Finite Group Databasebeta

← #27142813

Subgroups — #27142813

Order 12416.

426 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 91 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×#13095498C211yesmaximal, characteristic, complemented
2C22×#13095496C211yesmaximal, characteristic
3C2×#15690734C211yesmaximal, characteristic, complemented
4C2×#15690731C211yesmaximal, characteristic, complemented
5#20260637C211yesmaximal, characteristic, complemented
6#20260490C211yesmaximal, characteristic, complemented
7C2×#124×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095498—22no—
10C2×#13095498—22no—
11C2×#13095498—22no—
12C2×#13095496—22no—
13C2×#13095496—22no—
14C2×#13095496—22no—
15C22×#11715895C2211yescharacteristic
16C22×#11715895C2211yescharacteristic
17C2×C4⋊Dic97C2211yescharacteristic
18C2×#13095488—22no—
19#15690734—22no—
20#15690731—22no—
21C22×Q8×C97C2211yescharacteristic
22C22×D4×C97C2211yescharacteristic
23#124×C97—22no—
24#124×C97—22no—
25C2×C22⋊C4×C97—22no—
26C2×C42×C97C2211yescharacteristic
27C23×D97—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388—21no—
32C2×D388—21no—
33C2×D388D412yes—
34C22×Dic97—21no—
35#13095498—28no—
36#13095496—28no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C2×#11715895—21no—
40C2×#11715895—21no—
41C2×#11715895D412yes—
42C2×#11715895D412yes—
43C4⋊Dic97—21no—
44C4⋊Dic97—21no—
45#13095488—44no—
46C24×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97—21no—
49C2×Q8×C97—21no—
50C2×Q8×C97—21no—
51C2×Q8×C97—21no—
52C2×Q8×C97D412yes—
53C2×D4×C97—21no—
54C2×D4×C97—21no—
55C2×D4×C97—21no—
56C2×D4×C97—21no—
57C2×D4×C97—21no—
58C2×D4×C97D412yes—
59C22×C4×C97—21no—
60C22×C4×C97D412yes—
61C22×C4×C97C2311yescharacteristic
62C22⋊C4×C97—28no—
63C42×C97—21no—
64C42×C97—21no—
65C23×C97—22no—
66C23×C97—41no—
67C23×C97—41no—
68C23×C97—42no—
69C23×C97—42no—
70C23×C97C2×D411yescharacteristic
71C22×D97—22no—
72C22×D97—41no—
73C22×D97—41no—
74C22×D97—42no—
75C22×D97—42no—
76Q8×C97—22no—
77Q8×C97—22no—
78Q8×C97—41no—
79Q8×C97—41no—
80D4×C97—22no—
81D4×C97—22no—
82D4×C97—41no—
83D4×C97—41no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97—41no—
89C2×C4×C97—41no—
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97#4111yescharacteristic
96C2×Dic97—41no—
97C2×Dic97—42no—
98D388—22no—
99D388—22no—
100D388—41no—
101D388—41no—
102#11715895—22no—
103#11715895—22no—
104C22×C97—42no—
105C22×C97—44no—
106C22×C97—82no—
107C22×C97—81no—
108C22×C97—81no—
109C22×C97#12011yescharacteristic
110C22×C97#12111yescharacteristic
111C22×C97#12711yescharacteristic
112C22×C97#13512yes—
113C22×C97AGL(1,ℤ/8)12yes—
114C2×D97—42no—
115C2×D97—44no—
116C2×D97—82no—
117C2×D97—81no—
118C2×D97—81no—
119C4×C97—42no—
120C4×C97—42no—
121C4×C97#12011yescharacteristic
122C4×C97#12011yescharacteristic
123C4×C97#12111yescharacteristic
124C4×C97#12111yescharacteristic
125Dic97—81no—
126C2×C97—82no—
127C2×C97#39211yescharacteristic
128C2×C97#44912yes—
129C2×C97#46312yes—
130C2×C97#49511yescharacteristic
131C2×C97#49611yescharacteristic
132D97—82no—
133#2012—971nomaximal, Sylow
134C97#201211yescharacteristic, complemented, Sylow
135C22×SD16—971no—
136C22×D8—971no—
137C2×#124D9711yescharacteristic, complemented
138C2×#103—971no—
139C2×#102—971no—
140#390—971no—
141#336—971no—
142C22×Q8C2×D9711yescharacteristic
143C22×D4—971no—
144C22×D4C2×D9711yescharacteristic
145C2×SD16—1942no—
146C2×SD16—1942no—
147C2×SD16—1942no—
148C2×D8—1942no—
149C2×D8—1942no—
150C2×D8—1942no—
151C22×C8—971no—
152C22×C8—971no—
153#124—22no—
154#124—22no—
155C2×C4⋊C4—971no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—1942no—
158C2×C42C2×D9711yescharacteristic
159#103—1942no—
160#102—1942no—
161C24—21no—
162C24—1941no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8C22⋊D9712yes—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—972no—
175C2×D4—1941no—
176C2×D4—1941no—
177C2×D4—1941no—
178C2×D4—1941no—
179C2×D4—1941no—
180C2×D4C22⋊D9712yes—
181C22×C4—21no—
182C22×C4—1941no—
183C22×C4C22⋊D9712yes—
184C22×C4C22×D9711yescharacteristic
185SD16—1948no—
186D8—1948no—
187C2×C8—972no—
188C2×C8—972no—
189C2×C8—1941no—
190C2×C8—1941no—
191C2×C8—1941no—
192C2×C8—1941no—
193C4⋊C4—1941no—
194C4⋊C4—1941no—
195C22⋊C4—28no—
196C22⋊C4—3884no—
197C42—21no—
198C42—21no—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—3881no—
205C23—3881no—
206C23—3882no—
207C23—3882no—
208C23—1942no—
209C23C2×C22⋊D9711yescharacteristic
210Q8—22no—
211Q8—22no—
212Q8—41no—
213Q8—41no—
214D4—22no—
215D4—22no—
216D4—41no—
217D4—41no—
218D4—3881no—
219D4—3881no—
220D4—1942no—
221D4—1942no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3881no—
229C2×C4—3882no—
230C2×C4D4×D9711yescharacteristic
231C2×C4D4×D9711yescharacteristic
232C2×C4D4×D9711yescharacteristic
233C2×C4C2×C22⋊D9711yescharacteristic
234C2×C4C2×C22⋊D9711yescharacteristic
235C2×C4#1309551011yescharacteristic
236C8—1942no—
237C8—1942no—
238C22—44no—
239C22—42no—
240C22—81no—
241C22—82no—
242C22—81no—
243C22—7761no—
244C22—7762no—
245C22—7761no—
246C22—3882no—
247C22—3884no—
248C22#1569077911yescharacteristic
249C22#1569078111yescharacteristic
250C22#1569080611yescharacteristic
251C22#1569083012yes—
252C22#1569083512yes—
253C4—42no—
254C4—42no—
255C4—7761no—
256C4#1569077811yescharacteristic
257C4#1569077811yescharacteristic
258C4#1569078211yescharacteristic
259C4#1569078211yescharacteristic
260C2—82no—
261C2—7762no—
262C2#2026064311yescharacteristic
263C2#2026088012yes—
264C2#2026094712yes—
265C2#2026111711yescharacteristic
266C2#2026112511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#124×C97C2
socleC23×C97C2×D4
prime radicalC22×#13095496C2
hypercenterC2×#124D97
hypocenterC97#2012
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×#13095498C2semidirect
3. C2×#15690734C2semidirect
4. C2×#15690731C2semidirect
5. #20260637C2semidirect
6. #20260490C2semidirect
7. C2×#124×C97C2semidirect
134. C97#2012semidirect
137. C2×#124D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954982
C22×#130954962
C2×#156907342
C2×#156907312
#202606372
#202604902
C2×#124×C972
#201297