Finite Group Databasebeta

← #27142080

Subgroups — #27142080

Order 12416.

338 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 103 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×#13095501C211yesmaximal, characteristic
2C2×C4×Dic194C211yesmaximal, characteristic
3C2×#15690734C211yesmaximal, characteristic
4C2×#15690733C211yesmaximal, characteristic
5#20260497C211yesmaximal, characteristic
6#20260490C211yesmaximal, characteristic
7#391×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×#13095501—22no—
10C2×#13095501—22no—
11C2×#13095501C414yescomplemented
12C22×#11715895C2211yescharacteristic
13C22×#11715895C2211yescharacteristic
14C4×Dic194—22no—
15C4×Dic194—22no—
16C2×C4⋊Dic97C2211yescharacteristic
17C2×#13095490—21no—
18C2×C4×Dic97—21no—
19#15690734—22no—
20#15690733—22no—
21C22×Q8×C97C2211yescharacteristic
22C2×C4⋊C4×C97C2211yescharacteristic
23C2×C42×C97C2211yescharacteristic
24#95×C97—21no—
25#95×C97—21no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194C2×C412yes—
30C2×Dic194D412yes—
31C2×Dic194D412yes—
32C22×Dic97—21no—
33C22×Dic97—21no—
34#13095501—28no—
35C2×#11715895—21no—
36C2×#11715895—21no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C2×#11715895—21no—
40C2×#11715895C2×C412yes—
41C4⋊Dic97—21no—
42C4⋊Dic97D412yes—
43#13095490—24no—
44C4×Dic97—24no—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97—21no—
49C2×Q8×C97—21no—
50C2×Q8×C97C2×C412yes—
51C22×C4×C97—21no—
52C22×C4×C97—21no—
53C22×C4×C97D411yescharacteristic
54C22×C4×C97D411yescharacteristic
55C22×C4×C97C2311yescharacteristic
56C4⋊C4×C97—21no—
57C4⋊C4×C97—21no—
58C42×C97—21no—
59C42×C97—21no—
60C23×C97C2×D411yescharacteristic
61Q8×C97—22no—
62Q8×C97—22no—
63Q8×C97—41no—
64Q8×C97—41no—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—22no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97Q1612yescomplemented
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—22no—
84C2×Dic97—22no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—41no—
88C2×Dic97—41no—
89Dic194—22no—
90Dic194—22no—
91Dic194—22no—
92Dic194C22⋊C414yes—
93#11715895—22no—
94#11715895—41no—
95C22×C97C4×D411yescharacteristic
96C22×C97#12011yescharacteristic
97C22×C97#12111yescharacteristic
98C22×C97C2×Q1611yescharacteristic
99C22×C97#13511yescharacteristic
100C22×C97#13711yescharacteristic
101C22×C97#13711yescharacteristic
102C4×C97—22no—
103C4×C97—42no—
104C4×C97—41no—
105C4×C97—81no—
106C4×C97C2×C22⋊C411yescharacteristic
107C4×C97C4×D411yescharacteristic
108C4×C97#12111yescharacteristic
109C4×C97#12311yescharacteristic
110Dic97—24no—
111Dic97—42no—
112C2×C97#38611yescharacteristic
113C2×C97C4×Q1611yescharacteristic
114C2×C97#44111yescharacteristic
115C2×C97#45111yescharacteristic
116C2×C97#45211yescharacteristic
117C2×C97#46211yescharacteristic
118C2×C97#46411yescharacteristic
119#1878—971nomaximal, Sylow
120C97#187811yescharacteristic, complemented, Sylow
121C22×Q16—971no—
122C2×C4×Q8—971no—
123C2×#103—971no—
124C2×#103—971no—
125#391D9711yescharacteristic
126#340—971no—
127#336—971no—
128C22×Q8—971no—
129C22×Q8C2×D9711yescharacteristic
130C2×Q16—974no—
131C2×Q16—1942no—
132C2×Q16—1942no—
133C22×C8—971no—
134C22×C8—971no—
135C4×Q8—1942no—
136C4×Q8—1942no—
137C2×C4⋊C4—971no—
138C2×C4⋊C4—1941no—
139C2×C4⋊C4C2×D9711yescharacteristic
140C2×C42—1941no—
141C2×C42C2×D9711yescharacteristic
142#103—1942no—
143#103—1942no—
144#95—21no—
145#95—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—972no—
152C2×Q8—972no—
153C2×Q8—972no—
154C2×Q8—1941no—
155C2×Q8—1941no—
156C2×Q8—1941no—
157C2×Q8C4×D9712yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1941no—
161C22×C4—1941no—
162C22×C4C22⋊D9711yescharacteristic
163C22×C4C22⋊D9711yescharacteristic
164C22×C4C22×D9711yescharacteristic
165Q16—1948no—
166C2×C8—972no—
167C2×C8—1941no—
168C2×C8—1941no—
169C2×C8—1941no—
170C2×C8—1941no—
171C2×C8—1941no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—972no—
175C4⋊C4—1944no—
176C4⋊C4—1941no—
177C42—21no—
178C42—21no—
179C42—1944no—
180C23C2×C22⋊D9711yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—41no—
184Q8—41no—
185Q8—974no—
186Q8—1942no—
187Q8—1942no—
188Q8—1942no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—3881no—
201C2×C4—3881no—
202C2×C4—1942no—
203C2×C4—1942no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4#1309550112yescomplemented
207C2×C4C2×C4×D9711yescharacteristic
208C2×C4D4×D9711yescharacteristic
209C2×C4D4×D9711yescharacteristic
210C2×C4D4×D9711yescharacteristic
211C2×C4#1309551011yescharacteristic
212C2×C4#1309551111yescharacteristic
213C8—3881no—
214C8—1942no—
215C22C4×C22⋊D9711yescharacteristic
216C22#1569077811yescharacteristic
217C22#1569078111yescharacteristic
218C22C2×#1309550111yescharacteristic
219C22#1569083011yescharacteristic
220C22#1569084011yescharacteristic
221C22#1569084111yescharacteristic
222C4—22no—
223C4—42no—
224C4—41no—
225C4—81no—
226C4—3882no—
227C4—1944no—
228C4C22⋊C4×D9711yescharacteristic
229C4#1569077011yescharacteristic
230C4#1569078311yescharacteristic
231C4#1569079211yescharacteristic
232C2#2026061111yescharacteristic
233C2C4×#1309550111yescharacteristic
234C2#2026084011yescharacteristic
235C2#2026089011yescharacteristic
236C2#2026089511yescharacteristic
237C2#2026094211yescharacteristic
238C2#2026095411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#391×C97C2
socle, prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#391D97
hypocenterC97#1878
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
11. C2×#13095501C4semidirect
76. C2×C4×C97Q16semidirect
120. C97#1878semidirect
206. C2×C4#13095501semidirect

Maximal subgroups

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

subgroupindex
C22×#130955012
C2×C4×Dic1942
C2×#156907342
C2×#156907332
#202604972
#202604902
#391×C972
#187897