Finite Group Databasebeta

← #10596137⋊C3 #13153053

Subgroups — #10596137⋊C3 #13153053

Order 1728.

960 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#123C211yesmaximal, characteristic, complemented
2#64149⋊C3C211yesmaximal, characteristic, complemented
3#64146⋊C3C211yesmaximal, characteristic, complemented
4#64139⋊C3C212yesmaximal, complemented
5#64107⋊C3C212yesmaximal, complemented
6#64104⋊C3C212yesmaximal, complemented
7#64093⋊C3C212yesmaximal, complemented
8#64092⋊C3C212yesmaximal, complemented
9#64091⋊C3C212yesmaximal, complemented
10#10596137C311yesmaximal, characteristic, complemented
11C3×#5666—31nomaximal
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×#638⋊C3C2212yes—
16C2×#638⋊C3C2212yes—
17C2×#638⋊C3C2212yescomplemented
18C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
19C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
20C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
21C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
22C4⋊C4×H(3,𝔽3)C2212yescomplemented
23C22⋊C4×H(3,𝔽3)C2212yes—
24C22⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
25#3757⋊C3C2211yescharacteristic
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28#3752⋊C3C2212yes—
29#3752⋊C3C2212yescomplemented
30C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
31#3750⋊C3C2212yescomplemented
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
33C32×#123—21no—
34C32×#123D311yescharacteristic, complemented
35C32×#123C2×C311yescharacteristic, complemented
36#64149C2×C311yescharacteristic, complemented
37#64146C2×C311yescharacteristic, complemented
38#64139C2×C312yescomplemented
39#64107C2×C312yescomplemented
40#64104C2×C312yescomplemented
41#64093C2×C312yescomplemented
42#64092C2×C312yescomplemented
43#64091C2×C312yescomplemented
44C3×#940—31no—
45C3×#937—31no—
46C3×#930—32no—
47C3×#898—32no—
48C3×#895—32no—
49C3×#884—32no—
50C3×#883—32no—
51C3×#882—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C23×H(3,𝔽3)C2311yescharacteristic
54C22×(C3⋊D3)⋊C3—41no—
55C22×(C3⋊D3)⋊C3—42no—
56C22×(C3⋊D3)⋊C3C2312yes—
57C22×(C3⋊D3)⋊C3C2312yes—
58D4×H(3,𝔽3)—41no—
59C2×C4×H(3,𝔽3)—41no—
60C2×C4×H(3,𝔽3)C2312yes—
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63#638⋊C3—42no—
64#638⋊C3—42no—
65#638⋊C3—42no—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C4⋊((C3⋊D3)⋊C3)—42no—
69C4⋊((C3⋊D3)⋊C3)—42no—
70C4⋊((C3⋊D3)⋊C3)—41no—
71C4×(C3⋊D3)⋊C3—42no—
72C3×#560—91no—
73#5666—31no—
74#5666—91no—
75C23×C3⋊D3C22×C311yescharacteristic
76C23×C3×D3—31no—
77C2×C32×D4—21no—
78C2×C32×D4C2×D311yescharacteristic
79C2×C32×D4C22×C311yescharacteristic
80C22×C32×C4—21no—
81C22×C32×C4C2×D311yescharacteristic, complemented
82C22×C32×C4C22×C311yescharacteristic, complemented
83C2×#638C22×C312yes—
84C2×#638C22×C312yes—
85C2×#638C22×C312yescomplemented
86C2×C4⋊(C3⋊D3)C22×C312yes—
87C2×C4⋊(C3⋊D3)C22×C312yes—
88C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
89C2×C4×C3⋊D3C22×C312yescomplemented
90C2×C3×#67—32no—
91C2×C3×#67—32no—
92C2×C3×#67—32no—
93C2×C3×D12—32no—
94C2×C3×D12—32no—
95C2×C3×D12—31no—
96C2×C3×C4×D3—32no—
97C32×C4⋊C4—22no—
98C32×C4⋊C4C2×D312yescomplemented
99C32×C4⋊C4C22×C312yescomplemented
100C32×C22⋊C4—22no—
101C32×C22⋊C4—21no—
102C32×C22⋊C4C2×D312yes—
103C32×C22⋊C4C2×D311yescharacteristic, complemented
104C32×C22⋊C4C22×C312yes—
105C32×C22⋊C4C22×C311yescharacteristic, complemented
106#3757C22×C311yescharacteristic
107#3752C22×C312yes—
108#3752C22×C312yes—
109#3752C22×C312yes—
110#3752C22×C312yescomplemented
111C4⋊(C3⋊Dic3)C22×C312yescomplemented
112#3750C22×C312yescomplemented
113C4×C3⋊Dic3C22×C311yescharacteristic, complemented
114C3×#217—31no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×C4⋊Dic3—32no—
120C3×Dic3⋊C4—32no—
121C3×C4×Dic3—31no—
122C22×H(3,𝔽3)—21no—
123C22×H(3,𝔽3)—22no—
124C22×H(3,𝔽3)—22no—
125C22×H(3,𝔽3)—21no—
126C22×H(3,𝔽3)C2411yescharacteristic
127C2×(C3⋊D3)⋊C3—22no—
128C2×(C3⋊D3)⋊C3—22no—
129C2×(C3⋊D3)⋊C3—22no—
130C2×(C3⋊D3)⋊C3—22no—
131C2×(C3⋊D3)⋊C3—22no—
132C2×(C3⋊D3)⋊C3—22no—
133C2×(C3⋊D3)⋊C3—44no—
134C4×H(3,𝔽3)—22no—
135C4×H(3,𝔽3)—22no—
136C4×H(3,𝔽3)—21no—
137(C3⋊Dic3)⋊C3—22no—
138(C3⋊Dic3)⋊C3—22no—
139C3×#127—91no—
140C3×#126—92no—
141C3×#123—31no—
142C3×#123—31no—
143C3×#123—61no—
144C3×#123—92no—
145C3×#123C3×D311yescharacteristic
146C3×#121—92no—
147C3×#121—92no—
148C3×#121—92no—
149C3×#120—92no—
150C3×#120—91no—
151#940—31no—
152#940—91no—
153#937—31no—
154#937—91no—
155#930—32no—
156#930—92no—
157#898—32no—
158#898—92no—
159#895—32no—
160#895—92no—
161#884—32no—
162#884—92no—
163#883—32no—
164#883—92no—
165#882—32no—
166#882—92no—
167C23×C32—21no—
168C23×C32—21no—
169C23×C32C22×D311yescharacteristic
170C23×C32C22×D311yescharacteristic
171C23×C32C23×C311yescharacteristic
172C23×C32C23×C311yescharacteristic
173C22×C3⋊D3—41no—
174C22×C3⋊D3—42no—
175C22×C3⋊D3C23×C312yes—
176C22×C3⋊D3C23×C312yes—
177C22×C3×D3—32no—
178C22×C3×D3—32no—
179C22×C3×D3—121no—
180C22×C3×D3—122no—
181C32×D4—41no—
182C32×D4—41no—
183C32×D4—42no—
184C2×C32×C4—22no—
185C2×C32×C4—22no—
186C2×C32×C4—21no—
187C2×C32×C4—41no—
188C2×C32×C4—41no—
189C2×C32×C4—42no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D312yes—
192C2×C32×C4C22×D312yes—
193C2×C32×C4C23×C311yescharacteristic
194C2×C32×C4C23×C312yes—
195C2×C32×C4C23×C312yes—
196#638—42no—
197#638—42no—
198#638—42no—
199C2×C3⋊Dic3C23×C312yes—
200C2×C3⋊Dic3C23×C312yes—
201C4⋊(C3⋊D3)—42no—
202C4⋊(C3⋊D3)—42no—
203C4⋊(C3⋊D3)—41no—
204C4×C3⋊D3—42no—
205C3×#67—122no—
206C3×#67—122no—
207C3×#67—122no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C3×D12—122no—
211C3×D12—121no—
212C3×D12—122no—
213C3×C4×D3—122no—
214#560—91noSylow
215C2×H(3,𝔽3)—41no—
216C2×H(3,𝔽3)—41no—
217C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
218C2×H(3,𝔽3)H(5,𝔽2)12yes—
219(C3⋊D3)⋊C3—42no—
220(C3⋊D3)⋊C3—42no—
221C24×C3—91no—
222C23×D3—31no—
223C23×D3—91no—
224C2×C3×D4—31no—
225C2×C3×D4—31no—
226C2×C3×D4—61no—
227C2×C3×D4—92no—
228C2×C3×D4—92no—
229C2×C3×D4—91no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—92no—
233C2×C3×D4C2×C3×D311yescharacteristic
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—61no—
237C22×C3×C4—92no—
238C22×C3×C4C2×C3×D311yescharacteristic
239C2×#67—32no—
240C2×#67—32no—
241C2×#67—32no—
242C2×#67—92no—
243C2×#67—92no—
244C2×#67—92no—
245C2×D12—32no—
246C2×D12—32no—
247C2×D12—31no—
248C2×D12—92no—
249C2×D12—92no—
250C2×D12—91no—
251C2×C4×D3—32no—
252C2×C4×D3—92no—
253C3×C4⋊C4—32no—
254C3×C4⋊C4—32no—
255C3×C4⋊C4—62no—
256C3×C4⋊C4—92no—
257C3×C4⋊C4—92no—
258C3×C4⋊C4C2×C3×D312yes—
259C3×C22⋊C4—32no—
260C3×C22⋊C4—32no—
261C3×C22⋊C4—31no—
262C3×C22⋊C4—31no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—61no—
265C3×C22⋊C4—92no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4—91no—
270C3×C22⋊C4C2×C3×D312yes—
271C3×C22⋊C4C2×C3×D311yescharacteristic
272C3×C42—91no—
273#217—31no—
274#217—91no—
275(C2×D3)⋊C4—32no—
276(C2×D3)⋊C4—32no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283C4⋊Dic3—32no—
284C4⋊Dic3—92no—
285Dic3⋊C4—32no—
286Dic3⋊C4—92no—
287C4×Dic3—31no—
288C4×Dic3—91no—
289C22×C32—22no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—22no—
295C22×C32—22no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—42no—
300C22×C32—41no—
301C22×C32—42no—
302C22×C32C23×D311yescharacteristic
303C22×C32C24×C311yescharacteristic
304C2×C3⋊D3—22no—
305C2×C3⋊D3—22no—
306C2×C3⋊D3—22no—
307C2×C3⋊D3—22no—
308C2×C3⋊D3—22no—
309C2×C3⋊D3—22no—
310C2×C3⋊D3—44no—
311C2×C3×D3—62no—
312C2×C3×D3—62no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—124no—
318C32×C4—22no—
319C32×C4—21no—
320C32×C4—22no—
321C32×C4—22no—
322C32×C4—22no—
323C32×C4—21no—
324C32×C4—42no—
325C32×C4—42no—
326C32×C4—41no—
327C3⋊Dic3—22no—
328C3⋊Dic3—22no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331#127—91no—
332#126—92no—
333#123—92no—
334#123(C3⋊D3)⋊C311yescharacteristic, complemented
335#121—92no—
336#121—92no—
337#121—92no—
338#120—91no—
339#120—92no—
340H(3,𝔽3)#56011yescharacteristic, complemented, Sylow
341C23×C3—31no—
342C23×C3—31no—
343C23×C3—31no—
344C23×C3—31no—
345C23×C3—61no—
346C23×C3—61no—
347C23×C3—92no—
348C23×C3—92no—
349C23×C3—361no—
350C23×C3—362no—
351C23×C3C22×C3×D311yescharacteristic
352C23×C3C22×C3×D311yescharacteristic
353C22×D3—32no—
354C22×D3—32no—
355C22×D3—92no—
356C22×D3—92no—
357C22×D3—121no—
358C22×D3—122no—
359C22×D3—361no—
360C22×D3—362no—
361C3×D4—41no—
362C3×D4—121no—
363C3×D4—121no—
364C3×D4—122no—
365C3×D4—362no—
366C3×D4—362no—
367C3×D4—361no—
368C3×D4—362no—
369C3×D4—362no—
370C3×D4—362no—
371C2×C3×C4—31no—
372C2×C3×C4—32no—
373C2×C3×C4—32no—
374C2×C3×C4—31no—
375C2×C3×C4—32no—
376C2×C3×C4—32no—
377C2×C3×C4—41no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—61no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—121no—
384C2×C3×C4—121no—
385C2×C3×C4—122no—
386C2×C3×C4—362no—
387C2×C3×C4C22×C3×D311yescharacteristic
388C2×C3×C4C22×C3×D312yes—
389C2×C3×C4C22×C3×D312yes—
390#67—122no—
391#67—122no—
392#67—122no—
393#67—362no—
394#67—362no—
395#67—362no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—92no—
399C2×Dic3—92no—
400D12—122no—
401D12—121no—
402D12—122no—
403D12—362no—
404D12—362no—
405D12—361no—
406C4×D3—122no—
407C4×D3—362no—
408C2×C32—21no—
409C2×C32—22no—
410C2×C32—42no—
411C2×C32—41no—
412C2×C32—41no—
413C2×C32—41no—
414C2×C32—41no—
415C2×C32—81no—
416C2×C32#100411yescharacteristic
417C2×C32#100912yes—
418C2×C32C3×H(5,𝔽2)11yescharacteristic
419C2×C32C3×H(5,𝔽2)12yes—
420C3⋊D3—42no—
421C3⋊D3—42no—
422C3×D3—122no—
423C3×D3—122no—
424C24—91no—
425C2×D4—92no—
426C2×D4—92no—
427C2×D4—92no—
428C2×D4—92no—
429C2×D4—92no—
430C2×D4—91no—
431C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
432C22×C4—92no—
433C22×C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
434C4⋊C4—92no—
435C4⋊C4—92no—
436C4⋊C4C2×(C3⋊D3)⋊C312yescomplemented
437C22⋊C4—91no—
438C22⋊C4—92no—
439C22⋊C4—92no—
440C22⋊C4—92no—
441C22⋊C4—92no—
442C22⋊C4C2×(C3⋊D3)⋊C312yes—
443C22⋊C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
444C42—91no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—31no—
450C22×C3—31no—
451C22×C3—62no—
452C22×C3—62no—
453C22×C3—61no—
454C22×C3—61no—
455C22×C3—61no—
456C22×C3—61no—
457C22×C3—62no—
458C22×C3—62no—
459C22×C3—62no—
460C22×C3—61no—
461C22×C3—122no—
462C22×C3—121no—
463C22×C3—122no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—182no—
470C22×C3—364no—
471C22×C3C23×C3×D311yescharacteristic
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—124no—
479C2×D3—182no—
480C2×D3—182no—
481C2×D3—182no—
482C2×D3—182no—
483C2×D3—182no—
484C2×D3—182no—
485C2×D3—364no—
486C3×C4—22no—
487C3×C4—21no—
488C3×C4—22no—
489C3×C4—62no—
490C3×C4—62no—
491C3×C4—61no—
492C3×C4—62no—
493C3×C4—62no—
494C3×C4—61no—
495C3×C4—122no—
496C3×C4—121no—
497C3×C4—122no—
498C3×C4—182no—
499C3×C4—182no—
500Dic3—62no—
501Dic3—62no—
502Dic3—182no—
503Dic3—182no—
504C32—21no—
505C32#566611yescharacteristic, complemented
506C32C3×#56011yescharacteristic, complemented
507C23—92no—
508C23—92no—
509C23—361no—
510C23—362no—
511C23C22×(C3⋊D3)⋊C311yescharacteristic
512C23C22×(C3⋊D3)⋊C311yescharacteristic
513D4—41no—
514D4—362no—
515D4—361no—
516D4—362no—
517D4—362no—
518D4—362no—
519D4—362no—
520C2×C4—41no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—362no—
524C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
525C2×C4C22×(C3⋊D3)⋊C312yes—
526C2×C4C22×(C3⋊D3)⋊C312yes—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—32no—
530C2×C3—32no—
531C2×C3—41no—
532C2×C3—41no—
533C2×C3—61no—
534C2×C3—62no—
535C2×C3—121no—
536C2×C3—121no—
537C2×C3—121no—
538C2×C3—121no—
539C2×C3—122no—
540C2×C3—241no—
541C2×C3—362no—
542C2×C3—362no—
543C2×C3C3×#100411yescharacteristic
544C2×C3C3×#100912yes—
545D3—122no—
546D3—122no—
547D3—362no—
548D3—362no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—22no—
553C22—182no—
554C22—182no—
555C22—182no—
556C22—182no—
557C22—182no—
558C22—182no—
559C22—364no—
560C22C23×(C3⋊D3)⋊C311yescharacteristic
561C4—22no—
562C4—21no—
563C4—22no—
564C4—182no—
565C4—182no—
566C3—31no—
567C3—31no—
568C3—61no—
569C3C3×#566611yescharacteristic
570C2—41no—
571C2—41no—
572C2—362no—
573C2—362no—
574C2#64361⋊C311yescharacteristic
575C2#1172263812yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#123C2
hypercenter#123(C3⋊D3)⋊C3
hypocenterC32C3×#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#123C2semidirect
2. #64149⋊C3C2semidirect
3. #64146⋊C3C2semidirect
4. #64139⋊C3C2semidirect
5. #64107⋊C3C2semidirect
6. #64104⋊C3C2semidirect
7. #64093⋊C3C2semidirect
8. #64092⋊C3C2semidirect
9. #64091⋊C3C2semidirect
10. #10596137C3semidirect
14. C22×C4×H(3,𝔽3)C22semidirect
17. C2×#638⋊C3C22semidirect
20. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
21. C2×C4×(C3⋊D3)⋊C3C22semidirect
22. C4⋊C4×H(3,𝔽3)C22semidirect
24. C22⋊C4×H(3,𝔽3)C22semidirect
29. #3752⋊C3C22semidirect
30. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
31. #3750⋊C3C22semidirect
32. C4×(C3⋊Dic3)⋊C3C22semidirect
34. C32×#123D3semidirect
35. C32×#123C2×C3semidirect
36. #64149C2×C3semidirect
37. #64146C2×C3semidirect
38. #64139C2×C3semidirect
39. #64107C2×C3semidirect
40. #64104C2×C3semidirect
41. #64093C2×C3semidirect
42. #64092C2×C3semidirect
43. #64091C2×C3semidirect
81. C22×C32×C4C2×D3semidirect
82. C22×C32×C4C22×C3semidirect
85. C2×#638C22×C3semidirect
88. C2×C4⋊(C3⋊D3)C22×C3semidirect
89. C2×C4×C3⋊D3C22×C3semidirect
98. C32×C4⋊C4C2×D3semidirect
99. C32×C4⋊C4C22×C3semidirect
103. C32×C22⋊C4C2×D3semidirect
105. C32×C22⋊C4C22×C3semidirect
110. #3752C22×C3semidirect
111. C4⋊(C3⋊Dic3)C22×C3semidirect
112. #3750C22×C3semidirect
113. C4×C3⋊Dic3C22×C3semidirect
334. #123(C3⋊D3)⋊C3semidirect
340. H(3,𝔽3)#560semidirect
433. C22×C4C2×(C3⋊D3)⋊C3semidirect
436. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
443. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
505. C32#5666semidirect
506. C32C3×#560semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1232
#64149⋊C32
#64146⋊C32
#64139⋊C32
#64107⋊C32
#64104⋊C32
#64093⋊C32
#64092⋊C32
#64091⋊C32
#105961373
C3×#56663