Finite Group Databasebeta

← #35224498

Subgroups — #35224498

Order 20864.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782238C211yesmaximal, characteristic
2#23782146C211yesmaximal, characteristic
3#23782009C211yesmaximal, characteristic, complemented
4#23782007C211yesmaximal, characteristic, complemented
5#23782004C211yesmaximal, characteristic, complemented
6#23782003C211yesmaximal, characteristic, complemented
7#23782000C211yesmaximal, characteristic, complemented
8#23781999C211yesmaximal, characteristic, complemented
9#23781997C211yesmaximal, characteristic, complemented
10#23781704C211yesmaximal, characteristic, complemented
11#23781700C211yesmaximal, characteristic, complemented
12#23781672C211yesmaximal, characteristic, complemented
13#103×D163C211yesmaximal, characteristic, complemented
14#23781630C211yesmaximal, characteristic
15#483×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20#19104734—21no—
21#19104721—21no—
22#19104720—21no—
23#19104717—21no—
24#19104715C2211yescharacteristic
25#19104714C2211yescharacteristic
26#19104711C2211yescharacteristic
27#19104700—21no—
28#19104697—21no—
29C4×C22⋊D163—21no—
30C4×C22⋊D163—21no—
31#19104694C2211yescharacteristic
32#19104693C2211yescharacteristic
33#19104693C2211yescharacteristic
34#19104693C2211yescharacteristic
35C4⋊C4×D163C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104667C2211yescharacteristic
42#19104666C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104658C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52C22⋊Q8×C163C2211yescharacteristic
53C2×C4⋊C4×C163C2211yescharacteristic
54#106×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59#14579699—22no—
60#14579699—22no—
61C2×Dic326—21no—
62C2×Dic326D411yescharacteristic
63C2×Dic326C2311yescharacteristic
64Q8×D163—22no—
65Q8×D163—22no—
66C2×D652D411yescharacteristic
67C2×C22⋊D163D411yescharacteristic
68C2×C22⋊D163D411yescharacteristic
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163C2311yescharacteristic
73C2×C4×D163C2311yescharacteristic
74#14579682—22no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C8×D163—22no—
78C4⋊Dic163—21no—
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C2×Q8×C163C2311yescharacteristic
98C22×C4×C163—21no—
99C22×C4×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163—21no—
103C4⋊C4×C163—21no—
104C4⋊C4×C163C2311yescharacteristic
105C4⋊C4×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108C22×D163C2×D411yescharacteristic
109C22×D163C2×D411yescharacteristic
110Dic326—22no—
111Dic326—21no—
112Dic326—21no—
113Dic326—42no—
114D652—22no—
115C22⋊D163—22no—
116C22⋊D163—22no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C4×D163—21no—
124C4×D163—21no—
125C4×D163—42no—
126C4×D163—81no—
127C4×D163—81no—
128C4×D163#4112yes—
129C4×D163#4112yes—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C23×C163C2×D411yescharacteristic
133Q8×C163—21no—
134Q8×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—81no—
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2411yescharacteristic
144C8×C163—21no—
145C8×C163—21no—
146C22×C163—21no—
147C22×C163—21no—
148C22×C163—21no—
149C22×C163C22×D411yescharacteristic
150C2×D163—21no—
151C2×D163—21no—
152C2×D163—21no—
153C2×D163—21no—
154C2×D163—21no—
155C2×D163#12312yes—
156C4×C163—21no—
157C4×C163—41no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163C2×#4111yescharacteristic
162C4×C163C2×#4111yescharacteristic
163Dic163—21no—
164Dic163—41no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168Dic163#12312yes—
169C2×C163—41no—
170C2×C163C2×#12311yescharacteristic
171C2×C163#57711yescharacteristic
172C2×C163#57811yescharacteristic
173D163—22no—
174D163—41no—
175C163#309711yescharacteristic, complemented, Sylow
176#3097—1631nomaximal, Sylow
177#536—1631no—
178#520—1631no—
179#485—1631no—
180#485—1631no—
181#485—1631no—
182#485—1631no—
183#484—1631no—
184#484—1631no—
185#483—1631no—
186#483D16311yescharacteristic, complemented
187#428—1631no—
188#427—1631no—
189#419—1631no—
190C2×#103—1631no—
191#408—1631no—
192C2×#41—1631no—
193C22×Q8—1631no—
194C2×M16—1631no—
195C22×C8—1631no—
196C4⋊Q8—3261no—
197#124—3261no—
198#123—3261no—
199#123—3261no—
200C22⋊Q8—1631no—
201C22⋊Q8—1631no—
202C22⋊Q8—1631no—
203C22⋊Q8C2×D16311yescharacteristic
204C4×Q8—3261no—
205C4×D4—3261no—
206C4×D4—3261no—
207C4×D4—3261no—
208#117—1631no—
209#117—1631no—
210#117—1631no—
211#117—1631no—
212C2×C4⋊C4—1631no—
213C2×C4⋊C4—1631no—
214C2×C4⋊C4C2×D16311yescharacteristic
215#107—1631no—
216#107—1631no—
217#107—1631no—
218#107—1631no—
219#106—1631no—
220#106—1631no—
221#106C2×D16311yescharacteristic
222#106C2×D16311yescharacteristic
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103C2×D16311yescharacteristic
230#103C2×D16311yescharacteristic
231C22⋊C8—1631no—
232C22⋊C8—1631no—
233C22⋊C8—1631no—
234C22⋊C8C2×D16311yescharacteristic
235#41—3262no—
236#41—3262no—
237C2×Q8—3261no—
238C2×Q8—3262no—
239C2×Q8—3262no—
240C2×Q8—1631no—
241C2×Q8—1631no—
242C2×Q8C22×D16311yescharacteristic
243C2×D4—1631no—
244C2×D4—1631no—
245C2×D4—1631no—
246C22×C4—21no—
247C22×C4—3261no—
248C22×C4—3261no—
249C22×C4—3261no—
250C22×C4—1631no—
251C22×C4—1631no—
252C22×C4C22×D16311yescharacteristic
253M16—3262no—
254C2×C8—3262no—
255C2×C8—1631no—
256C2×C8—1631no—
257C2×C8C22×D16311yescharacteristic
258C2×C8C22×D16311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—3261no—
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—1631no—
267C4⋊C4—1631no—
268C4⋊C4—1631no—
269C4⋊C4C22×D16311yescharacteristic
270C4⋊C4C22×D16311yescharacteristic
271C4⋊C4C22×D16311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C42—3261no—
281C42—3261no—
282C42—3261no—
283C42—3261no—
284C23—1631no—
285C23—1631no—
286C23D4×D16311yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—3262no—
290Q8—3261no—
291Q8—3261no—
292Q8—6522no—
293D4—3262no—
294D4—3262no—
295D4—3262no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—13041no—
304C2×C4—13041no—
305C2×C4—3261no—
306C2×C4—3261no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—6522no—
312C2×C4—1631no—
313C2×C4—1631no—
314C2×C4—1632no—
315C2×C4—1632no—
316C2×C4D4×D16311yescharacteristic
317C2×C4C23×D16311yescharacteristic
318C8—21no—
319C8—21no—
320C8—3261no—
321C8—3261no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—3261no—
326C22—3261no—
327C22—3261no—
328C22—3261no—
329C22—3261no—
330C22—1632no—
331C22C2×D4×D16311yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3261no—
338C4—6521no—
339C4—6521no—
340C4—6521no—
341C4—6521no—
342C4—1632no—
343C4#41×D16311yescharacteristic
344C4#41×D16311yescharacteristic
345C2—41no—
346C2—3262no—
347C2—6521no—
348C2#123×D16311yescharacteristic
349C2#2378247111yescharacteristic
350C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#483×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579699C22
hypercenter#483D163
hypocenterC163#3097
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #23782009C2semidirect
4. #23782007C2semidirect
5. #23782004C2semidirect
6. #23782003C2semidirect
7. #23782000C2semidirect
8. #23781999C2semidirect
9. #23781997C2semidirect
10. #23781704C2semidirect
11. #23781700C2semidirect
12. #23781672C2semidirect
13. #103×D163C2semidirect
15. #483×C163C2semidirect
175. C163#3097semidirect
186. #483D163semidirect

Maximal subgroups

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

subgroupindex
#237822382
#237821462
#237820092
#237820072
#237820042
#237820032
#237820002
#237819992
#237819972
#237817042
#237817002
#237816722
#103×D1632
#237816302
#483×C1632
#3097163