Finite Group Databasebeta

← #27182863

Subgroups — #27182863

Order 12928.

494 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692276C212yesmaximal, complemented
2#20265849C211yesmaximal, characteristic, complemented
3#20265831C211yesmaximal, characteristic, complemented
4#20265550C211yesmaximal, characteristic, complemented
5#20265407C212yesmaximal, complemented
6#20265387C211yesmaximal, characteristic, complemented
7#20265372C211yesmaximal, characteristic, complemented
8#20265342C212yesmaximal, complemented
9#20265323C212yesmaximal, complemented
10C8×D404C211yesmaximal, characteristic, complemented
11C4×SD16×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2212yes—
13C2×#13108073C2212yes—
14#15692276—28no—
15C2×#13108064C2212yes—
16C2×#13108063C2212yes—
17C2×#13108063C2212yes—
18C2×#13108059C2212yes—
19C2×C8×D101C2212yes—
20#15692255C2211yescharacteristic
21#15692252—21no—
22#15692252C2211yescharacteristic
23#15692239—22no—
24#15692235C2212yes—
25#15692230—22no—
26#15692230C2212yes—
27#15692229—21no—
28C4×D404—21no—
29C4×D404C2211yescharacteristic
30C2×C4⋊Dic101—21no—
31C4⋊#11717579C2211yescharacteristic
32#15692178C2212yes—
33#15692175C2212yes—
34#15692167C2212yes—
35C2×SD16×C101C2211yescharacteristic
36C4×Q8×C101C2211yescharacteristic
37C4×D4×C101C2211yescharacteristic
38#106×C101C2211yescharacteristic
39#103×C101C2211yescharacteristic
40#102×C101C2211yescharacteristic
41C4×C8×C101C2211yescharacteristic
42#13108075—24no—
43#13108075—42no—
44#13108073—24no—
45#13108073—42no—
46C2×Dic202C2312yes—
47C2×D404—22no—
48C2×D404—21no—
49C2×D404—22no—
50C2×D404—21no—
51C2×D404D411yescharacteristic
52C2×D404C2312yes—
53C2×C22⋊D101—22no—
54C22×Dic101—22no—
55C2×C4×D101—22no—
56C2×C4×D101C2312yes—
57#13108064—24no—
58#13108063—24no—
59#13108063—42no—
60#13108059—24no—
61C2×#11717579C2312yes—
62C8×D101—24no—
63C4⋊Dic101—21no—
64C4⋊Dic101—21no—
65C4⋊Dic101—22no—
66C4⋊Dic101D411yescharacteristic
67#13108051—22no—
68#13108051—22no—
69#13108051D412yes—
70C2×Q8×C101C2311yescharacteristic
71C2×D4×C101C2311yescharacteristic
72C22×C4×C101—21no—
73SD16×C101D414yescomplemented
74C2×C8×C101C2311yescharacteristic
75C2×C8×C101C2311yescharacteristic
76C4⋊C4×C101—21no—
77C4⋊C4×C101C2311yescharacteristic
78C4⋊C4×C101C2311yescharacteristic
79C22⋊C4×C101—21no—
80C42×C101—21no—
81C42×C101C2311yescharacteristic
82C23×C101—21no—
83C22×D101—22no—
84C22×D101C2×D412yes—
85Q8×C101—21no—
86Q8×C101C2×D412yes—
87D4×C101—21no—
88D4×C101C2×D412yes—
89C2×C4×C101—21no—
90C2×C4×C101—21no—
91C2×C4×C101—21no—
92C2×C4×C101—42no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2411yescharacteristic
96C22⋊D101—44no—
97C2×Dic101—22no—
98C2×Dic101—44no—
99C2×Dic101C2×D412yes—
100D404—22no—
101D404—22no—
102D404—22no—
103D404—44no—
104D404—42no—
105D404—81no—
106D404—82no—
107C4×D101—22no—
108C4×D101—22no—
109C4×D101—44no—
110Dic202—22no—
111Dic202—22no—
112C8×C101—21no—
113C8×C101C2×D412yes—
114#11717579—22no—
115C22×C101—22no—
116C22×C101—41no—
117C22×C101—41no—
118C22×C101C22×D411yescharacteristic
119C2×D101—22no—
120C2×D101—22no—
121C2×D101—22no—
122C2×D101—42no—
123C2×D101—42no—
124C2×D101—42no—
125C4×C101—22no—
126C4×C101—21no—
127C4×C101—41no—
128C4×C101—41no—
129C4×C101#13512yescomplemented
130C4×C101C22×D411yescharacteristic
131C4×C101H(5,𝔽2)11yescharacteristic
132Dic101—22no—
133Dic101—42no—
134C2×C101—42no—
135C2×C101D4211yescharacteristic
136C2×C101C2×#13511yescharacteristic
137C2×C101#57711yescharacteristic
138D101—42no—
139D101—82no—
140#3284—1011nomaximal, Sylow
141C101#328411yescharacteristic, complemented, Sylow
142C2×#135—1012no—
143#550—1011no—
144#547—1011no—
145#492—1011no—
146#465—1012no—
147#462—1011no—
148#459—1011no—
149#452—1012no—
150#449—1012no—
151C4×SD16D10111yescharacteristic, complemented
152C8×D4—1011no—
153C2×#41—1012no—
154C2×#41—1012no—
155#135—2028no—
156C2×Q16—1012no—
157C2×SD16—1012no—
158C2×SD16—1012no—
159C2×SD16C2×D10111yescharacteristic
160C2×D8—1012no—
161C22×C8—1012no—
162C4⋊Q8—1011no—
163#127—1011no—
164#127—2021no—
165#123—2022no—
166C22⋊Q8—1012no—
167#121—1012no—
168#121—2022no—
169#121—2021no—
170C4×Q8C2×D10111yescharacteristic
171C4×D4—1011no—
172C4×D4—2021no—
173C4×D4C2×D10111yescharacteristic
174C2×C4⋊C4—2021no—
175#106C2×D10111yescharacteristic
176C4⋊C8—1011no—
177#103—1012no—
178#103C2×D10111yescharacteristic
179#102—1012no—
180#102C2×D10111yescharacteristic
181C22⋊C8—1012no—
182C4×C8C2×D10111yescharacteristic
183#41—4042no—
184#41—4042no—
185#41—2024no—
186#41—2024no—
187C2×Q8—1012no—
188C2×Q8C22×D10111yescharacteristic
189C2×D4—1011no—
190C2×D4—1012no—
191C2×D4—2022no—
192C2×D4—2022no—
193C2×D4—2022no—
194C2×D4—2021no—
195C2×D4—2021no—
196C2×D4C22×D10111yescharacteristic
197C22×C4—21no—
198C22×C4—1012no—
199C22×C4—2022no—
200C22×C4—2022no—
201Q16—2024no—
202SD16—4042no—
203SD16—2024no—
204SD16D40414yescomplemented
205D8—2024no—
206C2×C8—1012no—
207C2×C8—2024no—
208C2×C8C22×D10111yescharacteristic
209C2×C8C22×D10111yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—1011no—
212C4⋊C4—2022no—
213C4⋊C4—2021no—
214C4⋊C4—2021no—
215C4⋊C4C22×D10111yescharacteristic
216C4⋊C4C22×D10111yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—1012no—
219C22⋊C4—2022no—
220C22⋊C4—2022no—
221C42—21no—
222C42C22×D10111yescharacteristic
223C23—21no—
224C23—1012no—
225C23—2022no—
226Q8—21no—
227Q8—2022no—
228Q8—2022no—
229Q8C2×D40412yes—
230D4—21no—
231D4—8081no—
232D4—8082no—
233D4—4044no—
234D4—4044no—
235D4—4042no—
236D4—2022no—
237D4—2022no—
238D4—2022no—
239D4C2×D40412yes—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—1012no—
245C2×C4—4044no—
246C2×C4—4044no—
247C2×C4—2022no—
248C2×C4—2022no—
249C2×C4—2022no—
250C2×C4D4×D10111yescharacteristic
251C2×C4D4×D10111yescharacteristic
252C2×C4C23×D10111yescharacteristic
253C8—21no—
254C8—2022no—
255C8C2×D40412yes—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—4042no—
260C22—4042no—
261C22—4042no—
262C22—2022no—
263C22—2022no—
264C22—2022no—
265C22C2×D4×D10111yescharacteristic
266C4—21no—
267C4—22no—
268C4—41no—
269C4—41no—
270C4—4042no—
271C4—2022no—
272C4#1569227612yescomplemented
273C4C22×D40411yescharacteristic
274C4#1569230111yescharacteristic
275C2—42no—
276C2—8082no—
277C2—4042no—
278C2D4×D40411yescharacteristic
279C2C2×#1569227611yescharacteristic
280C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×SD16×C101C2
socleC22×C101C22×D4
hypercenterC4×SD16D101
hypocenterC101#3284
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15692276C2semidirect
2. #20265849C2semidirect
3. #20265831C2semidirect
4. #20265550C2semidirect
5. #20265407C2semidirect
6. #20265387C2semidirect
7. #20265372C2semidirect
8. #20265342C2semidirect
9. #20265323C2semidirect
10. C8×D404C2semidirect
11. C4×SD16×C101C2semidirect
73. SD16×C101D4semidirect
129. C4×C101#135semidirect
141. C101#3284semidirect
151. C4×SD16D101semidirect
204. SD16D404semidirect
272. C4#15692276semidirect

Maximal subgroups

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

subgroupindex
C2×#156922762
#202658492
#202658312
#202655502
#202654072
#202653872
#202653722
#202653422
#202653232
C8×D4042
C4×SD16×C1012
#3284101