Finite Group Databasebeta

← C2×#11722766 #13167098

Subgroups — C2×#11722766 #13167098

Order 1728.

408 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 55 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×#88810C211yesmaximal, characteristic
2C2×C3×#3718C211yesmaximal, characteristic, complemented
3C2×C3⋊#3708C211yesmaximal, characteristic, complemented
4#11722766C214yesmaximal, direct product factor
5C2×#63735—31nomaximal
6C2×C3×C4⋊(C3⋊D3)—21no—
7C2×C32×Dic6—21no—
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9#88810—21no—
10#88810C2212yes—
11C3×#3718—21no—
12C3×#3718C2212yescomplemented
13C3⋊#3708C2212yescomplemented
14C2×C3×#216—61no—
15C2×#3718D311yescharacteristic, complemented
16C2×#3714—61no—
17C2×#3713—31no—
18C2×#3708—31no—
19#63735—34no—
20C22×C3×C3⋊D3—41no—
21C2×C33×C4D411yescharacteristic
22C3×C4⋊(C3⋊D3)—22no—
23C3×C4⋊(C3⋊D3)—41no—
24C2×C32×Dic3—41no—
25C32×Dic6—22no—
26C32×Dic6—41no—
27C3×C3⋊(C3⋊C8)D411yescharacteristic
28C3×C3⋊(C3⋊C8)C2311yescharacteristic
29C2×#828—91nomaximal
30C2×C32×Q8—61no—
31C2×C4⋊(C3⋊D3)—21no—
32C2×C3×D12—61no—
33C2×C3×D12—61no—
34C2×C3×D12—121no—
35C2×C3×Dic6—21no—
36C2×C3×Dic6—21no—
37C2×C3×Dic6—41no—
38C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
39C3×#216—62no—
40C3×#216—62no—
41C2×C3×C3⋊C8—61no—
42C2×C3×C3⋊C8—61no—
43#3718—21no—
44#3718C2×D312yescomplemented
45#3714—62no—
46#3714—62no—
47#3713—32no—
48#3713—61no—
49#3708—32no—
50C22×C33D811yescharacteristic
51C2×C3×C3⋊D3—42no—
52C2×C3×C3⋊D3—82no—
53C33×C4D811yescharacteristic
54C33×C4C2×D411yescharacteristic
55C32×Dic3—42no—
56C2×C3×Q16—91no—
57C2×#216—61no—
58C2×#213—181no—
59C2×D24—91no—
60#828—94no—
61C2×C3⋊C16—91no—
62C22×C3⋊D3—41no—
63C22×C3×D3—121no—
64C22×C3×D3—122no—
65C22×C3×D3—121no—
66C32×Q8—62no—
67C32×Q8—121no—
68C2×C32×C4—21no—
69C2×C32×C4—21no—
70C2×C32×C4—41no—
71C2×C32×C4—41no—
72C2×C32×C4—121no—
73C2×C32×C4#6711yescharacteristic
74C4⋊(C3⋊D3)—22no—
75C4⋊(C3⋊D3)—41no—
76C2×C3×Dic3—41no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—41no—
79C3×D12—62no—
80C3×D12—62no—
81C3×D12—122no—
82C3×D12—121no—
83C3×D12—122no—
84C3×D12—121no—
85C3×Dic6—22no—
86C3×Dic6—22no—
87C3×Dic6—42no—
88C3×Dic6—41no—
89C3×Dic6—42no—
90C3×Dic6—41no—
91C3⋊(C3⋊C8)#6711yescharacteristic
92C3⋊(C3⋊C8)C22×D311yescharacteristic
93C3×C3⋊C8—61no—
94C3×C3⋊C8—61no—
95C3×C3⋊C8—61no—
96C3×C3⋊C8—61no—
97C2×SD32—271noSylow
98C2×C33SD3212yescomplemented
99C2×C33C2×D811yescharacteristic
100C3×C3⋊D3—82no—
101C2×C3×Q8—61no—
102C2×C3×Q8—61no—
103C2×C3×Q8—121no—
104C2×C3×D4—181no—
105C2×D12—61no—
106C2×D12—61no—
107C2×D12—121no—
108C2×Dic6—21no—
109C3×Q16—92no—
110C3×Q16—181no—
111C2×C3×C8—91no—
112#216—62no—
113#216—62no—
114#213—182no—
115#213—182no—
116C2×C3⋊C8—61no—
117C2×C3⋊C8—61no—
118D24—92no—
119D24—181no—
120C3⋊C16—92no—
121C22×C32—21no—
122C22×C32—21no—
123C22×C32—41no—
124C22×C32—41no—
125C22×C32#21311yescharacteristic
126C2×C3⋊D3—42no—
127C2×C3⋊D3—82no—
128C2×C3×D3—122no—
129C2×C3×D3—124no—
130C2×C3×D3—122no—
131C2×C3×D3—242no—
132C2×C3×D3—244no—
133C2×C3×D3—242no—
134C32×C4—21no—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4—41no—
139C32×C4—41no—
140C32×C4—41no—
141C32×C4—41no—
142C32×C4—122no—
143C32×C4#21311yescharacteristic
144C32×C4C2×#6711yescharacteristic
145C3×Dic3—42no—
146C3×Dic3—44no—
147C3×Dic3—42no—
148C2×Q16—91no—
149C2×D8—271no—
150SD32—274no—
151C2×C16—271no—
152C33C2×SD3211yescharacteristic, complemented, Sylow
153C23×C3—361no—
154C22×D3—121no—
155C22×D3—122no—
156C22×D3—121no—
157C3×Q8—62no—
158C3×Q8—62no—
159C3×Q8—122no—
160C3×Q8—121no—
161C3×Q8—122no—
162C3×Q8—121no—
163C3×D4—182no—
164C3×D4—361no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—121no—
170C2×C3×C4—122no—
171C2×C3×C4—121no—
172C2×C3×C4GO+(4,𝔽2)11yescharacteristic
173C2×Dic3—41no—
174D12—62no—
175D12—62no—
176D12—122no—
177D12—121no—
178D12—122no—
179D12—121no—
180Dic6—22no—
181Dic6—41no—
182C3×C8—91no—
183C3×C8—91no—
184C3⋊C8—61no—
185C3⋊C8—61no—
186C3⋊C8—61no—
187C3⋊C8—61no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—42no—
193C2×C32—42no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32#82812yescomplemented
197C2×C32C2×#21311yescharacteristic
198C3⋊D3—82no—
199C3×D3—242no—
200C3×D3—244no—
201C3×D3—242no—
202C2×Q8—61no—
203C2×D4—181no—
204Q16—92no—
205Q16—181no—
206D8—272no—
207D8—541no—
208C2×C8—91no—
209C16—272no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—362no—
215C22×C3—722no—
216C22×C3#377411yescharacteristic
217C2×D3—122no—
218C2×D3—124no—
219C2×D3—122no—
220C2×D3—242no—
221C2×D3—244no—
222C2×D3—242no—
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—41no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4—122no—
232C3×C4—124no—
233C3×C4—122no—
234C3×C4#377411yescharacteristic
235C3×C4C2×GO+(4,𝔽2)11yescharacteristic
236Dic3—42no—
237C32—21no—
238C32—21no—
239C32—41no—
240C32—41no—
241C32C2×#82811yescharacteristic, complemented
242C23—361no—
243Q8—62no—
244Q8—121no—
245D4—182no—
246D4—361no—
247C2×C4—121no—
248C2×C4D32⋊D311yescharacteristic
249C8—91no—
250C8—91no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—41no—
258C2×C3—41no—
259C2×C3—722no—
260C2×C3#6373512yescomplemented
261C2×C3C2×#377411yescharacteristic
262D3—242no—
263D3—244no—
264D3—242no—
265C22—362no—
266C22—722no—
267C22#8893511yescharacteristic
268C4—122no—
269C4#8893511yescharacteristic
270C4C2×D32⋊D311yescharacteristic
271C3—21no—
272C3—21no—
273C3—41no—
274C3—41no—
275C3C2×#6373511yescharacteristic, complemented
276C2—722no—
277C2#1172276612yesdirect product factor
278C2C2×#8893511yescharacteristic
subgroupG/H
centerC22#88935
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33D8
prime radicalC2×#88810C2
hypercenterC2×C4D32⋊D3
hypocenterC33C2×SD32
LCSG ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3718C2semidirect
3. C2×C3⋊#3708C2semidirect
4. #11722766C2direct (N × H)
12. C3×#3718C22semidirect
13. C3⋊#3708C22semidirect
15. C2×#3718D3semidirect
44. #3718C2×D3semidirect
98. C2×C33SD32semidirect
152. C33C2×SD32semidirect
196. C2×C32#828semidirect
241. C32C2×#828semidirect
260. C2×C3#63735semidirect
275. C3C2×#63735semidirect
277. C2#11722766direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#888102
C2×C3×#37182
C2×C3⋊#37082
#117227662
C2×#637353
C2×#8289