Finite Group Databasebeta

← C2×#11722768 #13167102

Subgroups — C2×#11722768 #13167102

Order 1728.

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

normal subgroups only

This group has 360 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
2C2×C3×#3718C211yesmaximal, characteristic
3C2×C3⋊#3708C211yesmaximal, characteristic
4#11722768C214yesmaximal, direct product factor
5C2×#63736—31nomaximal
6C2×C3×#634—21no—
7C2×C32×Dic6—21no—
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9#88812—21no—
10#88812C2212yes—
11C3×#3718—21no—
12C3×#3718C2212yes—
13C3⋊#3708C2212yes—
14C2×C3×#216—61no—
15C2×#3719—61no—
16C2×#3718—31no—
17C2×#3718D311yescharacteristic
18C2×#3708—31no—
19#63736—34no—
20C2×C33×C4D411yescharacteristic
21C2×C3×C3⋊Dic3—41no—
22C3×#634—22no—
23C3×#634—41no—
24C2×C32×Dic3—41no—
25C32×Dic6—22no—
26C32×Dic6—41no—
27C3×C3⋊(C3⋊C8)D411yescharacteristic
28C3×C3⋊(C3⋊C8)C2311yescharacteristic
29C2×#829—91nomaximal
30C2×C32×Q8—61no—
31C2×#634—21no—
32C2×C3×Dic6—21no—
33C2×C3×Dic6—21no—
34C2×C3×Dic6—41no—
35C2×C3×Dic6—61no—
36C2×C3×Dic6—61no—
37C2×C3×Dic6—121no—
38C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
39C3×#216—62no—
40C3×#216—62no—
41C2×C3×C3⋊C8—61no—
42C2×C3×C3⋊C8—61no—
43#3719—62no—
44#3719—62no—
45#3718—21no—
46#3718—32no—
47#3718—61no—
48#3718C2×D312yes—
49#3708—32no—
50C22×C33D811yescharacteristic
51C33×C4D811yescharacteristic
52C33×C4C2×D411yescharacteristic
53C3×C3⋊Dic3—42no—
54C32×Dic3—42no—
55C2×C3×Q16—91no—
56C2×#216—61no—
57C2×#216—181no—
58C2×Dic12—91no—
59#829—94no—
60C2×C3⋊C16—91no—
61C32×Q8—62no—
62C32×Q8—121no—
63C2×C32×C4—21no—
64C2×C32×C4—21no—
65C2×C32×C4—41no—
66C2×C32×C4—41no—
67C2×C32×C4—121no—
68C2×C32×C4#6711yescharacteristic
69C2×C3⋊Dic3—41no—
70#634—22no—
71#634—41no—
72C2×C3×Dic3—41no—
73C2×C3×Dic3—42no—
74C2×C3×Dic3—41no—
75C2×C3×Dic3—121no—
76C2×C3×Dic3—122no—
77C2×C3×Dic3—121no—
78C3×Dic6—22no—
79C3×Dic6—22no—
80C3×Dic6—42no—
81C3×Dic6—41no—
82C3×Dic6—42no—
83C3×Dic6—41no—
84C3×Dic6—62no—
85C3×Dic6—62no—
86C3×Dic6—122no—
87C3×Dic6—121no—
88C3×Dic6—122no—
89C3×Dic6—121no—
90C3⋊(C3⋊C8)#6711yescharacteristic
91C3⋊(C3⋊C8)C22×D311yescharacteristic
92C3×C3⋊C8—61no—
93C3×C3⋊C8—61no—
94C3×C3⋊C8—61no—
95C3×C3⋊C8—61no—
96C2×Q32—271noSylow
97C2×C33Q3212yescomplemented
98C2×C33C2×D811yescharacteristic
99C2×C3×Q8—61no—
100C2×C3×Q8—61no—
101C2×C3×Q8—121no—
102C2×C3×Q8—181no—
103C2×Dic6—21no—
104C2×Dic6—61no—
105C2×Dic6—61no—
106C2×Dic6—121no—
107C3×Q16—92no—
108C3×Q16—181no—
109C2×C3×C8—91no—
110#216—62no—
111#216—62no—
112#216—182no—
113#216—182no—
114C2×C3⋊C8—61no—
115C2×C3⋊C8—61no—
116Dic12—92no—
117Dic12—181no—
118C3⋊C16—92no—
119C22×C32—21no—
120C22×C32—21no—
121C22×C32—41no—
122C22×C32—41no—
123C22×C32#21311yescharacteristic
124C32×C4—21no—
125C32×C4—21no—
126C32×C4—21no—
127C32×C4—21no—
128C32×C4—41no—
129C32×C4—41no—
130C32×C4—41no—
131C32×C4—41no—
132C32×C4—122no—
133C32×C4#21311yescharacteristic
134C32×C4C2×#6711yescharacteristic
135C3⋊Dic3—42no—
136C3×Dic3—42no—
137C3×Dic3—44no—
138C3×Dic3—42no—
139C3×Dic3—122no—
140C3×Dic3—124no—
141C3×Dic3—122no—
142C2×Q16—91no—
143C2×Q16—271no—
144Q32—274no—
145C2×C16—271no—
146C33C2×Q3211yescharacteristic, complemented, Sylow
147C3×Q8—62no—
148C3×Q8—62no—
149C3×Q8—122no—
150C3×Q8—121no—
151C3×Q8—122no—
152C3×Q8—121no—
153C3×Q8—182no—
154C3×Q8—361no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—121no—
160C2×C3×C4—122no—
161C2×C3×C4—121no—
162C2×C3×C4—361no—
163C2×C3×C4GO+(4,𝔽2)11yescharacteristic
164C2×Dic3—41no—
165C2×Dic3—121no—
166C2×Dic3—122no—
167C2×Dic3—121no—
168Dic6—22no—
169Dic6—41no—
170Dic6—62no—
171Dic6—62no—
172Dic6—122no—
173Dic6—121no—
174Dic6—122no—
175Dic6—121no—
176C3×C8—91no—
177C3×C8—91no—
178C3⋊C8—61no—
179C3⋊C8—61no—
180C3⋊C8—61no—
181C3⋊C8—61no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—22no—
185C2×C32—22no—
186C2×C32—42no—
187C2×C32—42no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32#82912yescomplemented
191C2×C32C2×#21311yescharacteristic
192C2×Q8—61no—
193C2×Q8—181no—
194Q16—92no—
195Q16—181no—
196Q16—272no—
197Q16—541no—
198C2×C8—91no—
199C16—272no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3#377411yescharacteristic
205C3×C4—21no—
206C3×C4—21no—
207C3×C4—21no—
208C3×C4—21no—
209C3×C4—41no—
210C3×C4—41no—
211C3×C4—41no—
212C3×C4—41no—
213C3×C4—122no—
214C3×C4—124no—
215C3×C4—122no—
216C3×C4—362no—
217C3×C4#377411yescharacteristic
218C3×C4C2×GO+(4,𝔽2)11yescharacteristic
219Dic3—42no—
220Dic3—122no—
221Dic3—124no—
222Dic3—122no—
223C32—21no—
224C32—21no—
225C32—41no—
226C32—41no—
227C32C2×#82911yescharacteristic, complemented
228Q8—62no—
229Q8—121no—
230Q8—182no—
231Q8—361no—
232C2×C4—121no—
233C2×C4—361no—
234C2×C4D32⋊D311yescharacteristic
235C8—91no—
236C8—91no—
237C2×C3—22no—
238C2×C3—22no—
239C2×C3—21no—
240C2×C3—21no—
241C2×C3—42no—
242C2×C3—42no—
243C2×C3—41no—
244C2×C3—41no—
245C2×C3#6373612yescomplemented
246C2×C3C2×#377411yescharacteristic
247C22#8893511yescharacteristic
248C4—122no—
249C4—362no—
250C4#8893511yescharacteristic
251C4C2×D32⋊D311yescharacteristic
252C3—21no—
253C3—21no—
254C3—41no—
255C3—41no—
256C3C2×#6373611yescharacteristic, complemented
257C2#1172276812yesdirect product factor
258C2C2×#8893511yescharacteristic
subgroupG/H
centerC22#88935
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socle, prime radicalC22×C33D8
hypercenterC2×C4D32⋊D3
hypocenterC33C2×Q32
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
4. #11722768C2direct (N × H)
97. C2×C33Q32semidirect
146. C33C2×Q32semidirect
190. C2×C32#829semidirect
227. C32C2×#829semidirect
245. C2×C3#63736semidirect
256. C3C2×#63736semidirect
257. C2#11722768direct (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×C3×#37182
C2×C3⋊#37082
#117227682
C2×#637363
C2×#8299