Finite Group Databasebeta

← #35217696

Subgroups — #35217696

Order 20864.

464 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
1C2×#19104712C211yesmaximal, characteristic
2#23781570C211yesmaximal, characteristic, complemented
3#23781569C211yesmaximal, characteristic, complemented
4#23781563C211yesmaximal, characteristic, complemented
5#23781560C211yesmaximal, characteristic, complemented
6#23781558C211yesmaximal, characteristic
7#23781553C211yesmaximal, characteristic
8#23781553C211yesmaximal, characteristic
9#23781552C211yesmaximal, characteristic, complemented
10#23781549C211yesmaximal, characteristic, complemented
11#23781548C211yesmaximal, characteristic, complemented
12#23781547C211yesmaximal, characteristic, complemented
13#23781526C211yesmaximal, characteristic
14#23781523C211yesmaximal, characteristic
15#388×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18#19104712C418yescomplemented
19C2×C4⋊Dic163C2211yescharacteristic
20C2×C4⋊Dic163C2211yescharacteristic
21C2×Dic163⋊C4C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×C22⋊Dic163C2211yescharacteristic
28C2×C22⋊Dic163C2211yescharacteristic
29C2×C22⋊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
45C2×C4⋊C4×C163C2211yescharacteristic
46C2×C22⋊C4×C163C2211yescharacteristic
47C2×C22⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C42×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52C2×Dic326—41no—
53C2×Dic326—41no—
54C2×Dic326C2×C414yes—
55C22×Dic163—41no—
56C22×Dic163—41no—
57C22×Dic163C2×C414yes—
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C4⋊Dic163—41no—
67C4⋊Dic163C2×C414yes—
68Dic163⋊C4—41no—
69Dic163⋊C4—41no—
70Dic163⋊C4—41no—
71Dic163⋊C4—41no—
72Dic163⋊C4C2×C414yes—
73Dic163⋊C4C2×C414yes—
74C22⋊Dic163—41no—
75C22⋊Dic163—41no—
76C22⋊Dic163C2×C414yes—
77C4×Dic163—41no—
78C4×Dic163—41no—
79C4×Dic163—41no—
80C24×C163C2311yescharacteristic
81C22×C4×C163C2311yescharacteristic
82C22×C4×C163C2311yescharacteristic
83C22×C4×C163C2311yescharacteristic
84C22×C4×C163C2311yescharacteristic
85C22×C4×C163C2311yescharacteristic
86C22×C4×C163C2311yescharacteristic
87C4⋊C4×C163—41no—
88C22⋊C4×C163—41no—
89C22⋊C4×C163—41no—
90C22⋊C4×C163C2×C414yes—
91C42×C163—41no—
92Dic326—44no—
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×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—44no—
114C2×Dic163C22×C412yes—
115C2×Dic163C22×C412yes—
116C2×Dic163C22×C412yes—
117C2×Dic163C22×C412yes—
118C23×C163—21no—
119C23×C163—21no—
120C23×C163—21no—
121C23×C163—21no—
122C23×C163—21no—
123C23×C163—21no—
124C23×C163C22×C412yes—
125C23×C163C2411yescharacteristic
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×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163C22×C412yes—
143C2×C4×C163C22×C412yes—
144C22×C163—22no—
145C22×C163—22no—
146C22×C163—22no—
147C22×C163—41no—
148C22×C163—41no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163C23×C411yescharacteristic
152C22×C163H(5,𝔽2)11yescharacteristic
153C22×C163H(5,𝔽2)11yescharacteristic
154C22×C163H(5,𝔽2)11yescharacteristic
155C22×C1632−1+411yescharacteristic
156C22×C1632−1+411yescharacteristic
157C22×C1632−1+411yescharacteristic
158C4×C163—22no—
159C4×C163—22no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163—41no—
163C4×C163—41no—
164Dic163—22no—
165Dic163—22no—
166Dic163—22no—
167Dic163—22no—
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172C2×C163—42no—
173C2×C163#51811yescharacteristic
174C2×C163#51911yescharacteristic
175C2×C163#52011yescharacteristic
176C2×C163#56211yescharacteristic
177C2×C163#56211yescharacteristic
178C2×C163#56311yescharacteristic
179C2×C163#56311yescharacteristic
180C163#237811yescharacteristic, complemented, Sylow
181#2378—1631nomaximal, Sylow
182C2×C22⋊Q8—1631no—
183#391—1631no—
184#391—1631no—
185#389—1631no—
186#389—1631no—
187#388—1631no—
188#388—1631no—
189#388—1631no—
190#388D16311yescharacteristic
191#387—1631no—
192#387—1631no—
193#387—1631no—
194#387—1631no—
195#381—1631no—
196#380—1631no—
197C22×Q8—1631no—
198C23×C4—1631no—
199C22⋊Q8—1638no—
200C2×C4⋊C4—1631no—
201C2×C4⋊C4—1631no—
202C2×C4⋊C4—1631no—
203C2×C4⋊C4—1631no—
204C2×C4⋊C4—1631no—
205C2×C4⋊C4—1631no—
206C2×C4⋊C4—1631no—
207C2×C4⋊C4—1631no—
208C2×C4⋊C4C2×D16311yescharacteristic
209C2×C22⋊C4—1631no—
210C2×C22⋊C4—1631no—
211C2×C22⋊C4—1631no—
212C2×C22⋊C4C2×D16311yescharacteristic
213C2×C22⋊C4C2×D16311yescharacteristic
214C2×C22⋊C4C2×D16311yescharacteristic
215C2×C42—1631no—
216C2×C42—1631no—
217C2×C42—1631no—
218C2×C42C2×D16311yescharacteristic
219#95—1631no—
220#95—1631no—
221#95—1631no—
222#95—1631no—
223#95—1631no—
224#95—1631no—
225#95—1631no—
226#95—1631no—
227#95—1631no—
228#95—1631no—
229#95—1631no—
230#95—1631no—
231#95C2×D16311yescharacteristic
232#95C2×D16311yescharacteristic
233C24C22×D16311yescharacteristic
234C2×Q8—6521no—
235C2×Q8—6521no—
236C2×Q8—1634no—
237C22×C4—6521no—
238C22×C4—6521no—
239C22×C4—1631no—
240C22×C4—1631no—
241C22×C4—1631no—
242C22×C4—1631no—
243C22×C4—1631no—
244C22×C4—1631no—
245C22×C4—1631no—
246C22×C4—1631no—
247C22×C4—1634no—
248C22×C4C22×D16311yescharacteristic
249C22×C4C22×D16311yescharacteristic
250C22×C4C22×D16311yescharacteristic
251C22×C4C22×D16311yescharacteristic
252C22×C4C22×D16311yescharacteristic
253C22×C4C22×D16311yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—6521no—
256C4⋊C4—6521no—
257C4⋊C4—6521no—
258C4⋊C4—6521no—
259C4⋊C4—6521no—
260C4⋊C4—1634no—
261C4⋊C4—1634no—
262C4⋊C4—1634no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—6521no—
266C22⋊C4—6521no—
267C22⋊C4—1634no—
268C22⋊C4C4×D16314yes—
269C42—41no—
270C42—6521no—
271C42—6521no—
272C42—6521no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23C2×C4×D16312yes—
280C23C23×D16311yescharacteristic
281Q8—6524no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—3261no—
301C2×C4—3261no—
302C2×C4—3261no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—3261no—
306C2×C4—3261no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—6524no—
319C2×C4—1632no—
320C2×C4—1632no—
321C2×C4—1632no—
322C2×C4—1632no—
323C2×C4C2×C4×D16312yes—
324C2×C4C2×C4×D16312yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×C4×D16311yescharacteristic
333C22#1910478011yescharacteristic
334C22#1910478011yescharacteristic
335C22#1910478111yescharacteristic
336C22#1910478211yescharacteristic
337C22#1910478211yescharacteristic
338C22#1910478211yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—3262no—
346C4—3262no—
347C4—3262no—
348C4—3262no—
349C4—6521no—
350C4—6521no—
351C4—6521no—
352C4—6521no—
353C2—42no—
354C2#2378213711yescharacteristic
355C2#2378214211yescharacteristic
356C2#2378214411yescharacteristic
357C2#2378241011yescharacteristic
358C2#2378241211yescharacteristic
359C2#2378241511yescharacteristic
360C2#2378241711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#388×C163C2
socle, cosocleC23×C163C24
prime radicalC24×C163C23
hypercenter#388D163
hypocenterC163#2378
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #23781570C2semidirect
3. #23781569C2semidirect
4. #23781563C2semidirect
5. #23781560C2semidirect
9. #23781552C2semidirect
10. #23781549C2semidirect
11. #23781548C2semidirect
12. #23781547C2semidirect
18. #19104712C4semidirect
180. C163#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#191047122
#237815702
#237815692
#237815632
#237815602
#237815582
#237815532
#237815522
#237815492
#237815482
#237815472
#237815262
#237815232
#388×C1632
#2378163