Finite Group Databasebeta

← #35217279

Subgroups — #35217279

Order 20864.

460 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104733C211yesmaximal, characteristic
2#23781572C211yesmaximal, characteristic
3#23781569C211yesmaximal, characteristic
4#23781569C211yesmaximal, characteristic
5#23781562C211yesmaximal, characteristic
6#23781560C211yesmaximal, characteristic
7#23781552C211yesmaximal, characteristic
8#23781549C211yesmaximal, characteristic
9#23781547C211yesmaximal, characteristic
10#23781546C211yesmaximal, characteristic
11#23781529C211yesmaximal, characteristic
12C4×Dic163⋊C4C211yesmaximal, characteristic
13#23781516C211yesmaximal, characteristic
14#23781510C211yesmaximal, characteristic
15#391×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17#19104733C418yescomplemented
18C2×C4⋊Dic163C2211yescharacteristic
19C2×Dic163⋊C4C2211yescharacteristic
20C2×Dic163⋊C4C2211yescharacteristic
21C2×Dic163⋊C4C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×C4×Dic163C2211yescharacteristic
28C2×C4×Dic163C2211yescharacteristic
29C2×C4×Dic163C2211yescharacteristic
30C2×C4×Dic163C2211yescharacteristic
31C2×C4×Dic163C2211yescharacteristic
32C2×C4×Dic163C2211yescharacteristic
33C163⋊#95C2211yescharacteristic
34C163⋊#95C2211yescharacteristic
35C163⋊#95C2211yescharacteristic
36C163⋊#95C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C22×Q8×C163C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C42×C163C2211yescharacteristic
48#95×C163C2211yescharacteristic
49#95×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52C2×Dic326—41no—
53C2×Dic326—41no—
54C2×Dic326C2×C414yes—
55C22×Dic163C2311yescharacteristic
56C22×Dic163C2311yescharacteristic
57C22×Dic163C2311yescharacteristic
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C4⋊Dic163—41no—
64Dic163⋊C4—22no—
65Dic163⋊C4—22no—
66Dic163⋊C4—41no—
67Dic163⋊C4—41no—
68Dic163⋊C4C2×C414yes—
69Dic163⋊C4C2×C414yes—
70Dic163⋊C4C2×C414yes—
71Dic163⋊C4C2×C414yes—
72C4×Dic163—22no—
73C4×Dic163—41no—
74C4×Dic163—41no—
75C4×Dic163—41no—
76C4×Dic163C2×C414yes—
77C4×Dic163Q814yescomplemented
78C2×Q8×C163—41no—
79C2×Q8×C163—41no—
80C2×Q8×C163C2×C414yes—
81C22×C4×C163C2311yescharacteristic
82C22×C4×C163C2311yescharacteristic
83C22×C4×C163C2311yescharacteristic
84C22×C4×C163C2311yescharacteristic
85C22×C4×C163C2311yescharacteristic
86C22×C4×C163C2311yescharacteristic
87C22×C4×C163C2311yescharacteristic
88C4⋊C4×C163—41no—
89C42×C163—41no—
90Dic326—44no—
91C2×Dic163—21no—
92C2×Dic163—21no—
93C2×Dic163—21no—
94C2×Dic163—21no—
95C2×Dic163—21no—
96C2×Dic163—21no—
97C2×Dic163—21no—
98C2×Dic163—21no—
99C2×Dic163—21no—
100C2×Dic163—21no—
101C2×Dic163—21no—
102C2×Dic163—21no—
103C2×Dic163—21no—
104C2×Dic163—21no—
105C2×Dic163—21no—
106C2×Dic163—21no—
107C2×Dic163—21no—
108C2×Dic163C22×C412yes—
109C2×Dic163C22×C412yes—
110C2×Dic163C22×C412yes—
111C2×Dic163C22×C412yes—
112C2×Dic163C2×Q812yes—
113C2×Dic163C2×Q812yes—
114C2×Dic163#4112yes—
115C23×C163C2411yescharacteristic
116Q8×C163—44no—
117C2×C4×C163—21no—
118C2×C4×C163—21no—
119C2×C4×C163—21no—
120C2×C4×C163—21no—
121C2×C4×C163—21no—
122C2×C4×C163—21no—
123C2×C4×C163—21no—
124C2×C4×C163—21no—
125C2×C4×C163—21no—
126C2×C4×C163—21no—
127C2×C4×C163—21no—
128C2×C4×C163—21no—
129C2×C4×C163—21no—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163C22×C412yes—
135C2×C4×C163C22×C412yes—
136C2×C4×C163C22×C412yes—
137C2×C4×C163C2×Q812yes—
138C22×C163C23×C411yescharacteristic
139C22×C163C22×Q811yescharacteristic
140C22×C163C2×#4111yescharacteristic
141C22×C163H(5,𝔽2)11yescharacteristic
142C22×C163H(5,𝔽2)11yescharacteristic
143C22×C1632−1+411yescharacteristic
144C22×C1632−1+411yescharacteristic
145C4×C163—22no—
146C4×C163—22no—
147C4×C163—22no—
148C4×C163—22no—
149C4×C163—41no—
150C4×C163—41no—
151C4×C163—41no—
152Dic163—22no—
153Dic163—22no—
154Dic163—22no—
155Dic163—22no—
156Dic163—41no—
157Dic163—41no—
158Dic163—41no—
159Dic163C4×Q814yes—
160C2×C163C2×C4×Q811yescharacteristic
161C2×C163#51811yescharacteristic
162C2×C163#51911yescharacteristic
163C2×C163#54211yescharacteristic
164C2×C163#54211yescharacteristic
165C2×C163#54411yescharacteristic
166C2×C163#54411yescharacteristic
167C163#233111yescharacteristic, complemented, Sylow
168#2331—1631nomaximal, Sylow
169C2×C4⋊Q8—1631no—
170#391—1631no—
171#391—1631no—
172#391—1631no—
173#391D16311yescharacteristic
174#389—1631no—
175#389—1631no—
176#387—1631no—
177#387—1631no—
178#387—1631no—
179#387—1631no—
180#382—1631no—
181C4×C4⋊C4—1631no—
182C4×C4⋊C4—1631no—
183#376—1631no—
184C22×Q8—1631no—
185C22×Q8C2×D16311yescharacteristic
186C4⋊Q8—1638no—
187C2×C4⋊C4—1631no—
188C2×C4⋊C4—1631no—
189C2×C4⋊C4—1631no—
190C2×C4⋊C4—1631no—
191C2×C4⋊C4—1631no—
192C2×C4⋊C4—1631no—
193C2×C4⋊C4—1631no—
194C2×C4⋊C4—1631no—
195C2×C4⋊C4—1631no—
196C2×C4⋊C4C2×D16311yescharacteristic
197C2×C42—1631no—
198C2×C42—1631no—
199C2×C42—1631no—
200C2×C42—1631no—
201C2×C42—1631no—
202C2×C42—1631no—
203C2×C42C2×D16311yescharacteristic
204#95—1631no—
205#95—1631no—
206#95—1631no—
207#95—1631no—
208#95—1631no—
209#95—1631no—
210#95—1631no—
211#95—1631no—
212#95—1631no—
213#95—1631no—
214#95—1631no—
215#95—1631no—
216#95C2×D16311yescharacteristic
217#95C2×D16311yescharacteristic
218#95C2×D16311yescharacteristic
219#95C2×D16311yescharacteristic
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—6521no—
223C2×Q8—6521no—
224C2×Q8—1634no—
225C2×Q8C4×D16314yes—
226C22×C4—1631no—
227C22×C4—1631no—
228C22×C4—1631no—
229C22×C4—1631no—
230C22×C4—1631no—
231C22×C4—1631no—
232C22×C4—1631no—
233C22×C4—1631no—
234C22×C4C22×D16311yescharacteristic
235C22×C4C22×D16311yescharacteristic
236C22×C4C22×D16311yescharacteristic
237C22×C4C22×D16311yescharacteristic
238C22×C4C22×D16311yescharacteristic
239C22×C4C22×D16311yescharacteristic
240C22×C4C22×D16311yescharacteristic
241C4⋊C4—41no—
242C4⋊C4—3262no—
243C4⋊C4—3262no—
244C4⋊C4—6521no—
245C4⋊C4—6521no—
246C4⋊C4—6521no—
247C4⋊C4—1634no—
248C4⋊C4—1634no—
249C4⋊C4—1634no—
250C4⋊C4—1634no—
251C42—41no—
252C42—3262no—
253C42—6521no—
254C42—6521no—
255C42—6521no—
256C42—1634no—
257C42—1634no—
258C23C23×D16311yescharacteristic
259Q8—44no—
260Q8—6524no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—3261no—
279C2×C4—3261no—
280C2×C4—3261no—
281C2×C4—3261no—
282C2×C4—3261no—
283C2×C4—3261no—
284C2×C4—3261no—
285C2×C4—3261no—
286C2×C4—3261no—
287C2×C4—3261no—
288C2×C4—3261no—
289C2×C4—3261no—
290C2×C4—3261no—
291C2×C4—3261no—
292C2×C4—3261no—
293C2×C4—3261no—
294C2×C4—3261no—
295C2×C4—1632no—
296C2×C4—1632no—
297C2×C4—1632no—
298C2×C4—1632no—
299C2×C4—1632no—
300C2×C4—1632no—
301C2×C4—1632no—
302C2×C4C2×C4×D16312yes—
303C2×C4C2×C4×D16312yes—
304C2×C4C2×C4×D16312yes—
305C2×C4Q8×D16312yes—
306C22C22×C4×D16311yescharacteristic
307C22C2×Q8×D16311yescharacteristic
308C22#41×D16311yescharacteristic
309C22#1910478011yescharacteristic
310C22#1910478011yescharacteristic
311C22#1910478211yescharacteristic
312C22#1910478311yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—3262no—
321C4—3262no—
322C4—3262no—
323C4—3262no—
324C4—6521no—
325C4—6521no—
326C4—6521no—
327C4—1634no—
328C2C4×Q8×D16311yescharacteristic
329C2#2378213811yescharacteristic
330C2#2378214111yescharacteristic
331C2#2378227911yescharacteristic
332C2#2378228511yescharacteristic
333C2#2378229211yescharacteristic
334C2#2378229411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#391×C163C2
socle, prime radical, cosocleC23×C163C24
hypercenter#391D163
hypocenterC163#2331
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
17. #19104733C4semidirect
77. C4×Dic163Q8semidirect
167. C163#2331semidirect

Maximal subgroups

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

subgroupindex
C2×#191047332
#237815722
#237815692
#237815622
#237815602
#237815522
#237815492
#237815472
#237815462
#237815292
C4×Dic163⋊C42
#237815162
#237815102
#391×C1632
#2331163