Finite Group Databasebeta

← #35224821

Subgroups — #35224821

Order 20864.

382 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#23782268C211yesmaximal, characteristic
2#23782143C211yesmaximal, characteristic, complemented
3#23781990C211yesmaximal, characteristic, complemented
4#23781986C211yesmaximal, characteristic, complemented
5#23781974C211yesmaximal, characteristic, complemented
6#23781970C211yesmaximal, characteristic, complemented
7#23781863C211yesmaximal, characteristic, complemented
8#23781863C211yesmaximal, characteristic, complemented
9#23781804C211yesmaximal, characteristic, complemented
10#23781768C211yesmaximal, characteristic, complemented
11#23781766C211yesmaximal, characteristic, complemented
12SD16×Dic163C211yesmaximal, characteristic, complemented
13#23781750C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic, complemented
15#448×C163C211yesmaximal, characteristic, complemented
16C2×#14579686C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18#19104731C2211yescharacteristic
19#19104726C2211yescharacteristic
20#19104719—21no—
21#19104717—21no—
22#19104709C2211yescharacteristic
23#19104707—21no—
24#19104706—21no—
25Q8×Dic163—21no—
26Q8×Dic163—21no—
27Q8×Dic163C2211yescharacteristic
28Q8×Dic163C2211yescharacteristic
29Q8×Dic163C2211yescharacteristic
30Q8×Dic163C2211yescharacteristic
31D4×Dic163C2211yescharacteristic
32#19104697C2211yescharacteristic
33#19104694—21no—
34#19104694—21no—
35#19104694C2211yescharacteristic
36#19104694C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C22×Q8×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C2×SD16×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×D652C2311yescharacteristic
60C2×C22⋊D163—21no—
61C2×C22⋊D163—21no—
62C2×C22⋊D163—21no—
63C22×Dic163—21no—
64C2×C4×D163—21no—
65#14579686—22no—
66#14579686—22no—
67C2×C163⋊C8C2311yescharacteristic
68C2×C163⋊C8C2311yescharacteristic
69C4⋊Dic163—21no—
70C4⋊Dic163—21no—
71C4⋊Dic163—21no—
72C4⋊Dic163—21no—
73C4⋊Dic163—21no—
74C4⋊Dic163D411yescharacteristic
75C4⋊Dic163D411yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80C22⋊Dic163—21no—
81C22⋊Dic163—21no—
82C22⋊Dic163D411yescharacteristic
83C22⋊Dic163D411yescharacteristic
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86C4×Dic163—21no—
87C4×Dic163—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163C2311yescharacteristic
92C4×Dic163C2311yescharacteristic
93C2×Q8×C163—22no—
94C2×Q8×C163—22no—
95C2×Q8×C163—21no—
96C2×Q8×C163C2311yescharacteristic
97C2×Q8×C163C2311yescharacteristic
98C2×D4×C163—21no—
99C2×D4×C163C2311yescharacteristic
100C22×C4×C163—21no—
101C22×C4×C163C2311yescharacteristic
102SD16×C163—22no—
103SD16×C163—22no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108C22×D163—21no—
109D652—21no—
110D652—21no—
111C22⋊D163—81no—
112C22⋊D163—81no—
113C22⋊D163—81no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—81no—
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C4×D163—81no—
123C163⋊C8—21no—
124C163⋊C8—21no—
125C23×C163—21no—
126C23×C163C2×D411yescharacteristic
127Q8×C163—24no—
128Q8×C163—21no—
129Q8×C163—21no—
130Q8×C163#4112yes—
131Q8×C163#4112yes—
132D4×C163—21no—
133D4×C163—21no—
134D4×C163—81no—
135C2×C4×C163—24no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2411yescharacteristic
143C8×C163—21no—
144C8×C163—21no—
145C22×C163—21no—
146C22×C163—21no—
147C22×C163—21no—
148C22×C163—41no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163C22×D411yescharacteristic
152C2×D163—41no—
153C2×D163—41no—
154C2×D163—41no—
155C4×C163—22no—
156C4×C163—22no—
157C4×C163—21no—
158C4×C163—41no—
159C4×C163C2×#4111yescharacteristic
160C4×C163C2×#4111yescharacteristic
161Dic163—21no—
162Dic163—21no—
163Dic163—21no—
164Dic163—21no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168C2×C163—41no—
169C2×C163—81no—
170C2×C163#52511yescharacteristic
171C2×C163#57711yescharacteristic
172C2×C163#57711yescharacteristic
173D163—81no—
174C163#312611yescharacteristic, complemented, Sylow
175#3126—1631nomaximal, Sylow
176#540—1631no—
177#519—1631no—
178#482—1631no—
179#482—1631no—
180#479—1631no—
181#479—1631no—
182#460—1631no—
183#460—1631no—
184#448—1631no—
185#448D16311yescharacteristic, complemented
186#440—1631no—
187#440—1631no—
188C4×SD16—1631no—
189C4×SD16—1631no—
190#433—1631no—
191C22×Q8C2×D16311yescharacteristic
192C2×SD16—1631no—
193C2×SD16—1631no—
194C2×SD16C2×D16311yescharacteristic
195C2×SD16C2×D16311yescharacteristic
196#127—1631no—
197#125—1631no—
198#123—3261no—
199#123—3261no—
200#121—3261no—
201#121—3261no—
202#121—1631no—
203#121C2×D16311yescharacteristic
204C4×Q8—3261no—
205C4×Q8—3261no—
206C4×Q8—1631no—
207C4×Q8—1631no—
208C4×Q8—1631no—
209C4×Q8—1631no—
210C4×D4—1631no—
211C4×D4—1631no—
212#117—3261no—
213#117—3261no—
214#117—1631no—
215#117—1631no—
216#107—1631no—
217#107—1631no—
218#106—1631no—
219#106—1631no—
220C4⋊C8—1631no—
221C4⋊C8—1631no—
222#103—1631no—
223#103—1631no—
224#103C2×D16311yescharacteristic
225#103C2×D16311yescharacteristic
226#102—1631no—
227#102—1631no—
228#102—1631no—
229#102—1631no—
230C22⋊C8—1631no—
231C22⋊C8C2×D16311yescharacteristic
232#97—1631no—
233C4×C8—1631no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D16311yescharacteristic
238C2×Q8C22×D16311yescharacteristic
239C2×D4—21no—
240C2×D4—3261no—
241C2×D4—3261no—
242C2×D4—3261no—
243C2×D4—1631no—
244C2×D4C22×D16311yescharacteristic
245C22×C4—21no—
246C22×C4—3261no—
247C22×C4—3261no—
248C22×C4C22×D16311yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—3262no—
252SD16—3262no—
253C2×C8—1631no—
254C2×C8—1631no—
255C2×C8C22×D16311yescharacteristic
256C2×C8C22×D16311yescharacteristic
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—1631no—
264C4⋊C4—1631no—
265C4⋊C4—1631no—
266C4⋊C4—1631no—
267C4⋊C4—1631no—
268C4⋊C4C22×D16311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—3261no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—1631no—
275C22⋊C4—1631no—
276C42—3261no—
277C42—3261no—
278C42—3261no—
279C42—3261no—
280C42—3261no—
281C42—1631no—
282C42—1631no—
283C23—21no—
284C23—3261no—
285C23D4×D16311yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1457969812yes—
290Q8#1457969812yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—13041no—
295D4—13041no—
296D4—13041no—
297D4—3261no—
298D4—3261no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—13041no—
306C2×C4—13041no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—1631no—
311C2×C4—1631no—
312C2×C4—1631no—
313C2×C4—1631no—
314C2×C4D4×D16311yescharacteristic
315C2×C4C23×D16311yescharacteristic
316C8—21no—
317C8—21no—
318C8—3261no—
319C8—3261no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—6521no—
327C22—6521no—
328C22—6521no—
329C22C2×D4×D16311yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—3261no—
335C4—3261no—
336C4—3261no—
337C4—3261no—
338C4—6521no—
339C4—6521no—
340C4—6521no—
341C4C2×#1457969811yescharacteristic
342C4C2×#1457969811yescharacteristic
343C2—41no—
344C2—81no—
345C2—13041no—
346C2#2378217111yescharacteristic
347C2#2378247111yescharacteristic
348C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#448×C163C2
socleC22×C163C22×D4
prime radical#23782268C2
hypercenter#448D163
hypocenterC163#3126
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #23782143C2semidirect
3. #23781990C2semidirect
4. #23781986C2semidirect
5. #23781974C2semidirect
6. #23781970C2semidirect
7. #23781863C2semidirect
8. #23781863C2semidirect
9. #23781804C2semidirect
10. #23781768C2semidirect
11. #23781766C2semidirect
12. SD16×Dic163C2semidirect
13. #23781750C2semidirect
14. #23781723C2semidirect
15. #448×C163C2semidirect
174. C163#3126semidirect
185. #448D163semidirect

Maximal subgroups

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

subgroupindex
#237822682
#237821432
#237819902
#237819862
#237819742
#237819702
#237818632
#237818042
#237817682
#237817662
SD16×Dic1632
#237817502
#237817232
#448×C1632
#3126163