Finite Group Databasebeta

← #35228623

Subgroups — #35228623

Order 20864.

992 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#23782391C211yesmaximal, characteristic, complemented
2#23782390C213yesmaximal, complemented
3#23782386C213yesmaximal, complemented
4#23782357C213yesmaximal, complemented
5#23782355C213yesmaximal, complemented
6Q8×C22⋊D163C211yesmaximal, characteristic, complemented
7Q8×C22⋊D163C211yesmaximal, characteristic, complemented
8#23782276C213yesmaximal, complemented
9#23782275C213yesmaximal, complemented
10#23782217C213yesmaximal, complemented
11#23782195C213yesmaximal
12#23782143C211yesmaximal, characteristic, complemented
13#23782141C211yesmaximal, characteristic, complemented
14C2×Q8×Dic163C211yesmaximal, characteristic, complemented
15D4×Q8×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17#19104734C2213yes—
18#19104733C2213yes—
19#19104733C2213yes—
20#19104730C2213yes—
21#19104730C2216yes—
22#19104728C2213yes—
23#19104728C2216yes—
24#19104727C2213yes—
25#19104726C2216yes—
26#19104726C2213yes—
27#19104725C2213yes—
28#19104725C2216yes—
29#19104725C2216yes—
30#19104721C2213yes—
31#19104719C2213yes—
32#19104716C2213yes—
33#19104714C2213yes—
34#19104714C2213yes—
35#19104712C2213yes—
36#19104711C2213yes—
37#19104711C2213yes—
38#19104707C2211yescharacteristic
39Q8×Dic163C2213yes—
40Q8×Dic163C2213yes—
41Q8×Dic163C2211yescharacteristic
42Q8×Dic163C2211yescharacteristic
43Q8×Dic163C2213yes—
44Q8×Dic163C2213yes—
45Q8×Dic163C2211yescharacteristic
46C4×Dic326C2213yes—
47C4×Dic326C2213yes—
48#19104700C2213yes—
49#19104700C2216yes—
50C4×C22⋊D163C2213yes—
51C4×C22⋊D163C2213yes—
52#19104694C2213yes—
53#19104694C2213yes—
54#19104693C2213yes—
55#19104692C2213yes—
56#19104691C2216yes—
57C2×C4⋊Dic163C2213yes—
58C2×C4×Dic163C2213yes—
59C22×Q8×C163C2211yescharacteristic
60C22×Q8×C163C2211yescharacteristic
61C4⋊Q8×C163C2213yes—
62C22⋊Q8×C163C2213yes—
63C22⋊Q8×C163C2213yes—
64C4×Q8×C163C2211yescharacteristic
65C4×D4×C163C2213yes—
66C2×Dic326C2313yes—
67C2×Dic326C2313yes—
68Q8×D163—24no—
69C2×C22⋊D163C2311yescharacteristic
70C2×C22⋊D163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C22×Dic163C2313yes—
73C2×C4×D163C2313yes—
74C4⋊Dic163C2313yes—
75C4⋊Dic163C2313yes—
76C4⋊Dic163C2316yes—
77C4⋊Dic163C2313yes—
78C4⋊Dic163C2313yes—
79C4⋊Dic163C2313yes—
80C4⋊Dic163C2313yes—
81Dic163⋊C4C2316yes—
82Dic163⋊C4C2316yes—
83Dic163⋊C4C2313yes—
84Dic163⋊C4C2313yes—
85Dic163⋊C4C2313yes—
86Dic163⋊C4C2313yes—
87Dic163⋊C4C2316yes—
88Dic163⋊C4C2311yescharacteristic
89Dic163⋊C4C2313yes—
90C22⋊Dic163C2313yes—
91C22⋊Dic163C2311yescharacteristic
92C22⋊Dic163C2313yes—
93C22⋊Dic163C2313yes—
94(C2×D163)⋊C4C2311yescharacteristic
95(C2×D163)⋊C4C2313yes—
96(C2×D163)⋊C4C2313yes—
97(C2×D163)⋊C4C2313yes—
98C4×Dic163C2313yes—
99C4×Dic163C2313yes—
100C4×Dic163C2313yes—
101C4×Dic163C2313yes—
102C4×Dic163C2316yes—
103C4×Dic163C2316yes—
104C4×Dic163C2313yes—
105C2×Q8×C163—24no—
106C2×Q8×C163—24no—
107C2×Q8×C163C2313yes—
108C2×Q8×C163C2313yes—
109C2×Q8×C163C2311yescharacteristic
110C2×D4×C163C2311yescharacteristic
111C22×C4×C163C2313yes—
112C22×C4×C163C2313yes—
113C4⋊C4×C163C2313yes—
114C4⋊C4×C163C2313yes—
115C4⋊C4×C163C2316yes—
116C22⋊C4×C163C2313yes—
117C22⋊C4×C163C2313yes—
118C42×C163C2313yes—
119C22×D163C2411yescharacteristic
120Dic326—26no—
121Dic326—26no—
122C22⋊D163—22no—
123C22⋊D163—22no—
124C2×Dic163—26no—
125C2×Dic163—22no—
126C2×Dic163C2411yescharacteristic
127C2×Dic163C2413yes—
128C2×Dic163C2413yes—
129C2×Dic163C2411yescharacteristic
130C2×Dic163C2413yes—
131C2×Dic163C2413yes—
132C2×Dic163C2411yescharacteristic
133C4×D163—26no—
134C23×C163C2411yescharacteristic
135C23×C163C2411yescharacteristic
136Q8×C163—26no—
137Q8×C163—26no—
138Q8×C163#4114yes—
139D4×C163—22no—
140C2×C4×C163—26no—
141C2×C4×C163—26no—
142C2×C4×C163C2411yescharacteristic
143C2×C4×C163C2413yes—
144C2×C4×C163C2413yes—
145C2×C4×C163C2413yes—
146C2×C4×C163C2413yes—
147C22×C163—21no—
148C22×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C1632−1+412yes—
153C22×C163C2511yescharacteristic
154C2×D163—21no—
155C2×D163—21no—
156C2×D163—21no—
157C4×C163—23no—
158C4×C163—23no—
159C4×C163—21no—
160C4×C163—23no—
161C4×C163C2×#4116yes—
162Dic163—23no—
163Dic163—21no—
164Dic163—23no—
165Dic163—23no—
166Dic163—21no—
167Dic163—23no—
168Dic1632−1+412yes—
169C2×C163—22no—
170C2×C163—41no—
171C2×C163C22×#4111yescharacteristic
172C2×C163C2×2−1+411yescharacteristic
173C2×C163C2×2−1+411yescharacteristic
174D163—41no—
175C163#352311yescharacteristic, complemented, Sylow
176#3523—1631nomaximal, Sylow
177#557—1633no—
178#557—1631no—
179#556—1633no—
180#552—1633no—
181#552—1633no—
182D4×Q8—1631no—
183D4×Q8—1631no—
184D4×Q8D16311yescharacteristic, complemented
185#541—1633no—
186#541—1633no—
187#533—1633no—
188#529—1633no—
189#519—1631no—
190#519—1631no—
191C2×C4×Q8—1631no—
192C22×Q8—1631no—
193C22×Q8C2×D16311yescharacteristic
194C22×Q8C2×D16311yescharacteristic
195C4⋊Q8—1633no—
196C4⋊Q8—1633no—
197C4⋊Q8—1633no—
198C4⋊Q8C2×D16313yes—
199#126—1633no—
200#126—1636no—
201#126—1633no—
202#126—1636no—
203#125—1633no—
204#125—1636no—
205#125—1636no—
206#125—1636no—
207#125—1633no—
208#125—1633no—
209#124—1633no—
210#123—1633no—
211#123—1633no—
212C22⋊Q8—1633no—
213C22⋊Q8—1633no—
214C22⋊Q8—1633no—
215C22⋊Q8—1633no—
216C22⋊Q8—1633no—
217C22⋊Q8C2×D16313yes—
218C22⋊Q8C2×D16313yes—
219#121—1631no—
220C4×Q8—1631no—
221C4×Q8—1633no—
222C4×Q8—1631no—
223C4×Q8—1633no—
224C4×Q8—1633no—
225C4×Q8—1631no—
226C4×Q8—1633no—
227C4×Q8—1633no—
228C4×Q8—1636no—
229C4×Q8—1633no—
230C4×Q8—1633no—
231C4×Q8C2×D16311yescharacteristic
232C4×D4—1633no—
233C4×D4—1633no—
234C4×D4C2×D16313yes—
235#117—1633no—
236#117—1633no—
237#117—1633no—
238#117—1636no—
239#117—1633no—
240C2×C4⋊C4—1633no—
241C2×C42—1633no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—3264no—
245C2×Q8—1633no—
246C2×Q8—1633no—
247C2×Q8C22×D16313yes—
248C2×Q8C22×D16313yes—
249C2×Q8C22×D16311yescharacteristic
250C2×D4—1631no—
251C2×D4—1631no—
252C2×D4C22×D16311yescharacteristic
253C22×C4—1631no—
254C22×C4—1633no—
255C22×C4—1633no—
256C22×C4C22×D16313yes—
257C22×C4C22×D16313yes—
258C4⋊C4—1633no—
259C4⋊C4—1633no—
260C4⋊C4—1636no—
261C4⋊C4—1633no—
262C4⋊C4—1633no—
263C4⋊C4—1633no—
264C4⋊C4—1633no—
265C4⋊C4—1633no—
266C4⋊C4—1636no—
267C4⋊C4—1636no—
268C4⋊C4—1633no—
269C4⋊C4—1633no—
270C4⋊C4—1631no—
271C4⋊C4—1636no—
272C4⋊C4—1633no—
273C4⋊C4—1633no—
274C4⋊C4C22×D16313yes—
275C4⋊C4C22×D16313yes—
276C4⋊C4C22×D16316yes—
277C22⋊C4—1631no—
278C22⋊C4—1633no—
279C22⋊C4—1633no—
280C22⋊C4—1633no—
281C22⋊C4—1631no—
282C22⋊C4—1633no—
283C22⋊C4—1633no—
284C22⋊C4—1633no—
285C22⋊C4C22×D16313yes—
286C22⋊C4C22×D16313yes—
287C42—1633no—
288C42—1633no—
289C42—1633no—
290C42—1633no—
291C42—1636no—
292C42—1636no—
293C42—1633no—
294C42C22×D16313yes—
295C23—1631no—
296C23C23×D16311yescharacteristic
297C23C23×D16311yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—3266no—
301Q8—3266no—
302Q8#1457969814yes—
303D4—22no—
304D4—3262no—
305D4—3262no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—3266no—
309C2×C4—3262no—
310C2×C4—3266no—
311C2×C4—1631no—
312C2×C4—1633no—
313C2×C4—1633no—
314C2×C4—1631no—
315C2×C4—1633no—
316C2×C4—1633no—
317C2×C4—1631no—
318C2×C4C23×D16313yes—
319C2×C4C23×D16313yes—
320C2×C4C23×D16313yes—
321C2×C4C23×D16311yescharacteristic
322C2×C4C23×D16313yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—3261no—
329C22—3261no—
330C22—3261no—
331C22#1910478312yes—
332C22C24×D16311yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—3263no—
338C4—3261no—
339C4—3263no—
340C4—3263no—
341C4—3263no—
342C4—3261no—
343C4—1632no—
344C4C2×#1457969816yes—
345C2—22no—
346C2—41no—
347C2—6521no—
348C2C22×#1457969811yescharacteristic
349C22−1+4×D16311yescharacteristic
350C2C2×#1910478311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
FittingD4×Q8×C163C2
prime radical#19104707C22
hypercenterD4×Q8D163
hypocenterC163#3523
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #23782391C2semidirect
2. #23782390C2semidirect
3. #23782386C2semidirect
4. #23782357C2semidirect
5. #23782355C2semidirect
6. Q8×C22⋊D163C2semidirect
7. Q8×C22⋊D163C2semidirect
8. #23782276C2semidirect
9. #23782275C2semidirect
10. #23782217C2semidirect
12. #23782143C2semidirect
13. #23782141C2semidirect
14. C2×Q8×Dic163C2semidirect
15. D4×Q8×C163C2semidirect
175. C163#3523semidirect
184. D4×Q8D163semidirect

Maximal subgroups

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

subgroupindex
#237823912
#237823902
#237823862
#237823572
#237823552
Q8×C22⋊D1632
#237822762
#237822752
#237822172
#237821952
#237821432
#237821412
C2×Q8×Dic1632
D4×Q8×C1632
#3523163