Finite Group Databasebeta

← C4×#89042 #13178147

Subgroups — C4×#89042 #13178147

Order 1728.

1594 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1594 conjugacy classes of subgroups — too many to draw. Only its 287 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
1C2×#89042C211yesmaximal, characteristic
2C3×C42×C3⋊D3C213yesmaximal, complemented
3C4×#6296C212yesmaximal, complemented
4C4×#6295C216yesmaximal, complemented
5#11723673C213yesmaximal, complemented
6C4×#3798—33nomaximal
7C2×C3×C4×C3⋊D3C2216yes—
8C2×C3×C4×C3⋊D3C2213yes—
9C2×#6296C2211yescharacteristic
10C2×#6295C2213yes—
11#89042C418yesdirect product factor
12C33×C42C2211yescharacteristic
13C3×C4×C3⋊Dic3C2213yes—
14C3×C4×C3⋊Dic3C2216yes—
15#88815C2216yes—
16#88814C2216yes—
17C2×#3798—33no—
18C42×C3⋊D3D313yescomplemented
19C3×C42×D3—36no—
20C3×C42×D3—63no—
21C4×#627—36no—
22C4×#626—312no—
23C4×#625—36no—
24#63874—33no—
25#63855—36no—
26C22×C3×C3⋊D3C2313yes—
27C2×C33×C4C2311yescharacteristic
28C2×C33×C4C2312yes—
29C2×C3×C3⋊Dic3C2316yes—
30C2×C3×C3⋊Dic3C2313yes—
31C3×C4×C3⋊D3—212no—
32C3×C4×C3⋊D3C2×C4112yescomplemented
33#6296C2×C414yescomplemented
34#6295C2×C4112yescomplemented
35C4×#235—93no—
36C2×C4×C3⋊D3C2×D316yes—
37C2×C4×C3⋊D3C2×D313yes—
38C2×C3×C4×D3—36no—
39C2×C3×C4×D3—312no—
40C2×C3×C4×D3—63no—
41C2×C3×C4×D3—66no—
42C2×#627—33no—
43C2×#626—36no—
44C2×#625—33no—
45#3798—324no—
46C32×C42—23no—
47C32×C42—41no—
48C32×C42C2×D313yes—
49C4×C3⋊Dic3C2×D313yes—
50C4×C3⋊Dic3C2×D316yes—
51C3×C4×Dic3—36no—
52C3×C4×Dic3—312no—
53C3×C4×Dic3—66no—
54C3×C4×Dic3—63no—
55#3724—36no—
56Dic3⋊Dic3—312no—
57#3722—36no—
58(C2×D3)⋊Dic3—312no—
59C22×C33C2411yescharacteristic
60C2×C3×C3⋊D3—26no—
61C2×C3×C3⋊D3C22×C416yes—
62C33×C4C22×C412yes—
63C33×C4#4114yescomplemented
64C3×C3⋊Dic3—26no—
65C3×C3⋊Dic3C22×C416yes—
66C2×#235—93no—
67C2×C3×C42—93no—
68C4×#67—912no—
69#926—93no—
70C4×D12—96no—
71#872—96no—
72C42×D3—36no—
73C42×D3—63no—
74C4×Dic6—96no—
75C22×C3⋊D3C22×D313yes—
76C22×C3×D3—36no—
77C22×C3×D3—63no—
78C2×C32×C4—26no—
79C2×C32×C4—23no—
80C2×C32×C4—41no—
81C2×C32×C4—42no—
82C2×C32×C4C22×D313yes—
83C2×C32×C4C22×D316yes—
84C2×C3⋊Dic3C22×D316yes—
85C2×C3⋊Dic3C22×D313yes—
86C4×C3⋊D3—212no—
87C4×C3⋊D3C4×D3112yescomplemented
88C2×C3×Dic3—36no—
89C2×C3×Dic3—312no—
90C2×C3×Dic3—63no—
91C2×C3×Dic3—66no—
92C3×C4×D3—324no—
93C3×C4×D3—612no—
94C3×C4×D3—624no—
95C3×C4×D3—624no—
96#627—312no—
97#626—324no—
98#625—312no—
99C4×#41—271noSylow
100C2×C33C23×C411yescharacteristic
101C2×C33C2×#4112yes—
102C3×C3⋊D3—26no—
103C22×C3×C4—93no—
104C22×C3×C4—96no—
105C2×#67—96no—
106#235—924no—
107C2×D12—93no—
108C2×C4×D3—36no—
109C2×C4×D3—312no—
110C2×C4×D3—63no—
111C2×C4×D3—66no—
112C2×Dic6—93no—
113C3×C42—23no—
114C3×C42—41no—
115C3×C42—96no—
116C3×C42—93no—
117C3×C42D3213yes—
118#217—96no—
119(C2×D3)⋊C4—912no—
120C4⋊Dic3—96no—
121Dic3⋊C4—912no—
122C4×Dic3—312no—
123C4×Dic3—36no—
124C4×Dic3—63no—
125C4×Dic3—66no—
126C22×C32—23no—
127C22×C32—41no—
128C22×C32C23×D313yes—
129C2×C3⋊D3—26no—
130C2×C3⋊D3C2×C4×D316yes—
131C2×C3×D3—312no—
132C2×C3×D3—66no—
133C2×C3×D3—612no—
134C2×C3×D3—612no—
135C32×C4—212no—
136C32×C4—26no—
137C32×C4—42no—
138C32×C4—44no—
139C32×C4C2×C4×D316yes—
140C32×C4#235112yescomplemented
141C3⋊Dic3—26no—
142C3⋊Dic3C2×C4×D316yes—
143C3×Dic3—312no—
144C3×Dic3—66no—
145C3×Dic3—612no—
146C3×Dic3—612no—
147C2×#41—271no—
148C4×Q8—272no—
149C4×D4—276no—
150#117—273no—
151C2×C42—93no—
152C33C4×#4111yescharacteristic, complemented, Sylow
153C23×C3—93no—
154C22×D3—36no—
155C22×D3—63no—
156C2×C3×C4—26no—
157C2×C3×C4—23no—
158C2×C3×C4—42no—
159C2×C3×C4—41no—
160C2×C3×C4—93no—
161C2×C3×C4—912no—
162C2×C3×C4—96no—
163C2×C3×C4—1812no—
164C2×C3×C4C2×D3216yes—
165C2×C3×C4C2×D3213yes—
166#67—924no—
167C2×Dic3—36no—
168C2×Dic3—312no—
169C2×Dic3—63no—
170C2×Dic3—66no—
171D12—912no—
172C4×D3—324no—
173C4×D3—612no—
174C4×D3—624no—
175C4×D3—624no—
176Dic6—912no—
177C2×C32—26no—
178C2×C32—23no—
179C2×C32—41no—
180C2×C32—42no—
181C2×C32C22×C4×D313yes—
182C2×C32C2×#23516yes—
183C3⋊D3—26no—
184C3×D3—612no—
185C3×D3—612no—
186#41—278no—
187C2×Q8—271no—
188C2×D4—273no—
189C22×C4—93no—
190C22×C4—96no—
191C4⋊C4—276no—
192C22⋊C4—276no—
193C42—93no—
194C42—96no—
195C42#111311yescharacteristic
196C22×C3—23no—
197C22×C3—41no—
198C22×C3—96no—
199C22×C3—186no—
200C22×C3C22×D3213yes—
201C2×D3—312no—
202C2×D3—66no—
203C2×D3—612no—
204C2×D3—612no—
205C3×C4—212no—
206C3×C4—26no—
207C3×C4—44no—
208C3×C4—42no—
209C3×C4—96no—
210C3×C4—186no—
211C3×C4#3798112yescomplemented
212C3×C4C4×D3216yes—
213Dic3—312no—
214Dic3—66no—
215Dic3—612no—
216Dic3—612no—
217C32—23no—
218C32—41no—
219C32C4×#23513yescomplemented
220C23—93no—
221Q8—274no—
222D4—2712no—
223C2×C4—93no—
224C2×C4—912no—
225C2×C4—96no—
226C2×C4—1812no—
227C2×C4C2×#111312yes—
228C2×C4C2×#111311yescharacteristic
229C2×C3—26no—
230C2×C3—23no—
231C2×C3—42no—
232C2×C3—41no—
233C2×C3—186no—
234C2×C3C2×#379816yes—
235C2×C3C2×C4×D3213yes—
236D3—612no—
237D3—612no—
238C22—96no—
239C22—186no—
240C22C22×#111311yescharacteristic
241C4—96no—
242C4—186no—
243C4#8904214yesdirect product factor
244C4C4×#111312yes—
245C3—23no—
246C3—41no—
247C3C4×#379813yescomplemented
248C2—186no—
249C2C2×#8904212yes—
250C2C2×C4×#111311yescharacteristic
subgroupG/H
center, hypercenterC42#1113
commutatorC2×C33C23×C4
FrattiniC22C22×#1113
FittingC33×C42C22
socle, cosocleC22×C33C24
prime radicalC2×#89042C2
hypocenterC33C4×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C42×C3⋊D3C2semidirect
3. C4×#6296C2semidirect
4. C4×#6295C2semidirect
5. #11723673C2semidirect
11. #89042C4direct (N × H)
18. C42×C3⋊D3D3semidirect
32. C3×C4×C3⋊D3C2×C4semidirect
33. #6296C2×C4semidirect
34. #6295C2×C4semidirect
63. C33×C4#41semidirect
87. C4×C3⋊D3C4×D3semidirect
140. C32×C4#235semidirect
152. C33C4×#41semidirect
211. C3×C4#3798semidirect
219. C32C4×#235semidirect
243. C4#89042direct (N × H)
247. C3C4×#3798semidirect

Maximal subgroups

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

subgroupindex
C2×#890422
C3×C42×C3⋊D32
C4×#62962
C4×#62952
#117236732
C4×#37983