Finite Group Databasebeta

← #10596003⋊C3 #13152510

Subgroups — #10596003⋊C3 #13152510

Order 1728.

902 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3C4×#638⋊C3C211yesmaximal, characteristic, complemented
4#64104⋊C3C211yesmaximal, characteristic, complemented
5#64102⋊C3C211yesmaximal, characteristic, complemented
6#64101⋊C3C211yesmaximal, characteristic, complemented
7#64097⋊C3C211yesmaximal, characteristic, complemented
8#64094⋊C3C212yesmaximal, complemented
9#64093⋊C3C212yesmaximal, complemented
10#64084⋊C3C212yesmaximal, complemented
11#64081⋊C3C212yesmaximal, complemented
12#10596003C311yesmaximal, characteristic, complemented
13C3×#5532—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2211yescharacteristic
16C2×#638⋊C3C2212yes—
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
19C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
20C2×C4×(C3⋊D3)⋊C3C2212yes—
21C2×#634⋊C3C2212yes—
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C4⋊C4×H(3,𝔽3)C2211yescharacteristic
24C22⋊C4×H(3,𝔽3)C2212yes—
25C42×H(3,𝔽3)C2212yes—
26#3757⋊C3C2211yescharacteristic
27#3752⋊C3C2211yescharacteristic
28#3752⋊C3C2212yes—
29#3752⋊C3C2212yes—
30#3752⋊C3C2212yes—
31#3752⋊C3C2212yes—
32C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
33#3750⋊C3C2211yescharacteristic
34#3750⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#117—21no—
38C32×#117D311yescharacteristic, complemented
39C32×#117C2×C311yescharacteristic, complemented
40#64139C2×C311yescharacteristic, complemented
41C4×#638C2×C311yescharacteristic, complemented
42#64104C2×C311yescharacteristic, complemented
43#64102C2×C311yescharacteristic, complemented
44#64101C2×C311yescharacteristic, complemented
45#64097C2×C311yescharacteristic, complemented
46#64094C2×C312yescomplemented
47#64093C2×C312yescomplemented
48#64084C2×C312yescomplemented
49#64081C2×C312yescomplemented
50C3×#930—31no—
51C3×C4×#67—31no—
52C3×#895—31no—
53C3×#893—31no—
54C3×#892—31no—
55C3×#888—31no—
56C3×#885—32no—
57C3×#884—32no—
58C3×#875—32no—
59C3×#872—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C2×C4×H(3,𝔽3)—22no—
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2312yes—
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68#638⋊C3—22no—
69#638⋊C3—42no—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C4⋊((C3⋊D3)⋊C3)—22no—
75C4⋊((C3⋊D3)⋊C3)—42no—
76C4×(C3⋊D3)⋊C3—22no—
77C4×(C3⋊D3)⋊C3—24no—
78#634⋊C3—42no—
79C3×#555—91no—
80#5532—31no—
81#5532—91no—
82C22×C32×C4—21no—
83C22×C32×C4C2×D311yescharacteristic
84C22×C32×C4C22×C311yescharacteristic
85C2×#638C22×C311yescharacteristic
86C2×#638C22×C312yes—
87C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
88C2×C4⋊(C3⋊D3)C22×C312yes—
89C2×C4×C3⋊D3C22×C312yes—
90C2×C4×C3⋊D3C22×C311yescharacteristic
91C2×#634C22×C312yes—
92C2×C3×#67—31no—
93C2×C3×#67—32no—
94C2×C3×D12—31no—
95C2×C3×D12—32no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—31no—
98C2×C3×Dic6—32no—
99C32×C4⋊C4—21no—
100C32×C4⋊C4—21no—
101C32×C4⋊C4C2×D311yescharacteristic
102C32×C4⋊C4C2×D311yescharacteristic
103C32×C4⋊C4C22×C311yescharacteristic
104C32×C4⋊C4C22×C311yescharacteristic
105C32×C22⋊C4—22no—
106C32×C22⋊C4C2×D312yes—
107C32×C22⋊C4C22×C312yes—
108C32×C42—22no—
109C32×C42C2×D312yes—
110C32×C42C22×C312yes—
111#3757C22×C311yescharacteristic
112#3752C22×C312yes—
113#3752C22×C312yes—
114#3752C22×C311yescharacteristic
115#3752C22×C312yes—
116#3752C22×C312yes—
117C4⋊(C3⋊Dic3)C22×C311yescharacteristic
118#3750C22×C311yescharacteristic
119#3750C22×C312yes—
120C4×C3⋊Dic3C22×C312yes—
121C4×C3⋊Dic3C22×C311yescharacteristic
122C3×#217—31no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4—32no—
126C3×(C2×D3)⋊C4—31no—
127C3×(C2×D3)⋊C4—32no—
128C3×C4⋊Dic3—31no—
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—31no—
131C3×C4×Dic3—32no—
132C3×C4×Dic3—31no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)—21no—
135C22×H(3,𝔽3)—21no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—22no—
140C2×(C3⋊D3)⋊C3—21no—
141C2×(C3⋊D3)⋊C3—21no—
142C2×(C3⋊D3)⋊C3—22no—
143C4×H(3,𝔽3)—21no—
144C4×H(3,𝔽3)—22no—
145C4×H(3,𝔽3)—22no—
146C4×H(3,𝔽3)#4112yes—
147(C3⋊Dic3)⋊C3—21no—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—22no—
150(C3⋊Dic3)⋊C3#4112yes—
151C3×C4⋊Q8—91no—
152C3×#124—92no—
153C3×#124—92no—
154C3×#121—92no—
155C3×#121—91no—
156C3×#121—91no—
157C3×C4×D4—91no—
158C3×C4×D4—91no—
159C3×#117—31no—
160C3×#117—31no—
161C3×#117—61no—
162C3×#117—92no—
163C3×#117—91no—
164C3×#117C3×D311yescharacteristic
165#930—31no—
166#930—91no—
167C4×#67—31no—
168C4×#67—91no—
169#895—31no—
170#895—91no—
171#893—31no—
172#893—91no—
173#892—31no—
174#892—91no—
175#888—31no—
176#888—91no—
177#885—32no—
178#885—92no—
179#884—32no—
180#884—92no—
181#875—32no—
182#875—92no—
183#872—32no—
184#872—92no—
185C23×C32—21no—
186C23×C32C22×D311yescharacteristic
187C23×C32C23×C311yescharacteristic
188C22×C3⋊D3C23×C312yes—
189C22×C3⋊D3C23×C311yescharacteristic
190C22×C3×D3—31no—
191C22×C3×D3—32no—
192C2×C32×C4—24no—
193C2×C32×C4—22no—
194C2×C32×C4—22no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4—22no—
198C2×C32×C4—22no—
199C2×C32×C4C22×D312yes—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D312yes—
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C23×C312yes—
204C2×C32×C4C23×C311yescharacteristic
205C2×C32×C4C23×C312yes—
206C2×C32×C4C23×C311yescharacteristic
207#638—22no—
208#638—42no—
209C2×C3⋊Dic3C23×C311yescharacteristic
210C2×C3⋊Dic3C23×C312yes—
211C2×C3⋊Dic3C23×C311yescharacteristic
212C2×C3⋊Dic3C23×C311yescharacteristic
213C4⋊(C3⋊D3)—22no—
214C4⋊(C3⋊D3)—42no—
215C4×C3⋊D3—22no—
216C4×C3⋊D3—24no—
217#634—42no—
218C3×#67—62no—
219C3×#67—122no—
220C2×C3×Dic3—31no—
221C2×C3×Dic3—31no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—31no—
224C3×D12—62no—
225C3×D12—122no—
226C3×C4×D3—64no—
227C3×C4×D3—62no—
228C3×Dic6—122no—
229#555—91noSylow
230C2×H(3,𝔽3)—41no—
231C2×H(3,𝔽3)C2×#4111yescharacteristic
232C2×H(3,𝔽3)C2×#4111yescharacteristic
233C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
234(C3⋊D3)⋊C3—42no—
235(C3⋊D3)⋊C3—41no—
236C2×C3×Q8—92no—
237C2×C3×D4—91no—
238C2×C3×D4—92no—
239C2×C3×D4—91no—
240C2×C3×D4—92no—
241C22×C3×C4—31no—
242C22×C3×C4—31no—
243C22×C3×C4—61no—
244C22×C3×C4—91no—
245C22×C3×C4—92no—
246C22×C3×C4C2×C3×D311yescharacteristic
247C2×#67—31no—
248C2×#67—32no—
249C2×#67—91no—
250C2×#67—92no—
251C2×D12—31no—
252C2×D12—32no—
253C2×D12—91no—
254C2×D12—92no—
255C2×C4×D3—32no—
256C2×C4×D3—31no—
257C2×C4×D3—92no—
258C2×C4×D3—91no—
259C2×Dic6—32no—
260C2×Dic6—92no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—31no—
263C3×C4⋊C4—31no—
264C3×C4⋊C4—31no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—61no—
267C3×C4⋊C4—91no—
268C3×C4⋊C4—92no—
269C3×C4⋊C4—91no—
270C3×C4⋊C4C2×C3×D311yescharacteristic
271C3×C4⋊C4C2×C3×D311yescharacteristic
272C3×C22⋊C4—32no—
273C3×C22⋊C4—32no—
274C3×C22⋊C4—62no—
275C3×C22⋊C4—92no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4—92no—
278C3×C22⋊C4—91no—
279C3×C22⋊C4—92no—
280C3×C22⋊C4—91no—
281C3×C22⋊C4C2×C3×D312yes—
282C3×C42—32no—
283C3×C42—32no—
284C3×C42—62no—
285C3×C42—92no—
286C3×C42—91no—
287C3×C42C2×C3×D312yes—
288#217—31no—
289#217—91no—
290(C2×D3)⋊C4—32no—
291(C2×D3)⋊C4—32no—
292(C2×D3)⋊C4—31no—
293(C2×D3)⋊C4—32no—
294(C2×D3)⋊C4—32no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300C4⋊Dic3—31no—
301C4⋊Dic3—91no—
302Dic3⋊C4—31no—
303Dic3⋊C4—32no—
304Dic3⋊C4—91no—
305Dic3⋊C4—92no—
306C4×Dic3—32no—
307C4×Dic3—31no—
308C4×Dic3—92no—
309C4×Dic3—91no—
310C22×C32—22no—
311C22×C32—21no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—21no—
318C22×C32—41no—
319C22×C32—41no—
320C22×C32C23×D311yescharacteristic
321C22×C32C24×C311yescharacteristic
322C2×C3⋊D3—21no—
323C2×C3⋊D3—22no—
324C2×C3⋊D3—22no—
325C2×C3⋊D3—21no—
326C2×C3⋊D3—21no—
327C2×C3⋊D3—22no—
328C2×C3×D3—61no—
329C2×C3×D3—62no—
330C2×C3×D3—62no—
331C2×C3×D3—61no—
332C2×C3×D3—61no—
333C2×C3×D3—62no—
334C32×C4—22no—
335C32×C4—21no—
336C32×C4—22no—
337C32×C4—22no—
338C32×C4—22no—
339C32×C4—21no—
340C32×C4—22no—
341C32×C4—22no—
342C32×C4—42no—
343C32×C4—42no—
344C32×C4#23512yes—
345C32×C4C3×#4112yes—
346C3⋊Dic3—21no—
347C3⋊Dic3—21no—
348C3⋊Dic3—22no—
349C3⋊Dic3C3×#4112yes—
350C3×Dic3—32no—
351C3×Dic3—61no—
352C3×Dic3—61no—
353C3×Dic3—62no—
354C4⋊Q8—91no—
355#124—92no—
356#124—92no—
357#121—91no—
358#121—91no—
359#121—92no—
360C4×D4—91no—
361C4×D4—91no—
362#117—91no—
363#117—92no—
364#117(C3⋊D3)⋊C311yescharacteristic, complemented
365H(3,𝔽3)#55511yescharacteristic, complemented, Sylow
366C23×C3—31no—
367C23×C3—31no—
368C23×C3—61no—
369C23×C3—92no—
370C23×C3—91no—
371C23×C3C22×C3×D311yescharacteristic
372C22×D3—31no—
373C22×D3—32no—
374C22×D3—92no—
375C22×D3—91no—
376C3×Q8—362no—
377C3×D4—182no—
378C3×D4—182no—
379C3×D4—362no—
380C3×D4—362no—
381C2×C3×C4—22no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—32no—
386C2×C3×C4—32no—
387C2×C3×C4—32no—
388C2×C3×C4—31no—
389C2×C3×C4—32no—
390C2×C3×C4—62no—
391C2×C3×C4—62no—
392C2×C3×C4—64no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—62no—
396C2×C3×C4—62no—
397C2×C3×C4—91no—
398C2×C3×C4—91no—
399C2×C3×C4—91no—
400C2×C3×C4—92no—
401C2×C3×C4—182no—
402C2×C3×C4—184no—
403C2×C3×C4C22×C3×D312yes—
404C2×C3×C4C22×C3×D311yescharacteristic
405C2×C3×C4C22×C3×D312yes—
406C2×C3×C4C22×C3×D311yescharacteristic
407#67—62no—
408#67—122no—
409#67—182no—
410#67—362no—
411C2×Dic3—31no—
412C2×Dic3—31no—
413C2×Dic3—32no—
414C2×Dic3—31no—
415C2×Dic3—91no—
416C2×Dic3—91no—
417C2×Dic3—92no—
418C2×Dic3—91no—
419D12—62no—
420D12—122no—
421D12—182no—
422D12—362no—
423C4×D3—64no—
424C4×D3—62no—
425C4×D3—184no—
426C4×D3—182no—
427Dic6—122no—
428Dic6—362no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—42no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32C2×#23511yescharacteristic
436C2×C32D3×#4111yescharacteristic
437C2×C32#100911yescharacteristic
438C2×C32C2×C3×#4111yescharacteristic
439C2×C32C2×C3×#4111yescharacteristic
440C2×C32C3×H(5,𝔽2)11yescharacteristic
441C3⋊D3—42no—
442C3⋊D3—41no—
443C3×D3—121no—
444C3×D3—122no—
445C2×Q8—92no—
446C2×D4—91no—
447C2×D4—92no—
448C2×D4—91no—
449C2×D4—92no—
450C22×C4—92no—
451C22×C4—91no—
452C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
453C4⋊C4—91no—
454C4⋊C4—92no—
455C4⋊C4—91no—
456C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
457C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
458C22⋊C4—91no—
459C22⋊C4—92no—
460C22⋊C4—92no—
461C22⋊C4—92no—
462C22⋊C4—92no—
463C22⋊C4—91no—
464C22⋊C4C2×(C3⋊D3)⋊C312yes—
465C42—92no—
466C42—91no—
467C42C2×(C3⋊D3)⋊C312yes—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—31no—
472C22×C3—31no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—61no—
476C22×C3—61no—
477C22×C3—61no—
478C22×C3—61no—
479C22×C3—62no—
480C22×C3—61no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3—181no—
484C22×C3—182no—
485C22×C3—182no—
486C22×C3—181no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3C23×C3×D311yescharacteristic
490C2×D3—61no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—61no—
494C2×D3—61no—
495C2×D3—62no—
496C2×D3—181no—
497C2×D3—182no—
498C2×D3—182no—
499C2×D3—181no—
500C2×D3—182no—
501C2×D3—181no—
502C3×C4—21no—
503C3×C4—22no—
504C3×C4—22no—
505C3×C4—32no—
506C3×C4—32no—
507C3×C4—62no—
508C3×C4—62no—
509C3×C4—61no—
510C3×C4—61no—
511C3×C4—62no—
512C3×C4—62no—
513C3×C4—62no—
514C3×C4—62no—
515C3×C4—92no—
516C3×C4—122no—
517C3×C4—122no—
518C3×C4—181no—
519C3×C4—182no—
520C3×C4—181no—
521C3×C4C3×#23512yes—
522Dic3—32no—
523Dic3—61no—
524Dic3—61no—
525Dic3—62no—
526Dic3—92no—
527Dic3—181no—
528Dic3—182no—
529Dic3—181no—
530C32—21no—
531C32#553211yescharacteristic, complemented
532C32C3×#55511yescharacteristic, complemented
533C23—91no—
534C23—92no—
535C23C22×(C3⋊D3)⋊C311yescharacteristic
536Q8—362no—
537D4—182no—
538D4—182no—
539D4—362no—
540D4—362no—
541C2×C4—22no—
542C2×C4—91no—
543C2×C4—91no—
544C2×C4—91no—
545C2×C4—92no—
546C2×C4—184no—
547C2×C4—182no—
548C2×C4C22×(C3⋊D3)⋊C312yes—
549C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
550C2×C4C22×(C3⋊D3)⋊C312yes—
551C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
552C2×C3—31no—
553C2×C3—31no—
554C2×C3—31no—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—41no—
559C2×C3—61no—
560C2×C3—61no—
561C2×C3—61no—
562C2×C3—121no—
563C2×C3—121no—
564C2×C3—122no—
565C2×C3—361no—
566C2×C3—362no—
567C2×C3C2×C3×#23511yescharacteristic
568C2×C3C3×D3×#4111yescharacteristic
569C2×C3C3×#100911yescharacteristic
570D3—121no—
571D3—122no—
572D3—362no—
573D3—361no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—181no—
578C22—182no—
579C22—182no—
580C22—181no—
581C22—181no—
582C22—182no—
583C22C23×(C3⋊D3)⋊C311yescharacteristic
584C4—21no—
585C4—22no—
586C4—22no—
587C4—92no—
588C4—181no—
589C4—182no—
590C4—181no—
591C4#3828⋊C312yes—
592C3—31no—
593C3—31no—
594C3—61no—
595C3C3×#553211yescharacteristic
596C2—41no—
597C2—361no—
598C2—362no—
599C2C2×#3828⋊C311yescharacteristic
600C2#41×(C3⋊D3)⋊C311yescharacteristic
601C2#1172263811yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#555
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
2. #64139⋊C3C2semidirect
3. C4×#638⋊C3C2semidirect
4. #64104⋊C3C2semidirect
5. #64102⋊C3C2semidirect
6. #64101⋊C3C2semidirect
7. #64097⋊C3C2semidirect
8. #64094⋊C3C2semidirect
9. #64093⋊C3C2semidirect
10. #64084⋊C3C2semidirect
11. #64081⋊C3C2semidirect
12. #10596003C3semidirect
38. C32×#117D3semidirect
39. C32×#117C2×C3semidirect
40. #64139C2×C3semidirect
41. C4×#638C2×C3semidirect
42. #64104C2×C3semidirect
43. #64102C2×C3semidirect
44. #64101C2×C3semidirect
45. #64097C2×C3semidirect
46. #64094C2×C3semidirect
47. #64093C2×C3semidirect
48. #64084C2×C3semidirect
49. #64081C2×C3semidirect
364. #117(C3⋊D3)⋊C3semidirect
365. H(3,𝔽3)#555semidirect
531. C32#5532semidirect
532. C32C3×#555semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1172
#64139⋊C32
C4×#638⋊C32
#64104⋊C32
#64102⋊C32
#64101⋊C32
#64097⋊C32
#64094⋊C32
#64093⋊C32
#64084⋊C32
#64081⋊C32
#105960033
C3×#55323