Finite Group Databasebeta

← #10596004⋊C3 #13152511

Subgroups — #10596004⋊C3 #13152511

Order 1728.

818 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
2C4×#638⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#64106⋊C3C211yesmaximal, characteristic, complemented
5#64101⋊C3C211yesmaximal, characteristic, complemented
6#64097⋊C3C211yesmaximal, characteristic, complemented
7#64096⋊C3C211yesmaximal, characteristic, complemented
8#64094⋊C3C212yesmaximal
9#64088⋊C3C212yesmaximal, complemented
10#11721387C212yesmaximal, complemented
11C4×#634⋊C3C212yesmaximal, complemented
12#10596004C311yesmaximal, characteristic, complemented
13C3×#5533—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#634⋊C3C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C22⋊C4×H(3,𝔽3)C2212yes—
22C42×H(3,𝔽3)C2212yes—
23#3757⋊C3C2212yes—
24#3757⋊C3C2211yescharacteristic
25#3752⋊C3C2211yescharacteristic
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2211yescharacteristic
32#3750⋊C3C2212yes—
33#3750⋊C3C2212yes—
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#117—21no—
38C32×#117D311yescharacteristic, complemented
39C32×#117C2×C311yescharacteristic, complemented
40C4×#638C2×C311yescharacteristic
41#64133C2×C311yescharacteristic, complemented
42#64106C2×C311yescharacteristic, complemented
43#64101C2×C311yescharacteristic, complemented
44#64097C2×C311yescharacteristic, complemented
45#64096C2×C311yescharacteristic, complemented
46#64094C2×C312yes—
47#64088C2×C312yescomplemented
48#64085C2×C312yescomplemented
49C4×#634C2×C312yescomplemented
50C3×C4×#67—31no—
51C3×#924—31no—
52C3×#897—31no—
53C3×#892—31no—
54C3×#888—31no—
55C3×#887—31no—
56C3×#885—32no—
57C3×#879—32no—
58C3×#876—32no—
59C3×C4×Dic6—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67#638⋊C3—22no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4×(C3⋊D3)⋊C3—22no—
74#634⋊C3—22no—
75#634⋊C3—24no—
76C3×#556—91no—
77#5533—31no—
78#5533—91no—
79C22×C32×C4—21no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C22×C311yescharacteristic
82C2×#638C22×C311yescharacteristic
83C2×C4×C3⋊D3C22×C311yescharacteristic
84C2×#634C22×C312yes—
85C2×#634C22×C311yescharacteristic
86C2×C3×#67—31no—
87C2×C3×C4×D3—31no—
88C2×C3×Dic6—32no—
89C2×C3×Dic6—31no—
90C32×C4⋊C4—21no—
91C32×C4⋊C4—21no—
92C32×C4⋊C4C2×D311yescharacteristic
93C32×C4⋊C4C2×D311yescharacteristic
94C32×C4⋊C4C22×C311yescharacteristic
95C32×C4⋊C4C22×C311yescharacteristic
96C32×C22⋊C4—22no—
97C32×C22⋊C4C2×D312yes—
98C32×C22⋊C4C22×C312yes—
99C32×C42—22no—
100C32×C42C2×D312yes—
101C32×C42C22×C312yes—
102#3757C22×C311yescharacteristic
103#3757C22×C312yes—
104#3752C22×C311yescharacteristic
105#3752C22×C312yes—
106#3752C22×C312yes—
107C4⋊(C3⋊Dic3)C22×C312yes—
108C4⋊(C3⋊Dic3)C22×C311yescharacteristic
109#3750C22×C311yescharacteristic
110#3750C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113C4×C3⋊Dic3C22×C312yes—
114C4×C3⋊Dic3C22×C312yes—
115C4×C3⋊Dic3C22×C311yescharacteristic
116C3×#217—31no—
117C3×#217—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4—31no—
121C3×C4⋊Dic3—32no—
122C3×C4⋊Dic3—31no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—31no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—31no—
130C22×H(3,𝔽3)—21no—
131C22×H(3,𝔽3)—21no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)C2411yescharacteristic
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3—21no—
137C4×H(3,𝔽3)—21no—
138C4×H(3,𝔽3)—22no—
139C4×H(3,𝔽3)—22no—
140C4×H(3,𝔽3)#4112yes—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3#4112yes—
146C3×C4⋊Q8—91no—
147C3×#126—92no—
148C3×#126—92no—
149C3×#124—92no—
150C3×C22⋊Q8—91no—
151C3×C22⋊Q8—91no—
152C3×C4×Q8—92no—
153C3×C4×Q8—91no—
154C3×C4×D4—91no—
155C3×#117—31no—
156C3×#117—31no—
157C3×#117—61no—
158C3×#117—91no—
159C3×#117C3×D311yescharacteristic
160C4×#67—31no—
161C4×#67—91no—
162#924—31no—
163#924—91no—
164#897—31no—
165#897—91no—
166#892—31no—
167#892—91no—
168#888—31no—
169#888—91no—
170#887—31no—
171#887—91no—
172#885—32no—
173#885—92no—
174#879—32no—
175#879—92no—
176#876—32no—
177#876—92no—
178C4×Dic6—32no—
179C4×Dic6—92no—
180C23×C32—21no—
181C23×C32C22×D311yescharacteristic
182C23×C32C23×C311yescharacteristic
183C22×C3⋊D3C23×C311yescharacteristic
184C22×C3×D3—31no—
185C2×C32×C4—24no—
186C2×C32×C4—22no—
187C2×C32×C4—22no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—22no—
191C2×C32×C4—22no—
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C23×C312yes—
197C2×C32×C4C23×C311yescharacteristic
198C2×C32×C4C23×C312yes—
199C2×C32×C4C23×C311yescharacteristic
200#638—22no—
201C2×C3⋊Dic3C23×C311yescharacteristic
202C2×C3⋊Dic3C23×C311yescharacteristic
203C2×C3⋊Dic3C23×C312yes—
204C2×C3⋊Dic3C23×C312yes—
205C2×C3⋊Dic3C23×C311yescharacteristic
206C4×C3⋊D3—22no—
207#634—22no—
208#634—24no—
209C3×#67—62no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C3×C4×D3—62no—
216C3×Dic6—64no—
217C3×Dic6—62no—
218#556—91noSylow
219C2×H(3,𝔽3)—41no—
220C2×H(3,𝔽3)C2×#4111yescharacteristic
221C2×H(3,𝔽3)C2×#4111yescharacteristic
222C2×H(3,𝔽3)2−1+411yescharacteristic
223(C3⋊D3)⋊C3—41no—
224C2×C3×Q8—91no—
225C2×C3×Q8—92no—
226C2×C3×D4—91no—
227C22×C3×C4—31no—
228C22×C3×C4—31no—
229C22×C3×C4—61no—
230C22×C3×C4—91no—
231C22×C3×C4C2×C3×D311yescharacteristic
232C2×#67—31no—
233C2×#67—91no—
234C2×C4×D3—31no—
235C2×C4×D3—91no—
236C2×Dic6—32no—
237C2×Dic6—31no—
238C2×Dic6—92no—
239C2×Dic6—91no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4C2×C3×D311yescharacteristic
253C3×C4⋊C4C2×C3×D311yescharacteristic
254C3×C22⋊C4—32no—
255C3×C22⋊C4—32no—
256C3×C22⋊C4—62no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—91no—
259C3×C22⋊C4—92no—
260C3×C22⋊C4—91no—
261C3×C22⋊C4—92no—
262C3×C22⋊C4C2×C3×D312yes—
263C3×C42—32no—
264C3×C42—32no—
265C3×C42—62no—
266C3×C42—92no—
267C3×C42—91no—
268C3×C42—92no—
269C3×C42C2×C3×D312yes—
270#217—31no—
271#217—32no—
272#217—91no—
273#217—92no—
274(C2×D3)⋊C4—31no—
275(C2×D3)⋊C4—32no—
276(C2×D3)⋊C4—32no—
277(C2×D3)⋊C4—91no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280C4⋊Dic3—32no—
281C4⋊Dic3—31no—
282C4⋊Dic3—92no—
283C4⋊Dic3—91no—
284Dic3⋊C4—31no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—91no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292C4×Dic3—32no—
293C4×Dic3—32no—
294C4×Dic3—31no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C4×Dic3—91no—
298C22×C32—22no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32C23×D311yescharacteristic
309C22×C32C24×C311yescharacteristic
310C2×C3⋊D3—21no—
311C2×C3⋊D3—21no—
312C2×C3⋊D3—21no—
313C2×C3×D3—61no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C32×C4—22no—
317C32×C4—21no—
318C32×C4—22no—
319C32×C4—22no—
320C32×C4—22no—
321C32×C4—21no—
322C32×C4—22no—
323C32×C4—22no—
324C32×C4—42no—
325C32×C4—42no—
326C32×C4#23512yes—
327C32×C4C3×#4112yes—
328C3⋊Dic3—22no—
329C3⋊Dic3—22no—
330C3⋊Dic3—21no—
331C3⋊Dic3—21no—
332C3⋊Dic3C3×#4112yes—
333C3×Dic3—32no—
334C3×Dic3—62no—
335C3×Dic3—61no—
336C3×Dic3—62no—
337C3×Dic3—61no—
338C4⋊Q8—91no—
339#126—92no—
340#126—92no—
341#124—92no—
342C22⋊Q8—91no—
343C22⋊Q8—91no—
344C4×Q8—91no—
345C4×Q8—92no—
346C4×D4—91no—
347#117—91no—
348#117(C3⋊D3)⋊C311yescharacteristic, complemented
349H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
350C23×C3—31no—
351C23×C3—31no—
352C23×C3—61no—
353C23×C3—91no—
354C23×C3C22×C3×D311yescharacteristic
355C22×D3—31no—
356C22×D3—91no—
357C3×Q8—182no—
358C3×Q8—184no—
359C3×D4—182no—
360C2×C3×C4—22no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—32no—
365C2×C3×C4—32no—
366C2×C3×C4—32no—
367C2×C3×C4—31no—
368C2×C3×C4—32no—
369C2×C3×C4—62no—
370C2×C3×C4—62no—
371C2×C3×C4—64no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4—91no—
377C2×C3×C4—91no—
378C2×C3×C4—91no—
379C2×C3×C4—92no—
380C2×C3×C4—92no—
381C2×C3×C4—182no—
382C2×C3×C4C22×C3×D312yes—
383C2×C3×C4C22×C3×D311yescharacteristic
384C2×C3×C4C22×C3×D312yes—
385C2×C3×C4C22×C3×D311yescharacteristic
386#67—62no—
387#67—182no—
388C2×Dic3—31no—
389C2×Dic3—31no—
390C2×Dic3—31no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—91no—
394C2×Dic3—91no—
395C2×Dic3—91no—
396C2×Dic3—92no—
397C2×Dic3—92no—
398C4×D3—62no—
399C4×D3—182no—
400Dic6—64no—
401Dic6—62no—
402Dic6—184no—
403Dic6—182no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—42no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32C2×#23511yescharacteristic
411C2×C32D3×#4111yescharacteristic
412C2×C32#101011yescharacteristic
413C2×C32C2×C3×#4111yescharacteristic
414C2×C32C2×C3×#4111yescharacteristic
415C2×C32C3×2−1+411yescharacteristic
416C3⋊D3—41no—
417C3×D3—121no—
418C2×Q8—91no—
419C2×Q8—92no—
420C2×D4—91no—
421C22×C4—91no—
422C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
423C4⋊C4—91no—
424C4⋊C4—91no—
425C4⋊C4—92no—
426C4⋊C4—92no—
427C4⋊C4—92no—
428C4⋊C4—92no—
429C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
430C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
431C22⋊C4—91no—
432C22⋊C4—92no—
433C22⋊C4—92no—
434C22⋊C4—91no—
435C22⋊C4—92no—
436C22⋊C4C2×(C3⋊D3)⋊C312yes—
437C42—92no—
438C42—91no—
439C42—92no—
440C42C2×(C3⋊D3)⋊C312yes—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—62no—
453C22×C3—61no—
454C22×C3—121no—
455C22×C3—121no—
456C22×C3—181no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3C23×C3×D311yescharacteristic
460C2×D3—61no—
461C2×D3—61no—
462C2×D3—61no—
463C2×D3—181no—
464C2×D3—181no—
465C2×D3—181no—
466C3×C4—21no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—32no—
470C3×C4—32no—
471C3×C4—62no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—62no—
475C3×C4—62no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—62no—
479C3×C4—92no—
480C3×C4—122no—
481C3×C4—122no—
482C3×C4—181no—
483C3×C4—182no—
484C3×C4—182no—
485C3×C4—181no—
486C3×C4C3×#23512yes—
487Dic3—32no—
488Dic3—62no—
489Dic3—61no—
490Dic3—62no—
491Dic3—61no—
492Dic3—92no—
493Dic3—182no—
494Dic3—182no—
495Dic3—181no—
496Dic3—181no—
497C32—21no—
498C32#553311yescharacteristic, complemented
499C32C3×#55611yescharacteristic, complemented
500C23—91no—
501C23C22×(C3⋊D3)⋊C311yescharacteristic
502Q8—184no—
503Q8—182no—
504D4—182no—
505C2×C4—22no—
506C2×C4—91no—
507C2×C4—91no—
508C2×C4—91no—
509C2×C4—92no—
510C2×C4—92no—
511C2×C4—182no—
512C2×C4C22×(C3⋊D3)⋊C312yes—
513C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
514C2×C4C22×(C3⋊D3)⋊C312yes—
515C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
516C2×C3—31no—
517C2×C3—31no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—41no—
523C2×C3—61no—
524C2×C3—61no—
525C2×C3—61no—
526C2×C3—121no—
527C2×C3—121no—
528C2×C3—122no—
529C2×C3—361no—
530C2×C3C2×C3×#23511yescharacteristic
531C2×C3C3×D3×#4111yescharacteristic
532C2×C3C3×#101011yescharacteristic
533D3—121no—
534D3—361no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—181no—
539C22—181no—
540C22—181no—
541C22C23×(C3⋊D3)⋊C311yescharacteristic
542C4—21no—
543C4—22no—
544C4—22no—
545C4—92no—
546C4—181no—
547C4—182no—
548C4—182no—
549C4—181no—
550C4#3828⋊C312yes—
551C3—31no—
552C3—31no—
553C3—61no—
554C3C3×#553311yescharacteristic
555C2—41no—
556C2—361no—
557C2C2×#3828⋊C311yescharacteristic
558C2#41×(C3⋊D3)⋊C311yescharacteristic
559C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
prime radicalC2×#638⋊C3C22
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#556
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#117C2semidirect
3. #64133⋊C3C2semidirect
4. #64106⋊C3C2semidirect
5. #64101⋊C3C2semidirect
6. #64097⋊C3C2semidirect
7. #64096⋊C3C2semidirect
9. #64088⋊C3C2semidirect
10. #11721387C2semidirect
11. C4×#634⋊C3C2semidirect
12. #10596004C3semidirect
38. C32×#117D3semidirect
39. C32×#117C2×C3semidirect
41. #64133C2×C3semidirect
42. #64106C2×C3semidirect
43. #64101C2×C3semidirect
44. #64097C2×C3semidirect
45. #64096C2×C3semidirect
47. #64088C2×C3semidirect
48. #64085C2×C3semidirect
49. C4×#634C2×C3semidirect
348. #117(C3⋊D3)⋊C3semidirect
349. H(3,𝔽3)#556semidirect
498. C32#5533semidirect
499. C32C3×#556semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1172
C4×#638⋊C32
#64133⋊C32
#64106⋊C32
#64101⋊C32
#64097⋊C32
#64096⋊C32
#64094⋊C32
#64088⋊C32
#117213872
C4×#634⋊C32
#105960043
C3×#55333