Finite Group Databasebeta

← #35217811

Subgroups — #35217811

Order 20864.

528 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104700C211yesmaximal, characteristic, complemented
2C2×#19104691C211yesmaximal, characteristic
3#23781606C211yesmaximal, characteristic, complemented
4#23781601C211yesmaximal, characteristic
5#23781598C211yesmaximal, characteristic, complemented
6#23781595C211yesmaximal, characteristic, complemented
7#23781588C211yesmaximal, characteristic
8#23781585C211yesmaximal, characteristic, complemented
9#23781577C211yesmaximal, characteristic
10C4⋊C4×Dic163C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic163C211yesmaximal, characteristic, complemented
12#23781516C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic163C211yesmaximal, characteristic
14C22⋊C4×Dic163C211yesmaximal, characteristic
15#396×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18#19104700—24no—
19#19104691—24no—
20C2×C4⋊Dic163C2211yescharacteristic
21C2×C4⋊Dic163C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×C22⋊Dic163C2211yescharacteristic
29C2×C22⋊Dic163C2211yescharacteristic
30C2×C22⋊Dic163C2211yescharacteristic
31C2×C4×Dic163C2211yescharacteristic
32C2×C4×Dic163C2211yescharacteristic
33C2×C4×Dic163C2211yescharacteristic
34C2×C4×Dic163C2211yescharacteristic
35C2×C4×Dic163C2211yescharacteristic
36C2×C4×Dic163C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C4⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163C2211yescharacteristic
51C2×C22⋊C4×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C2×Dic326—22no—
54C2×Dic326—22no—
55C2×Dic326—22no—
56C22×Dic163—22no—
57C22×Dic163—22no—
58C22×Dic163Q814yescomplemented
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C4⋊Dic163—22no—
68C4⋊Dic163—22no—
69Dic163⋊C4—22no—
70Dic163⋊C4—22no—
71Dic163⋊C4—22no—
72Dic163⋊C4—22no—
73Dic163⋊C4—22no—
74Dic163⋊C4—22no—
75C22⋊Dic163—22no—
76C22⋊Dic163—22no—
77C22⋊Dic163—22no—
78C4×Dic163—22no—
79C4×Dic163—22no—
80C4×Dic163—22no—
81C4×Dic163—22no—
82C4×Dic163—22no—
83C4×Dic163D414yes—
84C24×C163C2311yescharacteristic
85C22×C4×C163C2311yescharacteristic
86C22×C4×C163C2311yescharacteristic
87C22×C4×C163C2311yescharacteristic
88C22×C4×C163C2311yescharacteristic
89C22×C4×C163C2311yescharacteristic
90C22×C4×C163C2311yescharacteristic
91C4⋊C4×C163—22no—
92C4⋊C4×C163—22no—
93C4⋊C4×C163—22no—
94C22⋊C4×C163—22no—
95C22⋊C4×C163—22no—
96C22⋊C4×C163—22no—
97Dic326—28no—
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—28no—
107C2×Dic163—21no—
108C2×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163C2×D412yes—
117C2×Dic163C2×D412yes—
118C2×Dic163C2×Q812yes—
119C2×Dic163C2×Q812yes—
120C2×Dic163#4112yes—
121C2×Dic163#4112yes—
122C2×Dic163#4112yes—
123C23×C163—21no—
124C23×C163—21no—
125C23×C163—21no—
126C23×C163—21no—
127C23×C163—21no—
128C23×C163—21no—
129C23×C163C2×Q812yes—
130C23×C163C2411yescharacteristic
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×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163C2×D412yes—
147C2×C4×C163#4112yes—
148C2×C4×C163#4112yes—
149C22×C163—22no—
150C22×C163—22no—
151C22×C163—22no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163C22×D411yescharacteristic
157C22×C163C22×Q811yescharacteristic
158C22×C163C2×#4111yescharacteristic
159C22×C163C2×#4111yescharacteristic
160C22×C163C2×#4111yescharacteristic
161C22×C163C2×#4111yescharacteristic
162C22×C163C2×#4111yescharacteristic
163C4×C163—22no—
164C4×C163—22no—
165C4×C163—22no—
166C4×C163—41no—
167C4×C163—41no—
168C4×C163—41no—
169Dic163—22no—
170Dic163—22no—
171Dic163—22no—
172Dic163—22no—
173Dic163—41no—
174Dic163—41no—
175Dic163—41no—
176Dic163C22⋊Q814yes—
177C2×C163—42no—
178C2×C163C2×C22⋊Q811yescharacteristic
179C2×C163#52511yescharacteristic
180C2×C163#52911yescharacteristic
181C2×C163#52911yescharacteristic
182C2×C163#53211yescharacteristic
183C2×C163#53311yescharacteristic
184C2×C163#53311yescharacteristic
185C163#239611yescharacteristic, complemented, Sylow
186#2396—1631nomaximal, Sylow
187C2×C4×Q8—1631no—
188C2×#117—1631no—
189#400—1631no—
190#399—1631no—
191#398—1631no—
192#398—1631no—
193#396—1631no—
194#396D16311yescharacteristic
195#395—1631no—
196#393—1631no—
197C4×C4⋊C4—1631no—
198C4×C4⋊C4—1631no—
199C4×C4⋊C4—1631no—
200C4×C22⋊C4—1631no—
201C4×C22⋊C4—1631no—
202C22×Q8—1631no—
203C23×C4—1631no—
204C4×Q8—3264no—
205#117—3264no—
206C2×C4⋊C4—1631no—
207C2×C4⋊C4—1631no—
208C2×C4⋊C4—1631no—
209C2×C4⋊C4—1631no—
210C2×C4⋊C4—1631no—
211C2×C4⋊C4—1631no—
212C2×C4⋊C4—1631no—
213C2×C4⋊C4—1631no—
214C2×C4⋊C4C2×D16311yescharacteristic
215C2×C4⋊C4C2×D16311yescharacteristic
216C2×C4⋊C4C2×D16311yescharacteristic
217C2×C22⋊C4—1631no—
218C2×C22⋊C4—1631no—
219C2×C22⋊C4—1631no—
220C2×C22⋊C4C2×D16311yescharacteristic
221C2×C22⋊C4C2×D16311yescharacteristic
222C2×C22⋊C4C2×D16311yescharacteristic
223C2×C42—1631no—
224C2×C42—1631no—
225C2×C42—1631no—
226C2×C42—1631no—
227C2×C42—1631no—
228C2×C42—1631no—
229#95—1631no—
230#95—1631no—
231#95—1631no—
232#95—1631no—
233#95—1631no—
234#95—1631no—
235#95—1631no—
236#95—1631no—
237#95—1631no—
238#95C2×D16311yescharacteristic
239C24C22×D16311yescharacteristic
240C2×Q8—3262no—
241C2×Q8—3262no—
242C2×Q8—3262no—
243C22×C4—3262no—
244C22×C4—3262no—
245C22×C4—1631no—
246C22×C4—1631no—
247C22×C4—1631no—
248C22×C4—1634no—
249C22×C4—1631no—
250C22×C4—1631no—
251C22×C4—1631no—
252C22×C4—1631no—
253C22×C4—1631no—
254C22×C4C22×D16311yescharacteristic
255C22×C4C22×D16311yescharacteristic
256C22×C4C22×D16311yescharacteristic
257C22×C4C22×D16311yescharacteristic
258C22×C4C22×D16311yescharacteristic
259C22×C4C22×D16311yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—3262no—
264C4⋊C4—3262no—
265C4⋊C4—3262no—
266C4⋊C4—3262no—
267C4⋊C4—3262no—
268C4⋊C4—3262no—
269C4⋊C4—3262no—
270C4⋊C4—3262no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—3262no—
275C22⋊C4—3262no—
276C22⋊C4—3262no—
277C42—3262no—
278C42—3262no—
279C42—3262no—
280C42—3262no—
281C42—3262no—
282C42—1634no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23Q8×D16312yes—
290C23C23×D16311yescharacteristic
291Q8—3268no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
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—3268no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—3261no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—1632no—
326C2×C4—1632no—
327C2×C4—1632no—
328C2×C4—1632no—
329C2×C4—1632no—
330C2×C4—1632no—
331C2×C4—1632no—
332C2×C4D4×D16312yes—
333C2×C4#1457969812yes—
334C2×C4#1457969812yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C2×D4×D16311yescharacteristic
343C22C2×Q8×D16311yescharacteristic
344C22#41×D16311yescharacteristic
345C22#41×D16311yescharacteristic
346C22#41×D16311yescharacteristic
347C22C2×#1457969811yescharacteristic
348C22C2×#1457969811yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3262no—
356C4—3262no—
357C4—3262no—
358C4—3262no—
359C4—6521no—
360C4—6521no—
361C4—6521no—
362C4—1634no—
363C2—42no—
364C2C22⋊Q8×D16311yescharacteristic
365C2#2378217511yescharacteristic
366C2#2378219611yescharacteristic
367C2#2378219811yescharacteristic
368C2#2378221011yescharacteristic
369C2#2378221711yescharacteristic
370C2#2378221711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#396×C163C2
prime radicalC24×C163C23
hypercenter#396D163
hypocenterC163#2396
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#19104700C2semidirect
3. #23781606C2semidirect
5. #23781598C2semidirect
6. #23781595C2semidirect
8. #23781585C2semidirect
10. C4⋊C4×Dic163C2semidirect
11. C4⋊C4×Dic163C2semidirect
12. #23781516C2semidirect
58. C22×Dic163Q8semidirect
185. C163#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#191047002
C2×#191046912
#237816062
#237816012
#237815982
#237815952
#237815882
#237815852
#237815772
C4⋊C4×Dic1632
#237815162
C22⋊C4×Dic1632
#396×C1632
#2396163