Finite Group Databasebeta

← #27142831

Subgroups — #27142831

Order 12416.

390 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13095501C211yesmaximal, characteristic, complemented
2C22×#13095500C211yesmaximal, characteristic
3C2×#15690799C211yesmaximal, characteristic
4C2×#15690730C211yesmaximal, characteristic
5#20260640C211yesmaximal, characteristic, complemented
6#20260489C211yesmaximal, characteristic, complemented
7C2×#103×C97C211yesmaximal, characteristic, complemented
8C22×Dic194C2211yescharacteristic
9C22×D388C2211yescharacteristic
10C2×#13095501—22no—
11C2×#13095501—22no—
12C2×#13095501—22no—
13C2×#13095500—22no—
14C2×#13095500—22no—
15C2×#13095500—22no—
16C22×#11715895C2211yescharacteristic
17#15690799—22no—
18#15690799—22no—
19C2×#13095488—22no—
20C2×C4×Dic97C2211yescharacteristic
21#15690730—22no—
22#15690717—22no—
23C22×Q8×C97C2211yescharacteristic
24C22×C8×C97C2211yescharacteristic
25C2×C4⋊C4×C97C2211yescharacteristic
26#103×C97—22no—
27C23×D97—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×Dic194—21no—
32C2×Dic194—21no—
33C2×Dic194D412yes—
34C2×D388—21no—
35C2×D388—21no—
36C2×D388—21no—
37C2×D388—21no—
38C2×D388—21no—
39C2×D388D412yes—
40C22×Dic97—21no—
41C22×Dic97D412yes—
42#13095501—28no—
43#13095500—28no—
44C2×#11715895—21no—
45C2×#11715895—21no—
46C2×#11715895D412yes—
47#13095488—28no—
48C4×Dic97—21no—
49C4×Dic97—21no—
50C2×Q8×C97—21no—
51C2×Q8×C97—21no—
52C2×Q8×C97—21no—
53C2×Q8×C97—21no—
54C2×Q8×C97—21no—
55C2×Q8×C97D412yes—
56C22×C4×C97—21no—
57C22×C4×C97—21no—
58C22×C4×C97C2311yescharacteristic
59C2×C8×C97—21no—
60C2×C8×C97—21no—
61C2×C8×C97D412yes—
62C4⋊C4×C97—21no—
63C4⋊C4×C97—21no—
64C23×C97C2×D411yescharacteristic
65C22×D97—22no—
66C22×D97—41no—
67C22×D97—41no—
68C22×D97—42no—
69C22×D97—42no—
70Q8×C97—22no—
71Q8×C97—22no—
72Q8×C97—41no—
73Q8×C97—41no—
74C2×C4×C97—22no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97—42no—
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97#4111yescharacteristic
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—22no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91D388—22no—
92D388—22no—
93D388—41no—
94D388—41no—
95Dic194—22no—
96Dic194—22no—
97Dic194—41no—
98Dic194—41no—
99C8×C97—22no—
100#11715895—22no—
101C22×C97#12011yescharacteristic
102C22×C97#12111yescharacteristic
103C22×C97#12711yescharacteristic
104C22×C97#13512yes—
105C22×C97#13712yes—
106C2×D97—42no—
107C2×D97—44no—
108C2×D97—82no—
109C2×D97—81no—
110C2×D97—81no—
111C4×C97—42no—
112C4×C97—81no—
113C4×C97#12011yescharacteristic
114C4×C97#12011yescharacteristic
115C4×C97#12111yescharacteristic
116C4×C97#12111yescharacteristic
117Dic97—42no—
118Dic97—42no—
119C2×C97#39211yescharacteristic
120C2×C97#45212yes—
121C2×C97#46412yes—
122C2×C97#49511yescharacteristic
123C2×C97#49711yescharacteristic
124D97—82no—
125#2015—971nomaximal, Sylow
126C97#201511yescharacteristic, complemented, Sylow
127C22×Q16—971no—
128C22×SD16—971no—
129C2×#124—971no—
130C2×#103D9711yescharacteristic, complemented
131C2×#102—971no—
132#391—971no—
133#336—971no—
134C22×Q8—971no—
135C22×Q8C2×D9711yescharacteristic
136C22×D4—971no—
137C2×Q16—1942no—
138C2×Q16—1942no—
139C2×Q16—1942no—
140C2×SD16—1942no—
141C2×SD16—1942no—
142C2×SD16—1942no—
143C22×C8—971no—
144C22×C8C2×D9711yescharacteristic
145#124—1942no—
146#124—1942no—
147C2×C4⋊C4C2×D9711yescharacteristic
148C2×C22⋊C4—1942no—
149C2×C42—971no—
150#103—22no—
151#102—1942no—
152#95—1942no—
153C24—1941no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—972no—
160C2×Q8—1941no—
161C2×Q8—1941no—
162C2×Q8—1941no—
163C2×Q8—1941no—
164C2×Q8—1941no—
165C2×Q8D38812yes—
166C2×D4—972no—
167C2×D4—1941no—
168C2×D4—1941no—
169C2×D4—1941no—
170C2×D4—1941no—
171C2×D4—1941no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—972no—
175C22×C4—1941no—
176C22×C4C22×D9711yescharacteristic
177Q16—1948no—
178SD16—1948no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—972no—
182C2×C8—1941no—
183C2×C8—1941no—
184C2×C8C22⋊D9712yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C22⋊C4—1948no—
188C42—1941no—
189C42—1941no—
190C23—3881no—
191C23—3881no—
192C23—3882no—
193C23—3882no—
194C23—1942no—
195C23D4×D9711yescharacteristic
196Q8—22no—
197Q8—22no—
198Q8—41no—
199Q8—41no—
200Q8—3881no—
201Q8—3881no—
202Q8—1942no—
203Q8—1942no—
204D4—3881no—
205D4—3881no—
206D4—1942no—
207D4—1942no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—3881no—
214C2×C4—3881no—
215C2×C4—1942no—
216C2×C4—1942no—
217C2×C4—1942no—
218C2×C4—1942no—
219C2×C4D4×D9711yescharacteristic
220C2×C4D4×D9711yescharacteristic
221C2×C4D4×D9711yescharacteristic
222C2×C4C2×C22⋊D9711yescharacteristic
223C2×C4C2×D38811yescharacteristic
224C2×C4#1309551111yescharacteristic
225C8—22no—
226C8—1942no—
227C22—7761no—
228C22—7762no—
229C22—7761no—
230C22—3882no—
231C22—3884no—
232C22#1569078011yescharacteristic
233C22#1569078311yescharacteristic
234C22#1569080611yescharacteristic
235C22#1569083112yes—
236C22#1569084212yes—
237C4—42no—
238C4—81no—
239C4—3882no—
240C4—3882no—
241C4#1569077811yescharacteristic
242C4#1569078011yescharacteristic
243C4#1569078311yescharacteristic
244C4#1569078411yescharacteristic
245C2—7762no—
246C2#2026064411yescharacteristic
247C2#2026090012yes—
248C2#2026096012yes—
249C2#2026111611yescharacteristic
250C2#2026112811yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#103×C97C2
socleC23×C97C2×D4
prime radicalC22×D388C22
hypercenterC2×#103D97
hypocenterC97#2015
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#13095501C2semidirect
5. #20260640C2semidirect
6. #20260489C2semidirect
7. C2×#103×C97C2semidirect
126. C97#2015semidirect
130. C2×#103D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955012
C22×#130955002
C2×#156907992
C2×#156907302
#202606402
#202604892
C2×#103×C972
#201597