Finite Group Databasebeta

← #27151269

Subgroups — #27151269

Order 12416.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690840C211yesmaximal, characteristic
2C22×Dic388C211yesmaximal, characteristic, complemented
3#20261306C211yesmaximal, characteristic
4#20261212C211yesmaximal, characteristic, complemented
5#20260956C214yesmaximal, complemented
6#20260944C214yesmaximal, complemented
7#20260761C211yesmaximal, characteristic, complemented
8#20260743C211yesmaximal, characteristic, complemented
9#416×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11C22×Dic194C2211yescharacteristic
12#15690840—24no—
13C2×Dic388—24no—
14C2×Dic388C2214yes—
15C2×#13095501C2212yes—
16C2×#13095499C2212yes—
17C2×#13095495C2211yescharacteristic
18#15690808C2212yes—
19#15690797C2212yes—
20#15690791—22no—
21#15690789—22no—
22C4×Dic194—22no—
23#15690775C2214yes—
24#15690769C2211yescharacteristic
25#15690766—22no—
26#15690739C2214yes—
27#15690733C2214yes—
28C2×#41×C97C2211yescharacteristic
29C22×C8×C97C2211yescharacteristic
30#117×C97C2211yescharacteristic
31#103×C97C2212yes—
32#102×C97C2212yes—
33C2×Dic194—21no—
34C2×Dic194—22no—
35C2×Dic194—22no—
36C2×Dic194—22no—
37C2×Dic194—21no—
38C2×Dic194D414yes—
39C2×Dic194C2312yes—
40C2×Dic194C2312yes—
41C22×Dic97—21no—
42C22×Dic97—21no—
43Dic388—28no—
44#13095501—24no—
45#13095499—24no—
46#13095495—22no—
47C2×#11715895C2312yes—
48C4⋊Dic97D412yes—
49#13095490—24no—
50#13095490—24no—
51C22⋊Dic97—24no—
52C22⋊Dic97D412yes—
53C4×Dic97—24no—
54C4×Dic97C2312yes—
55#41×C97—22no—
56#41×C97—22no—
57C2×Q8×C97C2311yescharacteristic
58C2×D4×C97—21no—
59C2×D4×C97C2311yescharacteristic
60C22×C4×C97—21no—
61C22×C4×C97C2311yescharacteristic
62C2×C8×C97—22no—
63C2×C8×C97C2312yes—
64C4⋊C4×C97C2312yes—
65C22⋊C4×C97—21no—
66C42×C97—21no—
67C23×C97—21no—
68C23×C97C2×D411yescharacteristic
69Q8×C97—21no—
70Q8×C97—21no—
71D4×C97—21no—
72D4×C97—21no—
73D4×C97—42no—
74C2×C4×C97—22no—
75C2×C4×C97—21no—
76C2×C4×C97—42no—
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D412yes—
79C2×C4×C97#4112yes—
80C2×C4×C97C2411yescharacteristic
81C2×Dic97—24no—
82C2×Dic97—22no—
83C2×Dic97—22no—
84C2×Dic97—42no—
85C2×Dic97C2×D414yes—
86Dic194—24no—
87Dic194—22no—
88Dic194—22no—
89Dic194—22no—
90Dic194—42no—
91Dic194C2×D414yes—
92C8×C97—22no—
93#11715895—22no—
94C22×C97—22no—
95C22×C97—41no—
96C22×C97—42no—
97C22×C97#12112yes—
98C22×C97C22×D411yescharacteristic
99C4×C97—41no—
100C4×C97—42no—
101C4×C97#12112yes—
102C4×C97C22×D411yescharacteristic
103C4×C97C2×#4111yescharacteristic
104Dic97—24no—
105Dic97—24no—
106Dic97—42no—
107C2×C97—22no—
108C2×C97—81no—
109C2×C97C2×#12111yescharacteristic
110C2×C97#57812yes—
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#424—971no—
120#419—971no—
121#416D9711yescharacteristic
122C2×#41C2×D9711yescharacteristic
123C22×Q8—971no—
124C22×Q8—971no—
125#137—1944no—
126C2×Q16—972no—
127C2×Q16—974no—
128C2×Q16—1944no—
129C2×SD16—972no—
130C2×M16—971no—
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⋊C8—974no—
142#103—974no—
143#103C2×D9712yes—
144#102C2×D9712yes—
145#41—22no—
146#41—22no—
147C2×Q8—974no—
148C2×Q8—972no—
149C2×Q8—972no—
150C2×Q8—1942no—
151C2×Q8—1942no—
152C2×Q8—1941no—
153C2×Q8—1941no—
154C2×Q8—1942no—
155C2×Q8C22×D9711yescharacteristic
156C2×D4—21no—
157C2×D4C22×D9711yescharacteristic
158C22×C4—21no—
159C22×C4—1941no—
160C22×C4—1941no—
161C22×C4C22×D9711yescharacteristic
162Q16—1948no—
163Q16—1944no—
164SD16—1944no—
165M16—1942no—
166C2×C8—22no—
167C2×C8—972no—
168C2×C8C22×D9712yes—
169C4⋊C4—972no—
170C4⋊C4—1944no—
171C4⋊C4—1944no—
172C4⋊C4C22×D9712yes—
173C22⋊C4—21no—
174C22⋊C4—972no—
175C22⋊C4—1944no—
176C42—21no—
177C42—972no—
178C42—1944no—
179C23—21no—
180C23D4×D9711yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—974no—
184Q8—3882no—
185Q8—1942no—
186Q8—1942no—
187Q8—1944no—
188Q8—1942no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—974no—
196C2×C4—3882no—
197C2×C4—1942no—
198C2×C4—1944no—
199C2×C4—1942no—
200C2×C4D4×D9711yescharacteristic
201C2×C4D4×D9712yes—
202C2×C4#1309551112yes—
203C2×C4C23×D9711yescharacteristic
204C8—22no—
205C8—1942no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1569078312yes—
210C22C2×D4×D9711yescharacteristic
211C4—41no—
212C4—42no—
213C4—3882no—
214C4—1944no—
215C4—1944no—
216C4#1569078312yes—
217C4C2×D4×D9711yescharacteristic
218C4C2×#1309551111yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1569078311yescharacteristic
222C2#2026154912yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#416×C97C2
socleC22×C97C22×D4
prime radicalC2×#41×C97C22
hypercenter#416D97
hypocenterC97#3044
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic388C2semidirect
4. #20261212C2semidirect
5. #20260956C2semidirect
6. #20260944C2semidirect
7. #20260761C2semidirect
8. #20260743C2semidirect
112. C97#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#156908402
C22×Dic3882
#202613062
#202612122
#202609562
#202609442
#202607612
#202607432
#416×C972
#304497