Finite Group Databasebeta

← #27142042

Subgroups — #27142042

Order 12416.

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

normal subgroups only

This group has 404 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×#15690790C211yesmaximal, characteristic
2C2×C4⋊C4×D97C211yesmaximal, characteristic
3C2×#15690735C211yesmaximal, characteristic, complemented
4C2×#15690733C211yesmaximal, characteristic, complemented
5C2×#15690722C211yesmaximal, characteristic
6#20260497C211yesmaximal, characteristic, complemented
7#340×C97C211yesmaximal, characteristic, complemented
8C22×Dic194C2211yescharacteristic
9C22×C4×D97—21no—
10C22×C4×D97C2211yescharacteristic
11C22×#11715895C2211yescharacteristic
12#15690790—22no—
13#15690790C414yescomplemented
14C2×C4⋊Dic97—21no—
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095490—21no—
17C4⋊C4×D97—22no—
18C4⋊C4×D97—22no—
19C2×#13095488—21no—
20#15690735—22no—
21#15690733—22no—
22#15690722—22no—
23C22×C8×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25C2×C4⋊C4×C97C2211yescharacteristic
26C23×D97D411yescharacteristic
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×Dic194—21no—
32C2×Dic194C2×C412yescomplemented
33C22×Dic97—21no—
34C22×Dic97—21no—
35C22×Dic97D411yescharacteristic
36C2×C4×D97—21no—
37C2×C4×D97—21no—
38C2×C4×D97—21no—
39C2×C4×D97—21no—
40C2×C4×D97—24no—
41C2×C4×D97—42no—
42C2×C4×D97—42no—
43C2×C4×D97C2×C412yes—
44C2×C4×D97D412yes—
45C2×#11715895—21no—
46C2×#11715895—21no—
47C2×#11715895—21no—
48C4⋊Dic97—21no—
49C4⋊Dic97—24no—
50C4⋊Dic97D412yes—
51#13095490—24no—
52#13095488—24no—
53C22×C4×C97—21no—
54C22×C4×C97—21no—
55C22×C4×C97C2311yescharacteristic
56C2×C8×C97—21no—
57C2×C8×C97—21no—
58C2×C8×C97—21no—
59C4⋊C4×C97—21no—
60C4⋊C4×C97—21no—
61C4⋊C4×C97C2×C412yescomplemented
62C4⋊C4×C97D412yes—
63C23×C97C2×D411yescharacteristic
64C22×D97—21no—
65C22×D97—21no—
66C22×D97—21no—
67C22×D97—21no—
68C22×D97C22⋊C412yes—
69C22×D97SD1612yes—
70C22×D97Q1612yes—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97C22×C411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97#4111yescharacteristic
82C2×C4×C97#4111yescharacteristic
83C2×Dic97—21no—
84C2×Dic97—21no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—41no—
88C2×Dic97—41no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91C2×Dic97C22⋊C412yes—
92C4×D97—22no—
93C4×D97—22no—
94C4×D97—22no—
95C4×D97—22no—
96C4×D97—48no—
97Dic194—22no—
98Dic194—22no—
99Dic194—41no—
100Dic194—41no—
101C8×C97—41no—
102#11715895—41no—
103C22×C97C2×C22⋊C411yescharacteristic
104C22×C97#12011yescharacteristic
105C22×C97#12311yescharacteristic
106C22×C97C2×SD1611yescharacteristic
107C22×C97C2×Q1611yescharacteristic
108C22×C97AGL(1,ℤ/8)11yescharacteristic
109C22×C97#13711yescharacteristic
110C2×D97—22no—
111C2×D97—22no—
112C2×D97—22no—
113C2×D97—22no—
114C2×D97—22no—
115C2×D97—22no—
116C2×D97#10314yescomplemented
117C4×C97—42no—
118C4×C97—42no—
119C4×C97C4×D411yescharacteristic
120C4×C97C4×D411yescharacteristic
121C4×C97#12111yescharacteristic
122C4×C97#12111yescharacteristic
123Dic97—22no—
124Dic97—42no—
125Dic97—42no—
126C2×C97#38611yescharacteristic
127C2×C97C2×#10311yescharacteristic
128C2×C97#41711yescharacteristic
129C2×C97#45011yescharacteristic
130C2×C97#45111yescharacteristic
131C2×C97#48311yescharacteristic
132C2×C97#48411yescharacteristic
133D97—24no—
134#1872—971nomaximal, Sylow
135C97#187211yescharacteristic, complemented, Sylow
136C2×C22⋊Q8—971no—
137C22×C4⋊C4—971no—
138C2×#103—971no—
139C2×#103—971no—
140C2×C22⋊C8—971no—
141#340—971no—
142#340D9711yescharacteristic, complemented
143C22×Q8—971no—
144C23×C4—971no—
145C23×C4—1941no—
146C22×C8—971no—
147C22×C8C2×D9711yescharacteristic
148C22⋊Q8—974no—
149C22⋊Q8—1942no—
150C2×C4⋊C4—971no—
151C2×C4⋊C4—1942no—
152C2×C4⋊C4—1941no—
153C2×C4⋊C4—1941no—
154C2×C4⋊C4—1942no—
155C2×C4⋊C4C2×D9711yescharacteristic
156C2×C4⋊C4C2×D9711yescharacteristic
157C2×C22⋊C4—1941no—
158#103—1942no—
159#103—1942no—
160C22⋊C8—1942no—
161C24—971no—
162C2×Q8—972no—
163C2×Q8—1941no—
164C2×Q8—1941no—
165C2×Q8—1941no—
166C2×Q8—1941no—
167C2×Q8—1941no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—971no—
171C22×C4—972no—
172C22×C4—972no—
173C22×C4—3882no—
174C22×C4—3882no—
175C22×C4—1944no—
176C22×C4—1941no—
177C22×C4—1941no—
178C22×C4—1941no—
179C22×C4—1941no—
180C22×C4—1941no—
181C22×C4—1941no—
182C22×C4C22×D9711yescharacteristic
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—1941no—
187C2×C8—1941no—
188C2×C8—1941no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—972no—
192C4⋊C4—1944no—
193C4⋊C4—1944no—
194C4⋊C4—1941no—
195C4⋊C4C4×D9712yescomplemented
196C4⋊C4D38812yes—
197C22⋊C4—1944no—
198C23—972no—
199C23—972no—
200C23—972no—
201C23—1941no—
202C23—1941no—
203C23—1941no—
204C23—1941no—
205C23D4×D9711yescharacteristic
206Q8—3881no—
207Q8—3881no—
208Q8—1942no—
209Q8—1942no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—972no—
217C2×C4—3888no—
218C2×C4—3881no—
219C2×C4—3881no—
220C2×C4—3881no—
221C2×C4—3881no—
222C2×C4—1942no—
223C2×C4—1942no—
224C2×C4—1941no—
225C2×C4—1941no—
226C2×C4—1942no—
227C2×C4—1942no—
228C2×C4—1942no—
229C2×C4—1942no—
230C2×C4C2×C4×D9711yescharacteristic
231C2×C4D4×D9711yescharacteristic
232C2×C4D4×D9711yescharacteristic
233C2×C4C2×D38811yescharacteristic
234C2×C4#1309551111yescharacteristic
235C2×C4#1309551211yescharacteristic
236C8—41no—
237C8—3881no—
238C22—974no—
239C22—1942no—
240C22—1942no—
241C22—1942no—
242C22—1942no—
243C22—1942no—
244C22—1942no—
245C22C22⋊C4×D9711yescharacteristic
246C22#1569078011yescharacteristic
247C22#1569079511yescharacteristic
248C22SD16×D9711yescharacteristic
249C22Q16×D9711yescharacteristic
250C22#1569083411yescharacteristic
251C22#1569083911yescharacteristic
252C4—42no—
253C4—42no—
254C4—3882no—
255C4—3882no—
256C4—1942no—
257C4#1569077211yescharacteristic
258C4C4×D38811yescharacteristic
259C4#1569078311yescharacteristic
260C4#1569078511yescharacteristic
261C2—1944no—
262C2#2026061311yescharacteristic
263C2#103×D9711yescharacteristic
264C2#2026073311yescharacteristic
265C2#2026088711yescharacteristic
266C2#2026089211yescharacteristic
267C2#2026106711yescharacteristic
268C2#2026107211yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#340×C97C2
socleC23×C97C2×D4
prime radicalC23×D97D4
cosocleC22×C4×C97C23
hypercenter#340D97
hypocenterC97#1872
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
3. C2×#15690735C2semidirect
4. C2×#15690733C2semidirect
6. #20260497C2semidirect
7. #340×C97C2semidirect
13. #15690790C4semidirect
32. C2×Dic194C2×C4semidirect
61. C4⋊C4×C97C2×C4semidirect
116. C2×D97#103semidirect
135. C97#1872semidirect
142. #340D97semidirect
195. C4⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907902
C2×C4⋊C4×D972
C2×#156907352
C2×#156907332
C2×#156907222
#202604972
#340×C972
#187297