Finite Group Databasebeta

← #27172815

Subgroups — #27172815

Order 12928.

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

normal subgroups only

This group has 250 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#20265161C211yesmaximal, characteristic, complemented
3#20264975C212yesmaximal, complemented
4#20264973C212yesmaximal, complemented
5#413×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C411yescharacteristic
7C2×#13108058—21no—
8#15692253C2211yescharacteristic
9#15692251C411yescharacteristic
10#15692244C2211yescharacteristic
11#15692242C411yescharacteristic
12#15692227—21no—
13#15692227—21no—
14C4×C22⋊D101—21no—
15C4×C22⋊D101—21no—
16C2×C4×Dic101C411yescharacteristic
17#15692174C2212yes—
18#15692172—21no—
19C2×#41×C101C2211yescharacteristic
20C2×M16×C101—21no—
21#101×C101C2212yes—
22#100×C101—21no—
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—
32#13108058—22no—
33#13108058—41no—
34C2×#11717579—41no—
35#13108053—21no—
36#13108053—21no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39#13108051—21no—
40#13108051—21no—
41C4×Dic101—21no—
42C4×Dic101C2×C411yescharacteristic
43C4×Dic101C2×C411yescharacteristic
44#41×C101—41no—
45#41×C101—41no—
46C2×Q8×C101C2311yescharacteristic
47C2×D4×C101C2×C411yescharacteristic
48C2×D4×C101D411yescharacteristic
49C2×D4×C101D411yescharacteristic
50C22×C4×C101C2×C411yescharacteristic
51C22×C4×C101D411yescharacteristic
52C22×C4×C101D411yescharacteristic
53M16×C101—22no—
54M16×C101—41no—
55C2×C8×C101—41no—
56C23×C101C22⋊C411yescharacteristic
57C23×C101C22⋊C411yescharacteristic
58C23×C101C22⋊C411yescharacteristic
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×C101C22⋊C411yescharacteristic
71C2×C4×C101C22×C411yescharacteristic
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2×D411yescharacteristic
74C22⋊D101—42no—
75C22⋊D101—42no—
76C22⋊D101—42no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—21no—
80C2×Dic101—42no—
81D404—41no—
82C4×D101—41no—
83C4×D101—41no—
84Dic202—41no—
85C8×C101—42no—
86#11717579—42no—
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#9911yescharacteristic
101C4×C101#9911yescharacteristic
102Dic101—22no—
103Dic101—41no—
104Dic101—41no—
105C2×C101—21no—
106C2×C101—41no—
107C2×C101—41no—
108C2×C101—41no—
109C2×C101C2×#9911yescharacteristic
110D101—81no—
111#2132—1011nomaximal, Sylow
112C101#213211yescharacteristic, complemented, Sylow
113#532—1011no—
114#413—1011no—
115#413D10111yescharacteristic, complemented
116#356—1012no—
117#355—1012no—
118C2×#41—1011no—
119C2×#41C2×D10111yescharacteristic
120C2×M16—21no—
121C2×M16—2021no—
122C4⋊Q8—1011no—
123#127—1011no—
124#124—1011no—
125#124—1011no—
126#121—2021no—
127#121—2021no—
128C4×D4—2021no—
129C4×D4—2021no—
130C2×C42—1011no—
131#101—1012no—
132#101C2×D10112yes—
133#100—21no—
134#100—2021no—
135#41—41no—
136#41—41no—
137#41—4041no—
138#41—4041no—
139C2×Q8—1011no—
140C2×Q8C22×D10111yescharacteristic
141C2×D4—1011no—
142C2×D4—2021no—
143C2×D4—2021no—
144C2×D4—2021no—
145C2×D4C4×D10111yescharacteristic
146C2×D4D40411yescharacteristic
147C2×D4C22⋊D10111yescharacteristic
148C22×C4—2021no—
149C22×C4—2021no—
150C22×C4C4×D10111yescharacteristic
151C22×C4D40411yescharacteristic
152C22×C4C22⋊D10111yescharacteristic
153M16—22no—
154M16—41no—
155M16—4041no—
156M16—2022no—
157C2×C8—41no—
158C2×C8—4041no—
159C4⋊C4—2021no—
160C4⋊C4—2021no—
161C22⋊C4—2021no—
162C22⋊C4—2021no—
163C22⋊C4—2021no—
164C22⋊C4—2021no—
165C42—1011no—
166C42—1011no—
167C42—2021no—
168C23—2021no—
169C23#1310805111yescharacteristic
170C23#1310805111yescharacteristic
171C23#1310805111yescharacteristic
172Q8—41no—
173Q8—4041no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—4041no—
178D4—4042no—
179D4—4042no—
180D4—4042no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—4042no—
188C2×C4—4041no—
189C2×C4—4041no—
190C2×C4—2021no—
191C2×C4—2021no—
192C2×C4—2021no—
193C2×C4#1310805111yescharacteristic
194C2×C4C2×C4×D10111yescharacteristic
195C2×C4C2×C22⋊D10111yescharacteristic
196C2×C4C2×D40411yescharacteristic
197C8—42no—
198C8—4042no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—4041no—
206C22—4041no—
207C22—4041no—
208C22C2×#1310805111yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—4041no—
213C4—4041no—
214C4—2022no—
215C4#1569216911yescharacteristic
216C4#1569216911yescharacteristic
217C2—21no—
218C2—41no—
219C2—41no—
220C2—41no—
221C2—8081no—
222C2C2×#1569216911yescharacteristic
subgroupG/H
centerC4#15692169
commutatorC2×C4×C101C22×C4
FrattiniC2×Q8C22×D101
Fitting#413×C101C2
socleC2×C101C2×#99
prime radical#20265723C2
cosocleC2×Q8×C101C23
hypercenter#413D101
hypocenterC101#2132
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #20265161C2semidirect
3. #20264975C2semidirect
4. #20264973C2semidirect
5. #413×C101C2semidirect
112. C101#2132semidirect
115. #413D101semidirect

Maximal subgroups

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

subgroupindex
#202657232
#202651612
#202649752
#202649732
#413×C1012
#2132101