Finite Group Databasebeta

← #27141694

Subgroups — #27141694

Order 12416.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15690858C211yesmaximal, characteristic
2C2×#15690738C212yesmaximal, complemented
3#20260734C212yesmaximal, complemented
4C2×#15690729C211yesmaximal, characteristic, complemented
5#391×C97C211yesmaximal, characteristic, complemented
6#15690858—24no—
7#15690858—22no—
8#15690858C414yescomplemented
9C2×#13095512—22no—
10C2×#13095511—22no—
11C2×#13095511C2212yes—
12C2×Q8×D97—22no—
13C2×#13095495C2212yes—
14#15690738—24no—
15#15690733—22no—
16#15690730—22no—
17#15690729—22no—
18C22×Q8×C97C2211yescharacteristic
19C2×C4⋊C4×C97C2211yescharacteristic
20C2×C42×C97C2211yescharacteristic
21#95×C97—22no—
22#13095512—28no—
23#13095512—28no—
24#13095511—28no—
25#13095511—28no—
26#13095511—22no—
27#13095511—22no—
28#13095511C2×C414yes—
29#13095511D414yes—
30C2×Dic194—22no—
31C2×Dic194D412yes—
32Q8×D97—28no—
33Q8×D97—28no—
34C2×D388—22no—
35C2×D388D412yes—
36C2×C22⋊D97—22no—
37C2×C4×D97—24no—
38C2×C4×D97—22no—
39#13095495—22no—
40#13095495—22no—
41C2×#11715895—22no—
42C2×Q8×C97—22no—
43C2×Q8×C97—22no—
44C2×Q8×C97C2×C412yes—
45C2×Q8×C97D412yes—
46C22×C4×C97—21no—
47C22×C4×C97—21no—
48C22×C4×C97D412yes—
49C22×C4×C97C2311yescharacteristic
50C4⋊C4×C97—22no—
51C42×C97—22no—
52C23×C97C2×D411yescharacteristic
53C22×D97—22no—
54Q8×C97—24no—
55Q8×C97—24no—
56C2×C4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97—42no—
62C2×C4×C97—41no—
63C2×C4×C97C22⋊C414yes—
64C2×C4×C97C22×C411yescharacteristic
65C2×C4×C97C2×D412yes—
66C2×C4×C97C2×D412yes—
67C2×C4×C97C2×D411yescharacteristic
68C22⋊D97—24no—
69C22⋊D97—24no—
70C2×Dic97—22no—
71D388—28no—
72D388—22no—
73D388C22⋊C414yes—
74C4×D97—216no—
75C4×D97—24no—
76C4×D97—24no—
77Dic194—28no—
78Dic194—22no—
79Dic194C22⋊C414yes—
80#11715895—42no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97C2×C22⋊C411yescharacteristic
84C22×C97#12011yescharacteristic
85C22×C97#12011yescharacteristic
86C2×D97—24no—
87C2×D97—42no—
88C2×D97—42no—
89C4×C97—24no—
90C4×C97—42no—
91C4×C97—81no—
92C4×C97C2×C22⋊C412yes—
93C4×C97#12012yes—
94Dic97—24no—
95C2×C97—21no—
96C2×C97—21no—
97C2×C97#37911yescharacteristic
98C2×C97#45411yescharacteristic
99C2×C97#45611yescharacteristic
100D97—44no—
101#1793—971nomaximal, Sylow
102C97#179311yescharacteristic, complemented, Sylow
103C2×2−1+4—971no—
104C2×#104—972no—
105#417—972no—
106C2×#101—971no—
107#391D9711yescharacteristic, complemented
1082−1+4—974no—
1092−1+4—1944no—
1102−1+4—1942no—
111C2×#41—972no—
112C2×#41—1942no—
113C2×#41—1942no—
114C22×Q8—1942no—
115C22×Q8C2×D9711yescharacteristic
116C2×M16—972no—
117C2×C4⋊C4C2×D9711yescharacteristic
118C2×C42C2×D9711yescharacteristic
119#104—1944no—
120#103—1942no—
121#102—1942no—
122#101—1942no—
123#95—22no—
124#41—974no—
125#41—974no—
126#41—1948no—
127#41—1948no—
128#41—1948no—
129#41—1948no—
130#41—1942no—
131#41—1942no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—972no—
135C2×Q8—1942no—
136C2×Q8—1948no—
137C2×Q8—1948no—
138C2×Q8C4×D9712yes—
139C2×Q8D38812yes—
140C2×D4—972no—
141C2×D4—1942no—
142C2×D4—1942no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—1944no—
146C22×C4—1942no—
147C22×C4C22⋊D9712yes—
148C22×C4C22×D9711yescharacteristic
149M16—1942no—
150M16—1942no—
151C2×C8—1942no—
152C4⋊C4—22no—
153C42—22no—
154C23—1942no—
155C23C2×C22⋊D9711yescharacteristic
156Q8—24no—
157Q8—24no—
158Q8—974no—
159Q8—1948no—
160Q8—1942no—
161D4—974no—
162D4—1948no—
163D4—1944no—
164D4—1944no—
165D4—1942no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—19416no—
174C2×C4—1942no—
175C2×C4—1944no—
176C2×C4—1944no—
177C2×C4#1309548814yes—
178C2×C4C2×C4×D9711yescharacteristic
179C2×C4D4×D9712yes—
180C2×C4D4×D9712yes—
181C2×C4C2×D38811yescharacteristic
182C8—3882no—
183C22—21no—
184C22—21no—
185C22—3882no—
186C22—3882no—
187C22—1944no—
188C22C2×#1309548811yescharacteristic
189C22#1569077811yescharacteristic
190C22#1569077811yescharacteristic
191C4—24no—
192C4—42no—
193C4—81no—
194C4—1944no—
195C4C22⋊C4×D9712yes—
196C4#1569078012yes—
197C2—21no—
198C2—21no—
199C2—3884no—
200C2#2026059011yescharacteristic
201C2#2026090711yescharacteristic
202C2#2026091711yescharacteristic
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#391×C97C2
socleC22×C97#120
prime radicalC2×#15690858C2
cosocleC22×C4×C97C23
hypercenter#391D97
hypocenterC97#1793
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#15690738C2semidirect
3. #20260734C2semidirect
4. C2×#15690729C2semidirect
5. #391×C97C2semidirect
8. #15690858C4semidirect
102. C97#1793semidirect
107. #391D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908582
C2×#156907382
#202607342
C2×#156907292
#391×C972
#179397