Finite Group Databasebeta

← #27142092

Subgroups — #27142092

Order 12416.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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
2C2×#15690775C211yesmaximal, characteristic, complemented
3C2×#15690733C211yesmaximal, characteristic, complemented
4C2×#15690730C211yesmaximal, characteristic
5#20260636C211yesmaximal, characteristic
6#20260489C211yesmaximal, characteristic, complemented
7#340×C97C211yesmaximal, characteristic, complemented
8C22×Dic194C2211yescharacteristic
9C22×D388C2211yescharacteristic
10C2×#13095500—22no—
11C2×#13095500—22no—
12C2×#13095500C414yescomplemented
13C22×#11715895C2211yescharacteristic
14#15690775—22no—
15#15690775—22no—
16C2×#13095490—21no—
17C2×#13095488—21no—
18C2×#13095488—21no—
19C2×C4×Dic97—21no—
20C2×C4×Dic97C2211yescharacteristic
21#15690733—22no—
22#15690730—22no—
23C22×C8×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25C2×C4⋊C4×C97C2211yescharacteristic
26C23×D97—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194C2×C412yes—
31C2×Dic194D412yes—
32C2×Dic194D412yes—
33C2×D388—21no—
34C2×D388—21no—
35C2×D388—21no—
36C2×D388—21no—
37C2×D388—21no—
38C2×D388C2×C412yes—
39C22×Dic97—21no—
40C22×Dic97D411yescharacteristic
41C22×Dic97D411yescharacteristic
42#13095500—28no—
43C2×#11715895—21no—
44C2×#11715895—21no—
45C2×#11715895—21no—
46#13095490—24no—
47#13095488—42no—
48#13095488—42no—
49C4×Dic97—24no—
50C4×Dic97—21no—
51C4×Dic97—21no—
52C22×C4×C97—21no—
53C22×C4×C97—21no—
54C22×C4×C97C2311yescharacteristic
55C2×C8×C97—21no—
56C2×C8×C97—21no—
57C2×C8×C97C2×C412yes—
58C4⋊C4×C97—21no—
59C4⋊C4×C97—21no—
60C4⋊C4×C97—21no—
61C4⋊C4×C97D412yes—
62C23×C97C2×D411yescharacteristic
63C22×D97—22no—
64C22×D97—41no—
65C22×D97—41no—
66C22×D97—41no—
67C22×D97—41no—
68C22×D97—41no—
69C22×D97—41no—
70C2×C4×C97—22no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97C22×C411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97#4111yescharacteristic
81C2×C4×C97#4111yescharacteristic
82C2×Dic97—22no—
83C2×Dic97—21no—
84C2×Dic97—22no—
85C2×Dic97—22no—
86C2×Dic97—21no—
87C2×Dic97—21no—
88C2×Dic97—21no—
89C2×Dic97—21no—
90C2×Dic97SD1612yescomplemented
91D388—22no—
92D388—22no—
93D388—41no—
94D388—41no—
95Dic194—22no—
96Dic194—22no—
97Dic194—22no—
98Dic194C22⋊C414yescomplemented
99C8×C97—22no—
100#11715895—41no—
101C22×C97C4×D411yescharacteristic
102C22×C97#12011yescharacteristic
103C22×C97#12111yescharacteristic
104C22×C97C2×SD1611yescharacteristic
105C22×C97#13511yescharacteristic
106C22×C97AGL(1,ℤ/8)11yescharacteristic
107C22×C97#13711yescharacteristic
108C2×D97—42no—
109C2×D97—42no—
110C2×D97—42no—
111C2×D97—81no—
112C2×D97—81no—
113C2×D97—81no—
114C2×D97—81no—
115C4×C97—42no—
116C4×C97—81no—
117C4×C97C2×C22⋊C411yescharacteristic
118C4×C97C4×D411yescharacteristic
119C4×C97#12111yescharacteristic
120C4×C97#12311yescharacteristic
121Dic97—24no—
122Dic97—22no—
123Dic97—41no—
124C2×C97#38611yescharacteristic
125C2×C97C4×SD1611yescharacteristic
126C2×C97#44011yescharacteristic
127C2×C97#44811yescharacteristic
128C2×C97#44911yescharacteristic
129C2×C97#46011yescharacteristic
130C2×C97#46411yescharacteristic
131D97—82no—
132#1880—971nomaximal, Sylow
133C97#188011yescharacteristic, complemented, Sylow
134C22×SD16—971no—
135C2×C4×Q8—971no—
136C2×#103—971no—
137C2×#102—971no—
138#390—971no—
139#340D9711yescharacteristic, complemented
140#336—971no—
141C22×Q8—971no—
142C22×D4—971no—
143C2×SD16—974no—
144C2×SD16—1942no—
145C2×SD16—1942no—
146C22×C8—971no—
147C22×C8C2×D9711yescharacteristic
148C4×Q8—1942no—
149C4×Q8—1942no—
150C2×C4⋊C4—1941no—
151C2×C4⋊C4C2×D9711yescharacteristic
152C2×C4⋊C4C2×D9711yescharacteristic
153C2×C22⋊C4—1941no—
154C2×C22⋊C4—1941no—
155C2×C42—971no—
156C2×C42—1941no—
157#103—1942no—
158#102—1942no—
159C24—1941no—
160C2×Q8—972no—
161C2×Q8—972no—
162C2×Q8—972no—
163C2×Q8—1941no—
164C2×Q8—1941no—
165C2×Q8—1941no—
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—971no—
175C22×C4—971no—
176C22×C4—1941no—
177C22×C4C22×D9711yescharacteristic
178SD16—1948no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—1941no—
182C2×C8—1941no—
183C2×C8—1941no—
184C2×C8C4×D9712yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—1944no—
189C4⋊C4D38812yes—
190C22⋊C4—3882no—
191C22⋊C4—3882no—
192C42—1944no—
193C42—1941no—
194C42—1941no—
195C23—3881no—
196C23—3881no—
197C23—3881no—
198C23—3881no—
199C23—3881no—
200C23—3881no—
201C23—1942no—
202C23D4×D9711yescharacteristic
203Q8—974no—
204Q8—1942no—
205Q8—1942no—
206Q8—1942no—
207D4—3881no—
208D4—3881no—
209D4—1942no—
210D4—1942no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—972no—
218C2×C4—1941no—
219C2×C4—1941no—
220C2×C4—1941no—
221C2×C4—1942no—
222C2×C4—1941no—
223C2×C4—1942no—
224C2×C4—1942no—
225C2×C4—1941no—
226C2×C4C2×C4×D9711yescharacteristic
227C2×C4D4×D9711yescharacteristic
228C2×C4D4×D9711yescharacteristic
229C2×C4C2×D38811yescharacteristic
230C2×C4#1309551111yescharacteristic
231C2×C4#1309551211yescharacteristic
232C8—22no—
233C8—3881no—
234C22—7761no—
235C22—7761no—
236C22—7761no—
237C22—7761no—
238C22—3882no—
239C22—3882no—
240C22—3882no—
241C22#1569077211yescharacteristic
242C22#1569078011yescharacteristic
243C22#1569078311yescharacteristic
244C22SD16×D9711yescharacteristic
245C22#1569083111yescharacteristic
246C22#1569083411yescharacteristic
247C22#1569084211yescharacteristic
248C4—42no—
249C4—81no—
250C4—3881no—
251C4—1944no—
252C4—1942no—
253C4C22⋊C4×D9711yescharacteristic
254C4C4×D38811yescharacteristic
255C4#1569078511yescharacteristic
256C4#1569079511yescharacteristic
257C2—7762no—
258C2#2026061311yescharacteristic
259C2#2026082211yescharacteristic
260C2#2026083411yescharacteristic
261C2#2026087511yescharacteristic
262C2#2026088111yescharacteristic
263C2#2026093411yescharacteristic
264C2#2026096011yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#340×C97C2
socleC23×C97C2×D4
prime radicalC22×D388C22
cosocleC22×C4×C97C23
hypercenter#340D97
hypocenterC97#1880
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#15690775C2semidirect
3. C2×#15690733C2semidirect
6. #20260489C2semidirect
7. #340×C97C2semidirect
12. C2×#13095500C4semidirect
90. C2×Dic97SD16semidirect
98. Dic194C22⋊C4semidirect
133. C97#1880semidirect
139. #340D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955002
C2×#156907752
C2×#156907332
C2×#156907302
#202606362
#202604892
#340×C972
#188097