Finite Group Databasebeta

← #27142996

Subgroups — #27142996

Order 12416.

276 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C22×#13095498C211yesmaximal, characteristic
2#20260762C211yesmaximal, characteristic, complemented
3#20260739C211yesmaximal, characteristic
4C2×#15690729C211yesmaximal, characteristic, complemented
5C2×#15690726C211yesmaximal, characteristic
6#20260499C211yesmaximal, characteristic, complemented
7#419×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095498—22no—
10C2×#13095498—22no—
11C2×#13095498—22no—
12C2×#13095495C2211yescharacteristic
13C2×#13095495C2211yescharacteristic
14C22×#11715895C2211yescharacteristic
15C4⋊#11715895—22no—
16#15690730—22no—
17#15690729—22no—
18#15690726—22no—
19C22×Q8×C97C2211yescharacteristic
20C2×M16×C97C2211yescharacteristic
21#117×C97C2211yescharacteristic
22#103×C97—22no—
23C23×D97—21no—
24C2×D388—21no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388D412yes—
30#13095498—28no—
31#13095495—21no—
32#13095495—21no—
33#13095495—21no—
34#13095495—21no—
35C2×#11715895—21no—
36C2×#11715895—21no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C2×#11715895D412yes—
40C2×Q8×C97—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97D412yes—
46C22×C4×C97—21no—
47C22×C4×C97C2311yescharacteristic
48M16×C97—21no—
49M16×C97—21no—
50C2×C8×C97—21no—
51C4⋊C4×C97D412yes—
52C22⋊C4×C97—21no—
53C42×C97—21no—
54C23×C97#4111yescharacteristic
55C22×D97—22no—
56C22×D97—41no—
57C22×D97—41no—
58C22×D97—82no—
59Q8×C97—22no—
60Q8×C97—22no—
61Q8×C97—41no—
62Q8×C97—41no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97#4111yescharacteristic
72C2×C4×C97#4111yescharacteristic
73D388—22no—
74D388—22no—
75D388—41no—
76D388—41no—
77C8×C97—41no—
78#11715895—22no—
79#11715895—41no—
80#11715895—41no—
81C22×C97—22no—
82C22×C97#12111yescharacteristic
83C22×C97#12311yescharacteristic
84C22×C97#12311yescharacteristic
85C2×D97—42no—
86C2×D97—82no—
87C2×D97—82no—
88C2×D97—82no—
89C4×C97—42no—
90C4×C97—42no—
91C4×C97#12011yescharacteristic
92C4×C97#12111yescharacteristic
93C4×C97#12111yescharacteristic
94C4×C97#12411yescharacteristic
95C2×C97—21no—
96C2×C97—21no—
97C2×C97#39411yescharacteristic
98C2×C97#47112yes—
99D97—82no—
100#2046—971nomaximal, Sylow
101C97#204611yescharacteristic, complemented, Sylow
102C22×SD16—971no—
103#424—971no—
104#419D9711yescharacteristic, complemented
105#418—971no—
106C2×#101—971no—
107C2×#100—971no—
108#341—971no—
109C22×Q8C2×D9711yescharacteristic
110C22×D4—971no—
111C2×SD16—1942no—
112C2×SD16—1942no—
113C2×SD16—1942no—
114C2×M16—971no—
115C2×M16—971no—
116C2×M16C2×D9711yescharacteristic
117C22×C8—971no—
118#117C2×D9711yescharacteristic
119C4⋊C8—1942no—
120#103—22no—
121#102—1942no—
122#101—1942no—
123#100—1942no—
124C24—1941no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8D38812yes—
131C2×D4—972no—
132C2×D4—1941no—
133C2×D4—1941no—
134C2×D4—1941no—
135C2×D4—1941no—
136C2×D4—1941no—
137C22×C4—21no—
138C22×C4C22×D9711yescharacteristic
139SD16—1948no—
140M16—21no—
141M16—21no—
142M16—1941no—
143M16—1941no—
144M16—1941no—
145M16—1941no—
146C2×C8—21no—
147C2×C8—972no—
148C2×C8—1941no—
149C2×C8—1941no—
150C2×C8—1941no—
151C2×C8—1941no—
152C4⋊C4C22⋊D9712yes—
153C22⋊C4—21no—
154C42—21no—
155C23—7762no—
156C23—3881no—
157C23—3881no—
158C23—1942no—
159C23#1309551011yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164D4—3881no—
165D4—3881no—
166D4—1942no—
167D4—1942no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4D4×D9711yescharacteristic
173C2×C4D4×D9711yescharacteristic
174C2×C4C2×C22⋊D9711yescharacteristic
175C2×C4C2×D38811yescharacteristic
176C2×C4#1309551011yescharacteristic
177C2×C4#1309551111yescharacteristic
178C8—41no—
179C8—3881no—
180C8—3881no—
181C8—1942no—
182C22—22no—
183C22—7762no—
184C22—7762no—
185C22—7762no—
186C22—3882no—
187C22#1569078211yescharacteristic
188C22#1569079211yescharacteristic
189C22#1569079411yescharacteristic
190C4—42no—
191C4—42no—
192C4#1569078011yescharacteristic
193C4#1569078111yescharacteristic
194C4#1569078411yescharacteristic
195C4#1569079611yescharacteristic
196C2—21no—
197C2—21no—
198C2—7762no—
199C2#2026065111yescharacteristic
200C2#2026100012yes—
subgroupG/H
centerC22#15690782
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#419×C97C2
socleC22×C97#121
prime radicalC22×D388C22
hypercenter#419D97
hypocenterC97#2046
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. #20260762C2semidirect
4. C2×#15690729C2semidirect
6. #20260499C2semidirect
7. #419×C97C2semidirect
101. C97#2046semidirect
104. #419D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954982
#202607622
#202607392
C2×#156907292
C2×#156907262
#202604992
#419×C972
#204697