Finite Group Databasebeta

← #35218036

Subgroups — #35218036

Order 20864.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104712C211yesmaximal, characteristic
2C2×C4×Dic326C211yesmaximal, characteristic, complemented
3#23781607C211yesmaximal, characteristic, complemented
4#23781605C211yesmaximal, characteristic, complemented
5#23781593C211yesmaximal, characteristic
6#23781590C211yesmaximal, characteristic
7#23781569C211yesmaximal, characteristic, complemented
8#23781560C211yesmaximal, characteristic, complemented
9#23781559C211yesmaximal, characteristic
10#23781551C211yesmaximal, characteristic, complemented
11#23781549C211yesmaximal, characteristic, complemented
12#23781529C211yesmaximal, characteristic, complemented
13#23781523C211yesmaximal, characteristic
14C22⋊C4×Dic163C211yesmaximal, characteristic
15#388×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18#19104712—24no—
19C4×Dic326—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
34C163⋊#95C2211yescharacteristic
35C163⋊#95C2211yescharacteristic
36C163⋊#95C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C22⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C42×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C2×Dic326—22no—
54C2×Dic326—22no—
55C2×Dic326D414yes—
56C22×Dic163—22no—
57C22×Dic163—22no—
58C22×Dic163—22no—
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—
81C24×C163C2311yescharacteristic
82C22×C4×C163C2311yescharacteristic
83C22×C4×C163C2311yescharacteristic
84C22×C4×C163C2311yescharacteristic
85C22×C4×C163C2311yescharacteristic
86C22×C4×C163C2311yescharacteristic
87C22×C4×C163C2311yescharacteristic
88C4⋊C4×C163—22no—
89C22⋊C4×C163—22no—
90C22⋊C4×C163—22no—
91C22⋊C4×C163—22no—
92C42×C163—22no—
93Dic326—28no—
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—21no—
114C2×Dic163—44no—
115C2×Dic163C2×D412yes—
116C2×Dic163C2×D412yes—
117C2×Dic163#4112yes—
118C2×Dic163#4112yes—
119C23×C163—21no—
120C23×C163—21no—
121C23×C163—21no—
122C23×C163—21no—
123C23×C163—21no—
124C23×C163—21no—
125C23×C163#4112yes—
126C23×C163C2411yescharacteristic
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×C163—21no—
143C2×C4×C163C2×D412yes—
144C2×C4×C163#4112yes—
145C22×C163—22no—
146C22×C163—22no—
147C22×C163—22no—
148C22×C163—41no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163C22×D411yescharacteristic
153C22×C163C2×#4111yescharacteristic
154C22×C163C2×#4111yescharacteristic
155C22×C163C2×#4111yescharacteristic
156C22×C163C2×#4111yescharacteristic
157C22×C1632−1+411yescharacteristic
158C22×C1632−1+411yescharacteristic
159C4×C163—22no—
160C4×C163—22no—
161C4×C163—41no—
162C4×C163—41no—
163C4×C163—41no—
164C4×C163—41no—
165Dic163—22no—
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172Dic163#12314yes—
173C2×C163—42no—
174C2×C163C2×#12311yescharacteristic
175C2×C163#52911yescharacteristic
176C2×C163#54111yescharacteristic
177C2×C163#54811yescharacteristic
178C2×C163#54811yescharacteristic
179C2×C163#55211yescharacteristic
180C2×C163#55611yescharacteristic
181C163#242111yescharacteristic, complemented, Sylow
182#2421—1631nomaximal, Sylow
183C2×C22⋊Q8—1631no—
184C2×C4×Q8—1631no—
185#400—1631no—
186#400—1631no—
187#397—1631no—
188#397—1631no—
189#391—1631no—
190#389—1631no—
191#388—1631no—
192#388D16311yescharacteristic
193#387—1631no—
194#387—1631no—
195#382—1631no—
196#380—1631no—
197C4×C22⋊C4—1631no—
198C22×Q8—1631no—
199C23×C4—1631no—
200C22⋊Q8—3264no—
201C4×Q8—3264no—
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⋊C4—1631no—
209C2×C4⋊C4—1631no—
210C2×C4⋊C4C2×D16311yescharacteristic
211C2×C22⋊C4—1631no—
212C2×C22⋊C4—1631no—
213C2×C22⋊C4—1631no—
214C2×C22⋊C4C2×D16311yescharacteristic
215C2×C22⋊C4C2×D16311yescharacteristic
216C2×C22⋊C4C2×D16311yescharacteristic
217C2×C42—1631no—
218C2×C42—1631no—
219C2×C42—1631no—
220C2×C42C2×D16311yescharacteristic
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#95—1631no—
232#95—1631no—
233#95C2×D16311yescharacteristic
234#95C2×D16311yescharacteristic
235C24C22×D16311yescharacteristic
236C2×Q8—3262no—
237C2×Q8—3262no—
238C2×Q8—1634no—
239C22×C4—3262no—
240C22×C4—3262no—
241C22×C4—3262no—
242C22×C4—1631no—
243C22×C4—1631no—
244C22×C4—1631no—
245C22×C4—1631no—
246C22×C4—1631no—
247C22×C4—1631no—
248C22×C4—1631no—
249C22×C4—1631no—
250C22×C4C22×D16311yescharacteristic
251C22×C4C22×D16311yescharacteristic
252C22×C4C22×D16311yescharacteristic
253C22×C4C22×D16311yescharacteristic
254C22×C4C22×D16311yescharacteristic
255C22×C4C22×D16311yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—3262no—
258C4⋊C4—3262no—
259C4⋊C4—3262no—
260C4⋊C4—3262no—
261C4⋊C4—3262no—
262C4⋊C4—3262no—
263C4⋊C4—3262no—
264C4⋊C4—3262no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—3262no—
269C22⋊C4—3262no—
270C22⋊C4—3262no—
271C42—22no—
272C42—3262no—
273C42—3262no—
274C42—3262no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23#1457969912yes—
282C23C23×D16311yescharacteristic
283Q8—3268no—
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—21no—
299C2×C4—21no—
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—3261no—
319C2×C4—3261no—
320C2×C4—6524no—
321C2×C4—1632no—
322C2×C4—1632no—
323C2×C4—1632no—
324C2×C4—1632no—
325C2×C4D4×D16312yes—
326C2×C4#1457969812yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22C2×D4×D16311yescharacteristic
335C22#41×D16311yescharacteristic
336C22#41×D16311yescharacteristic
337C22C2×#1457969811yescharacteristic
338C22C2×#1457969911yescharacteristic
339C22#1910478211yescharacteristic
340C22#1910478211yescharacteristic
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—3262no—
348C4—6521no—
349C4—6521no—
350C4—6521no—
351C4—6521no—
352C4—6521no—
353C4—6521no—
354C4—1634no—
355C2—42no—
356C2#123×D16311yescharacteristic
357C2#2378219511yescharacteristic
358C2#2378227211yescharacteristic
359C2#2378232111yescharacteristic
360C2#2378232411yescharacteristic
361C2#2378234811yescharacteristic
362C2#2378238411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#388×C163C2
prime radicalC24×C163C23
hypercenter#388D163
hypocenterC163#2421
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic326C2semidirect
3. #23781607C2semidirect
4. #23781605C2semidirect
7. #23781569C2semidirect
8. #23781560C2semidirect
10. #23781551C2semidirect
11. #23781549C2semidirect
12. #23781529C2semidirect
181. C163#2421semidirect

Maximal subgroups

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

subgroupindex
C2×#191047122
C2×C4×Dic3262
#237816072
#237816052
#237815932
#237815902
#237815692
#237815602
#237815592
#237815512
#237815492
#237815292
#237815232
C22⋊C4×Dic1632
#388×C1632
#2421163