Finite Group Databasebeta

← C2×#11724337 #13181131

Subgroups — C2×#11724337 #13181131

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×#89042C212yesmaximal, complemented
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3C2×#88943C211yesmaximal, characteristic, complemented
4C2×#88942C212yesmaximal, complemented
5C2×#88941C211yesmaximal, characteristic, complemented
6#11724337C218yesmaximal, direct product factor
7C2×#64223—31nomaximal
8C2×C3×C4×C3⋊D3—22no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6296C2212yes—
11C2×#6295—22no—
12C2×#6295C2212yes—
13#89042—24no—
14#89042C2218yescomplemented
15C2×C3×#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17C3×#3787C2214yescomplemented
18#88943C2214yescomplemented
19#88942C2218yescomplemented
20#88941C2214yescomplemented
21C2×#3798—32no—
22C2×#3798—62no—
23C2×#3787D311yescharacteristic, complemented
24C2×#3776—31no—
25C2×#3775—32no—
26C2×#3774—31no—
27#64223—38no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3D411yescharacteristic
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C3×C4×C3⋊D3—28no—
34C3×C4×C3⋊D3D412yes—
35C3×C4×C3⋊D3C2312yes—
36#6296—22no—
37#6296C2314yes—
38#6295—28no—
39#6295—22no—
40#6295C2314yes—
41C3×#622C2312yes—
42C3×#622C2312yes—
43C2×#904—91nomaximal
44C2×C4×C3⋊D3—22no—
45C2×C4×C3⋊D3C2×D311yescharacteristic
46C2×C3×C4×D3—62no—
47C2×C3×C4×D3—62no—
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—122no—
50C2×#627—32no—
51C2×#627—62no—
52C2×#626—62no—
53C2×#626—62no—
54C2×#626—62no—
55C2×#625—32no—
56C2×#625—62no—
57#3798—38no—
58#3798—68no—
59#3798—68no—
60#3798—64no—
61C2×#622C2×D311yescharacteristic
62C2×#622C2×D311yescharacteristic
63#3787C2×D314yescomplemented
64#3776—34no—
65#3775—38no—
66#3774—34no—
67C22×C33C2×D411yescharacteristic
68C2×C3×C3⋊D3—22no—
69C2×C3×C3⋊D3—24no—
70C2×C3×C3⋊D3—44no—
71C2×C3×C3⋊D3C2×D412yes—
72C33×C4C2×D412yes—
73C3×C3⋊Dic3—24no—
74C3×C3⋊Dic3C2×D411yescharacteristic
75C3×C3⋊Dic3C2411yescharacteristic
76C2×#235—92no—
77C2×#235—182no—
78C22×C3×C8—91no—
79C2×Dic12—91no—
80#904—98no—
81C2×D24—91no—
82C2×#204—92no—
83C22×C3⋊D3—22no—
84C22×C3⋊D3D1211yescharacteristic
85C22×C3×D3—62no—
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C22×C3×D3—122no—
89C2×C32×C4—22no—
90C2×C32×C4—42no—
91C2×C32×C4D1211yescharacteristic
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3C22×D311yescharacteristic
94C4×C3⋊D3—28no—
95C4×C3⋊D3D1212yes—
96C4×C3⋊D3C22×D312yes—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—122no—
101C3×C4×D3—64no—
102C3×C4×D3—64no—
103C3×C4×D3—68no—
104C3×C4×D3—68no—
105C3×C4×D3—128no—
106#627—34no—
107#627—64no—
108#627—64no—
109#627—62no—
110#626—64no—
111#626—64no—
112#626—68no—
113#626—68no—
114#625—34no—
115#625—68no—
116#625—62no—
117#622C22×D312yes—
118#622C22×D312yes—
119C2×#135—271noSylow
120C2×C33#13512yescomplemented
121C2×C33C22×D411yescharacteristic
122C3×C3⋊D3—22no—
123C3×C3⋊D3—44no—
124C22×C3×C4—91no—
125C22×C3×C4—182no—
126C2×#67—182no—
127C2×#67—182no—
128C2×#67—182no—
129#235—98no—
130#235—188no—
131#235—184no—
132#235—188no—
133C2×D12—92no—
134C2×D12—182no—
135C2×C4×D3—62no—
136C2×C4×D3—62no—
137C2×C4×D3—62no—
138C2×C4×D3—122no—
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—42no—
149C22×C32C2×D1211yescharacteristic
150C2×C3⋊D3—24no—
151C2×C3⋊D3—22no—
152C2×C3⋊D3—44no—
153C2×C3⋊D3C2×D1212yes—
154C2×C3×D3—64no—
155C2×C3×D3—64no—
156C2×C3×D3—64no—
157C2×C3×D3—124no—
158C2×C3×D3—124no—
159C2×C3×D3—124no—
160C2×C3×D3—128no—
161C2×C3×D3—124no—
162C32×C4—24no—
163C32×C4—44no—
164C32×C4C2×D1212yes—
165C3⋊Dic3—24no—
166C3⋊Dic3C2×D1211yescharacteristic
167C3⋊Dic3C23×D311yescharacteristic
168C3×Dic3—62no—
169C3×Dic3—62no—
170C3×Dic3—64no—
171C3×Dic3—64no—
172C3×Dic3—124no—
173C2×#41—272no—
174#135—278no—
175C2×Q16—271no—
176C2×SD16—272no—
177C2×D8—271no—
178C22×C8—91no—
179C33C2×#13511yescharacteristic, complemented, Sylow
180C23×C3—91no—
181C23×C3—182no—
182C22×D3—62no—
183C22×D3—62no—
184C22×D3—62no—
185C22×D3—122no—
186C2×C3×C4—22no—
187C2×C3×C4—42no—
188C2×C3×C4—91no—
189C2×C3×C4—92no—
190C2×C3×C4—92no—
191C2×C3×C4—188no—
192C2×C3×C4—182no—
193C2×C3×C4GO+(4,𝔽2)11yescharacteristic
194#67—184no—
195#67—188no—
196#67—184no—
197#67—188no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—122no—
202D12—94no—
203D12—182no—
204D12—188no—
205C4×D3—64no—
206C4×D3—64no—
207C4×D3—68no—
208C4×D3—68no—
209C4×D3—128no—
210Dic6—94no—
211Dic6—184no—
212Dic6—182no—
213Dic6—184no—
214C3×C8—92no—
215C3×C8—92no—
216C2×C32—22no—
217C2×C32—24no—
218C2×C32—44no—
219C2×C32—42no—
220C2×C32#90412yescomplemented
221C2×C32C22×D1211yescharacteristic
222C3⋊D3—22no—
223C3⋊D3—44no—
224C3×D3—124no—
225C3×D3—124no—
226C3×D3—128no—
227C3×D3—124no—
228#41—278no—
229#41—544no—
230C2×Q8—272no—
231C2×D4—272no—
232C2×D4—542no—
233C22×C4—91no—
234C22×C4—182no—
235Q16—274no—
236SD16—278no—
237D8—274no—
238C2×C8—91no—
239C2×C8—91no—
240C2×C8—94no—
241C22×C3—22no—
242C22×C3—42no—
243C22×C3—92no—
244C22×C3—184no—
245C22×C3—182no—
246C22×C3—364no—
247C22×C3C2×GO+(4,𝔽2)11yescharacteristic
248C2×D3—64no—
249C2×D3—64no—
250C2×D3—64no—
251C2×D3—124no—
252C2×D3—124no—
253C2×D3—124no—
254C2×D3—128no—
255C2×D3—124no—
256C3×C4—24no—
257C3×C4—44no—
258C3×C4—91no—
259C3×C4—91no—
260C3×C4—184no—
261C3×C4C2×GO+(4,𝔽2)12yes—
262Dic3—62no—
263Dic3—62no—
264Dic3—64no—
265Dic3—64no—
266Dic3—124no—
267C32—22no—
268C32—42no—
269C32C2×#90411yescharacteristic, complemented
270C23—91no—
271C23—182no—
272Q8—274no—
273Q8—542no—
274D4—274no—
275D4—542no—
276D4—548no—
277C2×C4—91no—
278C2×C4—92no—
279C2×C4—92no—
280C2×C4—182no—
281C2×C4—188no—
282C2×C4#157⋊D311yescharacteristic
283C8—92no—
284C8—92no—
285C2×C3—24no—
286C2×C3—22no—
287C2×C3—44no—
288C2×C3—42no—
289C2×C3—182no—
290C2×C3—364no—
291C2×C3#6422312yescomplemented
292C2×C3C22×GO+(4,𝔽2)11yescharacteristic
293D3—124no—
294D3—124no—
295D3—128no—
296D3—124no—
297C22—92no—
298C22—182no—
299C22—184no—
300C22—364no—
301C22C2×#157⋊D311yescharacteristic
302C4—91no—
303C4—91no—
304C4—184no—
305C4C2×#157⋊D312yes—
306C3—22no—
307C3—42no—
308C3C2×#6422311yescharacteristic, complemented
309C2—182no—
310C2—364no—
311C2#1172433712yesdirect product factor
312C2C22×#157⋊D311yescharacteristic
subgroupG/H
center, hypercenterC2×C4#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypocenterC33C2×#135
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89042C2semidirect
2. C2×C3×#3787C2semidirect
3. C2×#88943C2semidirect
4. C2×#88942C2semidirect
5. C2×#88941C2semidirect
6. #11724337C2direct (N × H)
14. #89042C22semidirect
17. C3×#3787C22semidirect
18. #88943C22semidirect
19. #88942C22semidirect
20. #88941C22semidirect
23. C2×#3787D3semidirect
63. #3787C2×D3semidirect
120. C2×C33#135semidirect
179. C33C2×#135semidirect
220. C2×C32#904semidirect
269. C32C2×#904semidirect
291. C2×C3#64223semidirect
308. C3C2×#64223semidirect
311. C2#11724337direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890422
C2×C3×#37872
C2×#889432
C2×#889422
C2×#889412
#117243372
C2×#642233
C2×#9049