Finite Group Databasebeta

← #27182999

Subgroups — #27182999

Order 12928.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×D404C211yesmaximal, characteristic, complemented
2#20265832C211yesmaximal, characteristic
3#20265528C211yesmaximal, characteristic, complemented
4#20265509C212yesmaximal, complemented
5#20265462C211yesmaximal, characteristic, complemented
6#20265405C212yesmaximal, complemented
7#20265382C211yesmaximal, characteristic, complemented
8#20265314C212yesmaximal, complemented
9C8×D404C211yesmaximal, characteristic, complemented
10#103×D101C212yesmaximal, complemented
11C4×SD16×C101C211yesmaximal, characteristic, complemented
12C2×#13108074—21no—
13C2×Q8×D101C2212yes—
14C2×#13108062C2212yes—
15C2×C8×D101C2212yes—
16#15692255—21no—
17#15692255C2211yescharacteristic
18#15692255C2211yescharacteristic
19#15692237—22no—
20#15692235—22no—
21#15692235C2212yes—
22#15692234—21no—
23#15692226C2212yes—
24C4×D404—21no—
25C4×D404C2211yescharacteristic
26C4⋊C4×D101C2212yes—
27#15692186C2212yes—
28C4⋊#11717579C2211yescharacteristic
29#15692180C2212yes—
30#15692180C2212yes—
31#15692179C2212yes—
32#15692178C2212yes—
33#15692167C2212yes—
34C2×SD16×C101C2211yescharacteristic
35C4×Q8×C101C2211yescharacteristic
36C4×D4×C101C2211yescharacteristic
37#106×C101C2211yescharacteristic
38#103×C101C2211yescharacteristic
39#102×C101C2211yescharacteristic
40C4×C8×C101C2211yescharacteristic
41#13108074—44no—
42C2×Dic202—21no—
43C2×Dic202—22no—
44C2×Dic202C2312yes—
45Q8×D101—24no—
46Q8×D101—42no—
47C2×D404D411yescharacteristic
48C2×C22⋊D101—22no—
49C2×C4×D101—22no—
50C2×C4×D101—22no—
51C2×C4×D101C2312yes—
52#13108062—42no—
53C2×#11717579C2312yes—
54C8×D101—42no—
55C4⋊Dic101—21no—
56C4⋊Dic101—21no—
57C4⋊Dic101—21no—
58C4⋊Dic101—22no—
59C4⋊Dic101D411yescharacteristic
60C4⋊Dic101C2312yes—
61#13108053—22no—
62C22⋊Dic101—22no—
63#13108051—22no—
64#13108051D412yes—
65C2×Q8×C101C2311yescharacteristic
66C2×D4×C101C2311yescharacteristic
67C22×C4×C101—21no—
68SD16×C101—22no—
69C2×C8×C101C2311yescharacteristic
70C2×C8×C101C2311yescharacteristic
71C4⋊C4×C101—21no—
72C4⋊C4×C101C2311yescharacteristic
73C4⋊C4×C101C2311yescharacteristic
74C22⋊C4×C101—21no—
75C42×C101—21no—
76C42×C101C2311yescharacteristic
77C23×C101—21no—
78C22×D101C2×D412yes—
79Q8×C101—21no—
80Q8×C101#4112yescomplemented
81D4×C101—21no—
82D4×C101—21no—
83C2×C4×C101—21no—
84C2×C4×C101—21no—
85C2×C4×C101—21no—
86C2×C4×C101—42no—
87C2×C4×C101C2×D411yescharacteristic
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101C2411yescharacteristic
90C22⋊D101—44no—
91C2×Dic101—22no—
92C2×Dic101—22no—
93C2×Dic101C2×D412yes—
94D404SD1614yescomplemented
95C4×D101—22no—
96C4×D101—22no—
97C4×D101—44no—
98C4×D101—44no—
99Dic202—22no—
100Dic202—22no—
101Dic202—44no—
102Dic202—42no—
103C8×C101—21no—
104C8×C101—21no—
105#11717579—22no—
106C22×C101—41no—
107C22×C101—41no—
108C22×C101—41no—
109C22×C101C22×D411yescharacteristic
110C2×D101—22no—
111C2×D101—22no—
112C2×D101C2×SD1614yes—
113C4×C101—22no—
114C4×C101—21no—
115C4×C101—41no—
116C4×C101—41no—
117C4×C101C2×SD1612yes—
118C4×C101C2×#4111yescharacteristic
119C4×C101H(5,𝔽2)11yescharacteristic
120Dic101—22no—
121Dic101—42no—
122Dic101—42no—
123C2×C101—81no—
124C2×C101#54611yescharacteristic
125C2×C101C22×SD1611yescharacteristic
126C2×C101#57811yescharacteristic
127D101—24no—
128#3298—1011nomaximal, Sylow
129C101#329811yescharacteristic, complemented, Sylow
130D4×Q8—1011no—
131#547—1011no—
132#487—1011no—
133#483—1012no—
134#475—1011no—
135#465—1012no—
136#461—1011no—
137#448—1012no—
138C4×SD16D10111yescharacteristic, complemented
139C8×D4—1011no—
140C2×#103—1012no—
141C2×#41—2021no—
142C22×Q8—1012no—
143C2×SD16—1012no—
144C2×SD16C2×D10111yescharacteristic
145C22×C8—1012no—
146C4⋊Q8—1011no—
147C4⋊Q8—1011no—
148C4⋊Q8—2021no—
149#123—2022no—
150C22⋊Q8—1012no—
151C22⋊Q8—2022no—
152C22⋊Q8—2021no—
153#121—1012no—
154C4×Q8C2×D10111yescharacteristic
155C4×D4—1011no—
156C4×D4—2021no—
157C4×D4C2×D10111yescharacteristic
158C2×C4⋊C4—1012no—
159#106—1012no—
160#106C2×D10111yescharacteristic
161C4⋊C8—1011no—
162#103—1012no—
163#103—1012no—
164#103—1012no—
165#103—1012no—
166#103C2×D10111yescharacteristic
167#102C2×D10111yescharacteristic
168C22⋊C8—1012no—
169C4×C8C2×D10111yescharacteristic
170#41—4044no—
171C2×Q8—1012no—
172C2×Q8—4042no—
173C2×Q8—2022no—
174C2×Q8—2024no—
175C2×Q8—2021no—
176C2×Q8C22×D10111yescharacteristic
177C2×D4—1011no—
178C2×D4—2022no—
179C2×D4C22×D10111yescharacteristic
180C22×C4—21no—
181C22×C4—1012no—
182C22×C4—2022no—
183C22×C4—2022no—
184SD16—22no—
185SD16—4042no—
186C2×C8—1012no—
187C2×C8—4042no—
188C2×C8C22×D10111yescharacteristic
189C2×C8C22×D10111yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—1011no—
192C4⋊C4—1012no—
193C4⋊C4—2022no—
194C4⋊C4—2022no—
195C4⋊C4—2021no—
196C4⋊C4—2021no—
197C4⋊C4—2021no—
198C4⋊C4C22×D10111yescharacteristic
199C4⋊C4C22×D10111yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—1012no—
202C22⋊C4—2022no—
203C22⋊C4—2022no—
204C42—21no—
205C42C22×D10111yescharacteristic
206C23—21no—
207C23—1012no—
208Q8—21no—
209Q8—4044no—
210Q8—4042no—
211Q8—2022no—
212Q8—2022no—
213Q8#1310807412yescomplemented
214D4—21no—
215D4—21no—
216D4—1014no—
217D4—4044no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—1012no—
223C2×C4—4044no—
224C2×C4—4044no—
225C2×C4—2022no—
226C2×C4—2022no—
227C2×C4—2022no—
228C2×C4—2022no—
229C2×C4D4×D10111yescharacteristic
230C2×C4D4×D10111yescharacteristic
231C2×C4C23×D10111yescharacteristic
232C8—21no—
233C8—21no—
234C8—2022no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—1014no—
239C22—2022no—
240C22—2022no—
241C22C2×D4×D10111yescharacteristic
242C4—21no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—4042no—
247C4—4042no—
248C4—2022no—
249C4SD16×D10112yes—
250C4C2×#1310807411yescharacteristic
251C4#1569230111yescharacteristic
252C2—81no—
253C2—2024no—
254C2#2026582311yescharacteristic
255C2C2×SD16×D10111yescharacteristic
256C2#2026599111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×SD16×C101C2
socleC22×C101C22×D4
prime radical#20265832C2
hypercenterC4×SD16D101
hypocenterC101#3298
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. Q8×D404C2semidirect
3. #20265528C2semidirect
4. #20265509C2semidirect
5. #20265462C2semidirect
6. #20265405C2semidirect
7. #20265382C2semidirect
8. #20265314C2semidirect
9. C8×D404C2semidirect
10. #103×D101C2semidirect
11. C4×SD16×C101C2semidirect
80. Q8×C101#41semidirect
94. D404SD16semidirect
129. C101#3298semidirect
138. C4×SD16D101semidirect
213. Q8#13108074semidirect

Maximal subgroups

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

subgroupindex
Q8×D4042
#202658322
#202655282
#202655092
#202654622
#202654052
#202653822
#202653142
C8×D4042
#103×D1012
C4×SD16×C1012
#3298101