Finite Group Databasebeta

← C2×#11724335 #13181130

Subgroups — C2×#11724335 #13181130

Order 1728.

968 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 175 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×#89040C212yesmaximal
2C2×C3×C4×#157C211yesmaximal, characteristic
3C2×#88940C214yesmaximal
4#11724335C218yesmaximal, direct product factor
5C2×#64222—31nomaximal
6C22×C3×#157C2212yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#634—22no—
9C2×#6296—22no—
10C2×#6294C2214yes—
11#89040—24no—
12#89040C2218yes—
13C3×C4×#157C2214yes—
14#88940C22116yes—
15C2×#3797—32no—
16C2×D3×Dic6—62no—
17C2×C4×#157D311yescharacteristic
18C2×#3773—34no—
19#64222—38no—
20C22×C3×C3⋊D3C2311yescharacteristic
21C2×C3×#157Q818yescomplemented
22C2×C3×#157C2314yes—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3—24no—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C3×C4×C3⋊D3D412yes—
27C3×C4×C3⋊D3C2312yes—
28C3×#634—28no—
29#6296—28no—
30#6294—24no—
31#6294C2318yes—
32C2×#869—91nomaximal
33C22×#157C2×D312yes—
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×#634—22no—
36C2×C3×C4×D3—62no—
37C2×C3×Dic6—62no—
38C2×C3×Dic6—62no—
39C2×C3×Dic6—122no—
40C2×#627—64no—
41C2×#627—62no—
42C2×#624—34no—
43C2×D3×Dic3—64no—
44#3797—38no—
45#3797—64no—
46D3×Dic6—68no—
47D3×Dic6—68no—
48C4×#157C2×D314yes—
49#3773—316no—
50C22×C33C2×D411yescharacteristic
51C2×C3×C3⋊D3C2×D411yescharacteristic
52C2×C3×C3⋊D3C2×Q814yes—
53C2×C3×C3⋊D3C2411yescharacteristic
54C3×#157C2×Q818yescomplemented
55C33×C4C2×D412yes—
56C3×C3⋊Dic3—28no—
57C3×C3⋊Dic3C2×D412yes—
58C2×D3×Q8—182no—
59C22×Dic6—92no—
60C2×C3×C42—91no—
61C2×C4⋊Dic3—94no—
62#869—98no—
63C22×C3⋊D3C22×D311yescharacteristic
64C22×C3×D3—62no—
65C2×#157Dic618yescomplemented
66C2×#157C22×D314yes—
67C2×C32×C4—22no—
68C2×C32×C4—42no—
69C2×C32×C4D1211yescharacteristic
70C2×C3⋊Dic3—24no—
71C2×C3⋊Dic3D1211yescharacteristic
72C4×C3⋊D3D1212yes—
73C4×C3⋊D3C22×D312yes—
74#634—28no—
75C2×C3×Dic3—62no—
76C2×C3×Dic3—64no—
77C2×C3×Dic3—64no—
78C2×C3×Dic3—124no—
79C3×C4×D3—64no—
80C3×C4×D3—64no—
81C3×Dic6—68no—
82C3×Dic6—68no—
83C3×Dic6—128no—
84#627—616no—
85#627—68no—
86#624—38no—
87#624—64no—
88D3×Dic3—68no—
89D3×Dic3—68no—
90C2×C4⋊Q8—271noSylow
91C2×C33C4⋊Q812yescomplemented
92C2×C33C22×D411yescharacteristic
93C3×C3⋊D3C4⋊Q812yescomplemented
94C3×C3⋊D3C22×Q812yes—
95C2×C3×Q8—182no—
96C22×C3×C4—92no—
97C22×C3×C4—91no—
98C22×Dic3—94no—
99D3×Q8—188no—
100D3×Q8—188no—
101C2×C4×D3—62no—
102C2×C4×D3—184no—
103C2×Dic6—62no—
104C2×Dic6—62no—
105C2×Dic6—98no—
106C2×Dic6—122no—
107C2×Dic6—184no—
108C2×Dic6—182no—
109C2×Dic6—184no—
110C3×C42—94no—
111C4⋊Dic3—916no—
112C22×C32—22no—
113C22×C32—42no—
114C22×C32C2×D1211yescharacteristic
115C2×C3⋊D3C2×Dic614yes—
116C2×C3⋊D3C2×D1211yescharacteristic
117C2×C3⋊D3C23×D311yescharacteristic
118C2×C3×D3—62no—
119C2×C3×D3—68no—
120C2×C3×D3—62no—
121#157C2×Dic618yescomplemented
122C32×C4—24no—
123C32×C4—44no—
124C32×C4C2×D1212yes—
125C3⋊Dic3—28no—
126C3⋊Dic3C2×D1212yes—
127C3×Dic3—64no—
128C3×Dic3—68no—
129C3×Dic3—68no—
130C3×Dic3—128no—
131C22×Q8—272no—
132C4⋊Q8—278no—
133C2×C4⋊C4—274no—
134C2×C42—91no—
135C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
136C23×C3—91no—
137C22×D3—62no—
138C3×Q8—188no—
139C2×C3×C4—22no—
140C2×C3×C4—42no—
141C2×C3×C4—98no—
142C2×C3×C4—94no—
143C2×C3×C4—91no—
144C2×C3×C4—92no—
145C2×C3×C4—92no—
146C2×C3×C4—184no—
147C2×C3×C4GO+(4,𝔽2)11yescharacteristic
148C2×Dic3—62no—
149C2×Dic3—64no—
150C2×Dic3—64no—
151C2×Dic3—98no—
152C2×Dic3—124no—
153C2×Dic3—184no—
154C4×D3—64no—
155C4×D3—64no—
156C4×D3—188no—
157C4×D3—188no—
158Dic6—68no—
159Dic6—68no—
160Dic6—128no—
161Dic6—188no—
162Dic6—1816no—
163C2×C32—22no—
164C2×C32—24no—
165C2×C32—44no—
166C2×C32—42no—
167C2×C32#86912yescomplemented
168C2×C32C22×D1211yescharacteristic
169C3⋊D3#86912yescomplemented
170C3⋊D3C22×Dic612yes—
171C3×D3—64no—
172C3×D3—64no—
173C2×Q8—182no—
174C2×Q8—278no—
175C2×Q8—542no—
176C2×Q8—544no—
177C22×C4—92no—
178C22×C4—91no—
179C22×C4—274no—
180C4⋊C4—2716no—
181C42—94no—
182C22×C3—22no—
183C22×C3—42no—
184C22×C3—91no—
185C22×C3—91no—
186C22×C3—94no—
187C22×C3C2×GO+(4,𝔽2)11yescharacteristic
188C2×D3—62no—
189C2×D3—62no—
190C2×D3—68no—
191C3×C4—24no—
192C3×C4—44no—
193C3×C4—98no—
194C3×C4—92no—
195C3×C4—188no—
196C3×C4C2×GO+(4,𝔽2)12yes—
197Dic3—64no—
198Dic3—68no—
199Dic3—68no—
200Dic3—128no—
201C32—22no—
202C32—42no—
203C32C2×#86911yescharacteristic, complemented
204C23—91no—
205Q8—188no—
206Q8—548no—
207C2×C4—94no—
208C2×C4—98no—
209C2×C4—91no—
210C2×C4—92no—
211C2×C4—92no—
212C2×C4—184no—
213C2×C4—278no—
214C2×C4—544no—
215C2×C4#157⋊D311yescharacteristic
216C2×C3—24no—
217C2×C3—22no—
218C2×C3—44no—
219C2×C3—42no—
220C2×C3—92no—
221C2×C3—92no—
222C2×C3#6422212yescomplemented
223C2×C3C22×GO+(4,𝔽2)11yescharacteristic
224D3—64no—
225D3—64no—
226C22—94no—
227C22—91no—
228C22—91no—
229C22C2×#157⋊D311yescharacteristic
230C4—98no—
231C4—92no—
232C4—188no—
233C4C2×#157⋊D312yes—
234C3—22no—
235C3—42no—
236C3C2×#6422211yescharacteristic, complemented
237C2—92no—
238C2—92no—
239C2#1172433512yesdirect product factor
240C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33C2×C4⋊Q8
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11724335C2direct (N × H)
21. C2×C3×#157Q8semidirect
54. C3×#157C2×Q8semidirect
65. C2×#157Dic6semidirect
91. C2×C33C4⋊Q8semidirect
93. C3×C3⋊D3C4⋊Q8semidirect
121. #157C2×Dic6semidirect
135. C33C2×C4⋊Q8semidirect
167. C2×C32#869semidirect
169. C3⋊D3#869semidirect
203. C32C2×#869semidirect
222. C2×C3#64222semidirect
236. C3C2×#64222semidirect
239. C2#11724335direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×C4×#1572
C2×#889402
#117243352
C2×#642223
C2×#8699