Finite Group Databasebeta

← C2×#11722789 #13167462

Subgroups — C2×#11722789 #13167462

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×#88812C211yesmaximal, characteristic, complemented
2C2×#88810C211yesmaximal, characteristic
3C2×C3×#3708C211yesmaximal, characteristic, complemented
4#11722789C214yesmaximal, direct product factor
5C2×#63735—31nomaximal
6C2×C3×C4⋊(C3⋊D3)—21no—
7C2×C3×#634—21no—
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9#88812—21no—
10#88812C2212yescomplemented
11#88810—21no—
12#88810C2212yes—
13C3×#3708C2212yescomplemented
14C2×#3719—61no—
15C2×#3718—31no—
16C2×#3714—61no—
17C2×#3713—31no—
18C2×#3708D311yescharacteristic, complemented
19#63735—34no—
20C22×C3×C3⋊D3—41no—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3—41no—
23C3×C4⋊(C3⋊D3)—22no—
24C3×C4⋊(C3⋊D3)—41no—
25C3×#634—22no—
26C3×#634—41no—
27C3×C3⋊(C3⋊C8)D411yescharacteristic
28C3×C3⋊(C3⋊C8)C2311yescharacteristic
29C2×#800—91nomaximal
30C2×C4⋊(C3⋊D3)—21no—
31C2×#634—21no—
32C2×C3×D12—61no—
33C2×C3×D12—61no—
34C2×C3×D12—121no—
35C2×C3×Dic6—61no—
36C2×C3×Dic6—61no—
37C2×C3×Dic6—121no—
38C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
39C2×C3×C3⋊C8—61no—
40C2×C3×C3⋊C8—61no—
41#3719—62no—
42#3719—62no—
43#3718—32no—
44#3718—61no—
45#3714—62no—
46#3714—62no—
47#3713—32no—
48#3713—61no—
49#3708C2×D312yescomplemented
50C22×C33D811yescharacteristic
51C2×C3×C3⋊D3—42no—
52C2×C3×C3⋊D3—82no—
53C33×C4D811yescharacteristic
54C33×C4C2×D411yescharacteristic
55C3×C3⋊Dic3—42no—
56C2×#216—181no—
57C2×#213—181no—
58C2×Dic12—91no—
59C2×D24—91no—
60C2×C3×C16—91no—
61#800—94no—
62C22×C3⋊D3—41no—
63C22×C3×D3—121no—
64C22×C3×D3—122no—
65C22×C3×D3—121no—
66C2×C32×C4—21no—
67C2×C32×C4—21no—
68C2×C32×C4—41no—
69C2×C32×C4—41no—
70C2×C32×C4D1211yescharacteristic
71C2×C3⋊Dic3—41no—
72C4⋊(C3⋊D3)—22no—
73C4⋊(C3⋊D3)—41no—
74#634—22no—
75#634—41no—
76C2×C3×Dic3—121no—
77C2×C3×Dic3—122no—
78C2×C3×Dic3—121no—
79C3×D12—62no—
80C3×D12—62no—
81C3×D12—122no—
82C3×D12—121no—
83C3×D12—122no—
84C3×D12—121no—
85C3×Dic6—62no—
86C3×Dic6—62no—
87C3×Dic6—122no—
88C3×Dic6—121no—
89C3×Dic6—122no—
90C3×Dic6—121no—
91C3⋊(C3⋊C8)D1211yescharacteristic
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—181no—
102C2×C3×D4—181no—
103C2×D12—61no—
104C2×D12—61no—
105C2×D12—121no—
106C2×Dic6—61no—
107C2×Dic6—61no—
108C2×Dic6—121no—
109C2×C3×C8—91no—
110#216—182no—
111#216—182no—
112#213—182no—
113#213—182no—
114C2×C3⋊C8—61no—
115C2×C3⋊C8—61no—
116Dic12—92no—
117Dic12—181no—
118D24—92no—
119D24—181no—
120C3×C16—92no—
121C22×C32—21no—
122C22×C32—21no—
123C22×C32—41no—
124C22×C32—41no—
125C22×C32D2411yescharacteristic
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×C4D2411yescharacteristic
143C32×C4C2×D1211yescharacteristic
144C3⋊Dic3—42no—
145C3×Dic3—122no—
146C3×Dic3—124no—
147C3×Dic3—122no—
148C2×Q16—271no—
149C2×D8—271no—
150SD32—274no—
151C2×C16—91no—
152C33C2×SD3211yescharacteristic, complemented, Sylow
153C23×C3—361no—
154C22×D3—121no—
155C22×D3—122no—
156C22×D3—121no—
157C3×Q8—182no—
158C3×Q8—361no—
159C3×D4—182no—
160C3×D4—361no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—361no—
166C2×C3×C4GO+(4,𝔽2)11yescharacteristic
167C2×Dic3—121no—
168C2×Dic3—122no—
169C2×Dic3—121no—
170D12—62no—
171D12—62no—
172D12—122no—
173D12—121no—
174D12—122no—
175D12—121no—
176Dic6—62no—
177Dic6—62no—
178Dic6—122no—
179Dic6—121no—
180Dic6—122no—
181Dic6—121no—
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#80012yescomplemented
197C2×C32C2×D2411yescharacteristic
198C3⋊D3—82no—
199C3×D3—242no—
200C3×D3—244no—
201C3×D3—242no—
202C2×Q8—181no—
203C2×D4—181no—
204Q16—272no—
205Q16—541no—
206D8—272no—
207D8—541no—
208C2×C8—91no—
209C16—92no—
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—362no—
232C3×C4#377411yescharacteristic
233C3×C4C2×GO+(4,𝔽2)11yescharacteristic
234Dic3—122no—
235Dic3—124no—
236Dic3—122no—
237C32—21no—
238C32—21no—
239C32—41no—
240C32—41no—
241C32C2×#80011yescharacteristic, complemented
242C23—361no—
243Q8—182no—
244Q8—361no—
245D4—182no—
246D4—361no—
247C2×C4—361no—
248C2×C4#157⋊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#8894111yescharacteristic
268C4—362no—
269C4#8894111yescharacteristic
270C4C2×#157⋊D311yescharacteristic
271C3—21no—
272C3—21no—
273C3—41no—
274C3—41no—
275C3C2×#6373511yescharacteristic, complemented
276C2—722no—
277C2#1172278912yesdirect product factor
278C2C2×#8894111yescharacteristic
subgroupG/H
centerC22#88941
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33D8
prime radicalC2×#88810C2
hypercenterC2×C4#157⋊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
1. C2×#88812C2semidirect
3. C2×C3×#3708C2semidirect
4. #11722789C2direct (N × H)
10. #88812C22semidirect
13. C3×#3708C22semidirect
18. C2×#3708D3semidirect
49. #3708C2×D3semidirect
98. C2×C33SD32semidirect
152. C33C2×SD32semidirect
196. C2×C32#800semidirect
241. C32C2×#800semidirect
260. C2×C3#63735semidirect
275. C3C2×#63735semidirect
277. C2#11722789direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#888122
C2×#888102
C2×C3×#37082
#117227892
C2×#637353
C2×#8009