Finite Group Databasebeta

← #13178265

Subgroups — #13178265

Order 1728.

1326 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 1326 conjugacy classes of subgroups — too many to draw. Only its 255 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×#6296C211yesmaximal, characteristic
2C3×#64086C213yesmaximal
3#11723715C213yesmaximal
4C4×#6296C212yesmaximal, complemented
5#11723665C216yesmaximal, complemented
6#10594245—33nomaximal
7C22×C3×C3⋊Dic3C2213yes—
8C2×#6296C418yescomplemented
9C2×#6296C2213yes—
10C2×#6296C2211yescharacteristic
11C33×C22⋊C4C2211yescharacteristic
12C3×#3757C2213yes—
13C3×#3750C2216yes—
14C3×C4×C3⋊Dic3C2216yes—
15#88815C2216yes—
16#88813C2216yes—
17C22×#627—33no—
18#64086D313yes—
19C3×#877—36no—
20C3×#877—63no—
21#63957—33no—
22#63955—36no—
23C4×#627—36no—
24#63838—36no—
25#63837—312no—
26C23×C33C2311yescharacteristic
27C2×C33×C4C2312yes—
28C2×C3×C3⋊Dic3C2×C4112yes—
29C2×C3×C3⋊Dic3C2316yes—
30C2×C3×C3⋊Dic3C2313yes—
31C2×C3×C3⋊Dic3C2313yes—
32#6296C2×C4112yes—
33#6296C2×C414yes—
34#5516—93no—
35C22×C3⋊Dic3C2×D313yes—
36C22×C3×Dic3—36no—
37C22×C3×Dic3—63no—
38C2×#627—324no—
39C2×#627—36no—
40C2×#627—33no—
41C2×#627—33no—
42C32×C22⋊C4—23no—
43C32×C22⋊C4—41no—
44C32×C22⋊C4C2×D313yes—
45#3757C2×D313yes—
46#3750C2×D316yes—
47C4×C3⋊Dic3C2×D316yes—
48C3×#217—36no—
49C3×#217—63no—
50C3×Dic3⋊C4—312no—
51C3×Dic3⋊C4—66no—
52C3×C4×Dic3—312no—
53C3×C4×Dic3—66no—
54#3724—36no—
55Dic3⋊Dic3—312no—
56Dic32—36no—
57Dic32—312no—
58C22×C33—22no—
59C22×C33C22×C412yes—
60C22×C33C2411yescharacteristic
61C33×C4—22no—
62C3×C3⋊Dic3—26no—
63C3×C3⋊Dic3C22×C416yes—
64C3×C3⋊Dic3C22×C416yes—
65C22×Dic6—93no—
66C3×#117—93no—
67#926—93no—
68#887—912no—
69#877—36no—
70#877—63no—
71#877—96no—
72C4×Dic6—96no—
73C4×Dic6—96no—
74C23×C32—23no—
75C23×C32—41no—
76C23×C32C22×D313yes—
77C2×C32×C4—26no—
78C2×C32×C4—42no—
79C2×C32×C4C22×D316yes—
80C2×C3⋊Dic3C4×D3112yes—
81C2×C3⋊Dic3C22×D316yes—
82C2×C3⋊Dic3C22×D313yes—
83C2×C3⋊Dic3C22×D313yes—
84C2×C3×Dic3—324no—
85C2×C3×Dic3—36no—
86C2×C3×Dic3—36no—
87C2×C3×Dic3—312no—
88C2×C3×Dic3—63no—
89C2×C3×Dic3—63no—
90C2×C3×Dic3—612no—
91C2×C3×Dic3—66no—
92#627—324no—
93#627—312no—
94#627—312no—
95#519—271noSylow
96C2×C33—22no—
97C2×C33C23×C411yescharacteristic
98C2×C332−1+412yes—
99C22×C3×C4—93no—
100C22×Dic3—36no—
101C22×Dic3—63no—
102C2×Dic6—93no—
103C2×Dic6—93no—
104C2×Dic6—96no—
105C2×Dic6—924no—
106C3×C4⋊C4—96no—
107C3×C22⋊C4—23no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—93no—
110C3×C22⋊C4D3213yes—
111C3×C42—96no—
112#217—36no—
113#217—63no—
114C4⋊Dic3—96no—
115Dic3⋊C4—312no—
116Dic3⋊C4—66no—
117Dic3⋊C4—912no—
118C4×Dic3—312no—
119C4×Dic3—66no—
120C4×Dic3—912no—
121C4×Dic3—96no—
122C22×C32—26no—
123C22×C32—23no—
124C22×C32—26no—
125C22×C32—41no—
126C22×C32—42no—
127C22×C32—46no—
128C22×C32—82no—
129C22×C32C2×C4×D316yes—
130C22×C32C23×D313yes—
131C32×C4—26no—
132C32×C4—46no—
133C32×C4—82no—
134C3⋊Dic3—26no—
135C3⋊Dic3C2×C4×D316yes—
136C3⋊Dic3C2×C4×D316yes—
137C3×Dic3—312no—
138C3×Dic3—312no—
139C3×Dic3—66no—
140C3×Dic3—66no—
141C3×Dic3—612no—
142C3×Dic3—126no—
143C22×Q8—271no—
144C4×Q8—272no—
145C4×Q8—276no—
146#117—93no—
147#117—273no—
148C33#51911yescharacteristic, complemented, Sylow
149C23×C3—23no—
150C23×C3—41no—
151C23×C3C2×D3213yes—
152C2×C3×C4—26no—
153C2×C3×C4—42no—
154C2×C3×C4—93no—
155C2×C3×C4—93no—
156C2×C3×C4—912no—
157C2×C3×C4—96no—
158C2×C3×C4C2×D3216yes—
159C2×Dic3—36no—
160C2×Dic3—36no—
161C2×Dic3—324no—
162C2×Dic3—312no—
163C2×Dic3—612no—
164C2×Dic3—63no—
165C2×Dic3—63no—
166C2×Dic3—66no—
167Dic6—912no—
168Dic6—912no—
169Dic6—924no—
170C2×C32—26no—
171C2×C32—23no—
172C2×C32—26no—
173C2×C32—41no—
174C2×C32—42no—
175C2×C32—46no—
176C2×C32—82no—
177C2×C32C22×C4×D313yes—
178C2×C32#101016yes—
179C2×Q8—271no—
180C2×Q8—273no—
181C2×Q8—278no—
182C22×C4—93no—
183C4⋊C4—96no—
184C4⋊C4—276no—
185C22⋊C4—93no—
186C22⋊C4#111311yescharacteristic
187C42—96no—
188C42—276no—
189C22×C3—26no—
190C22×C3—23no—
191C22×C3—26no—
192C22×C3—46no—
193C22×C3—41no—
194C22×C3—42no—
195C22×C3—82no—
196C22×C3C4×D3216yes—
197C22×C3C22×D3213yes—
198C3×C4—26no—
199C3×C4—46no—
200C3×C4—82no—
201C3×C4—96no—
202C3×C4—96no—
203C3×C4—186no—
204Dic3—312no—
205Dic3—312no—
206Dic3—66no—
207Dic3—66no—
208Dic3—612no—
209Dic3—126no—
210C32—23no—
211C32—41no—
212C32#551613yescomplemented
213C23C2×#111311yescharacteristic
214Q8—274no—
215Q8—2712no—
216C2×C4—93no—
217C2×C4—93no—
218C2×C4—912no—
219C2×C4—96no—
220C2×C4C2×#111312yes—
221C2×C3—26no—
222C2×C3—23no—
223C2×C3—26no—
224C2×C3—46no—
225C2×C3—41no—
226C2×C3—42no—
227C2×C3—82no—
228C2×C3C2×C4×D3213yes—
229C2×C3#6430916yes—
230C22—22no—
231C22C4×#111312yes—
232C22C22×#111311yescharacteristic
233C4—22no—
234C4—96no—
235C4—96no—
236C4—186no—
237C3—23no—
238C3—41no—
239C3#1059424513yescomplemented
240C2—22no—
241C2C2×C4×#111311yescharacteristic
242C2#1172462012yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutatorC2×C33C23×C4
FittingC33×C22⋊C4C22
socle, cosocleC22×C33C24
prime radicalC23×C33C23
hypercenterC22⋊C4#1113
hypocenterC33#519
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C4×#6296C2semidirect
5. #11723665C2semidirect
8. C2×#6296C4semidirect
148. C33#519semidirect
212. C32#5516semidirect
239. C3#10594245semidirect

Maximal subgroups

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

subgroupindex
C22×#62962
C3×#640862
#117237152
C4×#62962
#117236652
#105942453