Finite Group Databasebeta

← C2×#11724351 #13181166

Subgroups — C2×#11724351 #13181166

Order 1728.

968 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 127 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×#89046C212yesmaximal, complemented
2C22×C3×#622C211yesmaximal, characteristic, complemented
3C2×#88943C211yesmaximal, characteristic, complemented
4#11724351C218yesmaximal, direct product factor
5C2×#88942C212yesmaximal, complemented
6C2×#88941C211yesmaximal, characteristic, complemented
7C2×#64237—31nomaximal
8C2×C3×#638—22no—
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×#6296C2212yes—
11C2×#6295C2212yes—
12#89046—24no—
13#89046C2218yescomplemented
14C2×#6294—22no—
15C2×C3×#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17C2×C3×#622C2214yescomplemented
18#88943C2214yescomplemented
19#88942C2218yescomplemented
20#88941C2214yescomplemented
21C2×#3805—32no—
22C2×#3804—62no—
23C22×#622D311yescharacteristic, complemented
24C2×#3776—31no—
25C2×#3775—32no—
26#64237—38no—
27C2×#3774—31no—
28C23×C33D411yescharacteristic
29C22×C3×C3⋊D3—22no—
30C3×#638—28no—
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3D412yes—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C2×C3×C3⋊Dic3C2312yes—
36#6296—22no—
37#6296C2314yes—
38#6295—22no—
39#6295C2314yes—
40#6294—28no—
41C3×#622C2312yes—
42C3×#622C2312yes—
43C2×#904—91nomaximal
44C2×#638—22no—
45C22×C3⋊Dic3C2×D311yescharacteristic
46C2×C3×#67—62no—
47C2×C3×#67—62no—
48C2×C3×#67—122no—
49C22×C3×Dic3—62no—
50C2×#627—32no—
51C2×#627—62no—
52C2×#626—62no—
53C2×#625—32no—
54C2×#624—62no—
55#3805—38no—
56#3805—64no—
57#3804—68no—
58#3804—68no—
59C2×D3×Dic3—62no—
60C2×D3×Dic3—62no—
61C2×#622C2×D311yescharacteristic
62C2×#622C2×D311yescharacteristic
63C2×#622C2×D314yescomplemented
64#3776—34no—
65#3775—38no—
66#3774—34no—
67C22×C33—22no—
68C22×C33C2×D412yes—
69C22×C33C2×D411yescharacteristic
70C2×C3×C3⋊D3—24no—
71C2×C3×C3⋊D3—44no—
72C3×C3⋊Dic3—24no—
73C3×C3⋊Dic3C2×D412yes—
74C3×C3⋊Dic3C2×D411yescharacteristic
75C3×C3⋊Dic3C2411yescharacteristic
76C2×#237—182no—
77C2×#235—92no—
78C22×C3×C8—91no—
79C2×Dic12—91no—
80#904—98no—
81C2×D24—91no—
82C2×#204—92no—
83C23×C32—22no—
84C23×C32—42no—
85C23×C32D1211yescharacteristic
86C22×C3⋊D3—22no—
87C22×C3×D3—62no—
88C22×C3×D3—62no—
89C22×C3×D3—122no—
90#638—28no—
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3D1211yescharacteristic
93C2×C3⋊Dic3D1212yes—
94C2×C3⋊Dic3C22×D311yescharacteristic
95C2×C3⋊Dic3C22×D312yes—
96C3×#67—68no—
97C3×#67—68no—
98C3×#67—128no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—64no—
101C2×C3×Dic3—64no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—122no—
106#627—34no—
107#627—64no—
108#627—64no—
109#627—62no—
110#626—64no—
111#626—64no—
112#625—34no—
113#625—62no—
114#624—68no—
115D3×Dic3—68no—
116D3×Dic3—68no—
117#622C22×D312yes—
118#622C22×D312yes—
119C2×#135—271noSylow
120C2×C33—22no—
121C2×C33#13512yescomplemented
122C2×C33C22×D411yescharacteristic
123C3×C3⋊D3—44no—
124C2×C3×D4—182no—
125C22×C3×C4—91no—
126C2×#67—62no—
127C2×#67—62no—
128C2×#67—122no—
129C2×#67—182no—
130C22×Dic3—62no—
131C22×Dic3—182no—
132#237—188no—
133#237—188no—
134#235—98no—
135#235—184no—
136C2×D12—92no—
137C2×C4×D3—182no—
138C2×C4×D3—182no—
139C2×Dic6—92no—
140C2×Dic6—182no—
141C2×C3×C8—91no—
142C2×C3×C8—91no—
143C2×C3×C8—94no—
144Dic12—94no—
145D24—94no—
146#204—98no—
147C22×C32—22no—
148C22×C32—24no—
149C22×C32—22no—
150C22×C32—42no—
151C22×C32—44no—
152C22×C32—44no—
153C22×C32—84no—
154C22×C32C2×D1212yes—
155C22×C32C2×D1211yescharacteristic
156C2×C3⋊D3—24no—
157C2×C3⋊D3—44no—
158C2×C3×D3—64no—
159C2×C3×D3—64no—
160C2×C3×D3—124no—
161C2×C3×D3—124no—
162C2×C3×D3—124no—
163C2×C3×D3—244no—
164C3⋊Dic3—24no—
165C3⋊Dic3C2×D1212yes—
166C3⋊Dic3C2×D1211yescharacteristic
167C3⋊Dic3C23×D311yescharacteristic
168C3×Dic3—64no—
169C3×Dic3—62no—
170C3×Dic3—62no—
171C3×Dic3—64no—
172C3×Dic3—64no—
173C3×Dic3—124no—
174C2×#41—272no—
175#135—278no—
176C2×Q16—271no—
177C2×SD16—272no—
178C2×D8—271no—
179C22×C8—91no—
180C33C2×#13511yescharacteristic, complemented, Sylow
181C23×C3—22no—
182C23×C3—42no—
183C23×C3—182no—
184C23×C3GO+(4,𝔽2)11yescharacteristic
185C22×D3—62no—
186C22×D3—62no—
187C22×D3—122no—
188C3×D4—188no—
189C2×C3×C4—91no—
190C2×C3×C4—92no—
191C2×C3×C4—91no—
192C2×C3×C4—92no—
193C2×C3×C4—182no—
194#67—68no—
195#67—68no—
196#67—128no—
197#67—184no—
198#67—184no—
199C2×Dic3—62no—
200C2×Dic3—64no—
201C2×Dic3—62no—
202C2×Dic3—64no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—122no—
206C2×Dic3—188no—
207D12—94no—
208D12—182no—
209C4×D3—188no—
210C4×D3—188no—
211Dic6—94no—
212Dic6—184no—
213Dic6—182no—
214Dic6—184no—
215C3×C8—92no—
216C3×C8—92no—
217C2×C32—22no—
218C2×C32—24no—
219C2×C32—22no—
220C2×C32—44no—
221C2×C32—42no—
222C2×C32—44no—
223C2×C32—84no—
224C2×C32#90412yescomplemented
225C2×C32C22×D1211yescharacteristic
226C3⋊D3—44no—
227C3×D3—124no—
228C3×D3—124no—
229C3×D3—244no—
230#41—278no—
231#41—544no—
232C2×Q8—272no—
233C2×D4—182no—
234C2×D4—272no—
235C22×C4—91no—
236C22×C4—542no—
237Q16—274no—
238SD16—278no—
239D8—274no—
240C2×C8—91no—
241C2×C8—91no—
242C2×C8—94no—
243C22×C3—22no—
244C22×C3—24no—
245C22×C3—22no—
246C22×C3—44no—
247C22×C3—44no—
248C22×C3—42no—
249C22×C3—84no—
250C22×C3—184no—
251C22×C3—364no—
252C22×C3C2×GO+(4,𝔽2)12yes—
253C22×C3C2×GO+(4,𝔽2)11yescharacteristic
254C2×D3—64no—
255C2×D3—64no—
256C2×D3—124no—
257C2×D3—124no—
258C2×D3—124no—
259C2×D3—244no—
260C3×C4—91no—
261C3×C4—91no—
262C3×C4—92no—
263C3×C4—184no—
264Dic3—62no—
265Dic3—62no—
266Dic3—64no—
267Dic3—64no—
268Dic3—64no—
269Dic3—124no—
270C32—22no—
271C32—42no—
272C32C2×#90411yescharacteristic, complemented
273C23—182no—
274C23#157⋊D311yescharacteristic
275Q8—274no—
276Q8—542no—
277D4—188no—
278D4—274no—
279D4—542no—
280C2×C4—92no—
281C2×C4—92no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—182no—
285C2×C4—548no—
286C8—92no—
287C8—92no—
288C2×C3—22no—
289C2×C3—24no—
290C2×C3—22no—
291C2×C3—44no—
292C2×C3—44no—
293C2×C3—42no—
294C2×C3—84no—
295C2×C3—364no—
296C2×C3#6423712yescomplemented
297C2×C3C22×GO+(4,𝔽2)11yescharacteristic
298D3—124no—
299D3—124no—
300D3—244no—
301C22—22no—
302C22—184no—
303C22—364no—
304C22C2×#157⋊D312yes—
305C22C2×#157⋊D311yescharacteristic
306C4—91no—
307C4—92no—
308C4—91no—
309C4—184no—
310C3—22no—
311C3—42no—
312C3C2×#6423711yescharacteristic, complemented
313C2—22no—
314C2—364no—
315C2#1172435112yesdirect product factor
316C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×#157⋊D3
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23#157⋊D3
hypocenterC33C2×#135
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#89046C2semidirect
2. C22×C3×#622C2semidirect
3. C2×#88943C2semidirect
4. #11724351C2direct (N × H)
5. C2×#88942C2semidirect
6. C2×#88941C2semidirect
13. #89046C22semidirect
17. C2×C3×#622C22semidirect
18. #88943C22semidirect
19. #88942C22semidirect
20. #88941C22semidirect
23. C22×#622D3semidirect
63. C2×#622C2×D3semidirect
121. C2×C33#135semidirect
180. C33C2×#135semidirect
224. C2×C32#904semidirect
272. C32C2×#904semidirect
296. C2×C3#64237semidirect
312. C3C2×#64237semidirect
315. C2#11724351direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890462
C22×C3×#6222
C2×#889432
#117243512
C2×#889422
C2×#889412
C2×#642373
C2×#9049