Finite Group Databasebeta

← #27173701

Subgroups — #27173701

Order 12928.

588 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 237 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×#15692253C211yesmaximal, characteristic
2C2×Q8×Dic101C211yesmaximal, characteristic
3C2×#15692220C211yesmaximal, characteristic
4#20265083C211yesmaximal, characteristic
5#20265074C214yesmaximal
6#20265046C211yesmaximal, characteristic
7#20265028C214yesmaximal
8C42×Dic101C211yesmaximal, characteristic
9#391×C101C211yesmaximal, characteristic
10C22×Dic202C2211yescharacteristic
11#15692253C418yescomplemented
12Q8×Dic101—24no—
13#15692220—24no—
14C2×C4⋊Dic101C2211yescharacteristic
15C2×C4⋊Dic101C2212yes—
16C2×#13108053C2214yes—
17C2×#13108053C2212yes—
18C2×#13108053C2214yes—
19C2×C4×Dic101C2212yes—
20C2×C4×Dic101C2212yes—
21C2×C4×Dic101C2212yes—
22C2×C4×Dic101C2211yescharacteristic
23C2×C4×Dic101C2211yescharacteristic
24C2×C4×Dic101C2212yes—
25#15692162C2214yes—
26C22×Q8×C101C2211yescharacteristic
27C2×C4⋊C4×C101C2211yescharacteristic
28C2×C42×C101C2211yescharacteristic
29#95×C101C2214yes—
30C2×Dic202—22no—
31C2×Dic202—22no—
32C2×Dic202C2×C414yes—
33C22×Dic101C2312yes—
34C22×Dic101C2312yes—
35C22×Dic101C2311yescharacteristic
36C22×Dic101C2312yes—
37C22×Dic101C2311yescharacteristic
38C4⋊Dic101—22no—
39C4⋊Dic101—24no—
40#13108053—24no—
41#13108053—28no—
42#13108053C2×C4116yes—
43C4×Dic101—24no—
44C4×Dic101—24no—
45C4×Dic101—24no—
46C4×Dic101C2×C414yes—
47C4×Dic101D414yes—
48C4×Dic101Q818yescomplemented
49C2×Q8×C101—22no—
50C2×Q8×C101—22no—
51C2×Q8×C101C2×C414yes—
52C22×C4×C101C2312yes—
53C22×C4×C101C2312yes—
54C22×C4×C101C2312yes—
55C22×C4×C101C2311yescharacteristic
56C4⋊C4×C101—22no—
57C42×C101—22no—
58C23×C101C2411yescharacteristic
59Q8×C101—28no—
60C2×C4×C101—22no—
61C2×C4×C101—22no—
62C2×C4×C101—22no—
63C2×C4×C101—22no—
64C2×C4×C101—22no—
65C2×C4×C101—24no—
66C2×C4×C101C22×C414yes—
67C2×C4×C101C22×C412yes—
68C2×C4×C101C2×D412yes—
69C2×C4×C101C2×Q814yes—
70C2×C4×C101#4112yes—
71C2×Dic101—22no—
72C2×Dic101—22no—
73C2×Dic101—22no—
74C2×Dic101—22no—
75C2×Dic101—24no—
76C2×Dic101C22×C414yes—
77C2×Dic101C22×C414yes—
78C2×Dic101C2×D412yes—
79C2×Dic101C2×D412yes—
80C2×Dic101C2×Q814yes—
81C2×Dic101C2×Q814yes—
82C2×Dic101#4114yes—
83Dic202—28no—
84C22×C101C23×C411yescharacteristic
85C22×C101C22×D411yescharacteristic
86C22×C101C22×Q812yes—
87C22×C101C2×#4112yes—
88C22×C101C2×#4111yescharacteristic
89C4×C101—24no—
90C4×C101—24no—
91C4×C101—42no—
92C4×C101C4×D414yes—
93Dic101—24no—
94Dic101—22no—
95Dic101—42no—
96Dic101C4×Q818yescomplemented
97Dic101C4⋊Q814yescomplemented
98C2×C101C2×C4×D411yescharacteristic
99C2×C101C2×C4×Q812yes—
100C2×C101C2×C4⋊Q811yescharacteristic
101C2×C101#53211yescharacteristic
102C2×C101#53312yes—
103C4×C4⋊Q8—1011nomaximal, Sylow
104C101C4×C4⋊Q811yescharacteristic, complemented, Sylow
105C2×C4⋊Q8—1011no—
106C2×C4×Q8—1011no—
107C2×C4×Q8—1011no—
108#391—1011no—
109#391D10111yescharacteristic
110#389—1014no—
111#384—1011no—
112C4×C4⋊C4—1014no—
113C43—1011no—
114C22×Q8—1011no—
115C22×Q8C2×D10111yescharacteristic
116C4⋊Q8—1018no—
117C4×Q8—2024no—
118C4×Q8—2024no—
119C2×C4⋊C4—1014no—
120C2×C4⋊C4—1011no—
121C2×C4⋊C4—1012no—
122C2×C4⋊C4—1012no—
123C2×C4⋊C4—1014no—
124C2×C4⋊C4C2×D10111yescharacteristic
125C2×C42—1011no—
126C2×C42—1011no—
127C2×C42—1012no—
128C2×C42—1012no—
129C2×C42—1012no—
130C2×C42—1012no—
131C2×C42C2×D10111yescharacteristic
132#95—1014no—
133#95C2×D10114yes—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—1014no—
137C2×Q8—2022no—
138C2×Q8—2022no—
139C2×Q8C4×D10114yes—
140C22×C4—1011no—
141C22×C4—1012no—
142C22×C4—1012no—
143C22×C4—1011no—
144C22×C4—1012no—
145C22×C4C22×D10112yes—
146C22×C4C22×D10112yes—
147C22×C4C22×D10112yes—
148C22×C4C22×D10111yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—10116no—
151C4⋊C4—2028no—
152C4⋊C4—2024no—
153C4⋊C4—2024no—
154C4⋊C4—2022no—
155C42—22no—
156C42—1014no—
157C42—1014no—
158C42—1018no—
159C42—2024no—
160C42—2024no—
161C42—2024no—
162C23C23×D10111yescharacteristic
163Q8—28no—
164Q8—2028no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—1014no—
172C2×C4—1012no—
173C2×C4—1014no—
174C2×C4—1014no—
175C2×C4—1014no—
176C2×C4—1014no—
177C2×C4—1012no—
178C2×C4—2022no—
179C2×C4—2024no—
180C2×C4—2022no—
181C2×C4—2022no—
182C2×C4—2022no—
183C2×C4C2×C4×D10114yes—
184C2×C4C2×C4×D10112yes—
185C2×C4D4×D10112yes—
186C2×C4Q8×D10114yes—
187C2×C4#1310807312yes—
188C22C22×C4×D10111yescharacteristic
189C22C2×D4×D10111yescharacteristic
190C22C2×Q8×D10112yes—
191C22#41×D10112yes—
192C22C2×#1310807311yescharacteristic
193C4—24no—
194C4—24no—
195C4—42no—
196C4—1018no—
197C4—1014no—
198C4—4042no—
199C4—2024no—
200C4—2022no—
201C4#1569221514yes—
202C2C2×#1569221511yescharacteristic
203C2C4×Q8×D10112yes—
204C2C4⋊Q8×D10111yescharacteristic
205C2#2026572611yescharacteristic
206C2#2026572912yes—
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#391×C101C2
socle, prime radical, cosocleC23×C101C24
hypercenter#391D101
hypocenterC101C4×C4⋊Q8
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #15692253C4semidirect
48. C4×Dic101Q8semidirect
96. Dic101C4×Q8semidirect
97. Dic101C4⋊Q8semidirect
104. C101C4×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#156922532
C2×Q8×Dic1012
C2×#156922202
#202650832
#202650742
#202650462
#202650282
C42×Dic1012
#391×C1012
C4×C4⋊Q8101