Finite Group Databasebeta

← #35224664

Subgroups — #35224664

Order 20864.

380 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782219C211yesmaximal, characteristic
2C4×Q8×D163C211yesmaximal, characteristic
3#23781997C211yesmaximal, characteristic
4#23781994C211yesmaximal, characteristic
5#23781949C211yesmaximal, characteristic, complemented
6#23781948C211yesmaximal, characteristic, complemented
7#23781874C211yesmaximal, characteristic, complemented
8#23781873C211yesmaximal, characteristic, complemented
9#23781822C211yesmaximal, characteristic
10#23781819C211yesmaximal, characteristic
11#23781773C211yesmaximal, characteristic, complemented
12#23781770C211yesmaximal, characteristic, complemented
13#23781770C211yesmaximal, characteristic, complemented
14#23781720C211yesmaximal, characteristic
15#441×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×Dic652C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20#19104734C2211yescharacteristic
21#19104733C2211yescharacteristic
22#19104725—21no—
23#19104715C2211yescharacteristic
24#19104713—21no—
25#19104711C2211yescharacteristic
26Q8×Dic163C2211yescharacteristic
27C4×Dic326—21no—
28C4×Dic326—21no—
29C4×Dic326C2211yescharacteristic
30C4×Dic326C2211yescharacteristic
31#19104700—21no—
32#19104700C2211yescharacteristic
33#19104692—21no—
34C4⋊C4×D163—21no—
35C4⋊C4×D163C2211yescharacteristic
36C42×D163—21no—
37C42×D163C2211yescharacteristic
38#19104667C2211yescharacteristic
39#19104666C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104658C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104646C2211yescharacteristic
52C2×Q16×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326C2311yescharacteristic
62C2×Dic326C2311yescharacteristic
63Q8×D163—22no—
64Q8×D163—22no—
65C2×C4×D163—21no—
66C2×C4×D163—21no—
67C2×C4×D163D411yescharacteristic
68C2×C4×D163D411yescharacteristic
69C2×C4×D163C2311yescharacteristic
70Dic652—22no—
71#14579689—22no—
72#14579689—22no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163D411yescharacteristic
94C4×Dic163D411yescharacteristic
95C4×Dic163C2311yescharacteristic
96C2×Q8×C163C2311yescharacteristic
97C2×Q8×C163C2311yescharacteristic
98Q16×C163—22no—
99C2×C8×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C4⋊C4×C163—21no—
102C4⋊C4×C163—21no—
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C42×C163—21no—
106C42×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163C2×D411yescharacteristic
109Dic326—24no—
110Dic326—21no—
111Dic326—21no—
112Dic326—21no—
113Dic326—42no—
114Dic326#4112yes—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C4×D163—24no—
123C4×D163—22no—
124C4×D163—21no—
125C4×D163—21no—
126C4×D163—22no—
127C4×D163—42no—
128C163⋊C8—21no—
129C163⋊C8—21no—
130Q8×C163—21no—
131Q8×C163—21no—
132Q8×C163—21no—
133Q8×C163#4112yes—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2411yescharacteristic
141C8×C163—21no—
142C8×C163—21no—
143C22×C163C22×D411yescharacteristic
144C2×D163—21no—
145C2×D163—21no—
146C2×D163#13712yes—
147C4×C163—22no—
148C4×C163—21no—
149C4×C163—21no—
150C4×C163—41no—
151C4×C163—41no—
152C4×C163—41no—
153C4×C163C2×#4111yescharacteristic
154C4×C163C2×#4111yescharacteristic
155Dic163—22no—
156Dic163—21no—
157Dic163—21no—
158Dic163—41no—
159Dic163—41no—
160Dic163—41no—
161Dic163#13712yes—
162C2×C163#52511yescharacteristic
163C2×C163C2×#13711yescharacteristic
164C2×C163C2×#13711yescharacteristic
165D163—22no—
166C163#311311yescharacteristic, complemented, Sylow
167#3113—1631nomaximal, Sylow
168#533—1631no—
169C2×C4×Q8—1631no—
170#483—1631no—
171#483—1631no—
172#475—1631no—
173#475—1631no—
174#462—1631no—
175#462—1631no—
176#451—1631no—
177#451—1631no—
178#441—1631no—
179#441—1631no—
180#441—1631no—
181#441D16311yescharacteristic, complemented
182#432—1631no—
183C22×Q8—1631no—
184C2×Q16—1631no—
185C2×Q16—1631no—
186C2×Q16—1631no—
187C2×Q16C2×D16311yescharacteristic
188C4⋊Q8—1631no—
189C4⋊Q8—1631no—
190#125—3261no—
191C22⋊Q8—3261no—
192C22⋊Q8—1631no—
193C22⋊Q8—1631no—
194C4×Q8—3261no—
195C4×Q8—3261no—
196C4×Q8—3261no—
197C4×Q8—1631no—
198C4×Q8—1631no—
199C4×Q8—1631no—
200C4×Q8—1631no—
201C4×Q8C2×D16311yescharacteristic
202C4×Q8C2×D16311yescharacteristic
203#117—3261no—
204C2×C4⋊C4—3261no—
205C2×C4⋊C4—1631no—
206C2×C42—3261no—
207C2×C42—1631no—
208#106—1631no—
209#106—1631no—
210#106—1631no—
211#106C2×D16311yescharacteristic
212C4⋊C8—1631no—
213C4⋊C8—1631no—
214#103—1631no—
215#103—1631no—
216#103—1631no—
217#103—1631no—
218#103—1631no—
219#103—1631no—
220#103C2×D16311yescharacteristic
221#103C2×D16311yescharacteristic
222C22⋊C8—1631no—
223C22⋊C8—1631no—
224#97—1631no—
225#97C2×D16311yescharacteristic
226C2×Q8—3261no—
227C2×Q8—3261no—
228C2×Q8—3262no—
229C2×Q8—3262no—
230C2×Q8—1631no—
231C2×Q8—1631no—
232C2×Q8C22×D16311yescharacteristic
233C2×Q8C22×D16311yescharacteristic
234C22×C4—3261no—
235C22×C4—3261no—
236C22×C4—1631no—
237C22×C4—1631no—
238C22×C4—1631no—
239Q16—22no—
240Q16—3262no—
241Q16—3262no—
242Q16—3262no—
243C2×C8—1631no—
244C2×C8—1631no—
245C2×C8C22×D16311yescharacteristic
246C2×C8C22×D16311yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—3261no—
250C4⋊C4—3261no—
251C4⋊C4—3261no—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—3261no—
256C4⋊C4—3261no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—1631no—
260C4⋊C4—1631no—
261C4⋊C4C22×D16311yescharacteristic
262C4⋊C4C22×D16311yescharacteristic
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C42—21no—
266C42—21no—
267C42—3261no—
268C42—3261no—
269C42—3261no—
270C42—3261no—
271C42—1631no—
272C42—1631no—
273C42—1631no—
274C42C22×D16311yescharacteristic
275C23—1631no—
276Q8—21no—
277Q8—21no—
278Q8—21no—
279Q8—3264no—
280Q8—3261no—
281Q8—3261no—
282Q8—3261no—
283Q8—6522no—
284Q8—1632no—
285Q8#1457969912yes—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—3264no—
291C2×C4—3262no—
292C2×C4—3262no—
293C2×C4—3261no—
294C2×C4—3261no—
295C2×C4—3261no—
296C2×C4—3261no—
297C2×C4—3261no—
298C2×C4—3261no—
299C2×C4—6522no—
300C2×C4—1631no—
301C2×C4—1631no—
302C2×C4—1631no—
303C2×C4D4×D16311yescharacteristic
304C2×C4D4×D16311yescharacteristic
305C2×C4C23×D16311yescharacteristic
306C8—21no—
307C8—21no—
308C8—3261no—
309C8—3261no—
310C22—3261no—
311C22—3261no—
312C22—1632no—
313C22C2×D4×D16311yescharacteristic
314C4—22no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—3262no—
321C4—3261no—
322C4—3261no—
323C4—6521no—
324C4—6521no—
325C4—6521no—
326C4—1632no—
327C4#41×D16311yescharacteristic
328C4C2×#1457969911yescharacteristic
329C2—3262no—
330C2#2378217411yescharacteristic
331C2#137×D16311yescharacteristic
332C2#137×D16311yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#441×C163C2
socleC22×C163C22×D4
prime radicalC22×D163C2×D4
hypercenter#441D163
hypocenterC163#3113
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
5. #23781949C2semidirect
6. #23781948C2semidirect
7. #23781874C2semidirect
8. #23781873C2semidirect
11. #23781773C2semidirect
12. #23781770C2semidirect
13. #23781770C2semidirect
15. #441×C163C2semidirect
166. C163#3113semidirect
181. #441D163semidirect

Maximal subgroups

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

subgroupindex
#237822192
C4×Q8×D1632
#237819972
#237819942
#237819492
#237819482
#237818742
#237818732
#237818222
#237818192
#237817732
#237817702
#237817202
#441×C1632
#3113163