Finite Group Databasebeta

← C2×#11724127 #13180571

Subgroups — C2×#11724127 #13180571

Order 1728.

1416 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 124 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×#88954C212yesmaximal, complemented
2C2×#88941C211yesmaximal, characteristic, complemented
3C2×#88935C212yesmaximal, complemented
4C2×C3×#3774C211yesmaximal, characteristic, complemented
5C2×#88924C211yesmaximal, characteristic, complemented
6#11724127C218yesmaximal, direct product factor
7C2×#64225—31nomaximal
8C22×D3×C3⋊D3—22no—
9C2×#6295C2212yescomplemented
10C2×#6289C2211yescharacteristic
11C2×C3×#626—22no—
12C2×C3×#625C2212yescomplemented
13C2×C3⋊#622C2211yescharacteristic, complemented
14C2×C3×#622C2211yescharacteristic, complemented
15#88954—24no—
16#88954C2218yescomplemented
17#88941C2214yescomplemented
18#88935C2218yescomplemented
19C3×#3774C2214yescomplemented
20#88924C2214yescomplemented
21C2×#3811—62no—
22C2×#3802—32no—
23C2×#3787—31no—
24C2×#3774—32no—
25C2×#3774—31no—
26C2×#3774D311yescharacteristic, complemented
27#64225—38no—
28C22×C3⋊(C3⋊D3)D411yescharacteristic
29C22×C3×C3⋊D3—22no—
30C22×C32×D3—22no—
31C2×D3×C3⋊D3—28no—
32C2×D3×C3⋊D3—48no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3D411yescharacteristic
35#6295—22no—
36#6295C2314yescomplemented
37#6289D412yes—
38#6289C2312yes—
39C3×#626—28no—
40C3×#625—22no—
41C3×#625C2314yescomplemented
42C3⋊#622C2312yescomplemented
43C3×#622C2312yescomplemented
44C2×D3×D8—91nomaximal
45C23×C3⋊D3—62no—
46C22×D32—62no—
47C22×D32—62no—
48C22×D32—122no—
49C2×C4×C3⋊D3—31no—
50C2×C3×#67—62no—
51C2×C3×#67—62no—
52C2×C3×D12—62no—
53#3811—68no—
54#3811—68no—
55C2×#626—22no—
56C2×#626—62no—
57C2×#625—32no—
58C2×#625C2×D312yescomplemented
59C2×#624—62no—
60#3802—38no—
61#3802—64no—
62C2×#622—31no—
63C2×#622C2×D311yescharacteristic, complemented
64#3787—34no—
65#3774—38no—
66#3774—34no—
67#3774C2×D314yescomplemented
68C22×C33C2×D411yescharacteristic
69C2×C3⋊(C3⋊D3)D814yescomplemented
70C2×C3⋊(C3⋊D3)C2×D412yes—
71C2×C3×C3⋊D3—24no—
72C2×C3×C3⋊D3—44no—
73C2×C32×D3—24no—
74C2×C32×D3—44no—
75D3×C3⋊D3—416no—
76C3×C3⋊Dic3C2×D411yescharacteristic
77C3×C3⋊Dic3C2411yescharacteristic
78C32×Dic3C2×D412yes—
79C2×D3×D4—92no—
80C2×D3×D4—182no—
81C2×C3×D8—91no—
82C2×#213—92no—
83D3×D8—98no—
84C2×D24—91no—
85C2×D3×C8—91no—
86C23×C32—62no—
87C22×C3⋊D3—22no—
88C22×C3⋊D3—31no—
89C22×C3⋊D3—62no—
90C22×C3⋊D3—68no—
91C22×C3⋊D3—122no—
92C22×C3⋊D3—128no—
93C22×C3×D3—22no—
94C22×C3×D3—22no—
95C22×C3×D3—42no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C22×C3×D3—122no—
99C2×D32—68no—
100C2×D32—68no—
101C2×D32—128no—
102C2×D32—128no—
103C2×D32—128no—
104C2×D32—1216no—
105C2×C32×C4—31no—
106C2×C3⋊Dic3C22×D311yescharacteristic
107C4×C3⋊D3—32no—
108C4×C3⋊D3—32no—
109C3×#67—64no—
110C3×#67—68no—
111C3×#67—64no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—62no—
114C3×D12—68no—
115#626—28no—
116#626—64no—
117#626—64no—
118#625—22no—
119#625—34no—
120#625—62no—
121#625C22×D314yescomplemented
122#624—64no—
123#624—64no—
124#622—32no—
125#622C22×D312yescomplemented
126C22×D8—271noSylow
127C2×C33C2×D812yescomplemented
128C2×C33C22×D411yescharacteristic
129C3⋊(C3⋊D3)C2×D814yescomplemented
130C3×C3⋊D3—44no—
131C32×D3—44no—
132C23×D3—182no—
133C23×D3—182no—
134C23×D3—182no—
135C23×D3—362no—
136C2×C3×D4—92no—
137C2×C3×D4—182no—
138C2×#67—62no—
139C2×#67—62no—
140C2×#67—182no—
141D3×D4—98no—
142D3×D4—188no—
143D3×D4—184no—
144D3×D4—188no—
145C2×D12—62no—
146C2×D12—92no—
147C2×C4×D3—91no—
148C2×C4×D3—182no—
149C3×D8—94no—
150C2×C3×C8—91no—
151#213—98no—
152C2×C3⋊C8—91no—
153D24—94no—
154D3×C8—94no—
155C22×C32—22no—
156C22×C32—42no—
157C22×C32—64no—
158C22×C32—124no—
159C22×C32D3×D411yescharacteristic
160C2×C3⋊D3—24no—
161C2×C3⋊D3—34no—
162C2×C3⋊D3—32no—
163C2×C3⋊D3—44no—
164C2×C3⋊D3—68no—
165C2×C3⋊D3—64no—
166C2×C3⋊D3—128no—
167C2×C3⋊D3—124no—
168C2×C3⋊D3—1216no—
169C2×C3×D3—24no—
170C2×C3×D3—24no—
171C2×C3×D3—44no—
172C2×C3×D3—44no—
173C2×C3×D3—48no—
174C2×C3×D3—44no—
175C2×C3×D3—64no—
176C2×C3×D3—64no—
177C2×C3×D3—124no—
178C2×C3×D3—124no—
179C2×C3×D3—128no—
180C2×C3×D3—124no—
181D32—1216no—
182D32—1216no—
183D32—1232no—
184C32×C4—32no—
185C3⋊Dic3D3×D411yescharacteristic
186C3⋊Dic3C23×D311yescharacteristic
187C3×Dic3—24no—
188C3×Dic3—62no—
189C3×Dic3—62no—
190C22×D4—272no—
191C2×D8—91no—
192C2×D8—271no—
193C2×D8—278no—
194C2×D8—272no—
195C22×C8—271no—
196C33C22×D811yescharacteristic, complemented, Sylow
197C23×C3—62no—
198C23×C3—62no—
199C23×C3—122no—
200C23×C3—182no—
201C22×D3—22no—
202C22×D3—62no—
203C22×D3—62no—
204C22×D3—91no—
205C22×D3—122no—
206C22×D3—182no—
207C22×D3—188no—
208C22×D3—188no—
209C22×D3—188no—
210C22×D3—362no—
211C22×D3—368no—
212C22×D3—368no—
213C22×D3—368no—
214C22×D3—368no—
215C22×D3—3616no—
216C3×D4—94no—
217C3×D4—188no—
218C3×D4—182no—
219C2×C3×C4—62no—
220C2×C3×C4—91no—
221#67—64no—
222#67—64no—
223#67—68no—
224#67—184no—
225#67—184no—
226C2×Dic3—62no—
227C2×Dic3GO+(4,𝔽2)11yescharacteristic
228D12—68no—
229D12—94no—
230D12—182no—
231C4×D3—92no—
232C4×D3—92no—
233C4×D3—184no—
234C4×D3—184no—
235C3×C8—92no—
236C3⋊C8—92no—
237C2×C32—22no—
238C2×C32—24no—
239C2×C32—44no—
240C2×C32—42no—
241C2×C32—124no—
242C2×C32D3×D812yescomplemented
243C2×C32C2×D3×D411yescharacteristic
244C3⋊D3—34no—
245C3⋊D3—44no—
246C3⋊D3—68no—
247C3⋊D3—128no—
248C3×D3—44no—
249C3×D3—48no—
250C3×D3—44no—
251C3×D3—124no—
252C3×D3—128no—
253C3×D3—124no—
254C24—542no—
255C2×D4—92no—
256C2×D4—182no—
257C2×D4—272no—
258C2×D4—278no—
259C2×D4—544no—
260C22×C4—271no—
261D8—94no—
262D8—274no—
263D8—278no—
264C2×C8—91no—
265C2×C8—274no—
266C2×C8—271no—
267C22×C3—22no—
268C22×C3—42no—
269C22×C3—64no—
270C22×C3—64no—
271C22×C3—124no—
272C22×C3—124no—
273C22×C3—128no—
274C22×C3—124no—
275C22×C3—184no—
276C22×C3—364no—
277C22×C3C2×GO+(4,𝔽2)11yescharacteristic
278C2×D3—24no—
279C2×D3—44no—
280C2×D3—64no—
281C2×D3—64no—
282C2×D3—92no—
283C2×D3—94no—
284C2×D3—124no—
285C2×D3—124no—
286C2×D3—128no—
287C2×D3—124no—
288C2×D3—184no—
289C2×D3—188no—
290C2×D3—368no—
291C2×D3—364no—
292C2×D3—3616no—
293C2×D3—3616no—
294C2×D3—3616no—
295C2×D3—3632no—
296C3×C4—64no—
297C3×C4—91no—
298C3×C4—91no—
299Dic3—62no—
300Dic3—62no—
301Dic3C2×GO+(4,𝔽2)12yes—
302C32—22no—
303C32—42no—
304C32C2×D3×D811yescharacteristic, complemented
305C23—62no—
306C23—182no—
307C23—271no—
308C23—548no—
309C23—1088no—
310D4—94no—
311D4—188no—
312D4—182no—
313D4—274no—
314D4—542no—
315C2×C4—31no—
316C2×C4—91no—
317C2×C4—272no—
318C2×C4—272no—
319C8—92no—
320C8—272no—
321C2×C3—24no—
322C2×C3—22no—
323C2×C3—44no—
324C2×C3—42no—
325C2×C3—124no—
326C2×C3—128no—
327C2×C3—124no—
328C2×C3—364no—
329C2×C3#6422512yescomplemented
330C2×C3C22×GO+(4,𝔽2)11yescharacteristic
331D3—44no—
332D3—94no—
333D3—124no—
334D3—128no—
335D3—124no—
336D3—188no—
337D3—368no—
338C22—64no—
339C22—124no—
340C22—184no—
341C22—274no—
342C22—272no—
343C22—364no—
344C22—10816no—
345C22D3×GO+(4,𝔽2)11yescharacteristic
346C4—32no—
347C4—91no—
348C4—91no—
349C3—22no—
350C3—42no—
351C3C2×#6422511yescharacteristic, complemented
352C2—124no—
353C2—274no—
354C2—364no—
355C2#1172412712yesdirect product factor
356C2C2×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×#88954C2semidirect
2. C2×#88941C2semidirect
3. C2×#88935C2semidirect
4. C2×C3×#3774C2semidirect
5. C2×#88924C2semidirect
6. #11724127C2direct (N × H)
9. C2×#6295C22semidirect
12. C2×C3×#625C22semidirect
13. C2×C3⋊#622C22semidirect
14. C2×C3×#622C22semidirect
16. #88954C22semidirect
17. #88941C22semidirect
18. #88935C22semidirect
19. C3×#3774C22semidirect
20. #88924C22semidirect
26. C2×#3774D3semidirect
36. #6295C23semidirect
41. C3×#625C23semidirect
42. C3⋊#622C23semidirect
43. C3×#622C23semidirect
58. C2×#625C2×D3semidirect
63. C2×#622C2×D3semidirect
67. #3774C2×D3semidirect
69. C2×C3⋊(C3⋊D3)D8semidirect
121. #625C22×D3semidirect
125. #622C22×D3semidirect
127. C2×C33C2×D8semidirect
129. C3⋊(C3⋊D3)C2×D8semidirect
196. C33C22×D8semidirect
242. C2×C32D3×D8semidirect
304. C32C2×D3×D8semidirect
329. C2×C3#64225semidirect
351. C3C2×#64225semidirect
355. C2#11724127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889542
C2×#889412
C2×#889352
C2×C3×#37742
C2×#889242
#117241272
C2×#642253
C2×D3×D89