Finite Group Databasebeta

← C2×#11724134 #13180591

Subgroups — C2×#11724134 #13180591

Order 1728.

968 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 116 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×#88964C212yesmaximal, complemented
2C2×#88941C211yesmaximal, characteristic, complemented
3C2×#88936C212yesmaximal, complemented
4C2×C3×#3776C211yesmaximal, characteristic, complemented
5C2×D3×#622C211yesmaximal, characteristic, complemented
6#11724134C218yesmaximal, direct product factor
7C2×#64237—31nomaximal
8C2×#6295C2212yes—
9C2×Dic3×C3⋊D3—22no—
10C2×D3×C3⋊Dic3C2211yescharacteristic
11C2×C3×#627C2212yes—
12C2×C3×#626—22no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88964—24no—
16#88964C2218yescomplemented
17#88941C2214yescomplemented
18#88936C2218yescomplemented
19C3×#3776C2214yescomplemented
20D3×#622C2214yescomplemented
21C2×#3805—32no—
22C2×#3796—62no—
23C22×#622—31no—
24C2×#3776D311yescharacteristic, complemented
25C2×#3775—32no—
26#64237—38no—
27C2×#3774—31no—
28C22×C3×C3⋊D3—22no—
29C22×C32×D3D411yescharacteristic
30C2×C3⋊(C3⋊Dic3)D411yescharacteristic
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C2×C32×Dic3—22no—
33#6295—22no—
34#6295C2314yes—
35Dic3×C3⋊D3—28no—
36D3×C3⋊Dic3D412yes—
37D3×C3⋊Dic3C2312yes—
38C3×#627—22no—
39C3×#627C2314yes—
40C3×#626—28no—
41C3⋊#622C2312yes—
42C3×#622C2312yes—
43C2×#919—91nomaximal
44C22×C3⋊Dic3—31no—
45C2×C4×C3⋊D3—62no—
46C2×C3×#67—62no—
47C2×C3×D12—62no—
48C2×C3×Dic6—62no—
49C2×#627C2×D312yes—
50C2×#626—22no—
51C2×#626—62no—
52C2×#625—32no—
53#3805—38no—
54#3805—64no—
55C2×D3×Dic3—62no—
56C2×D3×Dic3—62no—
57C2×D3×Dic3—62no—
58C2×D3×Dic3—122no—
59#3796—68no—
60#3796—68no—
61C2×#622—31no—
62C2×#622—34no—
63C2×#622C2×D311yescharacteristic
64#3776C2×D314yescomplemented
65#3775—38no—
66#3774—34no—
67C22×C33C2×D411yescharacteristic
68C2×C3×C3⋊D3—24no—
69C2×C3×C3⋊D3—44no—
70C2×C32×D3—22no—
71C2×C32×D3C2×D412yes—
72C3⋊(C3⋊Dic3)C2×D412yes—
73C3×C3⋊Dic3C2×D411yescharacteristic
74C3×C3⋊Dic3C2411yescharacteristic
75C32×Dic3—24no—
76C2×#239—92no—
77C2×#237—182no—
78C2×C3×Q16—91no—
79C2×#215—92no—
80#919—98no—
81C2×D24—91no—
82C2×D3×C8—91no—
83C23×C32—31no—
84C22×C3⋊D3—22no—
85C22×C3×D3—22no—
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C22×C3×D3—122no—
89C2×C32×C4—62no—
90C2×C3⋊Dic3—31no—
91C2×C3⋊Dic3—32no—
92C2×C3⋊Dic3—32no—
93C2×C3⋊Dic3—62no—
94C2×C3⋊Dic3—122no—
95C2×C3⋊Dic3C22×D311yescharacteristic
96C4×C3⋊D3—68no—
97C3×#67—68no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3—22no—
100C2×C3×Dic3—42no—
101C2×C3×Dic3—62no—
102C3×D12—68no—
103C3×Dic6—64no—
104C3×Dic6—64no—
105#627—22no—
106#627C22×D314yes—
107#626—28no—
108#626—64no—
109#626—64no—
110#625—34no—
111#625—62no—
112D3×Dic3—64no—
113D3×Dic3—64no—
114D3×Dic3—68no—
115D3×Dic3—68no—
116D3×Dic3—128no—
117#622—32no—
118#622C22×D312yes—
119C2×#135—271noSylow
120C2×C33#13512yescomplemented
121C2×C33C22×D411yescharacteristic
122C3×C3⋊D3—44no—
123C32×D3—22no—
124C2×C3×Q8—92no—
125C2×C3×D4—182no—
126C2×#67—62no—
127C2×#67—182no—
128C22×Dic3—182no—
129C22×Dic3—182no—
130#239—98no—
131#239—184no—
132#237—188no—
133#237—188no—
134C2×D12—62no—
135C2×D12—92no—
136C2×C4×D3—91no—
137C2×C4×D3—182no—
138C2×C4×D3—182no—
139C2×C4×D3—362no—
140C2×Dic6—62no—
141C3×Q16—94no—
142C2×C3×C8—91no—
143#215—98no—
144C2×C3⋊C8—91no—
145D24—94no—
146D3×C8—94no—
147C22×C32—22no—
148C22×C32—32no—
149C22×C32—42no—
150C22×C32—62no—
151C22×C32D3×D411yescharacteristic
152C2×C3⋊D3—24no—
153C2×C3⋊D3—44no—
154C2×C3×D3—24no—
155C2×C3×D3—44no—
156C2×C3×D3—64no—
157C2×C3×D3—64no—
158C2×C3×D3—124no—
159C2×C3×D3—124no—
160C2×C3×D3—128no—
161C2×C3×D3—124no—
162C32×C4—64no—
163C3⋊Dic3—32no—
164C3⋊Dic3—64no—
165C3⋊Dic3—124no—
166C3⋊Dic3D3×D411yescharacteristic
167C3⋊Dic3C23×D311yescharacteristic
168C3×Dic3—24no—
169C3×Dic3—24no—
170C3×Dic3—44no—
171C3×Dic3—62no—
172C3×Dic3—62no—
173C2×#41—272no—
174#135—278no—
175C2×Q16—91no—
176C2×SD16—272no—
177C2×D8—271no—
178C22×C8—271no—
179C33C2×#13511yescharacteristic, complemented, Sylow
180C23×C3—62no—
181C23×C3—182no—
182C22×D3—62no—
183C22×D3—62no—
184C22×D3—122no—
185C22×D3GO+(4,𝔽2)11yescharacteristic
186C3×Q8—94no—
187C3×Q8—182no—
188C3×D4—188no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—91no—
192C2×C3×C4—122no—
193#67—68no—
194#67—184no—
195#67—184no—
196C2×Dic3—22no—
197C2×Dic3—62no—
198C2×Dic3—91no—
199C2×Dic3—184no—
200C2×Dic3—184no—
201C2×Dic3—182no—
202C2×Dic3—188no—
203C2×Dic3—362no—
204D12—68no—
205D12—94no—
206D12—182no—
207C4×D3—92no—
208C4×D3—92no—
209C4×D3—188no—
210C4×D3—188no—
211C4×D3—368no—
212Dic6—64no—
213Dic6—64no—
214C3×C8—92no—
215C3⋊C8—92no—
216C2×C32—22no—
217C2×C32—24no—
218C2×C32—44no—
219C2×C32—42no—
220C2×C32—62no—
221C2×C32#91912yescomplemented
222C2×C32C2×D3×D411yescharacteristic
223C3⋊D3—44no—
224C3×D3—44no—
225C3×D3—124no—
226C3×D3—128no—
227C3×D3—124no—
228#41—278no—
229#41—544no—
230C2×Q8—92no—
231C2×D4—182no—
232C2×D4—272no—
233C22×C4—271no—
234C22×C4—542no—
235Q16—94no—
236SD16—278no—
237D8—274no—
238C2×C8—91no—
239C2×C8—274no—
240C2×C8—271no—
241C22×C3—22no—
242C22×C3—42no—
243C22×C3—64no—
244C22×C3—124no—
245C22×C3—184no—
246C22×C3—364no—
247C22×C3C2×GO+(4,𝔽2)11yescharacteristic
248C2×D3—22no—
249C2×D3—64no—
250C2×D3—64no—
251C2×D3—124no—
252C2×D3—124no—
253C2×D3—128no—
254C2×D3—124no—
255C2×D3C2×GO+(4,𝔽2)12yes—
256C3×C4—64no—
257C3×C4—64no—
258C3×C4—91no—
259C3×C4—91no—
260C3×C4—124no—
261Dic3—24no—
262Dic3—62no—
263Dic3—62no—
264Dic3—92no—
265Dic3—184no—
266Dic3—364no—
267C32—22no—
268C32—42no—
269C32C2×#91911yescharacteristic, complemented
270C23—31no—
271C23—182no—
272Q8—94no—
273Q8—182no—
274D4—188no—
275D4—274no—
276D4—542no—
277C2×C4—62no—
278C2×C4—91no—
279C2×C4—272no—
280C2×C4—272no—
281C2×C4—271no—
282C2×C4—548no—
283C8—92no—
284C8—272no—
285C2×C3—24no—
286C2×C3—22no—
287C2×C3—44no—
288C2×C3—42no—
289C2×C3—124no—
290C2×C3—364no—
291C2×C3#6423712yescomplemented
292C2×C3C22×GO+(4,𝔽2)11yescharacteristic
293D3—22no—
294D3—124no—
295D3—128no—
296D3—124no—
297C22—32no—
298C22—62no—
299C22—184no—
300C22—364no—
301C22D3×GO+(4,𝔽2)11yescharacteristic
302C4—64no—
303C4—91no—
304C4—91no—
305C4—272no—
306C3—22no—
307C3—42no—
308C3C2×#6423711yescharacteristic, complemented
309C2—62no—
310C2—364no—
311C2#1172413412yesdirect product factor
312C2C2×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
hypocenterC33C2×#135
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88964C2semidirect
2. C2×#88941C2semidirect
3. C2×#88936C2semidirect
4. C2×C3×#3776C2semidirect
5. C2×D3×#622C2semidirect
6. #11724134C2direct (N × H)
16. #88964C22semidirect
17. #88941C22semidirect
18. #88936C22semidirect
19. C3×#3776C22semidirect
20. D3×#622C22semidirect
24. C2×#3776D3semidirect
64. #3776C2×D3semidirect
120. C2×C33#135semidirect
179. C33C2×#135semidirect
221. C2×C32#919semidirect
269. C32C2×#919semidirect
291. C2×C3#64237semidirect
308. C3C2×#64237semidirect
311. C2#11724134direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889642
C2×#889412
C2×#889362
C2×C3×#37762
C2×D3×#6222
#117241342
C2×#642373
C2×#9199