Finite Group Databasebeta

← #27151267

Subgroups — #27151267

Order 12416.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#15690843C211yesmaximal, characteristic
2C22×Dic388C211yesmaximal, characteristic, complemented
3#20261311C211yesmaximal, characteristic
4#20261212C211yesmaximal, characteristic, complemented
5#20260958C214yesmaximal, complemented
6#20260946C214yesmaximal, complemented
7#20260745C211yesmaximal, characteristic, complemented
8#20260732C211yesmaximal, characteristic
9#424×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11C22×Dic194C2211yescharacteristic
12C22×Dic194C2211yescharacteristic
13#15690843—24no—
14C2×Dic388—24no—
15C2×Dic388C2212yes—
16C2×Dic388C2214yes—
17C2×#13095500C2212yes—
18#15690810C2212yes—
19#15690798C2212yes—
20#15690794—22no—
21#15690790—22no—
22C4×Dic194C2214yes—
23#15690775—22no—
24#15690769C2211yescharacteristic
25#15690766—22no—
26#15690735C2212yes—
27#15690735C2214yes—
28#15690732C2212yes—
29C2×M16×C97C2211yescharacteristic
30C22×C8×C97C2211yescharacteristic
31#117×C97C2211yescharacteristic
32C4⋊C8×C97C2214yes—
33#13095511—22no—
34#13095511—22no—
35C2×Dic194—21no—
36C2×Dic194—22no—
37C2×Dic194—22no—
38C2×Dic194—22no—
39C2×Dic194—21no—
40C2×Dic194D414yes—
41C2×Dic194C2312yes—
42C2×Dic194C2311yescharacteristic
43C2×Dic194C2312yes—
44C2×D388C2311yescharacteristic
45C2×C22⋊D97—21no—
46C22×Dic97—21no—
47C22×Dic97—21no—
48C2×C4×D97—21no—
49Dic388—28no—
50Dic388—24no—
51#13095500—24no—
52C4⋊Dic97—24no—
53C4⋊Dic97C2312yes—
54#13095490—24no—
55C22⋊Dic97—21no—
56#13095488—24no—
57C4×Dic97—24no—
58C4×Dic97—21no—
59C22×C4×C97C2311yescharacteristic
60M16×C97—22no—
61C2×C8×C97—22no—
62C2×C8×C97C2312yes—
63C2×C8×C97C2312yes—
64C4⋊C4×C97D412yes—
65C22⋊C4×C97D412yes—
66C42×C97C2312yes—
67C23×C97C2×D411yescharacteristic
68C22×D97—21no—
69C2×C4×C97C2×D412yes—
70C2×C4×C97C2×D414yes—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97#4112yes—
73C2×C4×C97C2411yescharacteristic
74C22⋊D97—42no—
75C2×Dic97—24no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—22no—
79C2×Dic97—21no—
80C2×Dic97—42no—
81D388—21no—
82D388—21no—
83C4×D97—42no—
84Dic194—24no—
85Dic194—22no—
86Dic194—22no—
87Dic194—21no—
88Dic194—22no—
89Dic194—21no—
90Dic194—42no—
91Dic194C2×D414yes—
92C8×C97—22no—
93C8×C97—22no—
94C22×C97—22no—
95C22×C97#12112yes—
96C22×C97C22×D411yescharacteristic
97C2×D97—41no—
98C2×D97—42no—
99C4×C97—24no—
100C4×C97#12112yes—
101C4×C97C22×D411yescharacteristic
102C4×C97C2×#4111yescharacteristic
103Dic97—24no—
104Dic97—42no—
105Dic97—41no—
106Dic97—42no—
107C2×C97—22no—
108C2×C97C2×#12111yescharacteristic
109C2×C97#57812yes—
110D97—81no—
111#3044—971nomaximal, Sylow
112C97#304411yescharacteristic, complemented, Sylow
113C2×#137—971no—
114C22×Q16—971no—
115#536—971no—
116#519—971no—
117#464—974no—
118#462—974no—
119#424D9711yescharacteristic, complemented
120#419—971no—
121#416—971no—
122C2×#41—971no—
123C22×Q8—971no—
124C22×Q8—971no—
125#137—1944no—
126C2×Q16—974no—
127C2×Q16—972no—
128C2×Q16—1944no—
129C2×SD16—972no—
130C2×M16C2×D9711yescharacteristic
131C22×C8C2×D9711yescharacteristic
132C4⋊Q8—972no—
133#124—972no—
134#123—1942no—
135C22⋊Q8—1942no—
136C4×Q8—974no—
137C4×Q8—1942no—
138#117—971no—
139#117—1942no—
140#117C2×D9711yescharacteristic
141C4⋊C8C2×D9714yes—
142#103—972no—
143#103—974no—
144#102—972no—
145#41—1942no—
146#41—1942no—
147C2×Q8—974no—
148C2×Q8—972no—
149C2×Q8—972no—
150C2×Q8—971no—
151C2×Q8—1942no—
152C2×Q8—1942no—
153C2×Q8—1941no—
154C2×Q8—1941no—
155C2×Q8—1942no—
156C2×D4—971no—
157C2×D4—1941no—
158C22×C4—1941no—
159C22×C4—1941no—
160C22×C4—1941no—
161C22×C4C22×D9711yescharacteristic
162Q16—1948no—
163Q16—1944no—
164SD16—1944no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D9712yes—
168C2×C8C22×D9712yes—
169C4⋊C4—972no—
170C4⋊C4—1944no—
171C4⋊C4—1944no—
172C4⋊C4D38812yes—
173C22⋊C4—1941no—
174C22⋊C4—1944no—
175C22⋊C4D38812yes—
176C42—1941no—
177C42—1944no—
178C42C22×D9712yes—
179C23—1941no—
180C23C2×D38811yescharacteristic
181Q8—974no—
182Q8—3882no—
183Q8—1942no—
184Q8—1941no—
185Q8—1942no—
186Q8—1941no—
187Q8—1944no—
188Q8—1942no—
189D4—3882no—
190D4—1941no—
191D4—1941no—
192C2×C4—3882no—
193C2×C4—3882no—
194C2×C4—1942no—
195C2×C4—1942no—
196C2×C4—1941no—
197C2×C4—1944no—
198C2×C4—1942no—
199C2×C4D4×D9712yes—
200C2×C4C2×D38814yes—
201C2×C4C2×D38811yescharacteristic
202C2×C4#1309551212yes—
203C2×C4C23×D9711yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—3881no—
208C22—3882no—
209C22#1569078512yes—
210C22C22×D38811yescharacteristic
211C4—24no—
212C4—3881no—
213C4—3882no—
214C4—3882no—
215C4—1944no—
216C4#1569078512yes—
217C4C2×D4×D9711yescharacteristic
218C4C2×#1309551211yescharacteristic
219C2—22no—
220C2—7761no—
221C2C2×#1569078511yescharacteristic
222C2#2026155212yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#424×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095511C22
hypercenter#424D97
hypocenterC97#3044
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic388C2semidirect
4. #20261212C2semidirect
5. #20260958C2semidirect
6. #20260946C2semidirect
7. #20260745C2semidirect
9. #424×C97C2semidirect
112. C97#3044semidirect
119. #424D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908432
C22×Dic3882
#202613112
#202612122
#202609582
#202609462
#202607452
#202607322
#424×C972
#304497