Finite Group Databasebeta

← #27142052

Subgroups — #27142052

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×#15690785C211yesmaximal, characteristic
2C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
3C2×#15690732C211yesmaximal, characteristic, complemented
4C2×#15690730C211yesmaximal, characteristic, complemented
5C2×#15690722C211yesmaximal, characteristic, complemented
6#20260497C211yesmaximal, characteristic, complemented
7#340×C97C211yesmaximal, characteristic, complemented
8C22×D388—21no—
9C22×D388C2211yescharacteristic
10C22×C4×D97—21no—
11C22×C4×D97C2211yescharacteristic
12C22×#11715895C2211yescharacteristic
13#15690785—22no—
14#15690785C414yescomplemented
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095490—21no—
17C4⋊C4×D97—22no—
18C4⋊C4×D97—22no—
19C2×#13095488—21no—
20#15690732—22no—
21#15690730—22no—
22#15690722—22no—
23C22×C8×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25C2×C4⋊C4×C97C2211yescharacteristic
26C23×D97—21no—
27C23×D97D411yescharacteristic
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388—21no—
32C2×D388—24no—
33C2×D388—21no—
34C2×D388—42no—
35C2×D388—42no—
36C2×D388C2×C412yescomplemented
37C22×Dic97—21no—
38C22×Dic97D411yescharacteristic
39C2×C4×D97—21no—
40C2×C4×D97—21no—
41C2×C4×D97—21no—
42C2×C4×D97—21no—
43C2×C4×D97—24no—
44C2×C4×D97—42no—
45C2×C4×D97—42no—
46C2×C4×D97C2×C412yes—
47C2×C4×D97D412yes—
48C2×#11715895—21no—
49C2×#11715895—21no—
50C2×#11715895—21no—
51C4⋊Dic97—21no—
52C4⋊Dic97D412yescomplemented
53#13095490—24no—
54#13095488—24no—
55C22×C4×C97—21no—
56C22×C4×C97—21no—
57C22×C4×C97C2311yescharacteristic
58C2×C8×C97—21no—
59C2×C8×C97—21no—
60C2×C8×C97—21no—
61C4⋊C4×C97—21no—
62C4⋊C4×C97—21no—
63C4⋊C4×C97C2×C412yescomplemented
64C4⋊C4×C97D412yescomplemented
65C23×C97C2×D411yescharacteristic
66C22×D97—21no—
67C22×D97—21no—
68C22×D97—21no—
69C22×D97—21no—
70C22×D97—22no—
71C22×D97—41no—
72C22×D97—41no—
73C22×D97—41no—
74C22×D97—41no—
75C22×D97—41no—
76C22×D97—41no—
77C22×D97C22⋊C412yes—
78C22×D97D812yes—
79C22×D97SD1612yes—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—41no—
83C2×C4×C97—41no—
84C2×C4×C97—41no—
85C2×C4×C97—41no—
86C2×C4×C97C22×C411yescharacteristic
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97#4111yescharacteristic
91C2×C4×C97#4111yescharacteristic
92C2×Dic97—21no—
93C2×Dic97—21no—
94C2×Dic97—22no—
95C2×Dic97—41no—
96C2×Dic97—41no—
97C2×Dic97C22⋊C412yes—
98D388—22no—
99D388—22no—
100D388—48no—
101D388—41no—
102D388—41no—
103C4×D97—22no—
104C4×D97—22no—
105C4×D97—22no—
106C4×D97—22no—
107C4×D97—48no—
108C8×C97—41no—
109#11715895—41no—
110C22×C97C2×C22⋊C411yescharacteristic
111C22×C97#12011yescharacteristic
112C22×C97#12311yescharacteristic
113C22×C97C2×D811yescharacteristic
114C22×C97C2×SD1611yescharacteristic
115C22×C97AGL(1,ℤ/8)11yescharacteristic
116C22×C97#13711yescharacteristic
117C2×D97—22no—
118C2×D97—22no—
119C2×D97—22no—
120C2×D97—22no—
121C2×D97—22no—
122C2×D97—22no—
123C2×D97—42no—
124C2×D97—42no—
125C2×D97—42no—
126C2×D97—81no—
127C2×D97—81no—
128C2×D97—81no—
129C2×D97—81no—
130C2×D97#10214yescomplemented
131C4×C97—42no—
132C4×C97—42no—
133C4×C97C4×D411yescharacteristic
134C4×C97C4×D411yescharacteristic
135C4×C97#12111yescharacteristic
136C4×C97#12111yescharacteristic
137Dic97—22no—
138Dic97—42no—
139C2×C97#38611yescharacteristic
140C2×C97C2×#10211yescharacteristic
141C2×C97#41711yescharacteristic
142C2×C97#44711yescharacteristic
143C2×C97#44811yescharacteristic
144C2×C97#48011yescharacteristic
145C2×C97#48111yescharacteristic
146D97—24no—
147D97—82no—
148#1874—971nomaximal, Sylow
149C97#187411yescharacteristic, complemented, Sylow
150C2×#121—971no—
151C22×C4⋊C4—971no—
152C2×#102—971no—
153C2×#102—971no—
154C2×C22⋊C8—971no—
155#340—971no—
156#340D9711yescharacteristic, complemented
157C22×D4—971no—
158C22×D4—1941no—
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—1942no—
167C2×C4⋊C4—1941no—
168C2×C4⋊C4—1942no—
169C2×C4⋊C4C2×D9711yescharacteristic
170C2×C4⋊C4C2×D9711yescharacteristic
171C2×C22⋊C4—1941no—
172#102—1942no—
173#102—1942no—
174C22⋊C8—1942no—
175C24—971no—
176C24—1941no—
177C2×D4—972no—
178C2×D4—3882no—
179C2×D4—3882no—
180C2×D4—1944no—
181C2×D4—1941no—
182C2×D4—1941no—
183C2×D4—1941no—
184C2×D4—1941no—
185C2×D4—1941no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—971no—
189C22×C4—972no—
190C22×C4—972no—
191C22×C4—3882no—
192C22×C4—3882no—
193C22×C4—1944no—
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—21no—
208C4⋊C4—972no—
209C4⋊C4—1944no—
210C4⋊C4—1941no—
211C4⋊C4C4×D9712yescomplemented
212C4⋊C4D38812yescomplemented
213C22⋊C4—1944no—
214C23—972no—
215C23—972no—
216C23—972no—
217C23—3881no—
218C23—3881no—
219C23—3881no—
220C23—3881no—
221C23—3881no—
222C23—3881no—
223C23—1942no—
224C23—1941no—
225C23—1941no—
226C23—1941no—
227C23—1941no—
228C23D4×D9711yescharacteristic
229D4—3888no—
230D4—3881no—
231D4—3881no—
232D4—1942no—
233D4—1942no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—972no—
241C2×C4—3888no—
242C2×C4—3881no—
243C2×C4—3881no—
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×D38811yescharacteristic
255C2×C4#1309551111yescharacteristic
256C2×C4#1309551211yescharacteristic
257C8—41no—
258C8—3881no—
259C22—7761no—
260C22—7761no—
261C22—7761no—
262C22—7761no—
263C22—974no—
264C22—3882no—
265C22—3882no—
266C22—3882no—
267C22—1942no—
268C22—1942no—
269C22—1942no—
270C22—1942no—
271C22—1942no—
272C22—1942no—
273C22C22⋊C4×D9711yescharacteristic
274C22#1569078011yescharacteristic
275C22#1569079511yescharacteristic
276C22D8×D9711yescharacteristic
277C22SD16×D9711yescharacteristic
278C22#1569083711yescharacteristic
279C22#1569084211yescharacteristic
280C4—42no—
281C4—42no—
282C4—3882no—
283C4—1942no—
284C4#1569077211yescharacteristic
285C4C4×D38811yescharacteristic
286C4#1569078311yescharacteristic
287C4#1569078511yescharacteristic
288C2—7762no—
289C2—1944no—
290C2#2026061311yescharacteristic
291C2#102×D9711yescharacteristic
292C2#2026073511yescharacteristic
293C2#2026087011yescharacteristic
294C2#2026087511yescharacteristic
295C2#2026105011yescharacteristic
296C2#2026105511yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#340×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690785C2
cosocleC22×C4×C97C23
hypercenter#340D97
hypocenterC97#1874
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×C4⋊C4×D97C2semidirect
3. C2×#15690732C2semidirect
4. C2×#15690730C2semidirect
5. C2×#15690722C2semidirect
6. #20260497C2semidirect
7. #340×C97C2semidirect
14. #15690785C4semidirect
36. C2×D388C2×C4semidirect
52. C4⋊Dic97D4semidirect
63. C4⋊C4×C97C2×C4semidirect
64. C4⋊C4×C97D4semidirect
130. C2×D97#102semidirect
149. C97#1874semidirect
156. #340D97semidirect
211. C4⋊C4C4×D97semidirect
212. C4⋊C4D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156907852
C2×C4⋊C4×D972
C2×#156907322
C2×#156907302
C2×#156907222
#202604972
#340×C972
#187497