Finite Group Databasebeta

← #27142611

Subgroups — #27142611

Order 12416.

376 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 99 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×#13095496C211yesmaximal, characteristic
2C2×#15690741C211yesmaximal, characteristic, complemented
3C2×#15690731C211yesmaximal, characteristic, complemented
4C2×#15690730C211yesmaximal, characteristic, complemented
5C2×#15690721C211yesmaximal, characteristic, complemented
6#20260637C211yesmaximal, characteristic, complemented
7#390×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095496—22no—
10C2×#13095496—22no—
11C2×#13095496C414yescomplemented
12C22×#11715895C2211yescharacteristic
13C22×#11715895C2211yescharacteristic
14C2×C4⋊Dic97C2211yescharacteristic
15C2×#13095488—22no—
16#15690741—22no—
17#15690731—22no—
18#15690730—22no—
19#15690721—22no—
20C22×D4×C97C2211yescharacteristic
21C2×C4⋊C4×C97C2211yescharacteristic
22C2×C22⋊C4×C97—22no—
23C2×C42×C97C2211yescharacteristic
24C23×D97—21no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388C2×C412yes—
31C22×Dic97—21no—
32#13095496—28no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C2×#11715895C2×C412yes—
36C2×#11715895D412yes—
37C2×#11715895D412yes—
38C2×#11715895D412yes—
39C4⋊Dic97—21no—
40C4⋊Dic97—21no—
41#13095488—44no—
42C24×C97—21no—
43C2×D4×C97—21no—
44C2×D4×C97—21no—
45C2×D4×C97—21no—
46C2×D4×C97—21no—
47C2×D4×C97—21no—
48C2×D4×C97C2×C412yes—
49C22×C4×C97—21no—
50C22×C4×C97D412yes—
51C22×C4×C97C2311yescharacteristic
52C4⋊C4×C97—21no—
53C4⋊C4×C97—21no—
54C22⋊C4×C97—44no—
55C42×C97—21no—
56C42×C97—21no—
57C23×C97—22no—
58C23×C97—41no—
59C23×C97—41no—
60C23×C97—42no—
61C23×C97—42no—
62C23×C97C2×D411yescharacteristic
63C22×D97—22no—
64C22×D97—41no—
65C22×D97—41no—
66C22×D97—42no—
67C22×D97—42no—
68D4×C97—22no—
69D4×C97—22no—
70D4×C97—41no—
71D4×C97—41no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—41no—
76C2×C4×C97—42no—
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—41no—
84C2×Dic97—42no—
85D388—22no—
86D388—22no—
87D388—41no—
88D388—41no—
89#11715895—22no—
90#11715895C22⋊C414yescomplemented
91C22×C97—42no—
92C22×C97—44no—
93C22×C97—82no—
94C22×C97—81no—
95C22×C97—81no—
96C22×C97C4×D411yescharacteristic
97C22×C97#12111yescharacteristic
98C22×C97#12711yescharacteristic
99C22×C97AGL(1,ℤ/8)12yes—
100C22×C97AGL(1,ℤ/8)12yes—
101C2×D97—42no—
102C2×D97—44no—
103C2×D97—82no—
104C2×D97—81no—
105C2×D97—81no—
106C4×C97—42no—
107C4×C97—81no—
108C4×C97C2×C22⋊C411yescharacteristic
109C4×C97C4×D411yescharacteristic
110C4×C97#12111yescharacteristic
111C4×C97#12411yescharacteristic
112Dic97—81no—
113C2×C97—82no—
114C2×C97#39011yescharacteristic
115C2×C97#44212yes—
116C2×C97#46912yes—
117C2×C97#49611yescharacteristic
118C2×C97#49611yescharacteristic
119D97—82no—
120#1972—971nomaximal, Sylow
121C97#197211yescharacteristic, complemented, Sylow
122C22×D8—971no—
123C2×#106—971no—
124C2×#102—971no—
125C2×#102—971no—
126C2×#97—971no—
127#390—971no—
128#390D9711yescharacteristic, complemented
129C22×D4—971no—
130C22×D4C2×D9711yescharacteristic
131C2×D8—974no—
132C2×D8—1942no—
133C2×D8—1942no—
134C22×C8—971no—
135C22×C8—971no—
136C2×C4⋊C4—971no—
137C2×C4⋊C4C2×D9711yescharacteristic
138C2×C22⋊C4—22no—
139C2×C22⋊C4—1942no—
140C2×C42C2×D9711yescharacteristic
141#106—1942no—
142#102—1942no—
143#102—1942no—
144#97—1942no—
145C24—21no—
146C24—1941no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—972no—
153C2×D4—1941no—
154C2×D4—1941no—
155C2×D4—1941no—
156C2×D4—1941no—
157C2×D4—1941no—
158C2×D4C4×D9712yes—
159C22×C4—21no—
160C22×C4—1941no—
161C22×C4C22⋊D9712yes—
162C22×C4C22×D9711yescharacteristic
163D8—1948no—
164C2×C8—972no—
165C2×C8—972no—
166C2×C8—972no—
167C2×C8—972no—
168C2×C8—1941no—
169C2×C8—1941no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—1941no—
173C4⋊C4—1941no—
174C22⋊C4—44no—
175C22⋊C4—3884no—
176C42—21no—
177C42—21no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—42no—
182C23—42no—
183C23—3881no—
184C23—3881no—
185C23—3882no—
186C23—3882no—
187C23—1942no—
188C23C2×C22⋊D9711yescharacteristic
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193D4—3881no—
194D4—3881no—
195D4—1942no—
196D4—1942no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—3881no—
203C2×C4—3882no—
204C2×C4C2×C4×D9711yescharacteristic
205C2×C4D4×D9711yescharacteristic
206C2×C4D4×D9711yescharacteristic
207C2×C4D4×D9711yescharacteristic
208C2×C4#1309551011yescharacteristic
209C2×C4#1309551111yescharacteristic
210C8—974no—
211C8—1942no—
212C22—42no—
213C22—44no—
214C22—81no—
215C22—81no—
216C22—82no—
217C22—7761no—
218C22—7761no—
219C22—7762no—
220C22—3882no—
221C22—3884no—
222C22C4×C22⋊D9711yescharacteristic
223C22#1569078211yescharacteristic
224C22#1569080611yescharacteristic
225C22#1569083512yes—
226C22#1569083612yes—
227C4—42no—
228C4—81no—
229C4—7761no—
230C4C22⋊C4×D9711yescharacteristic
231C4#1569077011yescharacteristic
232C4#1569078311yescharacteristic
233C4#1569079611yescharacteristic
234C2—82no—
235C2—7762no—
236C2#2026063511yescharacteristic
237C2#2026084512yes—
238C2#2026098612yes—
239C2#2026112211yescharacteristic
240C2#2026112511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#390×C97C2
socleC23×C97C2×D4
prime radicalC22×#13095496C2
cosocleC22×C4×C97C23
hypercenter#390D97
hypocenterC97#1972
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#15690741C2semidirect
3. C2×#15690731C2semidirect
4. C2×#15690730C2semidirect
5. C2×#15690721C2semidirect
6. #20260637C2semidirect
7. #390×C97C2semidirect
11. C2×#13095496C4semidirect
90. #11715895C22⋊C4semidirect
121. C97#1972semidirect
128. #390D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954962
C2×#156907412
C2×#156907312
C2×#156907302
C2×#156907212
#202606372
#390×C972
#197297