Finite Group Databasebeta

← #27172795

Subgroups — #27172795

Order 12928.

258 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#20265723C211yesmaximal, characteristic
2#20265151C211yesmaximal, characteristic, complemented
3#20264971C212yesmaximal, complemented
4#20264969C212yesmaximal, complemented
5#410×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C411yescharacteristic, complemented
7#15692253C411yescharacteristic, complemented
8#15692251C2211yescharacteristic
9#15692244C411yescharacteristic, complemented
10#15692242C2211yescharacteristic
11#15692227—21no—
12#15692227—21no—
13C4×C22⋊D101—21no—
14C4×C22⋊D101—21no—
15#15692214—21no—
16C2×C4×Dic101C411yescharacteristic, complemented
17#15692170—21no—
18#15692170C2212yes—
19C2×#41×C101C2211yescharacteristic
20#117×C101—21no—
21#99×C101—21no—
22#99×C101C2212yes—
23#13108074—41no—
24#13108074—41no—
25C2×Dic202C2×C411yescharacteristic
26C2×D404C2×C411yescharacteristic
27C2×C22⋊D101—21no—
28C2×C22⋊D101—21no—
29C2×C22⋊D101—21no—
30C22×Dic101—21no—
31C2×C4×D101—21no—
32C4⋊Dic101—41no—
33#13108053—21no—
34#13108053—21no—
35C22⋊Dic101—21no—
36C22⋊Dic101—21no—
37C22⋊Dic101—22no—
38#13108051—21no—
39#13108051—21no—
40C4×Dic101—21no—
41C4×Dic101—41no—
42C4×Dic101C2×C411yescharacteristic
43C4×Dic101C2×C411yescharacteristic
44#41×C101—41no—
45#41×C101—41no—
46C2×Q8×C101C2×C411yescharacteristic
47C2×D4×C101D411yescharacteristic
48C2×D4×C101D411yescharacteristic
49C2×D4×C101C2311yescharacteristic
50C22×C4×C101C2×C411yescharacteristic
51C22×C4×C101D411yescharacteristic
52C22×C4×C101D411yescharacteristic
53C4⋊C4×C101—41no—
54C22⋊C4×C101—22no—
55C42×C101—41no—
56C23×C101C22⋊C411yescharacteristic
57C23×C101C2×D411yescharacteristic
58C23×C101C2×D411yescharacteristic
59C22×D101—21no—
60Q8×C101—41no—
61D4×C101—41no—
62D4×C101—41no—
63D4×C101—41no—
64C2×C4×C101—21no—
65C2×C4×C101—21no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—21no—
69C2×C4×C101—21no—
70C2×C4×C101—42no—
71C2×C4×C101C22⋊C411yescharacteristic
72C2×C4×C101C22⋊C411yescharacteristic
73C2×C4×C101C22⋊C411yescharacteristic
74C2×C4×C101C22×C411yescharacteristic
75C22⋊D101—42no—
76C22⋊D101—42no—
77C22⋊D101—42no—
78C2×Dic101—21no—
79C2×Dic101—21no—
80C2×Dic101—21no—
81C2×Dic101—42no—
82C2×Dic101—42no—
83D404—41no—
84C4×D101—41no—
85C4×D101—41no—
86Dic202—41no—
87C22×C101—21no—
88C22×C101—21no—
89C22×C101—21no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101C2×C22⋊C411yescharacteristic
94C2×D101—41no—
95C2×D101—41no—
96C2×D101—41no—
97C4×C101—21no—
98C4×C101—21no—
99C4×C101—21no—
100C4×C101—82no—
101C4×C101#9911yescharacteristic
102C4×C101#9911yescharacteristic
103Dic101—22no—
104Dic101—41no—
105Dic101—41no—
106Dic101—82no—
107C2×C101—21no—
108C2×C101—41no—
109C2×C101—41no—
110C2×C101—41no—
111C2×C101C2×#9911yescharacteristic
112D101—81no—
113#2126—1011nomaximal, Sylow
114C101#212611yescharacteristic, complemented, Sylow
115#532—1011no—
116#410—1011no—
117#410D10111yescharacteristic, complemented
118#354—1012no—
119#353—1012no—
120C2×#41—1011no—
121C2×#41C2×D10111yescharacteristic
122C4⋊Q8—1011no—
123#127—1011no—
124#124—1011no—
125#124—1011no—
126#121—2021no—
127#121—2021no—
128C4×D4—2021no—
129C4×D4—2021no—
130#117—21no—
131#117—2021no—
132C2×C42—1011no—
133#99—21no—
134#99—1012no—
135#99—2021no—
136#99C2×D10112yes—
137#41—41no—
138#41—41no—
139#41—4041no—
140#41—4041no—
141C2×Q8—1011no—
142C2×Q8C4×D10111yescharacteristic
143C2×D4—1011no—
144C2×D4—2021no—
145C2×D4—2021no—
146C2×D4—2021no—
147C2×D4D40411yescharacteristic
148C2×D4C22⋊D10111yescharacteristic
149C2×D4C22×D10111yescharacteristic
150C22×C4—2021no—
151C22×C4—2021no—
152C22×C4C4×D10111yescharacteristic
153C22×C4D40411yescharacteristic
154C22×C4C22⋊D10111yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—4041no—
157C4⋊C4—2021no—
158C4⋊C4—2021no—
159C22⋊C4—22no—
160C22⋊C4—2022no—
161C22⋊C4—2021no—
162C22⋊C4—2021no—
163C22⋊C4—2021no—
164C22⋊C4—2021no—
165C42—41no—
166C42—1011no—
167C42—1011no—
168C42—4041no—
169C42—2021no—
170C23—2021no—
171C23#1310805111yescharacteristic
172C23C2×C22⋊D10111yescharacteristic
173C23C2×D40411yescharacteristic
174Q8—41no—
175Q8—4041no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—4041no—
180D4—4042no—
181D4—4042no—
182D4—4042no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—4042no—
191C2×C4—4042no—
192C2×C4—4041no—
193C2×C4—4041no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4#1310805111yescharacteristic
198C2×C4#1310805111yescharacteristic
199C2×C4#1310805111yescharacteristic
200C2×C4C2×C4×D10111yescharacteristic
201C22—21no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—4041no—
208C22—4041no—
209C22—4041no—
210C22C2×#1310805111yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—8082no—
216C4—4041no—
217C4—4041no—
218C4—2022no—
219C4#1569216911yescharacteristic
220C4#1569216911yescharacteristic
221C2—21no—
222C2—41no—
223C2—41no—
224C2—41no—
225C2—8081no—
226C2C2×#1569216911yescharacteristic
subgroupG/H
centerC4#15692169
commutatorC2×C4×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#410×C101C2
socleC2×C101C2×#99
prime radical#20265723C2
cosocleC2×D4×C101C23
hypercenter#410D101
hypocenterC101#2126
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20265151C2semidirect
3. #20264971C2semidirect
4. #20264969C2semidirect
5. #410×C101C2semidirect
6. C2×#13108074C4semidirect
7. #15692253C4semidirect
9. #15692244C4semidirect
16. C2×C4×Dic101C4semidirect
114. C101#2126semidirect
117. #410D101semidirect

Maximal subgroups

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

subgroupindex
#202657232
#202651512
#202649712
#202649692
#410×C1012
#2126101