Finite Group Databasebeta

← C2×D3×#3774 #13180565

Subgroups — C2×D3×#3774 #13180565

Order 1728.

1256 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 132 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×D3×#625C212yesmaximal, complemented
2C2×#88941C211yesmaximal, characteristic, complemented
3C2×#88935C212yesmaximal, complemented
4C2×C3×#3774C211yesmaximal, characteristic, complemented
5C2×D3×#622C211yesmaximal, characteristic, complemented
6D3×#3774C218yesmaximal, direct product factor
7C22×#3774—31nomaximal
8C22×C3×D32—22no—
9C2×#6295C2212yescomplemented
10C2×#6290—22no—
11C2×D3×C3⋊Dic3C2211yescharacteristic
12C2×C3×#625C2212yescomplemented
13C2×C3⋊#622C2211yescharacteristic, complemented
14C2×C3×#622C2211yescharacteristic, complemented
15D3×#625—24no—
16D3×#625C2218yescomplemented
17#88941C2214yescomplemented
18#88935C2218yescomplemented
19C3×#3774C2214yescomplemented
20D3×#622C2214yescomplemented
21C2×D3×#67—62no—
22C22×#625—32no—
23C22×#622—31no—
24C2×#3774—32no—
25C2×#3774—31no—
26C2×#3774—38no—
27C2×#3774D311yescharacteristic, direct product factor
28C22×C3×C3⋊D3—22no—
29C22×C32×D3—22no—
30C22×C32×D3D411yescharacteristic
31C2×C3×D32—28no—
32C2×C3×D32—48no—
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35#6295—22no—
36#6295C2314yescomplemented
37#6290—28no—
38D3×C3⋊Dic3D412yes—
39D3×C3⋊Dic3C2312yes—
40C3×#625—22no—
41C3×#625C2314yescomplemented
42C3⋊#622C2312yescomplemented
43C3×#622C2312yescomplemented
44C2×D3×D8—91nomaximal
45C23×C3×D3—62no—
46C23×C3×D3—62no—
47C22×D32—22no—
48C2×#638—62no—
49C22×C3⋊Dic3—31no—
50C2×C3×#67—62no—
51D3×#67—68no—
52D3×#67—68no—
53C2×#626—62no—
54C2×#625—32no—
55C2×#625—38no—
56C2×#625—64no—
57C2×#625—62no—
58C2×#625—62no—
59C2×#625—122no—
60C2×#625C2×D312yescomplemented
61C2×D3×Dic3—62no—
62C2×#622—31no—
63C2×#622—34no—
64C2×#622C2×D311yescharacteristic, complemented
65#3774—38no—
66#3774—34no—
67#3774C2×D314yesdirect product factor
68C22×C33C2×D411yescharacteristic
69C2×C3×C3⋊D3—24no—
70C2×C3×C3⋊D3—44no—
71C2×C32×D3—24no—
72C2×C32×D3—44no—
73C2×C32×D3D814yescomplemented
74C2×C32×D3C2×D412yes—
75C3×D32—416no—
76C3⋊(C3⋊Dic3)C2×D412yes—
77C3×C3⋊Dic3C2×D411yescharacteristic
78C3×C3⋊Dic3C2411yescharacteristic
79C22×#67—182no—
80C2×D3×D4—92no—
81C2×C3×D8—91no—
82C2×#213—92no—
83D3×D8—98no—
84C2×D24—91no—
85C2×D3×C8—91no—
86C23×C32—31no—
87C23×C32—62no—
88C22×C3⋊D3—22no—
89C22×C3×D3—22no—
90C22×C3×D3—22no—
91C22×C3×D3—22no—
92C22×C3×D3—42no—
93C22×C3×D3—68no—
94C22×C3×D3—68no—
95C22×C3×D3—62no—
96C22×C3×D3—62no—
97C22×C3×D3—122no—
98C22×C3×D3—128no—
99C22×C3×D3—128no—
100C2×D32—28no—
101C2×D32—48no—
102#638—68no—
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—32no—
105C2×C3⋊Dic3—32no—
106C2×C3⋊Dic3—62no—
107C2×C3⋊Dic3—122no—
108C2×C3⋊Dic3C22×D311yescharacteristic
109C3×#67—64no—
110C3×#67—64no—
111C2×C3×Dic3—62no—
112#626—64no—
113#626—64no—
114#625—22no—
115#625—34no—
116#625—68no—
117#625—68no—
118#625—62no—
119#625—128no—
120#625C22×D314yescomplemented
121D3×Dic3—64no—
122D3×Dic3—64no—
123#622—32no—
124#622C22×D312yescomplemented
125C22×D8—271noSylow
126C2×C33C2×D812yescomplemented
127C2×C33C22×D411yescharacteristic
128C3×C3⋊D3—44no—
129C32×D3—44no—
130C32×D3C2×D814yescomplemented
131C24×C3—182no—
132C23×D3—62no—
133C23×D3—62no—
134C2×C3×D4—92no—
135C2×#67—62no—
136C2×#67—182no—
137C2×#67—188no—
138C2×#67—182no—
139C2×#67—188no—
140C2×#67—182no—
141C2×#67—182no—
142C2×#67—362no—
143C22×Dic3—182no—
144D3×D4—98no—
145D3×D4—184no—
146C2×D12—92no—
147C2×C4×D3—91no—
148C3×D8—94no—
149C2×C3×C8—91no—
150#213—98no—
151C2×C3⋊C8—91no—
152D24—94no—
153D3×C8—94no—
154C22×C32—22no—
155C22×C32—34no—
156C22×C32—32no—
157C22×C32—42no—
158C22×C32—64no—
159C22×C32—124no—
160C22×C32D3×D411yescharacteristic
161C2×C3⋊D3—24no—
162C2×C3⋊D3—44no—
163C2×C3×D3—24no—
164C2×C3×D3—24no—
165C2×C3×D3—24no—
166C2×C3×D3—28no—
167C2×C3×D3—44no—
168C2×C3×D3—44no—
169C2×C3×D3—44no—
170C2×C3×D3—64no—
171C2×C3×D3—64no—
172C2×C3×D3—84no—
173C2×C3×D3—124no—
174C2×C3×D3—1216no—
175C2×C3×D3—1216no—
176C2×C3×D3—124no—
177C2×C3×D3—124no—
178C2×C3×D3—244no—
179D32—416no—
180C3⋊Dic3—32no—
181C3⋊Dic3—64no—
182C3⋊Dic3—124no—
183C3⋊Dic3D3×D411yescharacteristic
184C3⋊Dic3C23×D311yescharacteristic
185C3×Dic3—62no—
186C3×Dic3—62no—
187C22×D4—272no—
188C2×D8—91no—
189C2×D8—271no—
190C2×D8—278no—
191C2×D8—272no—
192C22×C8—271no—
193C33C22×D811yescharacteristic, complemented, Sylow
194C23×C3—62no—
195C23×C3—62no—
196C23×C3—62no—
197C23×C3—122no—
198C23×C3—182no—
199C23×C3—188no—
200C23×C3—368no—
201C22×D3—22no—
202C22×D3—68no—
203C22×D3—62no—
204C22×D3—62no—
205C22×D3—68no—
206C22×D3—122no—
207C22×D3—128no—
208C22×D3—128no—
209C22×D3GO+(4,𝔽2)11yescharacteristic
210C3×D4—94no—
211C3×D4—182no—
212C2×C3×C4—91no—
213#67—64no—
214#67—64no—
215#67—184no—
216#67—184no—
217#67—188no—
218#67—188no—
219#67—188no—
220#67—368no—
221C2×Dic3—62no—
222C2×Dic3—91no—
223C2×Dic3—184no—
224C2×Dic3—184no—
225C2×Dic3—182no—
226C2×Dic3—362no—
227D12—94no—
228D12—182no—
229C4×D3—92no—
230C4×D3—92no—
231C3×C8—92no—
232C3⋊C8—92no—
233C2×C32—22no—
234C2×C32—24no—
235C2×C32—34no—
236C2×C32—44no—
237C2×C32—42no—
238C2×C32—124no—
239C2×C32D3×D812yescomplemented
240C2×C32C2×D3×D411yescharacteristic
241C3⋊D3—44no—
242C3×D3—28no—
243C3×D3—44no—
244C3×D3—44no—
245C3×D3—84no—
246C3×D3—124no—
247C3×D3—124no—
248C3×D3—244no—
249C24—182no—
250C2×D4—92no—
251C2×D4—272no—
252C2×D4—278no—
253C2×D4—544no—
254C2×D4—542no—
255C22×C4—271no—
256D8—94no—
257D8—274no—
258D8—278no—
259C2×C8—91no—
260C2×C8—274no—
261C2×C8—271no—
262C22×C3—22no—
263C22×C3—42no—
264C22×C3—64no—
265C22×C3—68no—
266C22×C3—64no—
267C22×C3—64no—
268C22×C3—124no—
269C22×C3—124no—
270C22×C3—124no—
271C22×C3—184no—
272C22×C3—244no—
273C22×C3—364no—
274C22×C3—3616no—
275C22×C3C2×GO+(4,𝔽2)11yescharacteristic
276C2×D3—24no—
277C2×D3—44no—
278C2×D3—64no—
279C2×D3—64no—
280C2×D3—124no—
281C2×D3—1216no—
282C2×D3—124no—
283C2×D3—124no—
284C2×D3—1216no—
285C2×D3—244no—
286C2×D3#377414yesdirect product factor
287C2×D3C2×GO+(4,𝔽2)12yes—
288C3×C4—91no—
289C3×C4—91no—
290Dic3—62no—
291Dic3—62no—
292Dic3—92no—
293Dic3—184no—
294Dic3—364no—
295C32—22no—
296C32—42no—
297C32C2×D3×D811yescharacteristic, complemented
298C23—31no—
299C23—62no—
300C23—188no—
301C23—182no—
302C23—368no—
303D4—94no—
304D4—182no—
305D4—274no—
306D4—542no—
307D4—548no—
308C2×C4—91no—
309C2×C4—272no—
310C2×C4—272no—
311C2×C4—271no—
312C8—92no—
313C8—272no—
314C2×C3—24no—
315C2×C3—22no—
316C2×C3—44no—
317C2×C3—42no—
318C2×C3—68no—
319C2×C3—124no—
320C2×C3—124no—
321C2×C3—244no—
322C2×C3—364no—
323C2×C3C2×#377412yescomplemented
324C2×C3C22×GO+(4,𝔽2)11yescharacteristic
325D3—44no—
326D3—124no—
327D3—124no—
328D3—244no—
329D3C2×#377414yesdirect product factor
330C22—34no—
331C22—32no—
332C22—64no—
333C22—124no—
334C22—184no—
335C22—3616no—
336C22—364no—
337C22D3×GO+(4,𝔽2)11yescharacteristic
338C4—91no—
339C4—91no—
340C4—272no—
341C3—22no—
342C3—42no—
343C3C22×#377411yescharacteristic, complemented
344C2—34no—
345C2—124no—
346C2—364no—
347C2D3×#377412yesdirect product factor
348C2C2×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
hypocenterC33C22×D8
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×D3×#625C2semidirect
2. C2×#88941C2semidirect
3. C2×#88935C2semidirect
4. C2×C3×#3774C2semidirect
5. C2×D3×#622C2semidirect
6. D3×#3774C2direct (N × H)
9. C2×#6295C22semidirect
12. C2×C3×#625C22semidirect
13. C2×C3⋊#622C22semidirect
14. C2×C3×#622C22semidirect
16. D3×#625C22semidirect
17. #88941C22semidirect
18. #88935C22semidirect
19. C3×#3774C22semidirect
20. D3×#622C22semidirect
27. C2×#3774D3direct (N × H)
36. #6295C23semidirect
41. C3×#625C23semidirect
42. C3⋊#622C23semidirect
43. C3×#622C23semidirect
60. C2×#625C2×D3semidirect
64. C2×#622C2×D3semidirect
67. #3774C2×D3direct (N × H)
73. C2×C32×D3D8semidirect
120. #625C22×D3semidirect
124. #622C22×D3semidirect
126. C2×C33C2×D8semidirect
130. C32×D3C2×D8semidirect
193. C33C22×D8semidirect
239. C2×C32D3×D8semidirect
286. C2×D3#3774direct (N × H)
297. C32C2×D3×D8semidirect
323. C2×C3C2×#3774semidirect
329. D3C2×#3774direct (N × H)
343. C3C22×#3774semidirect
347. C2D3×#3774direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×#6252
C2×#889412
C2×#889352
C2×C3×#37742
C2×D3×#6222
D3×#37742
C22×#37743
C2×D3×D89