4×Dic163C2213yes—
43D4×Dic163C2213yes—
44D4×Dic163C2213yes—
45C4×C22⋊D163C2213yes—
46C4×C22⋊D163C2213yes—
47(C22⋊D163)⋊C4C2213yes—
48(C22⋊D163)⋊C4C2211yescharacteristic
49#19104694C2213yes—
50#19104693C2213yes—
51#19104691C2216yes—
52#19104691C2213yes—
53#19104691C2213yes—
54C2×C4⋊Dic163C2213yes—
55C4⋊C4×D163C2213yes—
56C2×C4×Dic163C2213yes—
57C2×C4×Dic163C2213yes—
58C42×D163C2213yes—
59C2×#41×C163C2211yescharacteristic
60C22×Q8×C163C2211yescharacteristic
61#124×C163C2213yes—
62C22⋊Q8×C163C2213yes—
63#121×C163C2213yes—
64C4×Q8×C163C2211yescharacteristic
65C4×D4×C163C2213yes—
66#14579698—212no—
67C2×Dic326C2313yes—
68C2×Dic326C2313yes—
69Q8×D163—24no—
70C2×C22⋊D163C2313yes—
71C2×C22⋊D163C2311yescharacteristic
72C22×Dic163C2313yes—
73C22×Dic163C2313yes—
74C22×Dic163C2311yescharacteristic
75C22×Dic163C2313yes—
76C2×C4×D163C2313yes—
77C2×C4×D163C2313yes—
78C2×C4×D163C2311yescharacteristic
79C4⋊Dic163C2313yes—
80C4⋊Dic163C2313yes—
81C4⋊Dic163C2316yes—
82C4⋊Dic163C2313yes—
83C4⋊Dic163C2313yes—
84Dic163⋊C4C2316yes—
85Dic163⋊C4C2311yescharacteristic
86Dic163⋊C4C2313yes—
87Dic163⋊C4C2313yes—
88Dic163⋊C4C2313yes—
89Dic163⋊C4C2316yes—
90Dic163⋊C4C2316yes—
91C22⋊Dic163C2313yes—
92C22⋊Dic163C2313yes—
93C22⋊Dic163C2316yes—
94C22⋊Dic163C2316yes—
95(C2×D163)⋊C4C2311yescharacteristic
96(C2×D163)⋊C4C2313yes—
97C4×Dic163C2313yes—
98C4×Dic163C2313yes—
99C4×Dic163C2313yes—
100C4×Dic163C2311yescharacteristic
101C4×Dic163C2313yes—
102C4×Dic163C2313yes—
103C4×Dic163C2313yes—
104C4×Dic163C2313yes—
105C4×Dic163C2313yes—
106#41×C163—24no—
107C2×Q8×C163—24no—
108C2×Q8×C163C2313yes—
109C2×Q8×C163C2311yescharacteristic
110C2×D4×C163C2313yes—
111C2×D4×C163C2313yes—
112C22×C4×C163C2313yes—
113C22×C4×C163C2313yes—
114C4⋊C4×C163C2313yes—
115C4⋊C4×C163C2313yes—
116C22⋊C4×C163C2311yescharacteristic
117C22⋊C4×C163C2313yes—
118C22⋊C4×C163C2316yes—
119C42×C163C2313yes—
120C22×D163C2411yescharacteristic
121Dic326—26no—
122Dic326—26no—
123C22⋊D163—26no—
124C22⋊D163#4114yescomplemented
125C2×Dic163—26no—
126C2×Dic163—26no—
127C2×Dic163—22no—
128C2×Dic163—26no—
129C2×Dic163C2411yescharacteristic
130C2×Dic163C2413yes—
131C2×Dic163C2413yes—
132C2×Dic163C2413yes—
133C2×Dic163C2413yes—
134C2×Dic163C2411yescharacteristic
135C2×Dic163C2411yescharacteristic
136C4×D163—26no—
137C4×D163—26no—
138C4×D163—22no—
139C23×C163C2413yes—
140C23×C163C2411yescharacteristic
141Q8×C163—26no—
142Q8×C163#4114yescomplemented
143D4×C163—26no—
144D4×C163—26no—
145C2×C4×C163—26no—
146C2×C4×C163—26no—
147C2×C4×C163C2411yescharacteristic
148C2×C4×C163C2413yes—
149C2×C4×C163C2413yes—
150C2×C4×C163C2411yescharacteristic
151C2×C4×C163C2413yes—
152C22×C163—21no—
153C22×C163—21no—
154C22×C163—23no—
155C22×C163—23no—
156C22×C163—23no—
157C22×C163C2×#4112yes—
158C22×C163C2511yescharacteristic
159C2×D163—21no—
160C2×D163—21no—
161C2×D163C2×#4112yes—
162C4×C163—21no—
163C4×C163—23no—
164C4×C163—23no—
165C4×C163—21no—
166C4×C163C2×#4116yes—
167Dic163—23no—
168Dic163—23no—
169Dic163—23no—
170Dic163—21no—
171Dic163—23no—
172Dic163C2×#4112yes—
173Dic1632−1+412yes—
174C2×C163—22no—
175C2×C163—43no—
176C2×C163C22×#4111yescharacteristic
177C2×C163C22×#4111yescharacteristic
178C2×C163C2×2−1+411yescharacteristic
179D163—22no—
180C163#348211yescharacteristic, complemented, Sylow
181#3482—1631nomaximal, Sylow
182#557—1631no—
183#556—1633no—
184#552—1633no—
185#552—1633no—
186D4×Q8—1631no—
187#548—1631no—
188#548D16311yescharacteristic, complemented
189#547—1633no—
190#533—1633no—
191#529—1633no—
192#529—1633no—
193C4×#41—1631no—
194C4×#41—1633no—
195C2×C4×Q8—1631no—
196C2×C4×Q8—1631no—
197C2×#41—1633no—
198C2×#41C2×D16311yescharacteristic
199C22×Q8—1631no—
200C22×Q8C2×D16311yescharacteristic
201C4⋊Q8—1633no—
202C4⋊Q8—1633no—
203#126—1636no—
204#126—1636no—
205#125—1633no—
206#125—1636no—
207#125—1633no—
208#124—1633no—
209#124C2×D16313yes—
210#123—1636no—
211#123—1636no—
212C22⋊Q8—1633no—
213C22⋊Q8—1633no—
214C22⋊Q8—1633no—
215C22⋊Q8—1636no—
216C22⋊Q8C2×D16313yes—
217#121—1633no—
218#121C2×D16313yes—
219C4×Q8—1631no—
220C4×Q8—1633no—
221C4×Q8—1631no—
222C4×Q8—1633no—
223C4×Q8—1631no—
224C4×Q8—1633no—
225C4×Q8—1633no—
226C4×Q8—1633no—
227C4×Q8—1633no—
228C4×Q8C2×D16311yescharacteristic
229C4×D4—1633no—
230C4×D4—1633no—
231C4×D4—1633no—
232C4×D4—1633no—
233C4×D4—1633no—
234C4×D4—1631no—
235C4×D4—1633no—
236C4×D4C2×D16313yes—
237#117—1633no—
238#117—1633no—
239#117—1633no—
240#117—1633no—
241#117—1636no—
242C2×C4⋊C4—1633no—
243C2×C4⋊C4—1633no—
244C2×C42—1633no—
245C2×C42—1633no—
246C2×C42—1633no—
247#41—24no—
248#41—32612no—
249C2×Q8—24no—
250C2×Q8—3264no—
251C2×Q8—1633no—
252C2×Q8—1633no—
253C2×Q8C22×D16313yes—
254C2×Q8C22×D16311yescharacteristic
255C2×D4—1633no—
256C2×D4—1631no—
257C2×D4C22×D16313yes—
258C2×D4C22×D16313yes—
259C22×C4—1631no—
260C22×C4—1633no—
261C22×C4—1633no—
262C22×C4—1633no—
263C22×C4—1631no—
264C22×C4—1633no—
265C22×C4—1633no—
266C22×C4C22×D16313yes—
267C22×C4C22×D16313yes—
268C4⋊C4—1633no—
269C4⋊C4—1633no—
270C4⋊C4—1636no—
271C4⋊C4—1636no—
272C4⋊C4—1631no—
273C4⋊C4—1633no—
274C4⋊C4—1633no—
275C4⋊C4—1633no—
276C4⋊C4—1633no—
277C4⋊C4—1633no—
278C4⋊C4—1636no—
279C4⋊C4—1636no—
280C4⋊C4C22×D16313yes—
281C4⋊C4C22×D16313yes—
282C22⋊C4—1633no—
283C22⋊C4—1636no—
284C22⋊C4—1633no—
285C22⋊C4—1636no—
286C22⋊C4—1631no—
287C22⋊C4—1633no—
288C22⋊C4C22×D16311yescharacteristic
289C22⋊C4C22×D16313yes—
290C22⋊C4C22×D16316yes—
291C42—1633no—
292C42—1631no—
293C42—1633no—
294C42—1633no—
295C42—1633no—
296C42—1633no—
297C42—1633no—
298C42—1633no—
299C42—1633no—
300C42C22×D16313yes—
301C23—1631no—
302C23C23×D16313yes—
303C23C23×D16311yescharacteristic
304Q8—26no—
305Q8—3266no—
306Q8—3266no—
307Q8#1457969814yescomplemented
308D4—26no—
309D4—26no—
310D4—3266no—
311D4—1634no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—3266no—
315C2×C4—3262no—
316C2×C4—3266no—
317C2×C4—3266no—
318C2×C4—3262no—
319C2×C4—3266no—
320C2×C4—3266no—
321C2×C4—1631no—
322C2×C4—1633no—
323C2×C4—1633no—
324C2×C4—1633no—
325C2×C4—1633no—
326C2×C4—1631no—
327C2×C4—1631no—
328C2×C4C23×D16311yescharacteristic
329C2×C4C23×D16311yescharacteristic
330C2×C4C23×D16313yes—
331C2×C4C23×D16313yes—
332C2×C4C23×D16313yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—3261no—
339C22—3261no—
340C22—1632no—
341C22#41×D16312yes—
342C22C24×D16311yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—3263no—
348C4—3263no—
349C4—3263no—
350C4—3261no—
351C4—3263no—
352C4—1632no—
353C4—1632no—
354C4C2×#1457969816yes—
355C2—22no—
356C2—43no—
357C2—3262no—
358C2C2×#41×D16311yescharacteristic
359C2C22×#1457969811yescharacteristic
360C22−1+4×D16311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#548×C163C2
hypercenter#548D163
hypocenterC163#3482
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23782391C2semidirect
2. #23782384C2semidirect
3. #23782355C2semidirect
4. #23782349C2semidirect
5. Q8×C22⋊D163C2semidirect
6. #23782322C2semidirect
7. #23782315C2semidirect
8. #23782217C2semidirect
9. #23782196C2semidirect
10. #23782194C2semidirect
11. #41×Dic163C2semidirect
12. C4×#14579698C2semidirect
13. C2×Q8×Dic163C2semidirect
14. C4×Q8×D163C2semidirect
15. #548×C163C2semidirect
124. C22⋊D163#41semidirect
142. Q8×C163#41semidirect
180. C163#3482semidirect
188. #548D163semidirect
307. Q8#14579698semidirect

Maximal subgroups

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

subgroupindex
#237823912
#237823842
#237823552
#237823492
Q8×C22⋊D1632
#237823222
#237823152
#237822172
#237821962
#237821942
#41×Dic1632
C4×#145796982
C2×Q8×Dic1632
C4×Q8×D1632
#548×C1632
#3482163