Finite Group Databasebeta

← #27142551

Subgroups — #27142551

Order 12416.

344 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 99 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×#15690797C211yesmaximal, characteristic
2C2×#15690735C211yesmaximal, characteristic, complemented
3C2×#15690733C211yesmaximal, characteristic, complemented
4C2×#15690731C211yesmaximal, characteristic
5C2×#15690720C211yesmaximal, characteristic, complemented
6#20260604C211yesmaximal, characteristic, complemented
7C2×#102×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C22×#11715895C2211yescharacteristic
10#15690797—22no—
11#15690797C414yescomplemented
12C2×C4⋊Dic97C2211yescharacteristic
13C2×#13095490—22no—
14C2×C22⋊Dic97—22no—
15C2×C4×Dic97C2211yescharacteristic
16#15690735—22no—
17#15690733—22no—
18#15690731—22no—
19#15690720—22no—
20C22×D4×C97C2211yescharacteristic
21C22×C8×C97C2211yescharacteristic
22C2×C4⋊C4×C97C2211yescharacteristic
23#102×C97—22no—
24C2×Dic194—21no—
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194C2×C412yes—
30C22×Dic97—21no—
31C22×Dic97—21no—
32C22×Dic97D412yes—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C2×#11715895D412yes—
36C4⋊Dic97—21no—
37C4⋊Dic97—21no—
38#13095490—44no—
39C22⋊Dic97—28no—
40C4×Dic97C2×C412yes—
41C4×Dic97D412yes—
42C24×C97—21no—
43C2×D4×C97—21no—
44C2×D4×C97—21no—
45C2×D4×C97—21no—
46C2×D4×C97—21no—
47C2×D4×C97—21no—
48C2×D4×C97C2×C412yes—
49C22×C4×C97—21no—
50C22×C4×C97C2311yescharacteristic
51C2×C8×C97—21no—
52C2×C8×C97—21no—
53C2×C8×C97D412yes—
54C4⋊C4×C97—21no—
55C4⋊C4×C97—21no—
56C23×C97—22no—
57C23×C97—41no—
58C23×C97—41no—
59C23×C97—42no—
60C23×C97—42no—
61C23×C97C2×D411yescharacteristic
62D4×C97—22no—
63D4×C97—22no—
64D4×C97—41no—
65D4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—42no—
68C2×C4×C97C22×C411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97#4111yescharacteristic
73C2×C4×C97#4111yescharacteristic
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—41no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C2×Dic97—42no—
81C2×Dic97C22⋊C414yes—
82Dic194—22no—
83Dic194—22no—
84Dic194—41no—
85Dic194—41no—
86C8×C97—22no—
87#11715895—22no—
88C22×C97—42no—
89C22×C97—44no—
90C22×C97—82no—
91C22×C97—81no—
92C22×C97—81no—
93C22×C97C2×C22⋊C411yescharacteristic
94C22×C97#12411yescharacteristic
95C22×C97#12711yescharacteristic
96C22×C97AGL(1,ℤ/8)12yes—
97C22×C97#13712yes—
98C4×C97—81no—
99C4×C97C4×D411yescharacteristic
100C4×C97C4×D411yescharacteristic
101C4×C97#12111yescharacteristic
102C4×C97#12111yescharacteristic
103Dic97—24no—
104Dic97—42no—
105Dic97—81no—
106C2×C97—82no—
107C2×C97#39011yescharacteristic
108C2×C97#41712yes—
109C2×C97#48912yes—
110C2×C97#49611yescharacteristic
111C2×C97#49711yescharacteristic
112#1959—971nomaximal, Sylow
113C97#195911yescharacteristic, complemented, Sylow
114C2×#124—971no—
115C2×#103—971no—
116C2×#103—971no—
117C2×#102—971no—
118C2×#102D9711yescharacteristic
119C2×#97—971no—
120#384—971no—
121C22×Q8—971no—
122C22×D4C2×D9711yescharacteristic
123C22×C8—971no—
124C22×C8C2×D9711yescharacteristic
125#124—974no—
126#124—1942no—
127C2×C4⋊C4—971no—
128C2×C4⋊C4—1942no—
129C2×C4⋊C4C2×D9711yescharacteristic
130C2×C22⋊C4—1942no—
131C2×C42—971no—
132#103—1942no—
133#103—1942no—
134#102—22no—
135#102—1942no—
136#97—1942no—
137C24—21no—
138C2×Q8—972no—
139C2×Q8—1941no—
140C2×Q8—1941no—
141C2×Q8—1941no—
142C2×Q8—1941no—
143C2×Q8—1941no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C4×D9712yes—
150C22×C4—21no—
151C22×C4—972no—
152C22×C4—1941no—
153C22×C4—1941no—
154C22×C4C22×D9711yescharacteristic
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—972no—
158C2×C8—1941no—
159C2×C8—1941no—
160C2×C8C22⋊D9712yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—3884no—
164C4⋊C4—1941no—
165C4⋊C4—1941no—
166C22⋊C4—1948no—
167C42—972no—
168C42—972no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—42no—
173C23—42no—
174C23D4×D9711yescharacteristic
175Q8—3881no—
176Q8—3881no—
177Q8—1942no—
178Q8—1942no—
179D4—22no—
180D4—22no—
181D4—41no—
182D4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—974no—
186C2×C4—3881no—
187C2×C4—3882no—
188C2×C4—3881no—
189C2×C4—3881no—
190C2×C4—1942no—
191C2×C4—1942no—
192C2×C4—1942no—
193C2×C4C2×C4×D9711yescharacteristic
194C2×C4D4×D9711yescharacteristic
195C2×C4D4×D9711yescharacteristic
196C2×C4C2×C22⋊D9711yescharacteristic
197C2×C4#1309551011yescharacteristic
198C2×C4#1309551111yescharacteristic
199C8—22no—
200C8—1942no—
201C22—42no—
202C22—44no—
203C22—81no—
204C22—81no—
205C22—82no—
206C22C22⋊C4×D9711yescharacteristic
207C22#1569079611yescharacteristic
208C22#1569080611yescharacteristic
209C22#1569083412yes—
210C22#1569083912yes—
211C4—81no—
212C4—7761no—
213C4—3882no—
214C4—1944no—
215C4#1569077011yescharacteristic
216C4C4×C22⋊D9711yescharacteristic
217C4#1569078211yescharacteristic
218C4#1569078311yescharacteristic
219C2—82no—
220C2#2026063511yescharacteristic
221C2#2026073312yes—
222C2#2026109312yes—
223C2#2026112311yescharacteristic
224C2#2026112611yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#102×C97C2
socleC23×C97C2×D4
prime radicalC22×D4×C97C22
cosocleC22×C4×C97C23
hypercenterC2×#102D97
hypocenterC97#1959
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#15690735C2semidirect
3. C2×#15690733C2semidirect
5. C2×#15690720C2semidirect
6. #20260604C2semidirect
11. #15690797C4semidirect
113. C97#1959semidirect

Maximal subgroups

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

subgroupindex
C2×#156907972
C2×#156907352
C2×#156907332
C2×#156907312
C2×#156907202
#202606042
C2×#102×C972
#195997