Finite Group Databasebeta

← #27142778

Subgroups — #27142778

Order 12416.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#15690835C212yesmaximal, complemented
2C2×#15690737C212yesmaximal, complemented
3#20260636C211yesmaximal, characteristic, complemented
4#20260504C211yesmaximal, characteristic, complemented
5#537×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7#15690835—24no—
8#15690835—24no—
9C2×#13095498—22no—
10C2×#13095496—22no—
11C2×#13095495C2212yes—
12C2×#13095488—22no—
13C2×C4×Dic97C2211yescharacteristic
14#15690737—24no—
15C2×#41×C97C2212yes—
16C22⋊Q8×C97—22no—
17#121×C97—22no—
18#121×C97—22no—
19C2×C4⋊C4×C97C2211yescharacteristic
20C23×D97—21no—
21C2×D388—22no—
22C2×D388—22no—
23C2×D388—21no—
24C2×D388D412yescomplemented
25C22×Dic97—21no—
26#13095498—28no—
27#13095496—28no—
28#13095495—22no—
29#13095495D414yescomplemented
30C2×#11715895—22no—
31#13095488—44no—
32C4×Dic97—22no—
33#41×C97—22no—
34#41×C97—22no—
35#41×C97—22no—
36#41×C97—22no—
37C2×Q8×C97D412yes—
38C2×D4×C97—22no—
39C2×D4×C97—22no—
40C2×D4×C97D412yes—
41C22×C4×C97—22no—
42C22×C4×C97D412yes—
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—22no—
45C4⋊C4×C97—22no—
46C22⋊C4×C97—24no—
47C23×C97—22no—
48C23×C97C2×D411yescharacteristic
49C22×D97—22no—
50C22×D97—42no—
51C22×D97—42no—
52C22×D97—41no—
53C22×D97—41no—
54Q8×C97—22no—
55Q8×C97—22no—
56D4×C97—22no—
57D4×C97—22no—
58D4×C97—42no—
59D4×C97—42no—
60D4×C97—82no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—42no—
64C2×C4×C97—42no—
65C2×C4×C97—42no—
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D412yes—
69C2×C4×C97#4111yescharacteristic
70C2×Dic97—22no—
71C2×Dic97—41no—
72C2×Dic97—41no—
73D388—24no—
74D388—42no—
75#11715895—24no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97—42no—
79C22×C97—42no—
80C22×C97—42no—
81C22×C97#12011yescharacteristic
82C22×C97#12111yescharacteristic
83C22×C97#12711yescharacteristic
84C2×D97—44no—
85C2×D97—42no—
86C2×D97—82no—
87C2×D97—81no—
88C2×D97—81no—
89C4×C97—42no—
90C4×C97—42no—
91C4×C97#12012yes—
92C4×C97#12112yes—
93Dic97—42no—
94C2×C97—21no—
95C2×C97—21no—
96C2×C97—82no—
97C2×C97#39211yescharacteristic
98C2×C97#45311yescharacteristic
99C2×C97#45411yescharacteristic
100D97—82no—
101#2006—971nomaximal, Sylow
102C97#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—972no—
104#537D9711yescharacteristic, complemented
105C2×#104—972no—
106#390—971no—
107#343—971no—
108C2×#41C2×D9712yes—
109C22×D4—971no—
110AGL(1,ℤ/8)—1944no—
111AGL(1,ℤ/8)—1944no—
112C2×SD16—1942no—
113C2×D8—1942no—
114C2×M16—972no—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C2×D9711yescharacteristic
119C2×C22⋊C4—1942no—
120C2×C42—971no—
121#104—1944no—
122C24—1941no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8C22⋊D9712yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—972no—
131C2×D4—1942no—
132C2×D4—1942no—
133C2×D4—1941no—
134C2×D4C22⋊D9712yes—
135C22×C4—22no—
136C22×C4—1941no—
137C22×C4C22⋊D9712yes—
138C22×C4C22×D9711yescharacteristic
139SD16—1948no—
140D8—1948no—
141M16—974no—
142M16—1942no—
143C2×C8—1942no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—3884no—
148C42—1942no—
149C23—22no—
150C23—3882no—
151C23—3882no—
152C23—3881no—
153C23—3881no—
154C23—1942no—
155C23C2×C22⋊D9711yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—42no—
162D4—82no—
163D4—3882no—
164D4—1944no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—3881no—
171C2×C4—3881no—
172C2×C4—1942no—
173C2×C4D4×D9711yescharacteristic
174C2×C4D4×D9712yes—
175C2×C4C2×C22⋊D9712yes—
176C2×C4#1309551011yescharacteristic
177C8—1944no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—7762no—
184C22—7761no—
185C22—7761no—
186C22—3884no—
187C22—3882no—
188C22#1569077911yescharacteristic
189C22#1569078111yescharacteristic
190C22#1569080611yescharacteristic
191C4—42no—
192C4—42no—
193C4—3882no—
194C4#1569077812yes—
195C4#1569078212yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—7762no—
200C2#2026064311yescharacteristic
201C2#2026090311yescharacteristic
202C2#2026090811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#537×C97C2
socleC22×C97#120
hypercenter#537D97
hypocenterC97#2006
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×#15690737C2semidirect
3. #20260636C2semidirect
4. #20260504C2semidirect
5. #537×C97C2semidirect
24. C2×D388D4semidirect
29. #13095495D4semidirect
102. C97#2006semidirect
104. #537D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×#156907372
#202606362
#202605042
#537×C972
#200697