Finite Group Databasebeta

← #13172276

Subgroups — #13172276

Order 1728.

1326 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 1326 conjugacy classes of subgroups — too many to draw. Only its 252 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
2#11723595C213yesmaximal
3C3×#63957C213yesmaximal
4#11722945C212yesmaximal, complemented
5#11722944C216yesmaximal, complemented
6#10594245—33nomaximal
7C22×C3×C3⋊Dic3C2213yes—
8C2×#6296C418yescomplemented
9C2×#6296C2213yes—
10C2×#6296C2211yescharacteristic
11#88864C2211yescharacteristic
12C32×#217C2213yes—
13#88807C2216yes—
14Dic3×C3⋊Dic3C2216yes—
15C3×#3724C2216yes—
16C3×Dic32C2216yes—
17C22×#627—33no—
18#64086—33no—
19C3×#877—36no—
20#63957D313yes—
21#63955—36no—
22#63955—63no—
23C4×#627—36no—
24#63838—36no—
25#63837—312no—
26C23×C33C2311yescharacteristic
27C2×C3⋊(C3⋊Dic3)C2312yes—
28C2×C3×C3⋊Dic3C2×C4112yes—
29C2×C3×C3⋊Dic3C2313yes—
30C2×C3×C3⋊Dic3C2313yes—
31C2×C32×Dic3C2316yes—
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—33no—
43#3757—33no—
44#3757—63no—
45#3757—121no—
46#3750—36no—
47C4×C3⋊Dic3—36no—
48C3×#217—23no—
49C3×#217C2×D316yes—
50C3×Dic3⋊C4—312no—
51C3×C4×Dic3—312no—
52#3724C2×D316yes—
53Dic3⋊Dic3—312no—
54Dic3⋊Dic3—66no—
55Dic32—312no—
56Dic32—66no—
57Dic32C2×D316yes—
58C22×C33—22no—
59C22×C33C22×C412yes—
60C22×C33C2411yescharacteristic
61C3⋊(C3⋊Dic3)—22no—
62C3×C3⋊Dic3C22×C416yes—
63C3×C3⋊Dic3C22×C416yes—
64C32×Dic3—26no—
65C22×Dic6—93no—
66C3×#117—93no—
67#926—93no—
68#887—96no—
69#887—96no—
70#877—36no—
71#877—96no—
72#877—183no—
73C4×Dic6—912no—
74C23×C32—23no—
75C23×C32—41no—
76C23×C32C22×D313yes—
77C2×C32×C4—36no—
78C2×C3⋊Dic3—36no—
79C2×C3⋊Dic3—66no—
80C2×C3⋊Dic3—122no—
81C2×C3⋊Dic3C4×D3112yes—
82C2×C3⋊Dic3C22×D313yes—
83C2×C3⋊Dic3C22×D313yes—
84C2×C3×Dic3—26no—
85C2×C3×Dic3—324no—
86C2×C3×Dic3—36no—
87C2×C3×Dic3—36no—
88C2×C3×Dic3—63no—
89C2×C3×Dic3—63no—
90C2×C3×Dic3—612no—
91C2×C3×Dic3C22×D3112yes—
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—96no—
103C2×Dic6—93no—
104C2×Dic6—93no—
105C2×Dic6—924no—
106C3×C4⋊C4—96no—
107C3×C22⋊C4—36no—
108C3×C22⋊C4—63no—
109C3×C42—96no—
110#217—93no—
111#217—183no—
112#217—361no—
113#217D3213yes—
114C4⋊Dic3—96no—
115Dic3⋊C4—312no—
116Dic3⋊C4—912no—
117Dic3⋊C4—186no—
118C4×Dic3—312no—
119C4×Dic3—912no—
120C4×Dic3—96no—
121C4×Dic3—186no—
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—66no—
132C3⋊Dic3—66no—
133C3⋊Dic3—126no—
134C3⋊Dic3—242no—
135C3⋊Dic3C2×C4×D316yes—
136C3⋊Dic3C2×C4×D316yes—
137C3×Dic3—212no—
138C3×Dic3—312no—
139C3×Dic3—312no—
140C3×Dic3—46no—
141C3×Dic3—66no—
142C3×Dic3—66no—
143C22×Q8—271no—
144C4×Q8—276no—
145C4×Q8—272no—
146#117—93no—
147#117—273no—
148C33#51911yescharacteristic, complemented, Sylow
149C23×C3—23no—
150C23×C3—41no—
151C23×C3C2×D3213yes—
152C2×C3×C4—312no—
153C2×C3×C4—66no—
154C2×C3×C4—93no—
155C2×C3×C4—93no—
156C2×C3×C4—912no—
157C2×Dic3—36no—
158C2×Dic3—36no—
159C2×Dic3—324no—
160C2×Dic3—612no—
161C2×Dic3—63no—
162C2×Dic3—63no—
163C2×Dic3—96no—
164C2×Dic3—186no—
165C2×Dic3—362no—
166C2×Dic3C2×D3216yes—
167Dic6—924no—
168Dic6—912no—
169Dic6—912no—
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—273no—
180C2×Q8—271no—
181C2×Q8—278no—
182C22×C4—93no—
183C4⋊C4—96no—
184C4⋊C4—276no—
185C22⋊C4—33no—
186C22⋊C4—271no—
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—612no—
199C3×C4—96no—
200C3×C4—96no—
201C3×C4—126no—
202Dic3—26no—
203Dic3—312no—
204Dic3—312no—
205Dic3—66no—
206Dic3—66no—
207Dic3—186no—
208Dic3—366no—
209Dic3—722no—
210C32—23no—
211C32—41no—
212C32#551613yescomplemented
213C23D3311yescharacteristic
214Q8—2712no—
215Q8—274no—
216C2×C4—36no—
217C2×C4—93no—
218C2×C4—93no—
219C2×C4—912no—
220C2×C4—272no—
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—
231C22#8894912yes—
232C22C2×D3311yescharacteristic
233C4—66no—
234C4—96no—
235C4—96no—
236C4—542no—
237C3—23no—
238C3—41no—
239C3#1059424513yescomplemented
240C2—22no—
241C2C2×#8894911yescharacteristic
242C2#1172442812yes—
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
Fitting, prime radicalC23×C33C23
socle, cosocleC22×C33C24
hypercenterC23D33
hypocenterC33#519
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722945C2semidirect
5. #11722944C2semidirect
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
#117235952
C3×#639572
#117229452
#117229442
#105942453