Finite Group Databasebeta

← #10596301⋊C3 #13153720

Subgroups — #10596301⋊C3 #13153720

Order 1728.

1333 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3#64156⋊C3C212yesmaximal, complemented
4#64149⋊C3C212yesmaximal, complemented
5#64146⋊C3C214yesmaximal, complemented
6#64139⋊C3C214yesmaximal, complemented
7#64135⋊C3C211yesmaximal, characteristic, complemented
8#10596301C311yesmaximal, characteristic, complemented
9C3×#5830—31nomaximal
10C23×(C3⋊D3)⋊C3C2212yes—
11#41×H(3,𝔽3)—24no—
12C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
13C2×D4×H(3,𝔽3)C2212yes—
14C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C22×C4×H(3,𝔽3)C2212yescomplemented
17C2×#638⋊C3C2218yescomplemented
18C2×C4⋊((C3⋊D3)⋊C3)—24no—
19C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
20C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
21#3757⋊C3C2212yes—
22#3752⋊C3C2218yescomplemented
23C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
24C4×(C3⋊Dic3)⋊C3C2212yescomplemented
25C2×C32×#41—21no—
26C2×C32×#41D311yescharacteristic, complemented
27C2×C32×#41C2×C311yescharacteristic, complemented
28C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
29C22×C3×D12—31no—
30#64156C2×C312yescomplemented
31#64149C2×C312yescomplemented
32#64146C2×C314yescomplemented
33#64139C2×C314yescomplemented
34#64135C2×C311yescharacteristic, complemented
35C3×#947—32no—
36C3×#940—32no—
37C3×#937—34no—
38C3×#930—34no—
39C3×#926—31no—
40C23×H(3,𝔽3)C2312yes—
41C23×H(3,𝔽3)C2311yescharacteristic
42C22×(C3⋊D3)⋊C3—24no—
43C22×(C3⋊D3)⋊C3—44no—
44C22×(C3⋊D3)⋊C3C2314yes—
45Q8×H(3,𝔽3)—22no—
46D4×H(3,𝔽3)—24no—
47D4×H(3,𝔽3)—22no—
48C2×C4×H(3,𝔽3)—24no—
49C2×C4×H(3,𝔽3)D414yescomplemented
50C2×C4×H(3,𝔽3)C2312yes—
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53#638⋊C3—48no—
54C2×(C3⋊Dic3)⋊C3C2314yes—
55C4⋊((C3⋊D3)⋊C3)—24no—
56C4⋊((C3⋊D3)⋊C3)—24no—
57C3×#535—91no—
58#5830—31no—
59#5830—91no—
60C23×C3⋊D3C22×C312yes—
61C23×C3×D3—32no—
62C32×#41—28no—
63C32×#41—24no—
64C32×#41—24no—
65C2×C32×Q8—21no—
66C2×C32×Q8C2×D311yescharacteristic, complemented
67C2×C32×Q8C22×C311yescharacteristic, complemented
68C2×C32×D4—22no—
69C2×C32×D4—21no—
70C2×C32×D4C2×D312yes—
71C2×C32×D4C2×D311yescharacteristic, complemented
72C2×C32×D4C22×C312yes—
73C2×C32×D4C22×C311yescharacteristic, complemented
74C22×C32×C4—21no—
75C22×C32×C4—22no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C2×D312yescomplemented
78C22×C32×C4C22×C311yescharacteristic
79C22×C32×C4C22×C312yescomplemented
80C2×#638C22×C318yescomplemented
81C2×C4⋊(C3⋊D3)—24no—
82C2×C4⋊(C3⋊D3)C22×C312yes—
83C2×C4⋊(C3⋊D3)C22×C312yes—
84C2×C3×#67—38no—
85C2×C3×D12—32no—
86C2×C3×D12—32no—
87C2×C3×D12—64no—
88#3757C22×C312yes—
89#3752C22×C318yescomplemented
90C4⋊(C3⋊Dic3)C22×C312yescomplemented
91C4×C3⋊Dic3C22×C312yescomplemented
92C3×#217—32no—
93C3×(C2×D3)⋊C4—38no—
94C3×C4⋊Dic3—32no—
95C3×C4×Dic3—32no—
96C22×H(3,𝔽3)—22no—
97C22×H(3,𝔽3)—22no—
98C22×H(3,𝔽3)—24no—
99C22×H(3,𝔽3)C2×D412yes—
100C22×H(3,𝔽3)C2411yescharacteristic
101C2×(C3⋊D3)⋊C3—24no—
102C2×(C3⋊D3)⋊C3—28no—
103C2×(C3⋊D3)⋊C3—48no—
104C4×H(3,𝔽3)—22no—
105C4×H(3,𝔽3)C2×D412yes—
106C4×H(3,𝔽3)C2×D412yes—
107(C3⋊Dic3)⋊C3—24no—
108C2×C3×#41—31no—
109C2×C3×#41—31no—
110C2×C3×#41—61no—
111C2×C3×#41C3×D311yescharacteristic
112C22×C3×D4—91no—
113C22×D12—31no—
114C22×D12—91no—
115C3×#127—92no—
116C3×#124—92no—
117C3×#121—94no—
118C3×#120—94no—
119C3×#117—91no—
120#947—32no—
121#947—92no—
122#940—32no—
123#940—92no—
124#937—34no—
125#937—94no—
126#930—34no—
127#930—94no—
128#926—31no—
129#926—91no—
130C23×C32—22no—
131C23×C32—21no—
132C23×C32C22×D312yes—
133C23×C32C22×D311yescharacteristic
134C23×C32C23×C312yes—
135C23×C32C23×C311yescharacteristic
136C22×C3⋊D3—24no—
137C22×C3⋊D3—44no—
138C22×C3⋊D3C23×C314yes—
139C22×C3×D3—34no—
140C22×C3×D3—64no—
141C22×C3×D3—124no—
142C32×Q8—24no—
143C32×Q8—22no—
144C32×Q8—22no—
145C32×D4—28no—
146C32×D4—24no—
147C32×D4—24no—
148C32×D4—24no—
149C32×D4—22no—
150C32×D4—22no—
151C2×C32×C4—28no—
152C2×C32×C4—24no—
153C2×C32×C4—24no—
154C2×C32×C4—24no—
155C2×C32×C4—21no—
156C2×C32×C4—22no—
157C2×C32×C4—21no—
158C2×C32×C4#6714yescomplemented
159C2×C32×C4C3×D414yescomplemented
160C2×C32×C4C22×D312yes—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C23×C312yes—
164C2×C32×C4C23×C311yescharacteristic
165C2×C32×C4C23×C311yescharacteristic
166#638—48no—
167C2×C3⋊Dic3C23×C314yes—
168C4⋊(C3⋊D3)—24no—
169C4⋊(C3⋊D3)—24no—
170C3×#67—128no—
171C2×C3×Dic3—34no—
172C3×D12—64no—
173C3×D12—64no—
174#535—91noSylow
175C2×H(3,𝔽3)—22no—
176C2×H(3,𝔽3)—42no—
177C2×H(3,𝔽3)C22×D411yescharacteristic
178C2×H(3,𝔽3)H(5,𝔽2)12yes—
179(C3⋊D3)⋊C3—44no—
180C24×C3—92no—
181C23×D3—32no—
182C23×D3—92no—
183C3×#41—24no—
184C3×#41—64no—
185C3×#41—64no—
186C3×#41—68no—
187C2×C3×Q8—31no—
188C2×C3×Q8—31no—
189C2×C3×Q8—61no—
190C2×C3×Q8C2×C3×D311yescharacteristic
191C2×C3×D4—32no—
192C2×C3×D4—32no—
193C2×C3×D4—31no—
194C2×C3×D4—31no—
195C2×C3×D4—62no—
196C2×C3×D4—61no—
197C2×C3×D4—92no—
198C2×C3×D4—92no—
199C2×C3×D4—98no—
200C2×C3×D4—184no—
201C2×C3×D4C2×C3×D312yes—
202C2×C3×D4C2×C3×D311yescharacteristic
203C22×C3×C4—31no—
204C22×C3×C4—31no—
205C22×C3×C4—32no—
206C22×C3×C4—32no—
207C22×C3×C4—61no—
208C22×C3×C4—62no—
209C22×C3×C4C2×C3×D311yescharacteristic
210C22×C3×C4C2×C3×D312yes—
211C2×#67—38no—
212C2×#67—98no—
213C2×D12—32no—
214C2×D12—32no—
215C2×D12—64no—
216C2×D12—92no—
217C2×D12—92no—
218C2×D12—184no—
219C3×C4⋊C4—92no—
220C3×C22⋊C4—98no—
221C3×C22⋊C4—92no—
222C3×C42—92no—
223#217—32no—
224#217—92no—
225(C2×D3)⋊C4—38no—
226(C2×D3)⋊C4—98no—
227C4⋊Dic3—32no—
228C4⋊Dic3—92no—
229C4×Dic3—32no—
230C4×Dic3—92no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—24no—
234C22×C32—24no—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—24no—
238C22×C32—21no—
239C22×C32—22no—
240C22×C32—42no—
241C22×C32—44no—
242C22×C32C2×#6712yes—
243C22×C32C2×C3×D412yes—
244C22×C32C23×D311yescharacteristic
245C22×C32C24×C311yescharacteristic
246C2×C3⋊D3—24no—
247C2×C3⋊D3—28no—
248C2×C3⋊D3—48no—
249C2×C3×D3—68no—
250C2×C3×D3—64no—
251C2×C3×D3—128no—
252C32×C4—24no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4C2×#6712yes—
258C32×C4C2×#6712yes—
259C32×C4C2×C3×D412yes—
260C32×C4C2×C3×D412yes—
261C3⋊Dic3—24no—
262C3×Dic3—64no—
263C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
264C22×D4—91no—
265#127—92no—
266#124—92no—
267#121—94no—
268#120—94no—
269#117—91no—
270H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
271C23×C3—31no—
272C23×C3—31no—
273C23×C3—32no—
274C23×C3—32no—
275C23×C3—62no—
276C23×C3—61no—
277C23×C3—94no—
278C23×C3—184no—
279C23×C3—364no—
280C23×C3C22×C3×D312yes—
281C23×C3C22×C3×D311yescharacteristic
282C22×D3—34no—
283C22×D3—64no—
284C22×D3—94no—
285C22×D3—124no—
286C22×D3—184no—
287C22×D3—364no—
288C3×Q8—22no—
289C3×Q8—62no—
290C3×Q8—62no—
291C3×Q8—64no—
292C3×D4—24no—
293C3×D4—22no—
294C3×D4—62no—
295C3×D4—62no—
296C3×D4—64no—
297C3×D4—64no—
298C3×D4—68no—
299C3×D4—64no—
300C3×D4—184no—
301C3×D4—184no—
302C3×D4—368no—
303C2×C3×C4—24no—
304C2×C3×C4—34no—
305C2×C3×C4—34no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—32no—
311C2×C3×C4—32no—
312C2×C3×C4—64no—
313C2×C3×C4—64no—
314C2×C3×C4—68no—
315C2×C3×C4—64no—
316C2×C3×C4—61no—
317C2×C3×C4—62no—
318C2×C3×C4—61no—
319C2×C3×C4—94no—
320C2×C3×C4C3×#6714yes—
321C2×C3×C4C22×C3×D312yes—
322C2×C3×C4C22×C3×D311yescharacteristic
323C2×C3×C4C22×C3×D311yescharacteristic
324#67—128no—
325#67—368no—
326C2×Dic3—34no—
327C2×Dic3—94no—
328D12—64no—
329D12—64no—
330D12—184no—
331D12—184no—
332C2×C32—22no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—21no—
336C2×C32—42no—
337C2×C32—44no—
338C2×C32—42no—
339C2×C32—42no—
340C2×C32#100912yes—
341C2×C32C22×#6711yescharacteristic
342C2×C32C22×C3×D411yescharacteristic
343C2×C32C3×H(5,𝔽2)12yes—
344C3⋊D3—44no—
345C3×D3—124no—
346C24—92no—
347#41—24no—
348C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
349C2×D4—98no—
350C2×D4—92no—
351C2×D4—92no—
352C2×D4—184no—
353C2×D4C2×(C3⋊D3)⋊C312yes—
354C2×D4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
355C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
356C22×C4C2×(C3⋊D3)⋊C312yescomplemented
357C4⋊C4—92no—
358C22⋊C4—92no—
359C22⋊C4—98no—
360C42—92no—
361C22×C3—22no—
362C22×C3—24no—
363C22×C3—22no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—32no—
367C22×C3—32no—
368C22×C3—62no—
369C22×C3—62no—
370C22×C3—64no—
371C22×C3—62no—
372C22×C3—62no—
373C22×C3—64no—
374C22×C3—64no—
375C22×C3—61no—
376C22×C3—62no—
377C22×C3—122no—
378C22×C3—124no—
379C22×C3—184no—
380C22×C3—188no—
381C22×C3—368no—
382C22×C3C2×C3×#6712yes—
383C22×C3C23×C3×D311yescharacteristic
384C2×D3—68no—
385C2×D3—64no—
386C2×D3—128no—
387C2×D3—184no—
388C2×D3—188no—
389C2×D3—368no—
390C3×C4—22no—
391C3×C4—32no—
392C3×C4—32no—
393C3×C4—32no—
394C3×C4—32no—
395C3×C4—62no—
396C3×C4—62no—
397C3×C4—64no—
398C3×C4—62no—
399C3×C4—62no—
400C3×C4—184no—
401C3×C4C2×C3×#6712yes—
402C3×C4C2×C3×#6712yes—
403Dic3—64no—
404Dic3—184no—
405C32—21no—
406C32#583011yescharacteristic, complemented
407C32C3×#53511yescharacteristic, complemented
408C23—94no—
409C23—184no—
410C23—364no—
411C23C22×(C3⋊D3)⋊C312yes—
412C23C22×(C3⋊D3)⋊C311yescharacteristic
413Q8—22no—
414D4—24no—
415D4—22no—
416D4—184no—
417D4—184no—
418D4—368no—
419C2×C4—24no—
420C2×C4—94no—
421C2×C4#638⋊C314yescomplemented
422C2×C4C22×(C3⋊D3)⋊C312yes—
423C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
424C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
425C2×C3—22no—
426C2×C3—32no—
427C2×C3—32no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—42no—
431C2×C3—62no—
432C2×C3—62no—
433C2×C3—62no—
434C2×C3—61no—
435C2×C3—122no—
436C2×C3—122no—
437C2×C3—122no—
438C2×C3—124no—
439C2×C3—364no—
440C2×C3C3×#100912yes—
441C2×C3C22×C3×#6711yescharacteristic
442D3—124no—
443D3—364no—
444C22—22no—
445C22—24no—
446C22—22no—
447C22—188no—
448C22—184no—
449C22—368no—
450C22C2×#638⋊C312yes—
451C22C23×(C3⋊D3)⋊C311yescharacteristic
452C4—22no—
453C4—184no—
454C4C2×#638⋊C312yes—
455C4C2×#638⋊C312yes—
456C3—31no—
457C3—31no—
458C3—61no—
459C3C3×#583011yescharacteristic
460C2—22no—
461C2—42no—
462C2—364no—
463C2#1172263812yes—
464C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×#41×H(3,𝔽3)C2
hypercenterC2×#41(C3⋊D3)⋊C3
hypocenterC32C3×#535
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×H(3,𝔽3)C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #64156⋊C3C2semidirect
4. #64149⋊C3C2semidirect
5. #64146⋊C3C2semidirect
6. #64139⋊C3C2semidirect
7. #64135⋊C3C2semidirect
8. #10596301C3semidirect
12. C2×Q8×H(3,𝔽3)C22semidirect
14. C2×D4×H(3,𝔽3)C22semidirect
16. C22×C4×H(3,𝔽3)C22semidirect
17. C2×#638⋊C3C22semidirect
22. #3752⋊C3C22semidirect
23. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
24. C4×(C3⋊Dic3)⋊C3C22semidirect
26. C2×C32×#41D3semidirect
27. C2×C32×#41C2×C3semidirect
28. C22×C4⋊(C3⋊D3)C2×C3semidirect
30. #64156C2×C3semidirect
31. #64149C2×C3semidirect
32. #64146C2×C3semidirect
33. #64139C2×C3semidirect
34. #64135C2×C3semidirect
49. C2×C4×H(3,𝔽3)D4semidirect
66. C2×C32×Q8C2×D3semidirect
67. C2×C32×Q8C22×C3semidirect
71. C2×C32×D4C2×D3semidirect
73. C2×C32×D4C22×C3semidirect
77. C22×C32×C4C2×D3semidirect
79. C22×C32×C4C22×C3semidirect
80. C2×#638C22×C3semidirect
89. #3752C22×C3semidirect
90. C4⋊(C3⋊Dic3)C22×C3semidirect
91. C4×C3⋊Dic3C22×C3semidirect
158. C2×C32×C4#67semidirect
159. C2×C32×C4C3×D4semidirect
263. C2×#41(C3⋊D3)⋊C3semidirect
270. H(3,𝔽3)#535semidirect
348. C2×Q8C2×(C3⋊D3)⋊C3semidirect
354. C2×D4C2×(C3⋊D3)⋊C3semidirect
356. C22×C4C2×(C3⋊D3)⋊C3semidirect
406. C32#5830semidirect
407. C32C3×#535semidirect
421. C2×C4#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#41×H(3,𝔽3)2
C22×C4⋊((C3⋊D3)⋊C3)2
#64156⋊C32
#64149⋊C32
#64146⋊C32
#64139⋊C32
#64135⋊C32
#105963013
C3×#58303