Finite Group Databasebeta

← C2×#11724184 #13180775

Subgroups — C2×#11724184 #13180775

Order 1728.

1118 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 125 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×A4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic
3C2×C32×SL(2,ℤ/4)C211yesmaximal, characteristic
4#11724184C214yesmaximal, direct product factor
5C2×#64204—34nomaximal
6C2×#64087—31nomaximal
7C22×C32×A4C2211yescharacteristic
8C2×#6293—41nomaximal
9A4×C3⋊Dic3C2212yes—
10C3⋊(C3⋊SL(2,ℤ/4))C2212yes—
11C32×SL(2,ℤ/4)C2212yes—
12C23×C3⋊Dic3D311yescharacteristic
13C22×#634—31no—
14C2×Dic3×A4—34no—
15C2×C3⋊SL(2,ℤ/4)—34no—
16C2×C3⋊SL(2,ℤ/4)—64no—
17C2×C3×SL(2,ℤ/4)D314yes—
18#64204—316no—
19C2×C32×C22⋊C4—31no—
20C2×#3757—31no—
21C2×C4⋊(C3⋊Dic3)—31no—
22C2×#3750—31no—
23C2×#3750—31no—
24#64087—34no—
25#64087—34no—
26C2×C32×A4Q812yescomplemented
27C2×C32×A4C2311yescharacteristic
28C2×C3⋊(C3⋊Dic3)—41no—
29C2×C3×C3⋊Dic3—41no—
30C2×C32×Dic3—41no—
31#6293—44no—
32C2×A4⋊Q8—91no—
33C2×#878—94no—
34C24×C32C2×D311yescharacteristic
35C22×C3×A4—24no—
36C22×C3×A4C2×D314yes—
37C22×C32×C4—31no—
38C22×C32×C4—31no—
39C22×C3⋊Dic3—31no—
40C22×C3⋊Dic3—32no—
41C22×C3⋊Dic3—31no—
42C22×C3⋊Dic3—31no—
43C22×C3⋊Dic3—31no—
44C22×C3⋊Dic3C2×D312yes—
45C2×#634—34no—
46C2×#634—61no—
47C2×#634—62no—
48C2×#634—61no—
49C2×#627—124no—
50Dic3×A4—38no—
51C3⋊SL(2,ℤ/4)—38no—
52C3⋊SL(2,ℤ/4)—68no—
53C3×SL(2,ℤ/4)C2×D318yes—
54C32×C22⋊C4—32no—
55C32×C22⋊C4—32no—
56#3757—32no—
57#3757—32no—
58C4⋊(C3⋊Dic3)—34no—
59#3750—34no—
60#3750—34no—
61C22×C33—41no—
62C32×A4C2×Q811yescharacteristic, complemented
63C3⋊(C3⋊Dic3)—42no—
64C3×C3⋊Dic3—42no—
65C32×Dic3—42no—
66C23×Dic3—34no—
67C22×Dic6—94no—
68C2×C4×A4—91no—
69C2×SL(2,ℤ/4)—91no—
70C2×SL(2,ℤ/4)—184no—
71C2×SL(2,ℤ/4)C3⋊D311yescharacteristic
72A4⋊Q8—94no—
73C2×C3×C22⋊C4—34no—
74C2×#217—94no—
75C2×C4⋊Dic3—94no—
76C2×Dic3⋊C4—94no—
77C2×Dic3⋊C4—94no—
78#878—916no—
79#878—916no—
80C23×C32—31no—
81C23×C32—32no—
82C23×C32—31no—
83C23×C32Dic612yescomplemented
84C23×C32C22×D311yescharacteristic
85C2×C3×A4—28no—
86C2×C3×A4—24no—
87C2×C3×A4Dic618yescomplemented
88C2×C3×A4C22×D314yes—
89C2×C32×C4—32no—
90C2×C32×C4—32no—
91C2×C32×C4—61no—
92C2×C32×C4—62no—
93C2×C32×C4—61no—
94C2×C3⋊Dic3—31no—
95C2×C3⋊Dic3—32no—
96C2×C3⋊Dic3—32no—
97C2×C3⋊Dic3—32no—
98C2×C3⋊Dic3—32no—
99C2×C3⋊Dic3—32no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—61no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—124no—
105C2×C3⋊Dic3—244no—
106C2×C3⋊Dic3S411yescharacteristic
107#634—64no—
108#634—64no—
109C2×C3×Dic3—44no—
110C2×C3×Dic3—124no—
111#627—1216no—
112C2×C22⋊Q8—271noSylow
113C2×C33—42no—
114C2×C33—41no—
115C24×C3D3214yes—
116C22×A4—24no—
117C22×A4C2×C3⋊D311yescharacteristic
118C22×C3×C4—34no—
119C22×C3×C4—34no—
120C22×Dic3—38no—
121C22×Dic3—94no—
122C22×Dic3—98no—
123C22×Dic3—94no—
124C22×Dic3—94no—
125C22×Dic3—94no—
126C2×Dic6—916no—
127C2×Dic6—184no—
128C2×Dic6—188no—
129C2×Dic6—184no—
130C2×Dic6—361no—
131C4×A4—92no—
132SL(2,ℤ/4)—92no—
133SL(2,ℤ/4)—188no—
134SL(2,ℤ/4)C2×C3⋊D312yes—
135C3×C22⋊C4—38no—
136C3×C22⋊C4—38no—
137#217—98no—
138#217—98no—
139C4⋊Dic3—916no—
140Dic3⋊C4—916no—
141Dic3⋊C4—916no—
142C22×C32—32no—
143C22×C32—31no—
144C22×C32—32no—
145C22×C32—32no—
146C22×C32—31no—
147C22×C32—31no—
148C22×C32—44no—
149C22×C32—61no—
150C22×C32—84no—
151C22×C32C2×Dic611yescharacteristic, complemented
152C22×C32C2×S411yescharacteristic
153C3×A4—24no—
154C3×A4C2×Dic614yescomplemented
155C32×C4—62no—
156C32×C4—62no—
157C3⋊Dic3—32no—
158C3⋊Dic3—62no—
159C3⋊Dic3—62no—
160C3⋊Dic3—128no—
161C3⋊Dic3—248no—
162C3⋊Dic3C2×S412yes—
163C3×Dic3—48no—
164C3×Dic3—128no—
165C22×Q8—271no—
166C23×C4—91no—
167C22⋊Q8—274no—
168C22⋊Q8—274no—
169C2×C4⋊C4—271no—
170C2×C4⋊C4—271no—
171C2×C4⋊C4—271no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—271no—
174C33—41noSylow
175C23×C3—34no—
176C23×C3—34no—
177C23×C3—38no—
178C23×C3#62718yescomplemented
179C23×C3C2×D3214yes—
180C2×A4—28no—
181C2×A4—24no—
182C2×A4#63412yescomplemented
183C2×A4C22×C3⋊D311yescharacteristic
184C2×C3×C4—38no—
185C2×C3×C4—38no—
186C2×C3×C4—64no—
187C2×C3×C4—68no—
188C2×C3×C4—64no—
189C2×C3×C4—361no—
190C2×Dic3—34no—
191C2×Dic3—41no—
192C2×Dic3—98no—
193C2×Dic3—98no—
194C2×Dic3—94no—
195C2×Dic3—98no—
196C2×Dic3—98no—
197C2×Dic3—98no—
198C2×Dic3—184no—
199C2×Dic3—184no—
200C2×Dic3—188no—
201C2×Dic3—184no—
202C2×Dic3—361no—
203C2×Dic3—724no—
204Dic6—1816no—
205Dic6—1816no—
206Dic6—364no—
207C2×C32—31no—
208C2×C32—31no—
209C2×C32—32no—
210C2×C32—44no—
211C2×C32—48no—
212C2×C32—84no—
213C2×C32—88no—
214C2×C32A4⋊Q812yescomplemented
215C2×C32C22×S411yescharacteristic
216C24D3×C3⋊D311yescharacteristic
217C2×Q8—274no—
218C2×Q8—541no—
219C2×Q8—542no—
220C2×Q8—541no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—92no—
224C22×C4—271no—
225C22×C4—271no—
226C22×C4—272no—
227C22×C4—271no—
228C22×C4—271no—
229C4⋊C4—274no—
230C4⋊C4—274no—
231C4⋊C4—274no—
232C22⋊C4—32no—
233C22⋊C4—32no—
234C22⋊C4—272no—
235C22⋊C4—272no—
236C22×C3—38no—
237C22×C3—34no—
238C22×C3—34no—
239C22×C3—34no—
240C22×C3—38no—
241C22×C3—38no—
242C22×C3—41no—
243C22×C3—64no—
244C22×C3—84no—
245C22×C3C2×#62714yescomplemented
246C22×C3D3×S414yes—
247A4—24no—
248A4C2×#63411yescharacteristic, complemented
249C3×C4—68no—
250C3×C4—68no—
251C3×C4—362no—
252Dic3—38no—
253Dic3—42no—
254Dic3—98no—
255Dic3—188no—
256Dic3—188no—
257Dic3—362no—
258Dic3—728no—
259C32—44no—
260C32—84no—
261C32C2×A4⋊Q811yescharacteristic, complemented
262C23—31no—
263C23—31no—
264C23—32no—
265C23#629312yescomplemented
266C23C2×D3×C3⋊D311yescharacteristic
267Q8—544no—
268Q8—544no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—61no—
272C2×C4—62no—
273C2×C4—61no—
274C2×C4—91no—
275C2×C4—272no—
276C2×C4—272no—
277C2×C4—271no—
278C2×C4—272no—
279C2×C4—272no—
280C2×C4—272no—
281C2×C4—541no—
282C2×C4—541no—
283C2×C4—542no—
284C2×C4—541no—
285C2×C3—34no—
286C2×C3—34no—
287C2×C3—38no—
288C2×C3—42no—
289C2×C3—41no—
290C2×C3—88no—
291C2×C3—84no—
292C2×C3#6420418yescomplemented
293C2×C3C2×D3×S414yes—
294C22—32no—
295C22—31no—
296C22—32no—
297C22—32no—
298C22—31no—
299C22—31no—
300C22—61no—
301C22C2×#629311yescharacteristic, complemented
302C22C3⋊D3×S411yescharacteristic
303C4—62no—
304C4—62no—
305C4—92no—
306C4—272no—
307C4—542no—
308C4—542no—
309C3—41no—
310C3—84no—
311C3C2×#6420414yescomplemented
312C2—31no—
313C2—31no—
314C2—32no—
315C2#1172418412yesdirect product factor
316C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×S4
commutator, cosocleC2×C32×A4C23
FrattiniC2C2×C3⋊D3×S4
Fitting, socleC24×C32C2×D3
prime radicalC22×C32×A4C22
hypocenterC32×A4C2×Q8
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. #11724184C2direct (N × H)
26. C2×C32×A4Q8semidirect
62. C32×A4C2×Q8semidirect
83. C23×C32Dic6semidirect
87. C2×C3×A4Dic6semidirect
151. C22×C32C2×Dic6semidirect
154. C3×A4C2×Dic6semidirect
178. C23×C3#627semidirect
182. C2×A4#634semidirect
214. C2×C32A4⋊Q8semidirect
245. C22×C3C2×#627semidirect
248. A4C2×#634semidirect
261. C32C2×A4⋊Q8semidirect
265. C23#6293semidirect
292. C2×C3#64204semidirect
301. C22C2×#6293semidirect
311. C3C2×#64204semidirect
315. C2#11724184direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×C3⋊Dic32
C2×C3⋊(C3⋊SL(2,ℤ/4))2
C2×C32×SL(2,ℤ/4)2
#117241842
C2×#642043
C2×#640873
C2×#62934