Finite Group Databasebeta

← C2×D3×#3776 #13180577

Subgroups — C2×D3×#3776 #13180577

Order 1728.

904 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 132 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×D3×#627C212yesmaximal
2C2×#88943C211yesmaximal, characteristic, complemented
3C2×#88938C212yesmaximal, complemented
4C2×C3×#3776C211yesmaximal, characteristic, complemented
5C2×D3×#622C211yesmaximal, characteristic
6D3×#3776C218yesmaximal, direct product factor
7C22×#3776—31nomaximal
8C2×#6296C2212yes—
9C2×#6293—22no—
10C2×D3×C3⋊Dic3C2211yescharacteristic
11C2×C3×#627C2212yes—
12C2×C3×D3×Dic3—22no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15D3×#627—24no—
16D3×#627C2218yes—
17#88943C2214yescomplemented
18#88938C2218yescomplemented
19C3×#3776C2214yescomplemented
20D3×#622C2214yes—
21C22×#627—32no—
22C2×D3×Dic6—62no—
23C22×#622—31no—
24C2×#3776—32no—
25C2×#3776—31no—
26C2×#3776—38no—
27C2×#3776D311yescharacteristic, direct product factor
28C22×C32×D3D411yescharacteristic
29C2×C3⋊(C3⋊Dic3)D411yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C2×C32×Dic3—22no—
33#6296—22no—
34#6296C2314yes—
35#6293—28no—
36D3×C3⋊Dic3D412yes—
37D3×C3⋊Dic3C2312yes—
38C3×#627—22no—
39C3×#627C2314yes—
40C3×D3×Dic3—28no—
41C3⋊#622C2312yes—
42C3×#622C2312yes—
43C2×D3×Q16—91nomaximal
44C22×C3⋊Dic3—31no—
45C2×#634—62no—
46C22×C3×Dic3—62no—
47C2×C3×C4×D3—62no—
48C2×C3×Dic6—62no—
49C2×#627—32no—
50C2×#627—38no—
51C2×#627—64no—
52C2×#627—62no—
53C2×#627—62no—
54C2×#627—62no—
55C2×#627—122no—
56C2×#627C2×D312yes—
57C2×D3×Dic3—22no—
58C2×D3×Dic3—62no—
59D3×Dic6—68no—
60D3×Dic6—68no—
61C2×#622—31no—
62C2×#622—34no—
63C2×#622C2×D311yescharacteristic
64#3776—38no—
65#3776—34no—
66#3776C2×D314yesdirect product factor
67C22×C33C2×D411yescharacteristic
68C2×C32×D3Q1614yescomplemented
69C2×C32×D3C2×D412yes—
70C3⋊(C3⋊Dic3)C2×D412yes—
71C3×C3⋊Dic3—24no—
72C3×C3⋊Dic3C2×D411yescharacteristic
73C3×C3⋊Dic3C2411yescharacteristic
74C32×Dic3—24no—
75C2×D3×Q8—92no—
76C22×Dic6—182no—
77C2×C3×Q16—91no—
78C2×#216—92no—
79D3×Q16—98no—
80C2×Dic12—91no—
81C2×D3×C8—91no—
82C23×C32—31no—
83C22×C3×D3—22no—
84C2×C32×C4—62no—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—31no—
87C2×C3⋊Dic3—32no—
88C2×C3⋊Dic3—32no—
89C2×C3⋊Dic3—62no—
90C2×C3⋊Dic3—122no—
91C2×C3⋊Dic3C22×D311yescharacteristic
92#634—68no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—42no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—68no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—122no—
101C3×C4×D3—68no—
102C3×Dic6—64no—
103C3×Dic6—64no—
104#627—22no—
105#627—34no—
106#627—64no—
107#627—68no—
108#627—68no—
109#627—64no—
110#627—62no—
111#627—128no—
112#627C22×D314yes—
113D3×Dic3—28no—
114D3×Dic3—64no—
115D3×Dic3—64no—
116#622—32no—
117#622C22×D312yes—
118C22×Q16—271noSylow
119C2×C33C2×Q1612yescomplemented
120C2×C33C22×D411yescharacteristic
121C32×D3C2×Q1614yescomplemented
122C2×C3×Q8—92no—
123C22×C3×C4—182no—
124C22×Dic3—62no—
125C22×Dic3—182no—
126D3×Q8—98no—
127D3×Q8—184no—
128C2×C4×D3—62no—
129C2×C4×D3—91no—
130C2×Dic6—62no—
131C2×Dic6—92no—
132C2×Dic6—188no—
133C2×Dic6—182no—
134C2×Dic6—188no—
135C2×Dic6—182no—
136C2×Dic6—182no—
137C2×Dic6—182no—
138C2×Dic6—362no—
139C3×Q16—94no—
140C2×C3×C8—91no—
141#216—98no—
142C2×C3⋊C8—91no—
143Dic12—94no—
144D3×C8—94no—
145C22×C32—22no—
146C22×C32—34no—
147C22×C32—32no—
148C22×C32—42no—
149C22×C32D3×D411yescharacteristic
150C2×C3×D3—24no—
151C2×C3×D3—28no—
152C32×C4—64no—
153C3⋊Dic3—24no—
154C3⋊Dic3—32no—
155C3⋊Dic3—64no—
156C3⋊Dic3—124no—
157C3⋊Dic3D3×D411yescharacteristic
158C3⋊Dic3C23×D311yescharacteristic
159C3×Dic3—24no—
160C3×Dic3—24no—
161C3×Dic3—44no—
162C3×Dic3—62no—
163C3×Dic3—62no—
164C3×Dic3—64no—
165C3×Dic3—64no—
166C3×Dic3—124no—
167C22×Q8—272no—
168C2×Q16—91no—
169C2×Q16—271no—
170C2×Q16—278no—
171C2×Q16—272no—
172C22×C8—271no—
173C33C22×Q1611yescharacteristic, complemented, Sylow
174C23×C3—62no—
175C22×D3GO+(4,𝔽2)11yescharacteristic
176C3×Q8—94no—
177C3×Q8—182no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—91no—
181C2×C3×C4—122no—
182C2×C3×C4—188no—
183C2×C3×C4—182no—
184C2×Dic3—22no—
185C2×Dic3—62no—
186C2×Dic3—68no—
187C2×Dic3—62no—
188C2×Dic3—62no—
189C2×Dic3—91no—
190C2×Dic3—122no—
191C2×Dic3—184no—
192C2×Dic3—184no—
193C2×Dic3—182no—
194C2×Dic3—362no—
195C4×D3—68no—
196C4×D3—92no—
197C4×D3—92no—
198Dic6—64no—
199Dic6—64no—
200Dic6—94no—
201Dic6—184no—
202Dic6—188no—
203Dic6—182no—
204Dic6—184no—
205Dic6—188no—
206Dic6—188no—
207Dic6—368no—
208C3×C8—92no—
209C3⋊C8—92no—
210C2×C32—22no—
211C2×C32—24no—
212C2×C32—34no—
213C2×C32—44no—
214C2×C32—42no—
215C2×C32D3×Q1612yescomplemented
216C2×C32C2×D3×D411yescharacteristic
217C3×D3—28no—
218C2×Q8—92no—
219C2×Q8—272no—
220C2×Q8—278no—
221C2×Q8—542no—
222C2×Q8—544no—
223C22×C4—182no—
224C22×C4—271no—
225Q16—94no—
226Q16—274no—
227Q16—278no—
228C2×C8—91no—
229C2×C8—274no—
230C2×C8—271no—
231C22×C3—22no—
232C22×C3—42no—
233C22×C3—64no—
234C22×C3—68no—
235C22×C3C2×GO+(4,𝔽2)11yescharacteristic
236C2×D3#377614yesdirect product factor
237C2×D3C2×GO+(4,𝔽2)12yes—
238C3×C4—64no—
239C3×C4—64no—
240C3×C4—91no—
241C3×C4—91no—
242C3×C4—124no—
243C3×C4—184no—
244Dic3—24no—
245Dic3—62no—
246Dic3—62no—
247Dic3—64no—
248Dic3—64no—
249Dic3—92no—
250Dic3—124no—
251Dic3—184no—
252Dic3—364no—
253C32—22no—
254C32—42no—
255C32C2×D3×Q1611yescharacteristic, complemented
256C23—31no—
257Q8—94no—
258Q8—182no—
259Q8—274no—
260Q8—548no—
261Q8—542no—
262C2×C4—62no—
263C2×C4—91no—
264C2×C4—188no—
265C2×C4—182no—
266C2×C4—272no—
267C2×C4—272no—
268C2×C4—271no—
269C8—92no—
270C8—272no—
271C2×C3—24no—
272C2×C3—22no—
273C2×C3—44no—
274C2×C3—42no—
275C2×C3—68no—
276C2×C3C2×#377612yescomplemented
277C2×C3C22×GO+(4,𝔽2)11yescharacteristic
278D3C2×#377614yesdirect product factor
279C22—34no—
280C22—32no—
281C22D3×GO+(4,𝔽2)11yescharacteristic
282C4—64no—
283C4—91no—
284C4—91no—
285C4—184no—
286C4—272no—
287C3—22no—
288C3—42no—
289C3C22×#377611yescharacteristic, complemented
290C2—34no—
291C2D3×#377612yesdirect product factor
292C2C2×D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C2×D3×GO+(4,𝔽2)
Fitting, socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33C22×Q16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#88943C2semidirect
3. C2×#88938C2semidirect
4. C2×C3×#3776C2semidirect
6. D3×#3776C2direct (N × H)
17. #88943C22semidirect
18. #88938C22semidirect
19. C3×#3776C22semidirect
27. C2×#3776D3direct (N × H)
66. #3776C2×D3direct (N × H)
68. C2×C32×D3Q16semidirect
119. C2×C33C2×Q16semidirect
121. C32×D3C2×Q16semidirect
173. C33C22×Q16semidirect
215. C2×C32D3×Q16semidirect
236. C2×D3#3776direct (N × H)
255. C32C2×D3×Q16semidirect
276. C2×C3C2×#3776semidirect
278. D3C2×#3776direct (N × H)
289. C3C22×#3776semidirect
291. C2D3×#3776direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×#6272
C2×#889432
C2×#889382
C2×C3×#37762
C2×D3×#6222
D3×#37762
C22×#37763
C2×D3×Q169