Finite Group Databasebeta

← C2×#11724137 #13180590

Subgroups — C2×#11724137 #13180590

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×#88965C212yesmaximal, complemented
2C2×#88941C211yesmaximal, characteristic, complemented
3C2×#88936C212yesmaximal, complemented
4C2×C3×#3776C211yesmaximal, characteristic, complemented
5C2×#88924C211yesmaximal, characteristic, complemented
6#11724137C218yesmaximal, direct product factor
7C2×#64223—31nomaximal
8C2×#6295C2212yes—
9C2×#6292—22no—
10C2×#6289C2211yescharacteristic
11C2×C3×#627C2212yes—
12C2×C3×#624—22no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88965—24no—
16#88965C2218yescomplemented
17#88941C2214yescomplemented
18#88936C2218yescomplemented
19C3×#3776C2214yescomplemented
20#88924C2214yescomplemented
21C2×#3799—62no—
22C2×#3798—32no—
23C2×#3787—31no—
24C2×#3776D311yescharacteristic, complemented
25C2×#3775—32no—
26C2×#3774—31no—
27#64223—38no—
28C22×C3⋊(C3⋊D3)D411yescharacteristic
29C22×C3×C3⋊D3—22no—
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C2×C32×Dic3—22no—
32C2×C32×Dic3D411yescharacteristic
33#6295—22no—
34#6295C2314yes—
35#6292—28no—
36#6289D412yes—
37#6289C2312yes—
38C3×#627—22no—
39C3×#627C2314yes—
40C3×#624—28no—
41C3⋊#622C2312yes—
42C3×#622C2312yes—
43C2×#919—91nomaximal
44C2×C4⋊(C3⋊D3)—62no—
45C2×C4×C3⋊D3—31no—
46C2×C3×C4×D3—62no—
47C2×C3×C4×D3—62no—
48C2×C3×Dic6—62no—
49C2×#627C2×D312yes—
50C2×#626—62no—
51C2×#626—62no—
52C2×#626—62no—
53C2×#626—122no—
54C2×#625—32no—
55C2×#624—22no—
56C2×#624—62no—
57#3799—68no—
58#3799—68no—
59#3798—38no—
60#3798—64no—
61C2×#622—31no—
62C2×#622C2×D311yescharacteristic
63#3787—34no—
64#3776C2×D314yescomplemented
65#3775—38no—
66#3774—34no—
67C22×C33C2×D411yescharacteristic
68C2×C3⋊(C3⋊D3)—22no—
69C2×C3⋊(C3⋊D3)C2×D412yes—
70C2×C3×C3⋊D3—24no—
71C2×C3×C3⋊D3—44no—
72C3×C3⋊Dic3C2×D411yescharacteristic
73C3×C3⋊Dic3C2411yescharacteristic
74C32×Dic3—24no—
75C32×Dic3C2×D412yes—
76C2×#239—92no—
77C2×#235—182no—
78C2×C3×Q16—91no—
79C2×#215—92no—
80#919—98no—
81C2×D24—91no—
82C2×D3×C8—91no—
83C22×C3⋊D3—22no—
84C22×C3⋊D3—31no—
85C22×C3⋊D3—62no—
86C22×C3⋊D3—122no—
87C22×C3×D3—62no—
88C22×C3×D3—62no—
89C22×C3×D3—122no—
90C2×C32×C4—31no—
91C2×C32×C4—62no—
92C2×C3⋊Dic3C22×D311yescharacteristic
93C4⋊(C3⋊D3)—68no—
94C4×C3⋊D3—32no—
95C4×C3⋊D3—32no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3—42no—
100C2×C3×Dic3—62no—
101C3×C4×D3—68no—
102C3×C4×D3—68no—
103C3×Dic6—64no—
104C3×Dic6—64no—
105#627—22no—
106#627C22×D314yes—
107#626—64no—
108#626—68no—
109#626—68no—
110#626—64no—
111#626—128no—
112#625—34no—
113#625—62no—
114#624—28no—
115#624—64no—
116#624—64no—
117#622—32no—
118#622C22×D312yes—
119C2×#135—271noSylow
120C2×C33#13512yescomplemented
121C2×C33C22×D411yescharacteristic
122C3⋊(C3⋊D3)—22no—
123C3×C3⋊D3—44no—
124C2×C3×Q8—92no—
125C22×C3×C4—182no—
126C2×#67—182no—
127C2×#67—182no—
128#239—98no—
129#239—184no—
130#235—188no—
131#235—188no—
132C2×D12—92no—
133C2×D12—182no—
134C2×D12—182no—
135C2×D12—362no—
136C2×C4×D3—62no—
137C2×C4×D3—62no—
138C2×C4×D3—91no—
139C2×C4×D3—182no—
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—42no—
149C22×C32D3×D411yescharacteristic
150C2×C3⋊D3—24no—
151C2×C3⋊D3—32no—
152C2×C3⋊D3—44no—
153C2×C3⋊D3—64no—
154C2×C3⋊D3—62no—
155C2×C3⋊D3—124no—
156C2×C3⋊D3—124no—
157C2×C3⋊D3—244no—
158C2×C3×D3—64no—
159C2×C3×D3—64no—
160C2×C3×D3—124no—
161C2×C3×D3—124no—
162C2×C3×D3—124no—
163C2×C3×D3—244no—
164C32×C4—32no—
165C32×C4—64no—
166C3⋊Dic3D3×D411yescharacteristic
167C3⋊Dic3C23×D311yescharacteristic
168C3×Dic3—24no—
169C3×Dic3—24no—
170C3×Dic3—24no—
171C3×Dic3—44no—
172C3×Dic3—62no—
173C3×Dic3—62no—
174C2×#41—272no—
175#135—278no—
176C2×Q16—91no—
177C2×SD16—272no—
178C2×D8—271no—
179C22×C8—271no—
180C33C2×#13511yescharacteristic, complemented, Sylow
181C23×C3—182no—
182C22×D3—62no—
183C22×D3—62no—
184C22×D3—91no—
185C22×D3—122no—
186C22×D3—182no—
187C22×D3—362no—
188C3×Q8—94no—
189C3×Q8—182no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—91no—
194C2×C3×C4—122no—
195C2×C3×C4—188no—
196#67—184no—
197#67—188no—
198#67—184no—
199C2×Dic3—22no—
200C2×Dic3—62no—
201C2×Dic3GO+(4,𝔽2)11yescharacteristic
202D12—94no—
203D12—188no—
204D12—188no—
205D12—182no—
206D12—368no—
207C4×D3—68no—
208C4×D3—68no—
209C4×D3—92no—
210C4×D3—92no—
211C4×D3—184no—
212C4×D3—184no—
213Dic6—64no—
214Dic6—64no—
215C3×C8—92no—
216C3⋊C8—92no—
217C2×C32—22no—
218C2×C32—24no—
219C2×C32—44no—
220C2×C32—42no—
221C2×C32#91912yescomplemented
222C2×C32C2×D3×D411yescharacteristic
223C3⋊D3—44no—
224C3⋊D3—62no—
225C3⋊D3—124no—
226C3⋊D3—244no—
227C3×D3—124no—
228C3×D3—124no—
229C3×D3—244no—
230#41—278no—
231#41—544no—
232C2×Q8—92no—
233C2×D4—272no—
234C2×D4—542no—
235C22×C4—182no—
236C22×C4—271no—
237Q16—94no—
238SD16—278no—
239D8—274no—
240C2×C8—91no—
241C2×C8—274no—
242C2×C8—271no—
243C22×C3—22no—
244C22×C3—42no—
245C22×C3—184no—
246C22×C3—364no—
247C22×C3C2×GO+(4,𝔽2)11yescharacteristic
248C2×D3—64no—
249C2×D3—64no—
250C2×D3—92no—
251C2×D3—124no—
252C2×D3—124no—
253C2×D3—124no—
254C2×D3—184no—
255C2×D3—182no—
256C2×D3—244no—
257C2×D3—364no—
258C2×D3—364no—
259C2×D3—724no—
260C3×C4—64no—
261C3×C4—64no—
262C3×C4—64no—
263C3×C4—91no—
264C3×C4—91no—
265C3×C4—124no—
266Dic3—24no—
267Dic3—62no—
268Dic3—62no—
269Dic3C2×GO+(4,𝔽2)12yes—
270C32—22no—
271C32—42no—
272C32C2×#91911yescharacteristic, complemented
273C23—182no—
274C23—271no—
275Q8—94no—
276Q8—182no—
277D4—274no—
278D4—548no—
279D4—542no—
280C2×C4—31no—
281C2×C4—62no—
282C2×C4—91no—
283C2×C4—188no—
284C2×C4—272no—
285C2×C4—272no—
286C8—92no—
287C8—272no—
288C2×C3—24no—
289C2×C3—22no—
290C2×C3—44no—
291C2×C3—42no—
292C2×C3—364no—
293C2×C3#6422312yescomplemented
294C2×C3C22×GO+(4,𝔽2)11yescharacteristic
295D3—124no—
296D3—124no—
297D3—182no—
298D3—244no—
299D3—364no—
300D3—724no—
301C22—184no—
302C22—272no—
303C22—364no—
304C22—542no—
305C22D3×GO+(4,𝔽2)11yescharacteristic
306C4—32no—
307C4—64no—
308C4—91no—
309C4—91no—
310C3—22no—
311C3—42no—
312C3C2×#6422311yescharacteristic, complemented
313C2—364no—
314C2—542no—
315C2#1172413712yesdirect product factor
316C2C2×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×#88965C2semidirect
2. C2×#88941C2semidirect
3. C2×#88936C2semidirect
4. C2×C3×#3776C2semidirect
5. C2×#88924C2semidirect
6. #11724137C2direct (N × H)
16. #88965C22semidirect
17. #88941C22semidirect
18. #88936C22semidirect
19. C3×#3776C22semidirect
20. #88924C22semidirect
24. C2×#3776D3semidirect
64. #3776C2×D3semidirect
120. C2×C33#135semidirect
180. C33C2×#135semidirect
221. C2×C32#919semidirect
272. C32C2×#919semidirect
293. C2×C3#64223semidirect
312. C3C2×#64223semidirect
315. C2#11724137direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889652
C2×#889412
C2×#889362
C2×C3×#37762
C2×#889242
#117241372
C2×#642233
C2×#9199