Finite Group Databasebeta

← #10596189⋊C3 #13153271

Subgroups — #10596189⋊C3 #13153271

Order 1728.

1200 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3830⋊C3C212yesmaximal, complemented
2H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
3#64149⋊C3C212yesmaximal, complemented
4#64148⋊C3C214yesmaximal, complemented
5#64147⋊C3C212yesmaximal, complemented
6D4×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
7C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
8C4×#634⋊C3C211yesmaximal, characteristic, complemented
9#10596189C311yesmaximal, characteristic, complemented
10C3×#5718—31nomaximal
11C2×D4×H(3,𝔽3)C2212yes—
12C2×D4×H(3,𝔽3)C2212yes—
13C2×D4×H(3,𝔽3)C2212yes—
14C2×#638⋊C3C2214yes—
15C2×#638⋊C3C2214yes—
16C22×(C3⋊Dic3)⋊C3C2214yes—
17#3830⋊C3—28no—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3C2212yes—
20C2×#634⋊C3C2211yescharacteristic
21C42×H(3,𝔽3)C2211yescharacteristic
22#3757⋊C3C2214yes—
23#3752⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2212yes—
28C2×#3830C2×C312yescomplemented
29C2×C3×#237—32no—
30C32×#127—21no—
31C32×#127D311yescharacteristic, complemented
32C32×#127C2×C311yescharacteristic, complemented
33#64149C2×C312yescomplemented
34#64148C2×C314yescomplemented
35#64147C2×C312yescomplemented
36D4×C3⋊Dic3C2×C312yescomplemented
37C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
38C4×#634C2×C311yescharacteristic, complemented
39C3×#940—32no—
40C3×#939—34no—
41C3×#938—32no—
42C3×D4×Dic3—32no—
43C3×C4×D12—31no—
44C3×C4×Dic6—31no—
45C23×H(3,𝔽3)C2314yes—
46C22×(C3⋊D3)⋊C3C2312yes—
47D4×H(3,𝔽3)—24no—
48D4×H(3,𝔽3)—24no—
49D4×H(3,𝔽3)—42no—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53#638⋊C3—28no—
54#638⋊C3—44no—
55C2×(C3⋊Dic3)⋊C3—28no—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C2×(C3⋊Dic3)⋊C3C2312yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C4⋊((C3⋊D3)⋊C3)—22no—
60C4×(C3⋊D3)⋊C3—24no—
61#634⋊C3D414yescomplemented
62C3×#550—91no—
63#5718—31no—
64#5718—91no—
65C2×C32×D4—22no—
66C2×C32×D4—22no—
67C2×C32×D4—22no—
68C2×C32×D4C2×D312yes—
69C2×C32×D4C2×D312yes—
70C2×C32×D4C2×D312yes—
71C2×C32×D4C22×C312yes—
72C2×C32×D4C22×C312yes—
73C2×C32×D4C22×C312yes—
74C2×#638C22×C314yes—
75C2×#638C22×C314yes—
76C22×C3⋊Dic3C22×C314yes—
77#3830—28no—
78C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
79C2×C4×C3⋊D3C22×C312yes—
80C2×#634C22×C311yescharacteristic
81C2×C3×#67—34no—
82C2×C3×#67—34no—
83C22×C3×Dic3—34no—
84C3×#237—68no—
85C2×C3×D12—31no—
86C2×C3×C4×D3—32no—
87C2×C3×Dic6—31no—
88C32×C42—21no—
89C32×C42C2×D311yescharacteristic
90C32×C42C22×C311yescharacteristic
91#3757C22×C314yes—
92#3752C22×C312yes—
93C4⋊(C3⋊Dic3)C22×C311yescharacteristic
94C4⋊(C3⋊Dic3)C22×C311yescharacteristic
95#3750C22×C312yes—
96C4×C3⋊Dic3C22×C312yes—
97C3×#217—34no—
98C3×(C2×D3)⋊C4—32no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×Dic3⋊C4—32no—
102C3×C4×Dic3—32no—
103C22×H(3,𝔽3)—24no—
104C22×H(3,𝔽3)—24no—
105C22×H(3,𝔽3)—24no—
106C22×H(3,𝔽3)C2411yescharacteristic
107C2×(C3⋊D3)⋊C3—22no—
108C2×(C3⋊D3)⋊C3—22no—
109C2×(C3⋊D3)⋊C3—22no—
110C4×H(3,𝔽3)—21no—
111C4×H(3,𝔽3)C2×D412yes—
112C4×H(3,𝔽3)#4112yescomplemented
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3C2×D414yes—
116C2×C3×#41—92no—
117C2×#237—32no—
118C2×#237—92no—
119C3×#127—31no—
120C3×#127—31no—
121C3×#127—61no—
122C3×#127—92no—
123C3×#127C3×D311yescharacteristic
124C3×#121—92no—
125C3×#121—94no—
126C3×C4×Q8—91no—
127C3×C4×D4—91no—
128C3×C4×D4—92no—
129#940—32no—
130#940—92no—
131#939—34no—
132#939—94no—
133#938—32no—
134#938—92no—
135D4×Dic3—32no—
136D4×Dic3—92no—
137C4×D12—31no—
138C4×D12—91no—
139C4×Dic6—31no—
140C4×Dic6—91no—
141C23×C32—24no—
142C23×C32C22×D314yes—
143C23×C32C23×C314yes—
144C22×C3⋊D3C23×C312yes—
145C22×C3×D3—32no—
146C32×D4—24no—
147C32×D4—24no—
148C32×D4—24no—
149C32×D4—24no—
150C32×D4—44no—
151C32×D4—44no—
152C32×D4—42no—
153C32×D4—42no—
154C32×D4—44no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C23×C311yescharacteristic
162C2×C32×C4C23×C311yescharacteristic
163C2×C32×C4C23×C311yescharacteristic
164#638—28no—
165#638—44no—
166C2×C3⋊Dic3—28no—
167C2×C3⋊Dic3C23×C312yes—
168C2×C3⋊Dic3C23×C312yes—
169C2×C3⋊Dic3C23×C312yes—
170C4⋊(C3⋊D3)—22no—
171C4×C3⋊D3—24no—
172#634C3×D414yescomplemented
173C3×#67—68no—
174C3×#67—124no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—68no—
179C3×D12—62no—
180C3×C4×D3—64no—
181C3×Dic6—34no—
182#550—91noSylow
183C2×H(3,𝔽3)—44no—
184C2×H(3,𝔽3)C22×D411yescharacteristic
185C2×H(3,𝔽3)C2×#4111yescharacteristic
186C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
187(C3⋊D3)⋊C3—42no—
188C3×#41—188no—
189C2×C3×Q8—91no—
190C2×C3×D4—32no—
191C2×C3×D4—32no—
192C2×C3×D4—32no—
193C2×C3×D4—32no—
194C2×C3×D4—32no—
195C2×C3×D4—32no—
196C2×C3×D4—62no—
197C2×C3×D4—62no—
198C2×C3×D4—62no—
199C2×C3×D4—91no—
200C2×C3×D4—94no—
201C2×C3×D4—94no—
202C2×C3×D4C2×C3×D312yes—
203C2×C3×D4C2×C3×D312yes—
204C2×C3×D4C2×C3×D312yes—
205C22×C3×C4—92no—
206C22×C3×C4—94no—
207C2×#67—34no—
208C2×#67—34no—
209C2×#67—94no—
210C2×#67—94no—
211C22×Dic3—34no—
212C22×Dic3—94no—
213#237—68no—
214#237—188no—
215C2×D12—31no—
216C2×D12—91no—
217C2×C4×D3—32no—
218C2×C4×D3—92no—
219C2×Dic6—31no—
220C2×Dic6—91no—
221C3×C4⋊C4—92no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4—91no—
224C3×C22⋊C4—92no—
225C3×C22⋊C4—94no—
226C3×C42—31no—
227C3×C42—31no—
228C3×C42—61no—
229C3×C42—92no—
230C3×C42C2×C3×D311yescharacteristic
231#217—34no—
232#217—94no—
233(C2×D3)⋊C4—32no—
234(C2×D3)⋊C4—92no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—91no—
238C4⋊Dic3—91no—
239Dic3⋊C4—32no—
240Dic3⋊C4—92no—
241C4×Dic3—32no—
242C4×Dic3—92no—
243C22×C32—24no—
244C22×C32—24no—
245C22×C32—24no—
246C22×C32—24no—
247C22×C32—24no—
248C22×C32—24no—
249C22×C32—21no—
250C22×C32—44no—
251C22×C32—44no—
252C22×C32—44no—
253C22×C32C23×D311yescharacteristic
254C22×C32C24×C311yescharacteristic
255C2×C3⋊D3—22no—
256C2×C3⋊D3—22no—
257C2×C3⋊D3—22no—
258C2×C3×D3—62no—
259C2×C3×D3—62no—
260C2×C3×D3—62no—
261C32×C4—22no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—22no—
265C32×C4—22no—
266C32×C4D3×D412yes—
267C32×C4#23712yescomplemented
268C32×C4C2×C3×D412yes—
269C32×C4C3×#4112yescomplemented
270C3⋊Dic3—22no—
271C3⋊Dic3—22no—
272C3⋊Dic3C2×C3×D414yes—
273C3×Dic3—34no—
274C3×Dic3—62no—
275C3×Dic3—62no—
276C2×#41—92no—
277#127—92no—
278#127(C3⋊D3)⋊C311yescharacteristic, complemented
279#121—92no—
280#121—94no—
281C4×Q8—91no—
282C4×D4—92no—
283C4×D4—91no—
284H(3,𝔽3)#55011yescharacteristic, complemented, Sylow
285C23×C3—34no—
286C23×C3—34no—
287C23×C3—64no—
288C23×C3—92no—
289C23×C3C22×C3×D314yes—
290C22×D3—32no—
291C22×D3—92no—
292C3×Q8—94no—
293C3×D4—24no—
294C3×D4—24no—
295C3×D4—42no—
296C3×D4—64no—
297C3×D4—64no—
298C3×D4—64no—
299C3×D4—64no—
300C3×D4—124no—
301C3×D4—124no—
302C3×D4—122no—
303C3×D4—122no—
304C3×D4—124no—
305C3×D4—182no—
306C3×D4—188no—
307C3×D4—364no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—92no—
318C2×C3×C4—92no—
319C2×C3×C4—92no—
320C2×C3×C4—184no—
321C2×C3×C4—188no—
322C2×C3×C4C22×C3×D311yescharacteristic
323C2×C3×C4C22×C3×D311yescharacteristic
324C2×C3×C4C22×C3×D311yescharacteristic
325#67—68no—
326#67—124no—
327#67—188no—
328#67—364no—
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—32no—
332C2×Dic3—68no—
333C2×Dic3—92no—
334C2×Dic3—92no—
335C2×Dic3—92no—
336C2×Dic3—188no—
337D12—62no—
338D12—182no—
339C4×D3—64no—
340C4×D3—184no—
341Dic6—34no—
342Dic6—94no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—44no—
347C2×C32—44no—
348C2×C32—84no—
349C2×C32C2×D3×D411yescharacteristic
350C2×C32C2×#23711yescharacteristic
351C2×C32#100411yescharacteristic
352C2×C32C22×C3×D411yescharacteristic
353C2×C32C2×C3×#4111yescharacteristic
354C2×C32C3×H(5,𝔽2)11yescharacteristic
355C3⋊D3—42no—
356C3×D3—122no—
357#41—188no—
358C2×Q8—91no—
359C2×D4—94no—
360C2×D4—94no—
361C2×D4—91no—
362C2×D4C2×(C3⋊D3)⋊C312yes—
363C2×D4C2×(C3⋊D3)⋊C312yes—
364C2×D4C2×(C3⋊D3)⋊C312yes—
365C22×C4—94no—
366C22×C4—92no—
367C4⋊C4—91no—
368C4⋊C4—92no—
369C4⋊C4—91no—
370C22⋊C4—94no—
371C22⋊C4—92no—
372C42—92no—
373C42C2×(C3⋊D3)⋊C311yescharacteristic
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—64no—
380C22×C3—64no—
381C22×C3—64no—
382C22×C3—64no—
383C22×C3—64no—
384C22×C3—64no—
385C22×C3—61no—
386C22×C3—124no—
387C22×C3—124no—
388C22×C3—124no—
389C22×C3—182no—
390C22×C3—182no—
391C22×C3—182no—
392C22×C3C23×C3×D311yescharacteristic
393C2×D3—62no—
394C2×D3—62no—
395C2×D3—62no—
396C2×D3—182no—
397C2×D3—182no—
398C2×D3—182no—
399C3×C4—21no—
400C3×C4—32no—
401C3×C4—32no—
402C3×C4—32no—
403C3×C4—32no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—62no—
407C3×C4—62no—
408C3×C4—62no—
409C3×C4—94no—
410C3×C4—182no—
411C3×C4—182no—
412C3×C4C3×D3×D412yes—
413C3×C4C3×#23712yes—
414Dic3—34no—
415Dic3—62no—
416Dic3—62no—
417Dic3—94no—
418Dic3—182no—
419Dic3—182no—
420C32—21no—
421C32#571811yescharacteristic, complemented
422C32C3×#55011yescharacteristic, complemented
423C23—92no—
424C23C22×(C3⋊D3)⋊C314yes—
425Q8—94no—
426D4—24no—
427D4—24no—
428D4—42no—
429D4—188no—
430D4—182no—
431D4—364no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—92no—
435C2×C4—188no—
436C2×C4—184no—
437C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
438C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
439C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—44no—
447C2×C3—61no—
448C2×C3—61no—
449C2×C3—61no—
450C2×C3—124no—
451C2×C3—124no—
452C2×C3—244no—
453C2×C3—362no—
454C2×C3C2×C3×D3×D411yescharacteristic
455C2×C3C2×C3×#23711yescharacteristic
456C2×C3C3×#100411yescharacteristic
457D3—122no—
458D3—362no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—182no—
463C22—182no—
464C22—182no—
465C22C23×(C3⋊D3)⋊C311yescharacteristic
466C4—21no—
467C4—94no—
468C4—182no—
469C4—182no—
470C4D4×(C3⋊D3)⋊C312yes—
471C4#3830⋊C312yescomplemented
472C3—31no—
473C3—31no—
474C3—61no—
475C3C3×#571811yescharacteristic
476C2—44no—
477C2—362no—
478C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
479C2C2×#3830⋊C311yescharacteristic
480C2#64361⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#127C2
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×#550
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#3830⋊C3C2semidirect
2. H(3,𝔽3)×#127C2semidirect
3. #64149⋊C3C2semidirect
4. #64148⋊C3C2semidirect
5. #64147⋊C3C2semidirect
6. D4×(C3⋊Dic3)⋊C3C2semidirect
7. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
8. C4×#634⋊C3C2semidirect
9. #10596189C3semidirect
28. C2×#3830C2×C3semidirect
31. C32×#127D3semidirect
32. C32×#127C2×C3semidirect
33. #64149C2×C3semidirect
34. #64148C2×C3semidirect
35. #64147C2×C3semidirect
36. D4×C3⋊Dic3C2×C3semidirect
37. C4×C4⋊(C3⋊D3)C2×C3semidirect
38. C4×#634C2×C3semidirect
61. #634⋊C3D4semidirect
112. C4×H(3,𝔽3)#41semidirect
172. #634C3×D4semidirect
267. C32×C4#237semidirect
269. C32×C4C3×#41semidirect
278. #127(C3⋊D3)⋊C3semidirect
284. H(3,𝔽3)#550semidirect
421. C32#5718semidirect
422. C32C3×#550semidirect
471. C4#3830⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
H(3,𝔽3)×#1272
#64149⋊C32
#64148⋊C32
#64147⋊C32
D4×(C3⋊Dic3)⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#105961893
C3×#57183