Finite Group Databasebeta

← C2×#11723105 #13173404

Subgroups — C2×#11723105 #13173404

Order 1728.

712 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 93 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×C3×#634C211yesmaximal, characteristic
2C2×#88993C212yesmaximal
3#11723105C214yesmaximal, direct product factor
4C2×#63780—31nomaximal
5C22×C33×C4C411yescharacteristic
6C22×C3×C3⋊Dic3C411yescharacteristic
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3×#634—22no—
9C2×C3×#634—22no—
10C2×C3×#634C412yes—
11C2×C3×#634C2212yes—
12#88993—22no—
13#88993C2214yes—
14C2×C3⋊#622—22no—
15C22×#634D311yescharacteristic
16C22×C3×Dic6—62no—
17C2×#3792—32no—
18#63780—34no—
19C23×C33C2×C411yescharacteristic
20C2×C33×C4—21no—
21C2×C33×C4—21no—
22C2×C33×C4C2×C412yes—
23C2×C3×C3⋊Dic3—21no—
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3C2×C412yes—
26C2×C3×C3⋊Dic3C2×C411yescharacteristic
27C2×C3×C3⋊Dic3D412yes—
28C2×C3×C3⋊Dic3D412yes—
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×#634—24no—
31C3×#634—24no—
32C3⋊#622—24no—
33C2×#836—91nomaximal
34C22×C32×C4—22no—
35C22×C32×C4—42no—
36C22×C32×C4Dic311yescharacteristic
37C22×C3⋊Dic3Dic311yescharacteristic
38C22×C3⋊Dic3C2×D311yescharacteristic
39C2×#634—22no—
40C2×#634—22no—
41C2×#634Dic312yes—
42C2×#634C2×D312yes—
43C22×C3×Dic3—62no—
44C22×C3×Dic3—62no—
45C2×C3×Dic6—64no—
46C2×C3×Dic6—64no—
47C2×C3×Dic6—68no—
48C2×C3×Dic6—68no—
49#3792—34no—
50#3792—62no—
51C2×#622—62no—
52C22×C33—22no—
53C22×C33C22⋊C411yescharacteristic
54C22×C33C22⋊C411yescharacteristic
55C22×C33C22×C411yescharacteristic
56C33×C4—22no—
57C3×C3⋊Dic3—22no—
58C3×C3⋊Dic3C22⋊C412yes—
59C3×C3⋊Dic3C2×D412yes—
60C22×C3×Q8—91no—
61C22×Dic6—62no—
62C2×#208—92no—
63#836—94no—
64C23×C32—22no—
65C23×C32—42no—
66C23×C32C2×Dic311yescharacteristic
67C2×C32×C4—21no—
68C2×C32×C4—21no—
69C2×C32×C4—24no—
70C2×C32×C4—24no—
71C2×C32×C4—24no—
72C2×C32×C4—44no—
73C2×C32×C4—44no—
74C2×C32×C4—44no—
75C2×C32×C4C2×Dic312yes—
76C2×C3⋊Dic3—21no—
77C2×C3⋊Dic3—21no—
78C2×C3⋊Dic3C2×Dic312yes—
79C2×C3⋊Dic3C2×Dic311yescharacteristic
80C2×C3⋊Dic3#6712yes—
81C2×C3⋊Dic3#6712yes—
82C2×C3⋊Dic3C22×D311yescharacteristic
83#634—24no—
84#634—24no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—64no—
87C2×C3×Dic3—64no—
88C2×C3×Dic3—64no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—64no—
91C2×C3×Dic3—64no—
92C3×Dic6—616no—
93C3×Dic6—616no—
94#622—64no—
95C2×#101—271noSylow
96C2×C33—21no—
97C2×C33—21no—
98C2×C33#10112yescomplemented
99C2×C33C2×C22⋊C411yescharacteristic
100C2×C3×Q8—92no—
101C2×C3×Q8—92no—
102C2×C3×Q8—182no—
103C2×C3×Q8—182no—
104C22×C3×C4—22no—
105C22×C3×C4—42no—
106C22×C3×C4—91no—
107C22×C3×C4—91no—
108C22×C3×C4#15711yescharacteristic
109C22×Dic3—62no—
110C22×Dic3—62no—
111C2×Dic6—64no—
112C2×Dic6—64no—
113C2×Dic6—68no—
114C2×Dic6—68no—
115#208—94no—
116#208—182no—
117C2×C3⋊C8—182no—
118C22×C32—22no—
119C22×C32—28no—
120C22×C32—22no—
121C22×C32—22no—
122C22×C32—22no—
123C22×C32—48no—
124C22×C32—42no—
125C22×C32—42no—
126C22×C32—42no—
127C22×C32#21711yescharacteristic
128C22×C32#21711yescharacteristic
129C22×C32C22×Dic311yescharacteristic
130C32×C4—22no—
131C32×C4—28no—
132C32×C4—48no—
133C3⋊Dic3—22no—
134C3⋊Dic3#21712yes—
135C3⋊Dic3C2×#6712yes—
136C3×Dic3—64no—
137C3×Dic3—64no—
138C3×Dic3—68no—
139C22×Q8—91no—
140C2×M16—272no—
141#101—274no—
142C33C2×#10111yescharacteristic, complemented, Sylow
143C23×C3—22no—
144C23×C3—42no—
145C23×C3C2×#15711yescharacteristic
146C3×Q8—184no—
147C3×Q8—184no—
148C2×C3×C4—24no—
149C2×C3×C4—24no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—24no—
153C2×C3×C4—44no—
154C2×C3×C4—44no—
155C2×C3×C4—44no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158C2×C3×C4—92no—
159C2×C3×C4—92no—
160C2×C3×C4—92no—
161C2×C3×C4—181no—
162C2×C3×C4—181no—
163C2×C3×C4C2×#15712yes—
164C2×Dic3—62no—
165C2×Dic3—64no—
166C2×Dic3—64no—
167C2×Dic3—64no—
168C2×Dic3—62no—
169C2×Dic3—64no—
170C2×Dic3—64no—
171Dic6—616no—
172Dic6—616no—
173C3⋊C8—184no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—24no—
177C2×C32—24no—
178C2×C32—24no—
179C2×C32—22no—
180C2×C32—44no—
181C2×C32—44no—
182C2×C32—44no—
183C2×C32—42no—
184C2×C32#83612yescomplemented
185C2×C32C2×#21711yescharacteristic
186C2×Q8—92no—
187C2×Q8—92no—
188C2×Q8—182no—
189C2×Q8—182no—
190C22×C4—91no—
191C22×C4—91no—
192C22×C4C3⋊#15711yescharacteristic
193M16—274no—
194M16—542no—
195C2×C8—542no—
196C22×C3—28no—
197C22×C3—22no—
198C22×C3—22no—
199C22×C3—22no—
200C22×C3—22no—
201C22×C3—48no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3C22⋊#15711yescharacteristic
206C22×C3C22⋊#15711yescharacteristic
207C22×C3C22×#15711yescharacteristic
208C3×C4—28no—
209C3×C4—22no—
210C3×C4—48no—
211C3×C4—92no—
212C3×C4—92no—
213C3×C4—182no—
214Dic3—64no—
215Dic3—64no—
216Dic3—68no—
217C32—22no—
218C32—42no—
219C32C2×#83611yescharacteristic, complemented
220C23C2×C3⋊#15711yescharacteristic
221Q8—184no—
222Q8—184no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—91no—
226C2×C4—91no—
227C2×C4—92no—
228C2×C4—92no—
229C2×C4—92no—
230C2×C4—181no—
231C2×C4—181no—
232C2×C4C2×C3⋊#15712yes—
233C8—544no—
234C2×C3—24no—
235C2×C3—24no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—24no—
239C2×C3—22no—
240C2×C3—44no—
241C2×C3—44no—
242C2×C3—44no—
243C2×C3—42no—
244C2×C3#6378012yescomplemented
245C2×C3C2×C22⋊#15711yescharacteristic
246C22—22no—
247C22#8899411yescharacteristic
248C22#8899411yescharacteristic
249C22C22×C3⋊#15711yescharacteristic
250C4—22no—
251C4—92no—
252C4—92no—
253C4—182no—
254C3—22no—
255C3—42no—
256C3C2×#6378011yescharacteristic, complemented
257C2—21no—
258C2—21no—
259C2#1172310512yesdirect product factor
260C2C2×#8899411yescharacteristic
subgroupG/H
centerC22#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socleC22×C33C22⋊C4
prime radicalC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33C2×#101
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11723105C2direct (N × H)
98. C2×C33#101semidirect
142. C33C2×#101semidirect
184. C2×C32#836semidirect
219. C32C2×#836semidirect
244. C2×C3#63780semidirect
256. C3C2×#63780semidirect
259. C2#11723105direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C2×#889932
#117231052
C2×#637803
C2×#8369