Finite Group Databasebeta

← #35226592

Subgroups — #35226592

Order 20864.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782356C211yesmaximal, characteristic
2Q8×D652C211yesmaximal, characteristic
3#23782034C211yesmaximal, characteristic, complemented
4#23782009C211yesmaximal, characteristic, complemented
5#23781997C211yesmaximal, characteristic, complemented
6#23781972C211yesmaximal, characteristic, complemented
7#23781922C211yesmaximal, characteristic, complemented
8#23781905C211yesmaximal, characteristic, complemented
9#23781875C211yesmaximal, characteristic, complemented
10#23781819C211yesmaximal, characteristic, complemented
11#23781819C211yesmaximal, characteristic, complemented
12#23781738C211yesmaximal, characteristic
13#23781672C211yesmaximal, characteristic, complemented
14#103×D163C211yesmaximal, characteristic, complemented
15#441×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20C2×#14579682C2211yescharacteristic
21C2×C8×D163C2211yescharacteristic
22#19104735—21no—
23#19104735C2211yescharacteristic
24#19104729—21no—
25#19104727C2211yescharacteristic
26#19104725—21no—
27#19104720—21no—
28#19104715—21no—
29#19104715—21no—
30#19104715C2211yescharacteristic
31#19104715C2211yescharacteristic
32#19104711C2211yescharacteristic
33#19104711C2211yescharacteristic
34C4×D652—21no—
35C4×D652C2211yescharacteristic
36#19104693C2211yescharacteristic
37#19104692—21no—
38C4⋊C4×D163C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104666C2211yescharacteristic
41C4⋊(C163⋊C8)C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104660C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104659C2211yescharacteristic
48#19104658C2211yescharacteristic
49#19104658C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52C2×Q16×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326C2311yescharacteristic
62C2×Dic326C2311yescharacteristic
63Q8×D163—22no—
64Q8×D163—22no—
65Q8×D163—41no—
66Q8×D163—41no—
67C2×D652D411yescharacteristic
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163C2311yescharacteristic
72C2×C4×D163C2311yescharacteristic
73#14579689—41no—
74#14579689—41no—
75#14579682—41no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C8×D163—41no—
79C4⋊Dic163—21no—
80C4⋊Dic163—21no—
81C4⋊Dic163—21no—
82C4⋊Dic163—21no—
83C4⋊Dic163D411yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2311yescharacteristic
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4D411yescharacteristic
97(C2×D163)⋊C4D411yescharacteristic
98C4×Dic163—21no—
99C2×Q8×C163C2311yescharacteristic
100C2×Q8×C163C2311yescharacteristic
101Q16×C163—22no—
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C4⋊C4×C163—21no—
105C4⋊C4×C163—21no—
106C4⋊C4×C163C2311yescharacteristic
107C4⋊C4×C163C2311yescharacteristic
108C42×C163—21no—
109C42×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163C2×D411yescharacteristic
112C22×D163C2×D411yescharacteristic
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116Dic326—21no—
117Dic326—42no—
118Dic326—42no—
119D652—22no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C4×D163—21no—
127C4×D163—21no—
128C4×D163—21no—
129C4×D163—21no—
130C4×D163—42no—
131C4×D163—42no—
132C4×D163—42no—
133C163⋊C8—21no—
134C163⋊C8—21no—
135Q8×C163—21no—
136Q8×C163—21no—
137Q8×C163—21no—
138Q8×C163#4112yes—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2411yescharacteristic
146C8×C163—21no—
147C8×C163—21no—
148C22×C163C22×D411yescharacteristic
149C2×D163—21no—
150C2×D163—21no—
151C2×D163—21no—
152C2×D163—21no—
153C2×D163—21no—
154C2×D163#13712yes—
155C4×C163—22no—
156C4×C163—21no—
157C4×C163—21no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163C2×#4111yescharacteristic
162C4×C163H(5,𝔽2)11yescharacteristic
163Dic163—21no—
164Dic163—21no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169C2×C163#54611yescharacteristic
170C2×C163C2×#13711yescharacteristic
171C2×C163#57811yescharacteristic
172D163—22no—
173D163—41no—
174C163#331211yescharacteristic, complemented, Sylow
175#3312—1631nomaximal, Sylow
176#552—1631no—
177D4×Q8—1631no—
178#491—1631no—
179#485—1631no—
180#483—1631no—
181#479—1631no—
182#470—1631no—
183#467—1631no—
184#462—1631no—
185#451—1631no—
186#451—1631no—
187#441D16311yescharacteristic, complemented
188#435—1631no—
189#419—1631no—
190C2×#103—1631no—
191C22×Q8—1631no—
192C22×Q8—1631no—
193C2×Q16—1631no—
194C2×Q16—1631no—
195C2×Q16C2×D16311yescharacteristic
196C2×M16—1631no—
197C22×C8—1631no—
198C4⋊Q8—3261no—
199C4⋊Q8—1631no—
200#126—3261no—
201#125—3261no—
202#125—1631no—
203#123—3261no—
204C22⋊Q8—3261no—
205C22⋊Q8—3261no—
206C22⋊Q8—1631no—
207C22⋊Q8—1631no—
208C22⋊Q8—1631no—
209C22⋊Q8—1631no—
210C4×Q8C2×D16311yescharacteristic
211C4×Q8C2×D16311yescharacteristic
212C4×D4—3261no—
213C4×D4—1631no—
214#117—3261no—
215#117—1631no—
216C2×C4⋊C4—1631no—
217#107—1631no—
218#106—1631no—
219#106C2×D16311yescharacteristic
220C4⋊C8—1631no—
221#103—1631no—
222#103—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
231C22⋊C8—1631no—
232C22⋊C8—1631no—
233#97C2×D16311yescharacteristic
234C2×Q8—3261no—
235C2×Q8—3261no—
236C2×Q8—3262no—
237C2×Q8—3262no—
238C2×Q8—6521no—
239C2×Q8—6521no—
240C2×Q8—1631no—
241C2×Q8—1631no—
242C2×Q8C22×D16311yescharacteristic
243C2×Q8C22×D16311yescharacteristic
244C2×D4—1631no—
245C22×C4—3261no—
246C22×C4—3261no—
247C22×C4—3261no—
248C22×C4—1631no—
249C22×C4—1631no—
250Q16—22no—
251Q16—6521no—
252Q16—6521no—
253M16—6521no—
254C2×C8—6521no—
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—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4C22×D16311yescharacteristic
274C4⋊C4C22×D16311yescharacteristic
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—1631no—
281C22⋊C4—1631no—
282C42—21no—
283C42—21no—
284C42—3261no—
285C42C22×D16311yescharacteristic
286C23—1631no—
287C23—1631no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—3261no—
292Q8—3261no—
293Q8—3261no—
294Q8—3261no—
295Q8—6522no—
296Q8—6522no—
297Q8#1457969912yes—
298D4—3262no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
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—6522no—
312C2×C4—6522no—
313C2×C4—6522no—
314C2×C4—1631no—
315C2×C4—1631no—
316C2×C4D4×D16311yescharacteristic
317C2×C4D4×D16311yescharacteristic
318C2×C4C23×D16311yescharacteristic
319C8—21no—
320C8—21no—
321C8—3261no—
322C8—3261no—
323C22—3261no—
324C22—3261no—
325C22—3261no—
326C22—3261no—
327C22—3261no—
328C22—1632no—
329C22C2×D4×D16311yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—3261no—
337C4—3261no—
338C4—6521no—
339C4—6521no—
340C4—6521no—
341C4—6521no—
342C4C2×#1457969911yescharacteristic
343C4#1910478111yescharacteristic
344C2—3262no—
345C2—6521no—
346C2#2378231111yescharacteristic
347C2#137×D16311yescharacteristic
348C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#441×C163C2
socleC22×C163C22×D4
prime radicalC2×D652D4
hypercenter#441D163
hypocenterC163#3312
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. #23782034C2semidirect
4. #23782009C2semidirect
5. #23781997C2semidirect
6. #23781972C2semidirect
7. #23781922C2semidirect
8. #23781905C2semidirect
9. #23781875C2semidirect
10. #23781819C2semidirect
11. #23781819C2semidirect
13. #23781672C2semidirect
14. #103×D163C2semidirect
15. #441×C163C2semidirect
174. C163#3312semidirect
187. #441D163semidirect

Maximal subgroups

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

subgroupindex
#237823562
Q8×D6522
#237820342
#237820092
#237819972
#237819722
#237819222
#237819052
#237818752
#237818192
#237817382
#237816722
#103×D1632
#441×C1632
#3312163