Finite Group Databasebeta

← #35226407

Subgroups — #35226407

Order 20864.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782373C211yesmaximal, characteristic
2#23782328C211yesmaximal, characteristic
3#23782014C211yesmaximal, characteristic, complemented
4#23782002C211yesmaximal, characteristic, complemented
5#23781998C211yesmaximal, characteristic, complemented
6#23781959C211yesmaximal, characteristic, complemented
7#23781905C211yesmaximal, characteristic, complemented
8#23781873C211yesmaximal, characteristic, complemented
9#23781820C211yesmaximal, characteristic, complemented
10#23781819C211yesmaximal, characteristic, complemented
11#23781757C211yesmaximal, characteristic, complemented
12C8×C22⋊D163C211yesmaximal, characteristic
13C2×#19104660C211yesmaximal, characteristic, complemented
14#103×D163C211yesmaximal, characteristic, complemented
15#467×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21#19104734—21no—
22#19104734C2211yescharacteristic
23#19104733C2211yescharacteristic
24#19104725—21no—
25#19104715—21no—
26#19104715C2211yescharacteristic
27#19104713—21no—
28#19104712—21no—
29#19104712—21no—
30#19104711C2211yescharacteristic
31Q8×Dic163C2211yescharacteristic
32#19104700C2211yescharacteristic
33C4×C22⋊D163C2211yescharacteristic
34(C22⋊D163)⋊C4—21no—
35(C22⋊D163)⋊C4—21no—
36C2×C4⋊Dic163C2211yescharacteristic
37C4⋊C4×D163C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104668C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41#19104660C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104658C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×Q16×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326—22no—
62C2×Dic326—21no—
63C2×Dic326—41no—
64C2×Dic326C2311yescharacteristic
65C2×Dic326C2311yescharacteristic
66Q8×D163—22no—
67Q8×D163—41no—
68C2×C22⋊D163D411yescharacteristic
69C22×Dic163—21no—
70C22×Dic163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74#14579689—22no—
75#14579689—41no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C8×D163—41no—
79C4⋊Dic163—21no—
80C4⋊Dic163—21no—
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4D411yescharacteristic
91C22⋊Dic163D411yescharacteristic
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4D411yescharacteristic
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98C2×Q8×C163C2311yescharacteristic
99C2×Q8×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101Q16×C163—41no—
102C2×C8×C163—41no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C22⋊C4×C163—21no—
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—42no—
116Dic326—42no—
117Dic326—81no—
118Dic326#4112yes—
119C22⋊D163Q1614yescomplemented
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—42no—
125C2×Dic163—81no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—21no—
130C4×D163—21no—
131C4×D163—42no—
132C4×D163—81no—
133C163⋊C8—21no—
134C163⋊C8—21no—
135C23×C163C2×D411yescharacteristic
136Q8×C163—21no—
137Q8×C163—21no—
138Q8×C163—21no—
139Q8×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×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C8×C163—21no—
149C8×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163C2×Q1612yes—
153C22×C163C22×D411yescharacteristic
154C2×D163—21no—
155C2×D163—21no—
156C2×D163C2×Q1612yes—
157C4×C163—21no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163C2×#4111yescharacteristic
163C4×C163H(5,𝔽2)11yescharacteristic
164Dic163—21no—
165Dic163—22no—
166Dic163—21no—
167Dic163—41no—
168Dic163—41no—
169Dic163—41no—
170Dic163C2×Q1612yes—
171C2×C163—22no—
172C2×C163#54611yescharacteristic
173C2×C163C22×Q1611yescharacteristic
174C2×C163#57711yescharacteristic
175D163—22no—
176C163#329911yescharacteristic, complemented, Sylow
177#3299—1631nomaximal, Sylow
178#554—1631no—
179D4×Q8—1631no—
180#487—1631no—
181#484—1631no—
182#484—1631no—
183#477—1631no—
184#467—1631no—
185#467D16311yescharacteristic, complemented
186#462—1631no—
187#451—1631no—
188#451—1631no—
189C4×Q16—1631no—
190C8×D4—1631no—
191C2×#103—1631no—
192C2×#103—1631no—
193C22×Q8—1631no—
194C22×Q8—1631no—
195C2×Q16—1631no—
196C2×Q16—1631no—
197C2×Q16C2×D16311yescharacteristic
198C22×C8—1631no—
199C22×C8C2×D16311yescharacteristic
200C4⋊Q8—3261no—
201C4⋊Q8—1631no—
202C4⋊Q8—1631no—
203#125—3261no—
204C22⋊Q8—3261no—
205C22⋊Q8—3261no—
206C22⋊Q8—3261no—
207C22⋊Q8—3261no—
208C22⋊Q8—1631no—
209C22⋊Q8—1631no—
210C22⋊Q8C2×D16311yescharacteristic
211C22⋊Q8C2×D16311yescharacteristic
212C4×Q8—1631no—
213C4×Q8—1631no—
214C4×D4—3261no—
215C4×D4—3261no—
216C4×D4—1631no—
217C2×C4⋊C4—1631no—
218C2×C4⋊C4—1631no—
219#107—1631no—
220#107—1631no—
221#107C2×D16311yescharacteristic
222C4⋊C8—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103—1631no—
231#103C2×D16311yescharacteristic
232#103C2×D16311yescharacteristic
233C22⋊C8—1631no—
234C22⋊C8—1631no—
235C4×C8—1631no—
236C2×Q8—3261no—
237C2×Q8—3261no—
238C2×Q8—3262no—
239C2×Q8—3261no—
240C2×Q8—3262no—
241C2×Q8—6521no—
242C2×Q8—6521no—
243C2×Q8—1631no—
244C2×Q8—1631no—
245C2×Q8C22×D16311yescharacteristic
246C2×Q8C22×D16311yescharacteristic
247C2×D4—1631no—
248C22×C4—3261no—
249C22×C4—3261no—
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—1631no—
253C22×C4C22×D16311yescharacteristic
254Q16—41no—
255Q16—3262no—
256Q16—6521no—
257C2×C8—41no—
258C2×C8—6521no—
259C2×C8—1631no—
260C2×C8—1631no—
261C2×C8C22×D16311yescharacteristic
262C2×C8C22×D16311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—1631no—
275C4⋊C4—1631no—
276C4⋊C4—1631no—
277C4⋊C4C22×D16311yescharacteristic
278C4⋊C4C22×D16311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—1631no—
284C22⋊C4—1631no—
285C42—3261no—
286C42—3261no—
287C42—1631no—
288C23—1631no—
289C23D4×D16311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—13041no—
295Q8—3261no—
296Q8—3261no—
297Q8—3261no—
298Q8—6522no—
299Q8—6522no—
300Q8—1632no—
301D4—1634no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—13041no—
309C2×C4—13041no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—6522no—
317C2×C4—6522no—
318C2×C4—1631no—
319C2×C4—1631no—
320C2×C4—1631no—
321C2×C4D4×D16311yescharacteristic
322C2×C4C23×D16311yescharacteristic
323C8—21no—
324C8—21no—
325C8—3261no—
326C8—3261no—
327C22—21no—
328C22—21no—
329C22—3261no—
330C22—3261no—
331C22—1632no—
332C22Q16×D16312yes—
333C22C2×D4×D16311yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3261no—
340C4—3261no—
341C4—3262no—
342C4—6521no—
343C4—6521no—
344C4—6521no—
345C4—1632no—
346C4#41×D16311yescharacteristic
347C4#1910478011yescharacteristic
348C2—22no—
349C2—3262no—
350C2#2378230411yescharacteristic
351C2C2×Q16×D16311yescharacteristic
352C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#467×C163C2
socleC22×C163C22×D4
prime radicalC2×C22⋊D163D4
hypercenter#467D163
hypocenterC163#3299
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
3. #23782014C2semidirect
4. #23782002C2semidirect
5. #23781998C2semidirect
6. #23781959C2semidirect
7. #23781905C2semidirect
8. #23781873C2semidirect
9. #23781820C2semidirect
10. #23781819C2semidirect
11. #23781757C2semidirect
13. C2×#19104660C2semidirect
14. #103×D163C2semidirect
15. #467×C163C2semidirect
119. C22⋊D163Q16semidirect
176. C163#3299semidirect
185. #467D163semidirect

Maximal subgroups

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

subgroupindex
#237823732
#237823282
#237820142
#237820022
#237819982
#237819592
#237819052
#237818732
#237818202
#237818192
#237817572
C8×C22⋊D1632
C2×#191046602
#103×D1632
#467×C1632
#3299163