Finite Group Databasebeta

← #10595965⋊C3 #13152369

Subgroups — #10595965⋊C3 #13152369

Order 1728.

1007 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721702C212yesmaximal, complemented
3#64139⋊C3C211yesmaximal, characteristic, complemented
4C4×#638⋊C3C212yesmaximal, complemented
5#64133⋊C3C211yesmaximal, characteristic, complemented
6#64094⋊C3C212yesmaximal, complemented
7#64093⋊C3C212yesmaximal, complemented
8#64092⋊C3C212yesmaximal, complemented
9#64088⋊C3C212yesmaximal, complemented
10#10595965C311yesmaximal, characteristic, complemented
11C3×#5494—31nomaximal
12C24×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16C2×#638⋊C3C2212yes—
17C2×#638⋊C3C2212yes—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
19C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
20C2×#634⋊C3C2211yescharacteristic, complemented
21C22⋊C4×H(3,𝔽3)C2212yes—
22C22⋊C4×H(3,𝔽3)C2212yes—
23#3757⋊C3C2212yes—
24#3757⋊C3C2212yes—
25#3757⋊C3C2212yes—
26#3752⋊C3C2212yescomplemented
27#3752⋊C3C2212yescomplemented
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
30#3750⋊C3C2212yescomplemented
31#3750⋊C3C2212yescomplemented
32C4×(C3⋊Dic3)⋊C3C2212yescomplemented
33C2×C32×C22⋊C4—21no—
34C2×C32×C22⋊C4D311yescharacteristic, complemented
35C2×C32×C22⋊C4C2×C311yescharacteristic, complemented
36#64162C2×C312yescomplemented
37#64139C2×C311yescharacteristic, complemented
38C4×#638C2×C312yescomplemented
39#64133C2×C311yescharacteristic, complemented
40#64094C2×C312yescomplemented
41#64093C2×C312yescomplemented
42#64092C2×C312yescomplemented
43#64088C2×C312yescomplemented
44C3×#953—32no—
45C3×#930—31no—
46C3×C4×#67—32no—
47C3×#924—31no—
48C3×#885—32no—
49C3×#884—32no—
50C3×#883—32no—
51C3×#879—32no—
52C23×H(3,𝔽3)—22no—
53C23×H(3,𝔽3)—41no—
54C23×H(3,𝔽3)—41no—
55C23×H(3,𝔽3)C2312yes—
56C23×H(3,𝔽3)C2311yescharacteristic
57C22×(C3⋊D3)⋊C3C2312yes—
58C2×C4×H(3,𝔽3)—41no—
59C2×C4×H(3,𝔽3)—41no—
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63#638⋊C3—24no—
64#638⋊C3—42no—
65#638⋊C3—42no—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C4⋊((C3⋊D3)⋊C3)—41no—
70C4×(C3⋊D3)⋊C3—42no—
71#634⋊C3—41no—
72C3×#538—91no—
73#5494—31no—
74#5494—91no—
75C24×C32—21no—
76C24×C32C2×D311yescharacteristic
77C24×C32C22×C311yescharacteristic
78C22×C32×C4—21no—
79C22×C32×C4—21no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C22×C311yescharacteristic
83C22×C32×C4C22×C311yescharacteristic
84C2×#638C22×C312yes—
85C2×#638C22×C312yes—
86C2×#638C22×C312yes—
87C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
88C2×C4×C3⋊D3C22×C312yescomplemented
89C2×#634C22×C311yescharacteristic, complemented
90C2×C3×#67—32no—
91C2×C3×#67—32no—
92C2×C3×#67—32no—
93C2×C3×D12—31no—
94C2×C3×C4×D3—32no—
95C2×C3×Dic6—31no—
96C32×C22⋊C4—22no—
97C32×C22⋊C4—22no—
98C32×C22⋊C4C2×D312yes—
99C32×C22⋊C4C2×D312yes—
100C32×C22⋊C4C22×C312yes—
101C32×C22⋊C4C22×C312yes—
102#3757C22×C312yes—
103#3757C22×C312yes—
104#3757C22×C312yes—
105#3752C22×C312yescomplemented
106#3752C22×C312yescomplemented
107C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
108C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
109#3750C22×C312yescomplemented
110#3750C22×C312yescomplemented
111C4×C3⋊Dic3C22×C312yescomplemented
112C3×#217—32no—
113C3×#217—32no—
114C3×#217—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—32no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—31no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×C4×Dic3—32no—
122C22×H(3,𝔽3)—21no—
123C22×H(3,𝔽3)—21no—
124C22×H(3,𝔽3)—22no—
125C22×H(3,𝔽3)—22no—
126C22×H(3,𝔽3)—22no—
127C22×H(3,𝔽3)—44no—
128C22×H(3,𝔽3)#4112yes—
129C22×H(3,𝔽3)C2411yescharacteristic
130C2×(C3⋊D3)⋊C3—22no—
131C2×(C3⋊D3)⋊C3—22no—
132C2×(C3⋊D3)⋊C3—22no—
133C4×H(3,𝔽3)—21no—
134C4×H(3,𝔽3)—22no—
135C4×H(3,𝔽3)—21no—
136(C3⋊Dic3)⋊C3—22no—
137(C3⋊Dic3)⋊C3—22no—
138(C3⋊Dic3)⋊C3—22no—
139C3×#126—92no—
140C3×#124—92no—
141C3×#123—92no—
142C3×C22⋊Q8—91no—
143C3×#121—92no—
144C3×#121—91no—
145C3×#120—92no—
146C3×C4×D4—92no—
147C2×C3×C22⋊C4—31no—
148C2×C3×C22⋊C4—31no—
149C2×C3×C22⋊C4—61no—
150C2×C3×C22⋊C4C3×D311yescharacteristic
151#953—32no—
152#953—92no—
153#930—31no—
154#930—91no—
155C4×#67—32no—
156C4×#67—92no—
157#924—31no—
158#924—91no—
159#885—32no—
160#885—92no—
161#884—32no—
162#884—92no—
163#883—32no—
164#883—92no—
165#879—32no—
166#879—92no—
167C23×C32—24no—
168C23×C32—22no—
169C23×C32—22no—
170C23×C32—21no—
171C23×C32—22no—
172C23×C32—41no—
173C23×C32—41no—
174C23×C32—41no—
175C23×C32—41no—
176C23×C32—42no—
177C23×C32—42no—
178C23×C32C22×D312yes—
179C23×C32C22×D311yescharacteristic
180C23×C32C23×C312yes—
181C23×C32C23×C311yescharacteristic
182C22×C3⋊D3C23×C312yes—
183C22×C3×D3—32no—
184C2×C32×C4—22no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—41no—
188C2×C32×C4—41no—
189C2×C32×C4—41no—
190C2×C32×C4—41no—
191C2×C32×C4—42no—
192C2×C32×C4—42no—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C23×C312yes—
197C2×C32×C4C23×C311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199#638—24no—
200#638—42no—
201#638—42no—
202C2×C3⋊Dic3C23×C312yes—
203C2×C3⋊Dic3C23×C312yes—
204C2×C3⋊Dic3C23×C312yes—
205C4⋊(C3⋊D3)—41no—
206C4×C3⋊D3—42no—
207#634—41no—
208C3×#67—64no—
209C3×#67—122no—
210C3×#67—122no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C3×D12—121no—
215C3×C4×D3—122no—
216C3×Dic6—121no—
217#538—91noSylow
218C2×H(3,𝔽3)—22no—
219C2×H(3,𝔽3)—42no—
220C2×H(3,𝔽3)C2×#4111yescharacteristic
221C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
222C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
223(C3⋊D3)⋊C3—42no—
224C24×C3—31no—
225C24×C3—31no—
226C24×C3—61no—
227C24×C3C2×C3×D311yescharacteristic
228C2×C3×Q8—91no—
229C2×C3×D4—91no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—92no—
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—61no—
238C22×C3×C4—61no—
239C22×C3×C4—92no—
240C22×C3×C4C2×C3×D311yescharacteristic
241C22×C3×C4C2×C3×D311yescharacteristic
242C2×#67—32no—
243C2×#67—32no—
244C2×#67—32no—
245C2×#67—92no—
246C2×#67—92no—
247C2×#67—92no—
248C2×D12—31no—
249C2×D12—91no—
250C2×C4×D3—32no—
251C2×C4×D3—92no—
252C2×Dic6—31no—
253C2×Dic6—91no—
254C3×C4⋊C4—92no—
255C3×C4⋊C4—92no—
256C3×C4⋊C4—91no—
257C3×C4⋊C4—91no—
258C3×C22⋊C4—32no—
259C3×C22⋊C4—32no—
260C3×C22⋊C4—32no—
261C3×C22⋊C4—32no—
262C3×C22⋊C4—62no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—92no—
265C3×C22⋊C4—92no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4C2×C3×D312yes—
270C3×C22⋊C4C2×C3×D312yes—
271C3×C42—92no—
272#217—32no—
273#217—32no—
274#217—32no—
275#217—92no—
276#217—92no—
277#217—92no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282C4⋊Dic3—31no—
283C4⋊Dic3—31no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290C4×Dic3—32no—
291C4×Dic3—92no—
292C22×C32—22no—
293C22×C32—22no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—24no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32—22no—
302C22×C32—22no—
303C22×C32—22no—
304C22×C32—22no—
305C22×C32—21no—
306C22×C32—22no—
307C22×C32—42no—
308C22×C32—42no—
309C22×C32—48no—
310C22×C32—44no—
311C22×C32—44no—
312C22×C32#23512yes—
313C22×C32C3×#4112yes—
314C22×C32C23×D311yescharacteristic
315C22×C32C24×C311yescharacteristic
316C2×C3⋊D3—22no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3—22no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C32×C4—21no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—22no—
327C32×C4—21no—
328C32×C4—41no—
329C32×C4—42no—
330C32×C4—41no—
331C3⋊Dic3—22no—
332C3⋊Dic3—22no—
333C3⋊Dic3—22no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—62no—
337#126—92no—
338#124—92no—
339#123—92no—
340C22⋊Q8—91no—
341#121—91no—
342#121—92no—
343#120—92no—
344C4×D4—92no—
345C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
346H(3,𝔽3)#53811yescharacteristic, complemented, Sylow
347C23×C3—22no—
348C23×C3—31no—
349C23×C3—31no—
350C23×C3—32no—
351C23×C3—32no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—62no—
355C23×C3—62no—
356C23×C3—64no—
357C23×C3—61no—
358C23×C3—62no—
359C23×C3—92no—
360C23×C3—121no—
361C23×C3—121no—
362C23×C3—121no—
363C23×C3—121no—
364C23×C3—122no—
365C23×C3—122no—
366C23×C3C22×C3×D312yes—
367C23×C3C22×C3×D311yescharacteristic
368C22×D3—32no—
369C22×D3—92no—
370C3×Q8—361no—
371C3×D4—184no—
372C3×D4—361no—
373C3×D4—362no—
374C3×D4—362no—
375C2×C3×C4—32no—
376C2×C3×C4—31no—
377C2×C3×C4—31no—
378C2×C3×C4—32no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—62no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—92no—
387C2×C3×C4—92no—
388C2×C3×C4—92no—
389C2×C3×C4—121no—
390C2×C3×C4—121no—
391C2×C3×C4—121no—
392C2×C3×C4—121no—
393C2×C3×C4—122no—
394C2×C3×C4—122no—
395C2×C3×C4—362no—
396C2×C3×C4C22×C3×D312yes—
397C2×C3×C4C22×C3×D311yescharacteristic
398C2×C3×C4C22×C3×D311yescharacteristic
399#67—64no—
400#67—122no—
401#67—122no—
402#67—184no—
403#67—362no—
404#67—362no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—92no—
409C2×Dic3—92no—
410C2×Dic3—92no—
411D12—121no—
412D12—361no—
413C4×D3—122no—
414C4×D3—362no—
415Dic6—121no—
416Dic6—361no—
417C2×C32—24no—
418C2×C32—22no—
419C2×C32—22no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—44no—
424C2×C32—42no—
425C2×C32—42no—
426C2×C32C2×#23511yescharacteristic
427C2×C32#100411yescharacteristic
428C2×C32#100411yescharacteristic
429C2×C32C2×C3×#4111yescharacteristic
430C2×C32C3×H(5,𝔽2)11yescharacteristic
431C2×C32C3×H(5,𝔽2)11yescharacteristic
432C3⋊D3—42no—
433C3×D3—122no—
434C24C2×(C3⋊D3)⋊C311yescharacteristic
435C2×Q8—91no—
436C2×D4—92no—
437C2×D4—92no—
438C2×D4—92no—
439C2×D4—91no—
440C22×C4—92no—
441C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
442C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
443C4⋊C4—92no—
444C4⋊C4—92no—
445C4⋊C4—91no—
446C4⋊C4—91no—
447C22⋊C4—92no—
448C22⋊C4—92no—
449C22⋊C4—92no—
450C22⋊C4—92no—
451C22⋊C4—92no—
452C22⋊C4C2×(C3⋊D3)⋊C312yes—
453C22⋊C4C2×(C3⋊D3)⋊C312yes—
454C42—92no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—31no—
461C22×C3—31no—
462C22×C3—32no—
463C22×C3—32no—
464C22×C3—44no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—62no—
470C22×C3—62no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—62no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—62no—
477C22×C3—64no—
478C22×C3—61no—
479C22×C3—62no—
480C22×C3—122no—
481C22×C3—122no—
482C22×C3—124no—
483C22×C3—124no—
484C22×C3—128no—
485C22×C3—182no—
486C22×C3—182no—
487C22×C3—182no—
488C22×C3C3×#23512yes—
489C22×C3C23×C3×D311yescharacteristic
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—182no—
494C2×D3—182no—
495C2×D3—182no—
496C3×C4—21no—
497C3×C4—22no—
498C3×C4—21no—
499C3×C4—61no—
500C3×C4—62no—
501C3×C4—61no—
502C3×C4—62no—
503C3×C4—61no—
504C3×C4—61no—
505C3×C4—122no—
506C3×C4—121no—
507C3×C4—121no—
508C3×C4—182no—
509C3×C4—182no—
510C3×C4—182no—
511Dic3—62no—
512Dic3—62no—
513Dic3—62no—
514Dic3—182no—
515Dic3—182no—
516Dic3—182no—
517C32—21no—
518C32#549411yescharacteristic, complemented
519C32C3×#53811yescharacteristic, complemented
520C23—22no—
521C23—41no—
522C23—41no—
523C23—92no—
524C23C22×(C3⋊D3)⋊C312yes—
525C23C22×(C3⋊D3)⋊C311yescharacteristic
526Q8—361no—
527D4—184no—
528D4—361no—
529D4—362no—
530D4—362no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—92no—
534C2×C4—92no—
535C2×C4—92no—
536C2×C4—362no—
537C2×C4C22×(C3⋊D3)⋊C312yes—
538C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
539C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
540C2×C3—22no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—42no—
548C2×C3—62no—
549C2×C3—62no—
550C2×C3—64no—
551C2×C3—61no—
552C2×C3—61no—
553C2×C3—61no—
554C2×C3—122no—
555C2×C3—122no—
556C2×C3—124no—
557C2×C3—362no—
558C2×C3C2×C3×#23511yescharacteristic
559C2×C3C3×#100411yescharacteristic
560C2×C3C3×#100411yescharacteristic
561D3—122no—
562D3—362no—
563C22—21no—
564C22—21no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—44no—
569C22—182no—
570C22—182no—
571C22—182no—
572C22#3828⋊C312yes—
573C22C23×(C3⋊D3)⋊C311yescharacteristic
574C4—22no—
575C4—21no—
576C4—21no—
577C4—182no—
578C4—182no—
579C4—182no—
580C3—31no—
581C3—31no—
582C3—61no—
583C3C3×#549411yescharacteristic
584C2—22no—
585C2—42no—
586C2—362no—
587C2C2×#3828⋊C311yescharacteristic
588C2#64361⋊C311yescharacteristic
589C2#64361⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊C4×H(3,𝔽3)C2semidirect
2. #11721702C2semidirect
3. #64139⋊C3C2semidirect
4. C4×#638⋊C3C2semidirect
5. #64133⋊C3C2semidirect
6. #64094⋊C3C2semidirect
7. #64093⋊C3C2semidirect
8. #64092⋊C3C2semidirect
9. #64088⋊C3C2semidirect
10. #10595965C3semidirect
18. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. C2×C4×(C3⋊D3)⋊C3C22semidirect
20. C2×#634⋊C3C22semidirect
26. #3752⋊C3C22semidirect
27. #3752⋊C3C22semidirect
28. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
29. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
30. #3750⋊C3C22semidirect
31. #3750⋊C3C22semidirect
32. C4×(C3⋊Dic3)⋊C3C22semidirect
34. C2×C32×C22⋊C4D3semidirect
35. C2×C32×C22⋊C4C2×C3semidirect
36. #64162C2×C3semidirect
37. #64139C2×C3semidirect
38. C4×#638C2×C3semidirect
39. #64133C2×C3semidirect
40. #64094C2×C3semidirect
41. #64093C2×C3semidirect
42. #64092C2×C3semidirect
43. #64088C2×C3semidirect
87. C2×C4⋊(C3⋊D3)C22×C3semidirect
88. C2×C4×C3⋊D3C22×C3semidirect
89. C2×#634C22×C3semidirect
105. #3752C22×C3semidirect
106. #3752C22×C3semidirect
107. C4⋊(C3⋊Dic3)C22×C3semidirect
108. C4⋊(C3⋊Dic3)C22×C3semidirect
109. #3750C22×C3semidirect
110. #3750C22×C3semidirect
111. C4×C3⋊Dic3C22×C3semidirect
345. C2×C22⋊C4(C3⋊D3)⋊C3semidirect
346. H(3,𝔽3)#538semidirect
518. C32#5494semidirect
519. C32C3×#538semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
#117217022
#64139⋊C32
C4×#638⋊C32
#64133⋊C32
#64094⋊C32
#64093⋊C32
#64092⋊C32
#64088⋊C32
#105959653
C3×#54943