Finite Group Databasebeta

← #27142287

Subgroups — #27142287

Order 12416.

308 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 79 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#20261501C211yesmaximal, characteristic
2#20261311C211yesmaximal, characteristic
3#20260717C211yesmaximal, characteristic
4C2×#15690728C211yesmaximal, characteristic, complemented
5C2×#15690728C211yesmaximal, characteristic, complemented
6#20260505C211yesmaximal, characteristic, complemented
7#344×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9C22×Dic194C2211yescharacteristic
10#15690818—22no—
11#15690816—22no—
12C2×#13095495C2211yescharacteristic
13C2×#13095495C2211yescharacteristic
14M16×D97—22no—
15#15690810—21no—
16#15690798—21no—
17#15690794C414yes—
18#15690790—22no—
19#15690728—21no—
20#15690728—22no—
21#15690728—22no—
22#15690726—21no—
23C2×M16×C97C2211yescharacteristic
24C2×M16×C97C2211yescharacteristic
25#117×C97C2211yescharacteristic
26#13095511—21no—
27#13095511—21no—
28#13095511—21no—
29#13095511—21no—
30C2×Dic194—21no—
31C2×Dic194—21no—
32C2×Dic194—21no—
33C2×Dic194—21no—
34C2×Dic194—21no—
35C2×Dic194—21no—
36C2×Dic194D411yescharacteristic
37C2×D388D411yescharacteristic
38C2×C22⋊D97C2×C412yes—
39C22×Dic97C2×C412yes—
40C2×C4×D97D412yes—
41#13095495—21no—
42#13095495—21no—
43#13095495—21no—
44#13095495D412yes—
45#13095494—24no—
46C2×#11715895—21no—
47C2×#11715895—21no—
48C8×D97—24no—
49C4⋊Dic97—24no—
50#13095488—24no—
51C22×C4×C97C2311yescharacteristic
52M16×C97—21no—
53M16×C97—21no—
54M16×C97—21no—
55M16×C97D412yes—
56C2×C8×C97—21no—
57C2×C8×C97—21no—
58C4⋊C4×C97—21no—
59C22⋊C4×C97C2×C412yes—
60C42×C97—21no—
61C23×C97C22×C411yescharacteristic
62C22×D97C22⋊C412yes—
63C2×C4×C97—22no—
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97#4111yescharacteristic
69C2×C4×C97#4111yescharacteristic
70C22⋊D97—22no—
71C22⋊D97—22no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97C22⋊C412yes—
76D388—21no—
77D388—21no—
78C4×D97—22no—
79C4×D97—22no—
80Dic194—21no—
81Dic194—21no—
82Dic194—41no—
83Dic194—41no—
84Dic194—41no—
85Dic194—41no—
86C8×C97—22no—
87C8×C97—41no—
88#11715895—22no—
89#11715895—41no—
90C22×C97—41no—
91C22×C97C2×C22⋊C411yescharacteristic
92C22×C97C4×D411yescharacteristic
93C22×C97C4×D411yescharacteristic
94C2×D97—22no—
95C2×D97—42no—
96C4×C97—42no—
97C4×C97#12011yescharacteristic
98C4×C97#12111yescharacteristic
99C4×C97#12111yescharacteristic
100C4×C97#12311yescharacteristic
101Dic97—22no—
102Dic97—42no—
103C2×C97—21no—
104C2×C97—21no—
105C2×C97—21no—
106C2×C97#38611yescharacteristic
107D97—42no—
108#1915—971nomaximal, Sylow
109C97#191511yescharacteristic, complemented, Sylow
110#568—971no—
111#536—971no—
112#413—971no—
113C2×#101—971no—
114C2×#101—971no—
115#344—971no—
116#344D9711yescharacteristic, complemented
117C2×#41—971no—
118C22×Q8—971no—
119#131—1942no—
120#131—1942no—
121C2×M16—971no—
122C2×M16—971no—
123C2×M16—1942no—
124C2×M16C2×D9711yescharacteristic
125C2×M16C2×D9711yescharacteristic
126C4⋊Q8—1941no—
127#124—1941no—
128#123—974no—
129C22⋊Q8—1942no—
130#117C2×D9711yescharacteristic
131#101—1942no—
132#101—1941no—
133#101—1942no—
134#100—1941no—
135#41—1941no—
136#41—1941no—
137#41—1941no—
138#41—1941no—
139C2×Q8—971no—
140C2×Q8—1941no—
141C2×Q8—1941no—
142C2×Q8—1941no—
143C2×Q8—1941no—
144C2×Q8—1941no—
145C2×Q8—1941no—
146C2×D4—972no—
147C2×D4—971no—
148C22×C4—972no—
149C22×C4—972no—
150C22×C4C22×D9711yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16—972no—
155M16—1944no—
156M16—1941no—
157M16—1941no—
158M16—1941no—
159M16D38812yes—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—1944no—
163C2×C8—1941no—
164C2×C8—1941no—
165C4⋊C4—21no—
166C4⋊C4—1944no—
167C22⋊C4—1944no—
168C22⋊C4C4×D9712yes—
169C42—21no—
170C23—972no—
171C23C2×C4×D9711yescharacteristic
172Q8—3881no—
173Q8—3881no—
174Q8—3881no—
175Q8—3881no—
176Q8—1941no—
177Q8—1941no—
178D4—1942no—
179D4—1942no—
180D4—1941no—
181D4—1941no—
182C2×C4—22no—
183C2×C4—972no—
184C2×C4—1942no—
185C2×C4—1942no—
186C2×C4—1942no—
187C2×C4—1942no—
188C2×C4—1942no—
189C2×C4D4×D9711yescharacteristic
190C2×C4D4×D9711yescharacteristic
191C2×C4D4×D9711yescharacteristic
192C2×C4C2×D38811yescharacteristic
193C2×C4#1309551111yescharacteristic
194C2×C4#1309551211yescharacteristic
195C8—22no—
196C8—41no—
197C8—3881no—
198C8—1942no—
199C22—41no—
200C22—3882no—
201C22—1942no—
202C22C22⋊C4×D9711yescharacteristic
203C22#1569077211yescharacteristic
204C22C4×D38811yescharacteristic
205C4—42no—
206C4—3882no—
207C4—1942no—
208C4#1569078011yescharacteristic
209C4#1569078311yescharacteristic
210C4#1569078511yescharacteristic
211C4#1569079511yescharacteristic
212C2—21no—
213C2—21no—
214C2—21no—
215C2—3882no—
216C2#2026061311yescharacteristic
subgroupG/H
centerC2#20260613
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#344×C97C2
socleC2×C97#386
prime radicalC2×#13095511C22
cosocleC22×C4×C97C23
hypercenter#344D97
hypocenterC97#1915
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. C2×#15690728C2semidirect
5. C2×#15690728C2semidirect
6. #20260505C2semidirect
7. #344×C97C2semidirect
109. C97#1915semidirect
116. #344D97semidirect

Maximal subgroups

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

subgroupindex
#202615012
#202613112
#202607172
C2×#156907282
#202605052
#344×C972
#191597