Finite Group Databasebeta

← #35227611

Subgroups — #35227611

Order 20864.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782388C211yesmaximal, characteristic, complemented
2#23782060C211yesmaximal, characteristic, complemented
3#23782057C211yesmaximal, characteristic, complemented
4#23782047C211yesmaximal, characteristic
5#23781887C211yesmaximal, characteristic, complemented
6#23781884C211yesmaximal, characteristic, complemented
7#23781880C211yesmaximal, characteristic
8#23781877C211yesmaximal, characteristic
9#23781875C211yesmaximal, characteristic, complemented
10#23781872C211yesmaximal, characteristic, complemented
11#23781796C211yesmaximal, characteristic, complemented
12#23781765C211yesmaximal, characteristic, complemented
13#23781763C211yesmaximal, characteristic, complemented
14C4×#14579689C211yesmaximal, characteristic, complemented
15#556×C163C211yesmaximal, characteristic, complemented
16C2×#14579689C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20C2×#14579687C2211yescharacteristic
21C2×#14579687C2211yescharacteristic
22C2×#14579686C2211yescharacteristic
23C2×#14579686C2211yescharacteristic
24C2×#14579684C2211yescharacteristic
25#19104735C2211yescharacteristic
26#19104730—21no—
27#19104730—21no—
28#19104723C2211yescharacteristic
29#19104723C2211yescharacteristic
30#19104711—21no—
31C4×Dic326C2211yescharacteristic
32C4×Dic326C2211yescharacteristic
33C4×D652C2211yescharacteristic
34#19104693—21no—
35#19104668C2211yescharacteristic
36C4⋊(C163⋊C8)C2211yescharacteristic
37C4⋊(C163⋊C8)C2211yescharacteristic
38C4⋊(C163⋊C8)C2211yescharacteristic
39#19104659C2211yescharacteristic
40#19104659C2211yescharacteristic
41#19104658C2211yescharacteristic
42#19104658C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104655C2211yescharacteristic
45#19104646C2211yescharacteristic
46#19104646C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48C4⋊Q8×C163C2211yescharacteristic
49#126×C163—21no—
50#126×C163—21no—
51#124×C163C2211yescharacteristic
52#124×C163C2211yescharacteristic
53C22⋊Q8×C163—21no—
54C4×Q8×C163C2211yescharacteristic
55C4×Q8×C163C2211yescharacteristic
56C4×Q8×C163C2211yescharacteristic
57C4×D4×C163C2211yescharacteristic
58#117×C163—21no—
59C2×Dic326C2311yescharacteristic
60C2×Dic326C2311yescharacteristic
61C2×D652C2311yescharacteristic
62C2×C4×D163—21no—
63#14579689—22no—
64#14579689—41no—
65#14579689—41no—
66#14579689—41no—
67#14579687—22no—
68#14579687—41no—
69#14579686—22no—
70#14579686—41no—
71#14579684—41no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86C4×Dic163—21no—
87C4×Dic163—21no—
88C2×Q8×C163D411yescharacteristic
89C2×Q8×C163C2311yescharacteristic
90C2×Q8×C163C2311yescharacteristic
91C2×D4×C163C2311yescharacteristic
92C22×C4×C163—21no—
93C4⋊C4×C163—21no—
94C4⋊C4×C163—21no—
95C4⋊C4×C163—21no—
96C4⋊C4×C163—21no—
97C4⋊C4×C163D411yescharacteristic
98C4⋊C4×C163D411yescharacteristic
99C4⋊C4×C163D411yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C42×C163—21no—
107C42×C163—21no—
108C42×C163C2311yescharacteristic
109C42×C163C2311yescharacteristic
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116D652—21no—
117D652—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C4×D163—42no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124C163⋊C8—21no—
125C163⋊C8#4112yes—
126C23×C163—21no—
127Q8×C163—21no—
128Q8×C163—21no—
129Q8×C163—21no—
130Q8×C163—21no—
131Q8×C163—41no—
132D4×C163—21no—
133D4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—42no—
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2411yescharacteristic
145C22×C163—41no—
146C22×C163—41no—
147C22×C163—41no—
148C22×C163C22×D411yescharacteristic
149C2×D163—41no—
150C2×D163—41no—
151C2×D163—41no—
152C4×C163—21no—
153C4×C163—21no—
154C4×C163—21no—
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—21no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163C2×#4111yescharacteristic
162C4×C163H(5,𝔽2)11yescharacteristic
163Dic163—41no—
164Dic163—41no—
165Dic163—41no—
166C2×C163—81no—
167C2×C163#55011yescharacteristic
168C2×C163#57511yescharacteristic
169C2×C163#57811yescharacteristic
170D163—81no—
171C163#340211yescharacteristic, complemented, Sylow
172#3402—1631nomaximal, Sylow
173#556—1631no—
174#556D16311yescharacteristic, complemented
175#497—1631no—
176#497—1631no—
177#495—1631no—
178#464—1631no—
179#464—1631no—
180#463—1631no—
181#463—1631no—
182#462—1631no—
183#462—1631no—
184#446—1631no—
185#440—1631no—
186#440—1631no—
187C4×Q16—1631no—
188C2×Q16—1631no—
189C2×Q16—1631no—
190C2×Q16—1631no—
191C2×Q16—1631no—
192C2×SD16—1631no—
193C2×SD16—1631no—
194C2×SD16—1631no—
195C2×SD16—1631no—
196C2×D8—1631no—
197C4⋊Q8—1631no—
198C4⋊Q8C2×D16311yescharacteristic
199#126—21no—
200#126—21no—
201#126—3261no—
202#126—3261no—
203#124—1631no—
204#124—1631no—
205#124C2×D16311yescharacteristic
206#124C2×D16311yescharacteristic
207C22⋊Q8—21no—
208C22⋊Q8—3261no—
209C4×Q8—1631no—
210C4×Q8—1631no—
211C4×Q8C2×D16311yescharacteristic
212C4×Q8C2×D16311yescharacteristic
213C4×Q8C2×D16311yescharacteristic
214C4×D4—1631no—
215C4×D4C2×D16311yescharacteristic
216#117—21no—
217#117—3261no—
218#107—1631no—
219C4⋊C8—1631no—
220C4⋊C8—1631no—
221C4⋊C8—1631no—
222#103—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#102—1631no—
227#102—1631no—
228#97—1631no—
229#97—1631no—
230C4×C8—1631no—
231C2×Q8—1631no—
232C2×Q8—1631no—
233C2×Q8C22⋊D16311yescharacteristic
234C2×Q8C22×D16311yescharacteristic
235C2×Q8C22×D16311yescharacteristic
236C2×D4—1631no—
237C2×D4C22×D16311yescharacteristic
238C22×C4—21no—
239C22×C4—3261no—
240Q16—3262no—
241Q16—6521no—
242Q16—6521no—
243Q16—6521no—
244SD16—3262no—
245SD16—3262no—
246SD16—6521no—
247SD16—6521no—
248D8—6521no—
249C2×C8—1631no—
250C2×C8—1631no—
251C2×C8—1631no—
252C2×C8—1631no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—1631no—
262C4⋊C4C22⋊D16311yescharacteristic
263C4⋊C4C22⋊D16311yescharacteristic
264C4⋊C4C22⋊D16311yescharacteristic
265C4⋊C4C22×D16311yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C22⋊C4—3261no—
276C42—21no—
277C42—21no—
278C42—3261no—
279C42—3261no—
280C42C22×D16311yescharacteristic
281C42C22×D16311yescharacteristic
282C42C22×D16311yescharacteristic
283C23—21no—
284C23—3261no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—41no—
290Q8—3261no—
291Q8—3261no—
292Q8—3261no—
293Q8—3261no—
294D4—21no—
295D4—21no—
296D4—3261no—
297D4—3261no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—3261no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—6522no—
306C2×C4C2×C22⋊D16311yescharacteristic
307C2×C4C2×C22⋊D16311yescharacteristic
308C2×C4C2×C22⋊D16311yescharacteristic
309C2×C4C2×C22⋊D16311yescharacteristic
310C2×C4C2×C22⋊D16311yescharacteristic
311C2×C4C2×C22⋊D16311yescharacteristic
312C2×C4C23×D16311yescharacteristic
313C8—3261no—
314C8—3261no—
315C8—3261no—
316C8—1632no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—6521no—
321C22—6521no—
322C22—6521no—
323C22C22×C22⋊D16311yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—6521no—
334C4—6521no—
335C4—6521no—
336C4#41×D16311yescharacteristic
337C4#1910478111yescharacteristic
338C2—81no—
339C2—13041no—
340C2#2378233411yescharacteristic
341C2#2378246311yescharacteristic
342C2#2378248011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#556×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579684C22
hypercenter#556D163
hypocenterC163#3402
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #23782388C2semidirect
2. #23782060C2semidirect
3. #23782057C2semidirect
5. #23781887C2semidirect
6. #23781884C2semidirect
9. #23781875C2semidirect
10. #23781872C2semidirect
11. #23781796C2semidirect
12. #23781765C2semidirect
13. #23781763C2semidirect
14. C4×#14579689C2semidirect
15. #556×C163C2semidirect
171. C163#3402semidirect
174. #556D163semidirect

Maximal subgroups

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

subgroupindex
#237823882
#237820602
#237820572
#237820472
#237818872
#237818842
#237818802
#237818772
#237818752
#237818722
#237817962
#237817652
#237817632
C4×#145796892
#556×C1632
#3402163