Finite Group Databasebeta

← #27143049

Subgroups — #27143049

Order 12416.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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×#15690739C211yesmaximal, characteristic, complemented
2C2×#15690734C211yesmaximal, characteristic, complemented
3C2×#15690731C211yesmaximal, characteristic
4#20260642C211yesmaximal, characteristic, complemented
5#20260638C211yesmaximal, characteristic
6#20260498C211yesmaximal, characteristic, complemented
7C22×SD16×C97C211yesmaximal, characteristic
8C22×#11715895C2211yescharacteristic
9C2×C4⋊Dic97C2211yescharacteristic
10C2×C4⋊Dic97C2211yescharacteristic
11C2×C22⋊Dic97—21no—
12C2×C22⋊Dic97—21no—
13C2×C4×Dic97C2211yescharacteristic
14#15690739—22no—
15#15690734—22no—
16#15690731—22no—
17#15690717—21no—
18#15690717—21no—
19C22×Q8×C97C2211yescharacteristic
20C22×D4×C97C2211yescharacteristic
21C2×SD16×C97—22no—
22C2×SD16×C97—22no—
23C2×SD16×C97—22no—
24C22×C8×C97C2211yescharacteristic
25C22×Dic97—21no—
26C22×Dic97—21no—
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29C2×#11715895—21no—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C22⋊Dic97—42no—
37C22⋊Dic97—42no—
38C4×Dic97—21no—
39C4×Dic97—21no—
40C24×C97—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97D412yes—
47C2×D4×C97—21no—
48C2×D4×C97—21no—
49C2×D4×C97—21no—
50C2×D4×C97—21no—
51C2×D4×C97—21no—
52C2×D4×C97D412yes—
53C22×C4×C97—21no—
54C22×C4×C97C2311yescharacteristic
55SD16×C97—28no—
56C2×C8×C97—21no—
57C2×C8×C97—21no—
58C2×C8×C97D412yes—
59C23×C97—22no—
60C23×C97—41no—
61C23×C97—41no—
62C23×C97—41no—
63C23×C97—41no—
64C23×C97—41no—
65C23×C97—41no—
66C23×C97C2×D411yescharacteristic
67Q8×C97—22no—
68Q8×C97—22no—
69Q8×C97—41no—
70Q8×C97—41no—
71D4×C97—22no—
72D4×C97—22no—
73D4×C97—41no—
74D4×C97—41no—
75C2×C4×C97—22no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97#4111yescharacteristic
82C2×C4×C97#4111yescharacteristic
83C2×C4×C97#4111yescharacteristic
84C2×Dic97—21no—
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—21no—
88C2×Dic97—21no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91C2×Dic97—41no—
92C2×Dic97—41no—
93C2×Dic97—41no—
94C2×Dic97—41no—
95C2×Dic97SD1612yescomplemented
96C8×C97—22no—
97#11715895—41no—
98C22×C97—42no—
99C22×C97—42no—
100C22×C97—42no—
101C22×C97—81no—
102C22×C97—81no—
103C22×C97—81no—
104C22×C97—81no—
105C22×C97#12111yescharacteristic
106C22×C97#12111yescharacteristic
107C22×C97#12111yescharacteristic
108C22×C97C2×SD1611yescharacteristic
109C22×C97#13511yescharacteristic
110C22×C97AGL(1,ℤ/8)11yescharacteristic
111C22×C97#13711yescharacteristic
112C4×C97—42no—
113C4×C97#12011yescharacteristic
114C4×C97#12311yescharacteristic
115C4×C97#12311yescharacteristic
116C4×C97#12411yescharacteristic
117Dic97—22no—
118Dic97—41no—
119Dic97—81no—
120Dic97—81no—
121C2×C97—82no—
122C2×C97#39411yescharacteristic
123C2×C97#46011yescharacteristic
124C2×C97#46111yescharacteristic
125C2×C97#46311yescharacteristic
126C2×C97#46411yescharacteristic
127C2×C97#46511yescharacteristic
128C2×C97#46811yescharacteristic
129#2055—971nomaximal, Sylow
130C97#205511yescharacteristic, complemented, Sylow
131C22×SD16D9711yescharacteristic
132C2×C4⋊C8—971no—
133C2×#103—971no—
134C2×#102—971no—
135#391—971no—
136#390—971no—
137#340—971no—
138C22×Q8C2×D9711yescharacteristic
139C22×D4C2×D9711yescharacteristic
140C2×SD16—22no—
141C2×SD16—22no—
142C2×SD16—22no—
143C22×C8—971no—
144C22×C8C2×D9711yescharacteristic
145C2×C4⋊C4—971no—
146C2×C4⋊C4—971no—
147C2×C22⋊C4—1941no—
148C2×C22⋊C4—1941no—
149C2×C42—971no—
150C4⋊C8—1942no—
151#103—1942no—
152#102—1942no—
153#95—1941no—
154#95—1941no—
155C24—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8C22⋊D9712yes—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4C22⋊D9712yes—
168C22×C4—21no—
169C22×C4—971no—
170C22×C4—971no—
171C22×C4—1941no—
172C22×C4—1941no—
173C22×C4C22×D9711yescharacteristic
174SD16—28no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—1941no—
178C2×C8—1941no—
179C2×C8—1941no—
180C2×C8C22⋊D9712yes—
181C4⋊C4—1941no—
182C4⋊C4—1941no—
183C4⋊C4—1941no—
184C4⋊C4—1941no—
185C22⋊C4—3882no—
186C22⋊C4—3882no—
187C42—1941no—
188C42—1941no—
189C23—22no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23D4×D9711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201D4—22no—
202D4—22no—
203D4—41no—
204D4—41no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—972no—
209C2×C4—3881no—
210C2×C4—3881no—
211C2×C4—3881no—
212C2×C4—3881no—
213C2×C4—3881no—
214C2×C4—3881no—
215C2×C4—1941no—
216C2×C4—1941no—
217C2×C4—1941no—
218C2×C4—1941no—
219C2×C4—1941no—
220C2×C4C2×C22⋊D9711yescharacteristic
221C2×C4C2×C22⋊D9711yescharacteristic
222C2×C4C2×C22⋊D9711yescharacteristic
223C2×C4#1309551011yescharacteristic
224C2×C4#1309551011yescharacteristic
225C2×C4#1309551011yescharacteristic
226C8—22no—
227C8—3881no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22#1569078111yescharacteristic
236C22#1569078211yescharacteristic
237C22#1569078211yescharacteristic
238C22SD16×D9711yescharacteristic
239C22#1569083111yescharacteristic
240C22#1569083711yescharacteristic
241C22#1569083911yescharacteristic
242C4—42no—
243C4—7761no—
244C4—7761no—
245C4—3881no—
246C4—1942no—
247C4#1569077911yescharacteristic
248C4#1569079111yescharacteristic
249C4#1569079111yescharacteristic
250C4#1569079711yescharacteristic
251C2—82no—
252C2#2026064911yescharacteristic
253C2#2026093311yescharacteristic
254C2#2026093811yescharacteristic
255C2#2026095211yescharacteristic
256C2#2026095511yescharacteristic
257C2#2026096111yescharacteristic
258C2#2026097811yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC22×SD16×C97C2
socleC23×C97C2×D4
prime radicalC22×D4×C97C22
hypercenterC22×SD16D97
hypocenterC97#2055
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15690739C2semidirect
2. C2×#15690734C2semidirect
4. #20260642C2semidirect
6. #20260498C2semidirect
95. C2×Dic97SD16semidirect
130. C97#2055semidirect

Maximal subgroups

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

subgroupindex
C2×#156907392
C2×#156907342
C2×#156907312
#202606422
#202606382
#202604982
C22×SD16×C972
#205597