Finite Group Databasebeta

← D4×#6296 #13178684

Subgroups — D4×#6296 #13178684

Order 1728.

1456 conjugacy classes of subgroups; only the 263 normal subgroups are shown.

normal subgroups only

This group has 1456 conjugacy classes of subgroups — too many to draw. Only its 263 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×#6296C212yesmaximal, complemented
2C3×D4×C3⋊Dic3C213yesmaximal
3#11723721C216yesmaximal, complemented
4C4⋊#6296C213yesmaximal, complemented
5C4×#6296C211yesmaximal, characteristic, complemented
6D4×#627—33nomaximal
7C2×C33×D4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2216yes—
9C2×#6296—28no—
10C2×#6296C2216yes—
11C2×#6296C2211yescharacteristic
12C3×#3757C2216yes—
13C3×C4⋊(C3⋊Dic3)C2213yes—
14C3×C4×C3⋊Dic3C2213yes—
15#88815C2213yes—
16#88815C2216yes—
17C22×#627—36no—
18D4×C3⋊Dic3D313yes—
19C3×D4×Dic3—36no—
20C3×D4×Dic3—63no—
21#63982—36no—
22#63964—312no—
23#63919—33no—
24#63918—36no—
25C4×#627—33no—
26C23×C33C2312yes—
27C33×D4Q814yescomplemented
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3—212no—
30C2×C3×C3⋊Dic3C2313yes—
31C2×C3×C3⋊Dic3C2316yes—
32C2×C3×C3⋊Dic3C2313yes—
33#6296—212no—
34#6296D414yesdirect product factor
35D4×Dic6—93no—
36C2×C32×D4—23no—
37C2×C32×D4—41no—
38C2×C32×D4C2×D313yes—
39C22×C3⋊Dic3C2×D316yes—
40C22×C3×Dic3—312no—
41C22×C3×Dic3—66no—
42C2×#627—33no—
43C2×#627—312no—
44C2×#627—36no—
45C2×#627—624no—
46#3757C2×D316yes—
47C4⋊(C3⋊Dic3)C2×D313yes—
48C4×C3⋊Dic3C2×D313yes—
49C3×#217—312no—
50C3×#217—66no—
51C3×C4⋊Dic3—36no—
52C3×C4⋊Dic3—63no—
53C3×C4×Dic3—36no—
54C3×C4×Dic3—63no—
55#3724—33no—
56#3724—36no—
57Dic3⋊Dic3—36no—
58Dic3⋊Dic3—312no—
59C22×C33—22no—
60C22×C33—22no—
61C22×C33C2×Q814yescomplemented
62C22×C33C2411yescharacteristic
63C33×C4C2×Q812yescomplemented
64C3×C3⋊Dic3—26no—
65C3×C3⋊Dic3—23no—
66C3×C3⋊Dic3C2×D416yes—
67C22×Dic6—96no—
68C3×C4×D4—93no—
69D4×Dic3—36no—
70D4×Dic3—63no—
71#924—96no—
72#888—96no—
73#878—912no—
74#869—93no—
75C4×Dic6—93no—
76C23×C32—26no—
77C23×C32—42no—
78C23×C32C22×D316yes—
79C32×D4—212no—
80C32×D4—44no—
81C32×D4Dic6112yescomplemented
82C2×C32×C4—23no—
83C2×C32×C4—41no—
84C2×C32×C4C22×D313yes—
85C2×C3⋊Dic3—212no—
86C2×C3⋊Dic3C22×D313yes—
87C2×C3⋊Dic3C22×D316yes—
88C2×C3⋊Dic3C22×D313yes—
89C2×C3×Dic3—36no—
90C2×C3×Dic3—36no—
91C2×C3×Dic3—312no—
92C2×C3×Dic3—63no—
93C2×C3×Dic3—63no—
94C2×C3×Dic3—66no—
95C2×C3×Dic3—624no—
96C2×C3×Dic3—1212no—
97#627—312no—
98#627—624no—
99#627—612no—
100D4×Q8—271noSylow
101C2×C33—24no—
102C2×C33C22×D411yescharacteristic
103C2×C33C22×Q811yescharacteristic
104C2×C332−1+411yescharacteristic
105C2×C3×D4—23no—
106C2×C3×D4—41no—
107C2×C3×D4D3213yes—
108C22×C3×C4—96no—
109C22×Dic3—312no—
110C22×Dic3—66no—
111C2×Dic6—93no—
112C2×Dic6—96no—
113C2×Dic6—912no—
114C2×Dic6—1824no—
115C3×C4⋊C4—93no—
116C3×C22⋊C4—96no—
117C3×C42—93no—
118#217—312no—
119#217—66no—
120C4⋊Dic3—36no—
121C4⋊Dic3—63no—
122C4⋊Dic3—93no—
123C4⋊Dic3—96no—
124Dic3⋊C4—96no—
125Dic3⋊C4—912no—
126C4×Dic3—36no—
127C4×Dic3—63no—
128C22×C32—26no—
129C22×C32—26no—
130C22×C32—23no—
131C22×C32—212no—
132C22×C32—41no—
133C22×C32—44no—
134C22×C32—46no—
135C22×C32—46no—
136C22×C32—82no—
137C22×C32—82no—
138C22×C32C2×Dic6112yescomplemented
139C22×C32C23×D313yes—
140C32×C4—26no—
141C32×C4—42no—
142C32×C4C2×Dic616yescomplemented
143C3⋊Dic3—26no—
144C3⋊Dic3—23no—
145C3⋊Dic3D3×D416yes—
146C3×Dic3—312no—
147C3×Dic3—66no—
148C3×Dic3—612no—
149C3×Dic3—66no—
150C3×Dic3—123no—
151C3×Dic3—126no—
152C22×Q8—272no—
153C4⋊Q8—273no—
154C22⋊Q8—276no—
155C4×Q8—271no—
156C4×D4—93no—
157C33D4×Q811yescharacteristic, complemented, Sylow
158C23×C3—26no—
159C23×C3—42no—
160C23×C3C2×D3216yes—
161C3×D4—212no—
162C3×D4—44no—
163C3×D4#627112yescomplemented
164C2×C3×C4—23no—
165C2×C3×C4—41no—
166C2×C3×C4—93no—
167C2×C3×C4—93no—
168C2×C3×C4—96no—
169C2×C3×C4—1812no—
170C2×C3×C4C2×D3213yes—
171C2×Dic3—36no—
172C2×Dic3—36no—
173C2×Dic3—312no—
174C2×Dic3—63no—
175C2×Dic3—63no—
176C2×Dic3—66no—
177C2×Dic3—624no—
178C2×Dic3—1212no—
179Dic6—912no—
180Dic6—1824no—
181Dic6—1812no—
182C2×C32—212no—
183C2×C32—23no—
184C2×C32—23no—
185C2×C32—23no—
186C2×C32—41no—
187C2×C32—41no—
188C2×C32—41no—
189C2×C32—412no—
190C2×C32—84no—
191C2×C32C22×Dic613yes—
192C2×C32C2×D3×D413yes—
193C2×C32#101013yes—
194C2×Q8—271no—
195C2×Q8—276no—
196C2×Q8—548no—
197C2×D4#111311yescharacteristic
198C22×C4—96no—
199C4⋊C4—93no—
200C4⋊C4—273no—
201C4⋊C4—276no—
202C22⋊C4—96no—
203C42—93no—
204C22×C3—26no—
205C22×C3—26no—
206C22×C3—23no—
207C22×C3—212no—
208C22×C3—46no—
209C22×C3—46no—
210C22×C3—41no—
211C22×C3—44no—
212C22×C3—82no—
213C22×C3—82no—
214C22×C3C2×#627112yescomplemented
215C22×C3C22×D3213yes—
216C3×C4—26no—
217C3×C4—42no—
218C3×C4—96no—
219C3×C4—183no—
220C3×C4—186no—
221C3×C4C2×#62716yescomplemented
222Dic3—312no—
223Dic3—66no—
224Dic3—66no—
225Dic3—612no—
226Dic3—126no—
227Dic3—123no—
228C32—23no—
229C32—41no—
230C32D4×Dic613yescomplemented
231C23C2×#111312yes—
232Q8—274no—
233Q8—5412no—
234D4#629614yesdirect product factor
235C2×C4—93no—
236C2×C4—93no—
237C2×C4—96no—
238C2×C4—1812no—
239C2×C4C2×#111311yescharacteristic
240C2×C3—212no—
241C2×C3—23no—
242C2×C3—23no—
243C2×C3—23no—
244C2×C3—412no—
245C2×C3—41no—
246C2×C3—41no—
247C2×C3—41no—
248C2×C3—84no—
249C2×C3#6430913yes—
250C2×C3D32×D413yes—
251C2×C3C22×#62713yes—
252C22—22no—
253C22—22no—
254C22C2×#629614yescomplemented
255C22C22×#111311yescharacteristic
256C4—96no—
257C4—183no—
258C4—186no—
259C4C2×#629612yescomplemented
260C3—23no—
261C3—41no—
262C3D4×#62713yescomplemented
263C2—24no—
264C2#1172462011yescharacteristic
265C2D4×#111311yescharacteristic
266C2C22×#629611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33D4×Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C22×#6296C2semidirect
3. #11723721C2semidirect
4. C4⋊#6296C2semidirect
5. C4×#6296C2semidirect
27. C33×D4Q8semidirect
34. #6296D4direct (N × H)
61. C22×C33C2×Q8semidirect
63. C33×C4C2×Q8semidirect
81. C32×D4Dic6semidirect
138. C22×C32C2×Dic6semidirect
142. C32×C4C2×Dic6semidirect
157. C33D4×Q8semidirect
163. C3×D4#627semidirect
214. C22×C3C2×#627semidirect
221. C3×C4C2×#627semidirect
230. C32D4×Dic6semidirect
234. D4#6296direct (N × H)
254. C22C2×#6296semidirect
259. C4C2×#6296semidirect
262. C3D4×#627semidirect

Maximal subgroups

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

subgroupindex
C22×#62962
C3×D4×C3⋊Dic32
#117237212
C4⋊#62962
C4×#62962
D4×#6273