Finite Group Databasebeta

← #27142050

Subgroups — #27142050

Order 12416.

456 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 107 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×#15690781C211yesmaximal, characteristic
2C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
3C2×#15690731C211yesmaximal, characteristic, complemented
4C2×#15690722C211yesmaximal, characteristic, complemented
5#20260498C211yesmaximal, characteristic, complemented
6#20260497C211yesmaximal, characteristic, complemented
7C2×#102×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97—21no—
9C22×C4×D97—21no—
10C22×C4×D97C2211yescharacteristic
11C22×#11715895C2211yescharacteristic
12#15690781—22no—
13#15690781C414yescomplemented
14C2×C4⋊Dic97C2211yescharacteristic
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095490—21no—
17C4⋊C4×D97—22no—
18C4⋊C4×D97—22no—
19C2×C22⋊Dic97—21no—
20#15690731—22no—
21#15690722—22no—
22C22×D4×C97C2211yescharacteristic
23C22×C8×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25#102×C97—22no—
26C23×D97D411yescharacteristic
27C2×C22⋊D97—24no—
28C2×C22⋊D97—42no—
29C2×C22⋊D97—42no—
30C22×Dic97—21no—
31C22×Dic97—21no—
32C22×Dic97D411yescharacteristic
33C2×C4×D97—21no—
34C2×C4×D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97—21no—
37C2×C4×D97—24no—
38C2×C4×D97—42no—
39C2×C4×D97—42no—
40C2×C4×D97C2×C412yes—
41C2×C4×D97D412yes—
42C2×#11715895—21no—
43C2×#11715895—21no—
44C2×#11715895—21no—
45C4⋊Dic97—21no—
46C4⋊Dic97—21no—
47C4⋊Dic97C2×C412yescomplemented
48C4⋊Dic97D412yescomplemented
49#13095490—24no—
50C22⋊Dic97—24no—
51C24×C97—21no—
52C2×D4×C97—21no—
53C2×D4×C97—21no—
54C2×D4×C97—21no—
55C2×D4×C97—21no—
56C2×D4×C97—21no—
57C2×D4×C97C2×C412yescomplemented
58C22×C4×C97—21no—
59C22×C4×C97C2311yescharacteristic
60C2×C8×C97—21no—
61C2×C8×C97—21no—
62C2×C8×C97—21no—
63C4⋊C4×C97—21no—
64C4⋊C4×C97D412yescomplemented
65C23×C97—22no—
66C23×C97—41no—
67C23×C97—41no—
68C23×C97—41no—
69C23×C97—41no—
70C23×C97—41no—
71C23×C97—41no—
72C23×C97C2×D411yescharacteristic
73C22×D97—21no—
74C22×D97—21no—
75C22×D97—21no—
76C22×D97—21no—
77C22×D97C22⋊C412yes—
78C22×D97D812yes—
79C22×D97SD1612yes—
80D4×C97—22no—
81D4×C97—22no—
82D4×C97—41no—
83D4×C97—41no—
84C2×C4×C97—22no—
85C2×C4×C97—41no—
86C2×C4×C97—41no—
87C2×C4×C97C22×C411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97#4111yescharacteristic
92C2×C4×C97#4111yescharacteristic
93C22⋊D97—48no—
94C2×Dic97—21no—
95C2×Dic97—21no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97—41no—
99C2×Dic97—41no—
100C2×Dic97—41no—
101C2×Dic97—41no—
102C2×Dic97C22⋊C412yes—
103C4×D97—22no—
104C4×D97—22no—
105C4×D97—22no—
106C4×D97—22no—
107C4×D97—48no—
108C8×C97—41no—
109#11715895—41no—
110C22×C97—42no—
111C22×C97—42no—
112C22×C97—42no—
113C22×C97—81no—
114C22×C97—81no—
115C22×C97—81no—
116C22×C97—81no—
117C22×C97C2×C22⋊C411yescharacteristic
118C22×C97#12011yescharacteristic
119C22×C97#12311yescharacteristic
120C22×C97C2×D811yescharacteristic
121C22×C97C2×SD1611yescharacteristic
122C22×C97AGL(1,ℤ/8)11yescharacteristic
123C22×C97#13711yescharacteristic
124C2×D97—22no—
125C2×D97—22no—
126C2×D97—22no—
127C2×D97—22no—
128C2×D97—22no—
129C2×D97—22no—
130C2×D97#10214yescomplemented
131C4×C97—42no—
132C4×C97C4×D411yescharacteristic
133C4×C97C4×D411yescharacteristic
134C4×C97#12111yescharacteristic
135C4×C97#12111yescharacteristic
136Dic97—22no—
137Dic97—42no—
138Dic97—42no—
139C2×C97—82no—
140C2×C97#38611yescharacteristic
141C2×C97C2×#10211yescharacteristic
142C2×C97#41711yescharacteristic
143C2×C97#44711yescharacteristic
144C2×C97#44811yescharacteristic
145C2×C97#48011yescharacteristic
146C2×C97#48111yescharacteristic
147D97—24no—
148#1874—971nomaximal, Sylow
149C97#187411yescharacteristic, complemented, Sylow
150C2×#121—971no—
151C22×C4⋊C4—971no—
152C2×#102—971no—
153C2×#102D9711yescharacteristic, complemented
154C2×C22⋊C8—971no—
155#340—971no—
156#340—971no—
157C22×D4—1941no—
158C22×D4C2×D9711yescharacteristic
159C23×C4—971no—
160C23×C4—1941no—
161C22×C8—971no—
162C22×C8C2×D9711yescharacteristic
163#121—974no—
164#121—1942no—
165C2×C4⋊C4—971no—
166C2×C4⋊C4—971no—
167C2×C4⋊C4—1942no—
168C2×C4⋊C4—1941no—
169C2×C4⋊C4—1942no—
170C2×C4⋊C4C2×D9711yescharacteristic
171C2×C22⋊C4—1941no—
172#102—22no—
173#102—1942no—
174C22⋊C8—1942no—
175C24—21no—
176C24—971no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—3882no—
183C2×D4—3882no—
184C2×D4—1944no—
185C2×D4C4×D9712yescomplemented
186C22×C4—21no—
187C22×C4—971no—
188C22×C4—972no—
189C22×C4—972no—
190C22×C4—3882no—
191C22×C4—3882no—
192C22×C4—1944no—
193C22×C4—1941no—
194C22×C4—1941no—
195C22×C4—1941no—
196C22×C4—1941no—
197C22×C4—1941no—
198C22×C4—1941no—
199C22×C4C22×D9711yescharacteristic
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—1941no—
204C2×C8—1941no—
205C2×C8—1941no—
206C4⋊C4—21no—
207C4⋊C4—972no—
208C4⋊C4—972no—
209C4⋊C4—1944no—
210C4⋊C4—1941no—
211C4⋊C4—1941no—
212C4⋊C4C22⋊D9712yescomplemented
213C22⋊C4—1944no—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—972no—
222C23—972no—
223C23—972no—
224C23—1941no—
225C23—1941no—
226C23—1941no—
227C23—1941no—
228C23D4×D9711yescharacteristic
229D4—22no—
230D4—22no—
231D4—41no—
232D4—41no—
233D4—3888no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—972no—
238C2×C4—3888no—
239C2×C4—3881no—
240C2×C4—3881no—
241C2×C4—3881no—
242C2×C4—3881no—
243C2×C4—1942no—
244C2×C4—1942no—
245C2×C4—1941no—
246C2×C4—1941no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4—1942no—
251C2×C4C2×C4×D9711yescharacteristic
252C2×C4D4×D9711yescharacteristic
253C2×C4D4×D9711yescharacteristic
254C2×C4C2×C22⋊D9711yescharacteristic
255C2×C4#1309551011yescharacteristic
256C2×C4#1309551111yescharacteristic
257C8—41no—
258C8—3881no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—81no—
266C22—974no—
267C22—1942no—
268C22—1942no—
269C22—1942no—
270C22—1942no—
271C22—1942no—
272C22—1942no—
273C22C22⋊C4×D9711yescharacteristic
274C22#1569077811yescharacteristic
275C22#1569079211yescharacteristic
276C22D8×D9711yescharacteristic
277C22SD16×D9711yescharacteristic
278C22#1569083411yescharacteristic
279C22#1569083911yescharacteristic
280C4—42no—
281C4—3882no—
282C4—3882no—
283C4—1942no—
284C4#1569077011yescharacteristic
285C4C4×C22⋊D9711yescharacteristic
286C4#1569078111yescharacteristic
287C4#1569078311yescharacteristic
288C2—82no—
289C2—1944no—
290C2#2026061111yescharacteristic
291C2#102×D9711yescharacteristic
292C2#2026073311yescharacteristic
293C2#2026086711yescharacteristic
294C2#2026087211yescharacteristic
295C2#2026104711yescharacteristic
296C2#2026105211yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#102×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690781C2
cosocleC22×C4×C97C23
hypercenterC2×#102D97
hypocenterC97#1874
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D97C2semidirect
3. C2×#15690731C2semidirect
4. C2×#15690722C2semidirect
5. #20260498C2semidirect
6. #20260497C2semidirect
7. C2×#102×C97C2semidirect
13. #15690781C4semidirect
47. C4⋊Dic97C2×C4semidirect
48. C4⋊Dic97D4semidirect
57. C2×D4×C97C2×C4semidirect
64. C4⋊C4×C97D4semidirect
130. C2×D97#102semidirect
149. C97#1874semidirect
153. C2×#102D97semidirect
185. C2×D4C4×D97semidirect
212. C4⋊C4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907812
C2×C4⋊C4×D972
C2×#156907312
C2×#156907222
#202604982
#202604972
C2×#102×C972
#187497