Finite Group Databasebeta

← C2×#11722765 #13167094

Subgroups — C2×#11722765 #13167094

Order 1728.

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

normal subgroups only

This group has 456 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, complemented
2C2×C3×#3713C211yesmaximal, characteristic, complemented
3C2×C3⋊#3708C211yesmaximal, characteristic, complemented
4#11722765C214yesmaximal, direct product factor
5C2×#63734—31nomaximal
6C2×C3×C4⋊(C3⋊D3)—21no—
7C2×C32×D12—21no—
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9#88810—21no—
10#88810C2212yescomplemented
11C3×#3713—21no—
12C3×#3713C2212yescomplemented
13C3⋊#3708C2212yescomplemented
14C2×C3×#213—61no—
15C2×#3714—61no—
16C2×#3713—31no—
17C2×#3713D311yescharacteristic, complemented
18C2×#3708—31no—
19#63734—34no—
20C22×C3×C3⋊D3—41no—
21C22×C32×D3—41no—
22C2×C33×C4D411yescharacteristic
23C3×C4⋊(C3⋊D3)—22no—
24C3×C4⋊(C3⋊D3)—41no—
25C32×D12—22no—
26C32×D12—41no—
27C3×C3⋊(C3⋊C8)D411yescharacteristic
28C3×C3⋊(C3⋊C8)C2311yescharacteristic
29C2×#826—91nomaximal
30C2×C32×D4—61no—
31C2×C4⋊(C3⋊D3)—21no—
32C2×C3×D12—21no—
33C2×C3×D12—21no—
34C2×C3×D12—41no—
35C2×C3×D12—61no—
36C2×C3×D12—61no—
37C2×C3×D12—121no—
38C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
39C3×#213—62no—
40C3×#213—62no—
41C2×C3×C3⋊C8—61no—
42C2×C3×C3⋊C8—61no—
43#3714—62no—
44#3714—62no—
45#3713—21no—
46#3713—32no—
47#3713—61no—
48#3713C2×D312yescomplemented
49#3708—32no—
50C22×C33D811yescharacteristic
51C2×C3×C3⋊D3—42no—
52C2×C3×C3⋊D3—82no—
53C2×C32×D3—42no—
54C2×C32×D3—82no—
55C33×C4D811yescharacteristic
56C33×C4C2×D411yescharacteristic
57C2×C3×D8—91no—
58C2×#213—61no—
59C2×#213—181no—
60C2×D24—91no—
61#826—94no—
62C2×C3⋊C16—91no—
63C23×C32—121no—
64C22×C3⋊D3—41no—
65C22×C3×D3—41no—
66C22×C3×D3—42no—
67C22×C3×D3—41no—
68C22×C3×D3—121no—
69C22×C3×D3—122no—
70C22×C3×D3—121no—
71C32×D4—62no—
72C32×D4—121no—
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4—41no—
76C2×C32×C4—41no—
77C2×C32×C4#6711yescharacteristic
78C4⋊(C3⋊D3)—22no—
79C4⋊(C3⋊D3)—41no—
80C3×D12—22no—
81C3×D12—22no—
82C3×D12—42no—
83C3×D12—41no—
84C3×D12—42no—
85C3×D12—41no—
86C3×D12—62no—
87C3×D12—62no—
88C3×D12—122no—
89C3×D12—121no—
90C3×D12—122no—
91C3×D12—121no—
92C3⋊(C3⋊C8)#6711yescharacteristic
93C3⋊(C3⋊C8)C22×D311yescharacteristic
94C3×C3⋊C8—61no—
95C3×C3⋊C8—61no—
96C3×C3⋊C8—61no—
97C3×C3⋊C8—61no—
98C2×D16—271noSylow
99C2×C33D1612yescomplemented
100C2×C33C2×D811yescharacteristic
101C3×C3⋊D3—82no—
102C32×D3—82no—
103C2×C3×D4—61no—
104C2×C3×D4—61no—
105C2×C3×D4—121no—
106C2×C3×D4—181no—
107C2×D12—21no—
108C2×D12—61no—
109C2×D12—61no—
110C2×D12—121no—
111C3×D8—92no—
112C3×D8—181no—
113C2×C3×C8—91no—
114#213—62no—
115#213—62no—
116#213—182no—
117#213—182no—
118C2×C3⋊C8—61no—
119C2×C3⋊C8—61no—
120D24—92no—
121D24—181no—
122C3⋊C16—92no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—122no—
128C22×C32—242no—
129C22×C32#21311yescharacteristic
130C2×C3⋊D3—42no—
131C2×C3⋊D3—82no—
132C2×C3×D3—42no—
133C2×C3×D3—44no—
134C2×C3×D3—42no—
135C2×C3×D3—82no—
136C2×C3×D3—84no—
137C2×C3×D3—82no—
138C2×C3×D3—122no—
139C2×C3×D3—124no—
140C2×C3×D3—122no—
141C2×C3×D3—242no—
142C2×C3×D3—244no—
143C2×C3×D3—242no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—41no—
149C32×C4—41no—
150C32×C4—41no—
151C32×C4—41no—
152C32×C4#21311yescharacteristic
153C32×C4C2×#6711yescharacteristic
154C2×D8—91no—
155C2×D8—271no—
156D16—274no—
157C2×C16—271no—
158C33C2×D1611yescharacteristic, complemented, Sylow
159C23×C3—121no—
160C23×C3—122no—
161C23×C3—121no—
162C23×C3—361no—
163C22×D3—41no—
164C22×D3—121no—
165C22×D3—122no—
166C22×D3—121no—
167C3×D4—62no—
168C3×D4—62no—
169C3×D4—122no—
170C3×D4—121no—
171C3×D4—122no—
172C3×D4—121no—
173C3×D4—182no—
174C3×D4—361no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4GO+(4,𝔽2)11yescharacteristic
180D12—22no—
181D12—41no—
182D12—62no—
183D12—62no—
184D12—122no—
185D12—121no—
186D12—122no—
187D12—121no—
188C3×C8—91no—
189C3×C8—91no—
190C3⋊C8—61no—
191C3⋊C8—61no—
192C3⋊C8—61no—
193C3⋊C8—61no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—22no—
197C2×C32—22no—
198C2×C32—42no—
199C2×C32—42no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—242no—
203C2×C32#82612yescomplemented
204C2×C32C2×#21311yescharacteristic
205C3⋊D3—82no—
206C3×D3—82no—
207C3×D3—84no—
208C3×D3—82no—
209C3×D3—242no—
210C3×D3—244no—
211C3×D3—242no—
212C2×D4—61no—
213C2×D4—181no—
214D8—92no—
215D8—181no—
216D8—272no—
217D8—541no—
218C2×C8—91no—
219C16—272no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—122no—
225C22×C3—124no—
226C22×C3—122no—
227C22×C3—242no—
228C22×C3—244no—
229C22×C3—242no—
230C22×C3—362no—
231C22×C3—722no—
232C22×C3#377411yescharacteristic
233C2×D3—42no—
234C2×D3—82no—
235C2×D3—122no—
236C2×D3—124no—
237C2×D3—122no—
238C2×D3—242no—
239C2×D3—244no—
240C2×D3—242no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—41no—
249C3×C4#377411yescharacteristic
250C3×C4C2×GO+(4,𝔽2)11yescharacteristic
251C32—21no—
252C32—21no—
253C32—41no—
254C32—41no—
255C32C2×#82611yescharacteristic, complemented
256C23—121no—
257C23—361no—
258D4—62no—
259D4—121no—
260D4—182no—
261D4—361no—
262C2×C4D32⋊D311yescharacteristic
263C8—91no—
264C8—91no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—42no—
270C2×C3—42no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—242no—
274C2×C3—244no—
275C2×C3—242no—
276C2×C3—722no—
277C2×C3#6373412yescomplemented
278C2×C3C2×#377411yescharacteristic
279D3—82no—
280D3—242no—
281D3—244no—
282D3—242no—
283C22—122no—
284C22—242no—
285C22—362no—
286C22—722no—
287C22#8893511yescharacteristic
288C4#8893511yescharacteristic
289C4C2×D32⋊D311yescharacteristic
290C3—21no—
291C3—21no—
292C3—41no—
293C3—41no—
294C3C2×#6373411yescharacteristic, complemented
295C2—242no—
296C2—722no—
297C2#1172276512yesdirect product factor
298C2C2×#8893511yescharacteristic
subgroupG/H
centerC22#88935
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33D8
hypercenterC2×C4D32⋊D3
hypocenterC33C2×D16
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
1. C2×#88810C2semidirect
2. C2×C3×#3713C2semidirect
3. C2×C3⋊#3708C2semidirect
4. #11722765C2direct (N × H)
10. #88810C22semidirect
12. C3×#3713C22semidirect
13. C3⋊#3708C22semidirect
17. C2×#3713D3semidirect
48. #3713C2×D3semidirect
99. C2×C33D16semidirect
158. C33C2×D16semidirect
203. C2×C32#826semidirect
255. C32C2×#826semidirect
277. C2×C3#63734semidirect
294. C3C2×#63734semidirect
297. C2#11722765direct (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×#37132
C2×C3⋊#37082
#117227652
C2×#637343
C2×#8269