Finite Group Databasebeta

← #27174220

Subgroups — #27174220

Order 12928.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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
2#20265084C211yesmaximal, characteristic
3#20265081C212yesmaximal
4#20265074C212yesmaximal
5#20265072C212yesmaximal
6C4×#13108053C212yesmaximal
7#20265028C212yesmaximal
8#20265028C212yesmaximal
9#391×C101C211yesmaximal, characteristic
10C22×Dic202C2211yescharacteristic
11#15692253C418yescomplemented
12C2×C4⋊Dic101C2211yescharacteristic
13C2×#13108053C2212yes—
14C2×#13108053C2212yes—
15C2×#13108053C2212yes—
16C2×#13108053C2212yes—
17C2×#13108053C2212yes—
18C2×C4×Dic101C2212yes—
19C2×C4×Dic101C2212yes—
20C2×C4×Dic101C2211yescharacteristic
21C2×C4×Dic101C2212yes—
22C2×C4×Dic101C2211yescharacteristic
23#15692162C2212yes—
24#15692162C2212yes—
25#15692162C2212yes—
26#15692162C2212yes—
27C22×Q8×C101C2211yescharacteristic
28C2×C4⋊C4×C101C2211yescharacteristic
29C2×C42×C101C2211yescharacteristic
30#95×C101C2212yes—
31#95×C101C2212yes—
32C2×Dic202—41no—
33C2×Dic202—41no—
34C2×Dic202C2×C414yes—
35C22×Dic101C2312yes—
36C22×Dic101C2312yes—
37C22×Dic101C2312yes—
38C22×Dic101C2312yes—
39C4⋊Dic101—41no—
40#13108053—24no—
41#13108053—42no—
42#13108053C2×C418yes—
43#13108053C2×C418yes—
44#13108053Q818yescomplemented
45C4×Dic101—24no—
46C4×Dic101—24no—
47C4×Dic101—41no—
48C4×Dic101—42no—
49C4×Dic101C2×C414yes—
50C2×Q8×C101—41no—
51C2×Q8×C101—41no—
52C2×Q8×C101C2×C414yes—
53C22×C4×C101C2312yes—
54C22×C4×C101C2311yescharacteristic
55C22×C4×C101C2311yescharacteristic
56C22×C4×C101C2311yescharacteristic
57C22×C4×C101C2312yes—
58C4⋊C4×C101—41no—
59C42×C101—41no—
60C23×C101C2411yescharacteristic
61Q8×C101—44no—
62C2×C4×C101—22no—
63C2×C4×C101—21no—
64C2×C4×C101—22no—
65C2×C4×C101—21no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—21no—
69C2×C4×C101—21no—
70C2×C4×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101—22no—
73C2×C4×C101C22×C412yes—
74C2×C4×C101C22×C412yes—
75C2×C4×C101C22×C412yes—
76C2×C4×C101C2×Q814yes—
77C2×Dic101—22no—
78C2×Dic101—22no—
79C2×Dic101—22no—
80C2×Dic101—22no—
81C2×Dic101—22no—
82C2×Dic101—22no—
83C2×Dic101—22no—
84C2×Dic101C22×C414yes—
85C2×Dic101C22×C414yes—
86C2×Dic101C2×Q814yes—
87C2×Dic101C2×Q814yes—
88C2×Dic101#4114yes—
89Dic202—44no—
90C22×C101C23×C411yescharacteristic
91C22×C101C22×Q812yes—
92C22×C101C2×#4112yes—
93C22×C101H(5,𝔽2)11yescharacteristic
94C22×C101H(5,𝔽2)11yescharacteristic
95C4×C101—22no—
96C4×C101—22no—
97C4×C101—22no—
98C4×C101—24no—
99C4×C101—42no—
100Dic101—24no—
101Dic101—24no—
102Dic101—42no—
103Dic101C4×Q818yes—
104C2×C101C2×C4×Q812yes—
105C2×C101#51811yescharacteristic
106C2×C101#55412yes—
107C2×C101#55511yescharacteristic
108C2×C101Q8211yescharacteristic
109#2369—1011nomaximal, Sylow
110C101#236911yescharacteristic, complemented, Sylow
111C2×C4⋊Q8—1011no—
112#391—1011no—
113#391—1012no—
114#391D10111yescharacteristic
115#389—1012no—
116#389—1012no—
117C4×C4⋊C4—1012no—
118C4×C4⋊C4—1012no—
119C4×C4⋊C4—1012no—
120C22×Q8—1011no—
121C22×Q8C2×D10111yescharacteristic
122C4⋊Q8—1018no—
123C2×C4⋊C4—1012no—
124C2×C4⋊C4—1012no—
125C2×C4⋊C4—1012no—
126C2×C4⋊C4—1012no—
127C2×C4⋊C4—1011no—
128C2×C4⋊C4—1012no—
129C2×C4⋊C4C2×D10111yescharacteristic
130C2×C42—1011no—
131C2×C42—1011no—
132C2×C42—1012no—
133C2×C42—1012no—
134C2×C42—1012no—
135C2×C42C2×D10111yescharacteristic
136#95—1012no—
137#95—1012no—
138#95—1012no—
139#95—1012no—
140#95C2×D10112yes—
141#95C2×D10112yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—1014no—
145C2×Q8—4041no—
146C2×Q8—4041no—
147C2×Q8C4×D10114yes—
148C22×C4—1012no—
149C22×C4—1012no—
150C22×C4—1012no—
151C22×C4—1012no—
152C22×C4C22×D10112yes—
153C22×C4C22×D10111yescharacteristic
154C22×C4C22×D10111yescharacteristic
155C22×C4C22×D10111yescharacteristic
156C22×C4C22×D10112yes—
157C4⋊C4—41no—
158C4⋊C4—1018no—
159C4⋊C4—1018no—
160C4⋊C4—1018no—
161C4⋊C4—4042no—
162C4⋊C4—4041no—
163C4⋊C4—2024no—
164C42—41no—
165C42—1014no—
166C42—4042no—
167C42—4041no—
168C42—2024no—
169C42—2024no—
170C23C23×D10111yescharacteristic
171Q8—44no—
172Q8—4044no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—1014no—
185C2×C4—1014no—
186C2×C4—1014no—
187C2×C4—1014no—
188C2×C4—1014no—
189C2×C4—2022no—
190C2×C4—2022no—
191C2×C4—2022no—
192C2×C4—2022no—
193C2×C4—2022no—
194C2×C4—2022no—
195C2×C4—2022no—
196C2×C4C2×C4×D10112yes—
197C2×C4C2×C4×D10112yes—
198C2×C4C2×C4×D10112yes—
199C2×C4Q8×D10114yes—
200C22C22×C4×D10111yescharacteristic
201C22C2×Q8×D10112yes—
202C22#41×D10112yes—
203C22#1569230011yescharacteristic
204C22#1569230011yescharacteristic
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—1018no—
211C4—4042no—
212C4—2024no—
213C4—2024no—
214C2C4×Q8×D10112yes—
215C2#2026565011yescharacteristic
216C2#2026588112yes—
217C2#2026588911yescharacteristic
218C2#2026590711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#391×C101C2
socle, prime radical, cosocleC23×C101C24
hypercenter#391D101
hypocenterC101#2369
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #15692253C4semidirect
44. #13108053Q8semidirect
110. C101#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#156922532
#202650842
#202650812
#202650742
#202650722
C4×#131080532
#202650282
#391×C1012
#2369101