Finite Group Databasebeta

← #27142820

Subgroups — #27142820

Order 12416.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13095500C211yesmaximal, characteristic
2C22×#13095498C211yesmaximal, characteristic
3C2×#15690808C211yesmaximal, characteristic, complemented
4C2×#15690733C211yesmaximal, characteristic, complemented
5#20260636C211yesmaximal, characteristic
6#20260489C211yesmaximal, characteristic, complemented
7C2×#103×C97C211yesmaximal, characteristic, complemented
8C22×Dic194C2211yescharacteristic
9C22×D388C2211yescharacteristic
10C2×#13095500—22no—
11C2×#13095500—22no—
12C2×#13095500—22no—
13C2×#13095498—22no—
14C2×#13095498—22no—
15C2×#13095498—22no—
16C22×#11715895C2211yescharacteristic
17#15690808—22no—
18#15690808—22no—
19C2×#13095490—22no—
20C2×#13095488—22no—
21C2×C4×Dic97C2211yescharacteristic
22#15690733—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#13095500—28no—
43#13095498—28no—
44C2×#11715895—21no—
45C2×#11715895—21no—
46C2×#11715895D412yes—
47#13095490—28no—
48#13095488—44no—
49C4×Dic97—21no—
50C4×Dic97—21no—
51C2×Q8×C97—21no—
52C2×Q8×C97—21no—
53C2×Q8×C97—21no—
54C2×Q8×C97—21no—
55C2×Q8×C97—21no—
56C2×Q8×C97D412yes—
57C22×C4×C97—21no—
58C22×C4×C97—21no—
59C22×C4×C97C2311yescharacteristic
60C2×C8×C97—21no—
61C2×C8×C97—21no—
62C2×C8×C97D412yes—
63C4⋊C4×C97—21no—
64C4⋊C4×C97—21no—
65C23×C97C2×D411yescharacteristic
66C22×D97—22no—
67C22×D97—41no—
68C22×D97—41no—
69C22×D97—42no—
70C22×D97—42no—
71Q8×C97—22no—
72Q8×C97—22no—
73Q8×C97—41no—
74Q8×C97—41no—
75C2×C4×C97—22no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97—41no—
79C2×C4×C97—42no—
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97#4111yescharacteristic
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—22no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91C2×Dic97SD1614yescomplemented
92D388—22no—
93D388—22no—
94D388—41no—
95D388—41no—
96Dic194—22no—
97Dic194—22no—
98Dic194—41no—
99Dic194—41no—
100C8×C97—22no—
101#11715895—22no—
102C22×C97#12011yescharacteristic
103C22×C97#12111yescharacteristic
104C22×C97#12711yescharacteristic
105C22×C97C2×SD1612yes—
106C22×C97#13712yes—
107C2×D97—42no—
108C2×D97—44no—
109C2×D97—82no—
110C2×D97—81no—
111C2×D97—81no—
112C4×C97—42no—
113C4×C97—81no—
114C4×C97#12011yescharacteristic
115C4×C97#12011yescharacteristic
116C4×C97#12111yescharacteristic
117C4×C97#12111yescharacteristic
118Dic97—24no—
119Dic97—42no—
120C2×C97#39211yescharacteristic
121C2×C97#44812yes—
122C2×C97#46112yes—
123C2×C97#49211yescharacteristic
124C2×C97#49711yescharacteristic
125D97—82no—
126#2013—971nomaximal, Sylow
127C97#201311yescharacteristic, complemented, Sylow
128C22×SD16—971no—
129C22×SD16—971no—
130C2×C4⋊Q8—971no—
131C2×#103—971no—
132C2×#103D9711yescharacteristic, complemented
133#390—971no—
134#336—971no—
135C22×Q8—971no—
136C22×Q8C2×D9711yescharacteristic
137C22×D4—971no—
138C2×SD16—1942no—
139C2×SD16—1942no—
140C2×SD16—1942no—
141C2×SD16—1942no—
142C2×SD16—1942no—
143C2×SD16—1942no—
144C22×C8—971no—
145C22×C8C2×D9711yescharacteristic
146C4⋊Q8—1942no—
147C4⋊Q8—1942no—
148C2×C4⋊C4—1942no—
149C2×C4⋊C4C2×D9711yescharacteristic
150C2×C22⋊C4—1942no—
151C2×C42—971no—
152#103—22no—
153#103—1942no—
154C24—1941no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—972no—
161C2×Q8—1941no—
162C2×Q8—1941no—
163C2×Q8—1941no—
164C2×Q8—1941no—
165C2×Q8—1941no—
166C2×Q8D38812yes—
167C2×D4—972no—
168C2×D4—1941no—
169C2×D4—1941no—
170C2×D4—1941no—
171C2×D4—1941no—
172C2×D4—1941no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—972no—
176C22×C4—1941no—
177C22×C4C22×D9711yescharacteristic
178SD16—1948no—
179SD16—1948no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—972no—
183C2×C8—1941no—
184C2×C8—1941no—
185C2×C8C22⋊D9712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—1948no—
189C22⋊C4—3884no—
190C42—1941no—
191C42—1941no—
192C23—3881no—
193C23—3881no—
194C23—3882no—
195C23—3882no—
196C23—1942no—
197C23D4×D9711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—3881no—
203Q8—3881no—
204Q8—1942no—
205Q8—1942no—
206D4—3881no—
207D4—3881no—
208D4—1942no—
209D4—1942no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—974no—
216C2×C4—3881no—
217C2×C4—3881no—
218C2×C4—1942no—
219C2×C4—1942no—
220C2×C4—1942no—
221C2×C4D4×D9711yescharacteristic
222C2×C4D4×D9711yescharacteristic
223C2×C4D4×D9711yescharacteristic
224C2×C4C2×C22⋊D9711yescharacteristic
225C2×C4C2×D38811yescharacteristic
226C2×C4#1309551111yescharacteristic
227C8—22no—
228C8—1942no—
229C22—7761no—
230C22—7762no—
231C22—7761no—
232C22—3882no—
233C22—3884no—
234C22#1569078011yescharacteristic
235C22#1569078311yescharacteristic
236C22#1569080611yescharacteristic
237C22SD16×D9712yes—
238C22#1569084212yes—
239C4—42no—
240C4—81no—
241C4—3882no—
242C4—1944no—
243C4#1569077811yescharacteristic
244C4#1569078011yescharacteristic
245C4#1569078311yescharacteristic
246C4#1569078411yescharacteristic
247C2—7762no—
248C2#2026064411yescharacteristic
249C2#2026087512yes—
250C2#2026093912yes—
251C2#2026110611yescharacteristic
252C2#2026112811yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#103×C97C2
socleC23×C97C2×D4
prime radicalC22×D388C22
hypercenterC2×#103D97
hypocenterC97#2013
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
3. C2×#15690808C2semidirect
4. C2×#15690733C2semidirect
6. #20260489C2semidirect
7. C2×#103×C97C2semidirect
91. C2×Dic97SD16semidirect
127. C97#2013semidirect
132. C2×#103D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955002
C22×#130954982
C2×#156908082
C2×#156907332
#202606362
#202604892
C2×#103×C972
#201397