Finite Group Databasebeta

← #35226408

Subgroups — #35226408

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#23782370C211yesmaximal, characteristic
2Q8×C22⋊D163C211yesmaximal, characteristic
3#23782014C211yesmaximal, characteristic, complemented
4#23782000C211yesmaximal, characteristic, complemented
5#23781999C211yesmaximal, characteristic, complemented
6#23781958C211yesmaximal, characteristic, complemented
7#23781906C211yesmaximal, characteristic, complemented
8#23781905C211yesmaximal, characteristic, complemented
9#23781874C211yesmaximal, characteristic, complemented
10#23781822C211yesmaximal, characteristic, complemented
11#23781821C211yesmaximal, characteristic, complemented
12#23781757C211yesmaximal, characteristic, complemented
13C8×C22⋊D163C211yesmaximal, characteristic
14#103×D163C211yesmaximal, characteristic, complemented
15C2×#103×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×Dic652C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104733—21no—
23#19104733C2211yescharacteristic
24#19104725—21no—
25#19104715C2211yescharacteristic
26#19104714—21no—
27#19104714—21no—
28#19104714C2211yescharacteristic
29#19104714C2211yescharacteristic
30#19104713—21no—
31#19104711—21no—
32#19104711C2211yescharacteristic
33Q8×Dic163C2211yescharacteristic
34#19104700C2211yescharacteristic
35C4×C22⋊D163—21no—
36C4×C22⋊D163—21no—
37C4×C22⋊D163C2211yescharacteristic
38C4⋊C4×D163C2211yescharacteristic
39#19104669C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104668C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104658C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C22×Q8×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C2×C4⋊C4×C163C2211yescharacteristic
55#103×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326C2311yescharacteristic
62C2×Dic326C2311yescharacteristic
63Q8×D163—22no—
64Q8×D163—41no—
65C2×C22⋊D163D411yescharacteristic
66C2×C4×D163—21no—
67C2×C4×D163—21no—
68C2×C4×D163C2311yescharacteristic
69Dic652—41no—
70#14579689—22no—
71#14579689—41no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C8×D163—41no—
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4D411yescharacteristic
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163D411yescharacteristic
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163C2311yescharacteristic
97C2×Q8×C163—21no—
98C2×Q8×C163—22no—
99C2×Q8×C163—41no—
100C2×Q8×C163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C22×C4×C163—21no—
103C22×C4×C163—21no—
104C22×C4×C163C2311yescharacteristic
105C2×C8×C163—41no—
106C2×C8×C163C2311yescharacteristic
107C2×C8×C163C2311yescharacteristic
108C4⋊C4×C163—21no—
109C4⋊C4×C163C2311yescharacteristic
110C4⋊C4×C163C2311yescharacteristic
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116Dic326—42no—
117Dic326—81no—
118C22⋊D163Q1614yescomplemented
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C4×D163—21no—
127C4×D163—21no—
128C4×D163—42no—
129C4×D163—81no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C23×C163C2×D411yescharacteristic
133Q8×C163—21no—
134Q8×C163—21no—
135Q8×C163—21no—
136Q8×C163—42no—
137Q8×C163#4112yes—
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—42no—
145C2×C4×C163—81no—
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—22no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163C2×#4111yescharacteristic
163C4×C163H(5,𝔽2)11yescharacteristic
164Dic163—21no—
165Dic163—21no—
166Dic163—41no—
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#467—1631no—
186#462—1631no—
187#451—1631no—
188#451—1631no—
189C4×Q16—1631no—
190C8×D4—1631no—
191C2×#103—1631no—
192C2×#103D16311yescharacteristic, complemented
193C22×Q8—1631no—
194C22×Q8C2×D16311yescharacteristic
195C2×Q16—1631no—
196C2×Q16—1631no—
197C2×Q16—1631no—
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⋊Q8—1631no—
211C22⋊Q8—1631no—
212C4×Q8—1631no—
213C4×Q8—1631no—
214C4×D4—3261no—
215C4×D4—3261no—
216C4×D4—1631no—
217C2×C4⋊C4—1631no—
218C2×C4⋊C4C2×D16311yescharacteristic
219#107—1631no—
220#107—1631no—
221#107—1631no—
222C4⋊C8—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103C2×D16311yescharacteristic
230#103C2×D16311yescharacteristic
231#103C2×D16311yescharacteristic
232#103C2×D16311yescharacteristic
233C22⋊C8—1631no—
234C22⋊C8—1631no—
235C4×C8—1631no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—3261no—
240C2×Q8—3261no—
241C2×Q8—3262no—
242C2×Q8—6521no—
243C2×Q8—1631no—
244C2×Q8—1631no—
245C2×Q8C22×D16311yescharacteristic
246C2×Q8C22×D16311yescharacteristic
247C2×D4—1631no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—1631no—
253C22×C4C22×D16311yescharacteristic
254Q16—3262no—
255Q16—6521no—
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—3261no—
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—3261no—
280C22⋊C4—3261no—
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—42no—
294Q8—13041no—
295Q8—3261no—
296Q8—3261no—
297Q8—3261no—
298Q8—3261no—
299Q8—6522no—
300Q8#1457969912yes—
301D4—1634no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—13041no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
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—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3261no—
340C4—3261no—
341C4—6521no—
342C4—6521no—
343C4—6521no—
344C4—6521no—
345C4—1632no—
346C4C2×#1457969911yescharacteristic
347C4#1910478011yescharacteristic
348C2—22no—
349C2—3262no—
350C2#2378230511yescharacteristic
351C2C2×Q16×D16311yescharacteristic
352C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#103×C163C2
socleC22×C163C22×D4
prime radicalC2×C22⋊D163D4
hypercenterC2×#103D163
hypocenterC163#3299
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #23782014C2semidirect
4. #23782000C2semidirect
5. #23781999C2semidirect
6. #23781958C2semidirect
7. #23781906C2semidirect
8. #23781905C2semidirect
9. #23781874C2semidirect
10. #23781822C2semidirect
11. #23781821C2semidirect
12. #23781757C2semidirect
14. #103×D163C2semidirect
15. C2×#103×C163C2semidirect
118. C22⋊D163Q16semidirect
176. C163#3299semidirect
192. C2×#103D163semidirect

Maximal subgroups

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

subgroupindex
#237823702
Q8×C22⋊D1632
#237820142
#237820002
#237819992
#237819582
#237819062
#237819052
#237818742
#237818222
#237818212
#237817572
C8×C22⋊D1632
#103×D1632
C2×#103×C1632
#3299163