Finite Group Databasebeta

← #13156040

Subgroups — #13156040

Order 1728.

720 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723439C211yesmaximal, characteristic, complemented
2C2×#88794C211yesmaximal, characteristic, complemented
3C3×Dic3×D12C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic
5#11722075C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591468—31nomaximal
9#10590755—31nomaximal
10#10590366—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3×D3×Dic3—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88803—21no—
20#88803—21no—
21#88794—22no—
22C3×(C2×D3)⋊Dic3—21no—
23C3×#3714—22no—
24C2×#3754—31no—
25#64084—31no—
26C3×D4×Dic3—31no—
27C2×C3×#213—31no—
28C2×C3×D24—31no—
29C3×C4×D12—31no—
30Dic3×D12D311yescharacteristic, complemented
31#63860—31no—
32#63860—31no—
33#63860—61no—
34C2×#3716—31no—
35C2×#3716—31no—
36C2×#3716—61no—
37C2×#3714D311yescharacteristic
38#63643—31no—
39C3×Dic3⋊C8—31no—
40C3×C4⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
42#63567—31no—
43#63567—31no—
44#63567—61no—
45#63564—31no—
46#63563—31no—
47#63562—31no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52#6308—21no—
53#6308—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12D412yes—
61C3×D3×Dic3—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5153—91no—
65#5016—91no—
66#4844—91no—
67C2×C32×D4—31no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×#634—31no—
70C2×#634—31no—
71C2×#634—31no—
72C2×#634—61no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—121no—
76C22×C3×Dic3—61no—
77C2×C3×D12—21no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—61no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×C4×D3—61no—
84C2×D3×Dic3—21no—
85C2×C32×C8—31no—
86C32×C42—31no—
87#3754—62no—
88#3752—61no—
89#3752—61no—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#217—61no—
93C3×#213—62no—
94C3×(C2×D3)⋊C4—61no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—61no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3C2×D311yescharacteristic
100C3×C4×Dic3C2×D311yescharacteristic
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C3×D24—62no—
108(C2×D3)⋊Dic3—21no—
109(C2×D3)⋊Dic3—62no—
110(C2×D3)⋊Dic3—62no—
111(C2×D3)⋊Dic3—61no—
112(C2×D3)⋊Dic3—61no—
113(C2×D3)⋊Dic3—61no—
114(C2×D3)⋊Dic3—61no—
115#3716—64no—
116#3716—62no—
117#3716—62no—
118#3714—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C32×D3—22no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3⋊(C3⋊Dic3)—41no—
129C3×C3⋊Dic3—41no—
130C32×Dic3—21no—
131C32×Dic3—21no—
132C2×C3×D8—91no—
133C3×C4×D4—91no—
134#936—91no—
135D4×Dic3—31no—
136C2×#214—91no—
137C2×#214—91no—
138C2×#214—181no—
139C2×#213—31no—
140C2×D24—31no—
141C2×#204—91no—
142#875—91no—
143#875—91no—
144#875—181no—
145C4×D12—31no—
146C3×C4⋊C8—91no—
147#832—91no—
148#821—91no—
149#816—91no—
150#816—91no—
151#816—181no—
152Dic3⋊C8—31no—
153#810—91no—
154#809—91no—
155C4⋊(C3⋊C8)—31no—
156C23×C32—61no—
157C22×C3⋊D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—41no—
161C22×C3×D3—62no—
162C22×C3×D3—61no—
163C22×C3×D3—61no—
164C32×D4—32no—
165C32×D4—61no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—31no—
170C2×C32×C4—31no—
171C2×C32×C4—41no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—62no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—121no—
181C2×C3⋊Dic3—121no—
182C2×C3⋊Dic3—122no—
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)—21no—
185#634—61no—
186#634—61no—
187#634—62no—
188#634—62no—
189#634—61no—
190#634—61no—
191#634—61no—
192#634—61no—
193#634—121no—
194#634—121no—
195#634—121no—
196#634—121no—
197#634—122no—
198#634—122no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—121no—
204C2×C3×Dic3—122no—
205C2×C3×Dic3D1211yescharacteristic
206C2×C3×Dic3D1211yescharacteristic
207C2×C3×Dic3#6711yescharacteristic
208C2×C3×Dic3#6711yescharacteristic
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—22no—
212C3×D12—41no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—121no—
218C3×D12—121no—
219C3×D12D1212yes—
220C3×D12#6712yes—
221C3×C4×D3—122no—
222D3×Dic3—42no—
223C32×C8—61no—
224C3⋊(C3⋊C8)—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—41no—
228C3×C3⋊C8—62no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231#463—271noSylow
232C2×C33#12111yescharacteristic
233C2×C33#13511yescharacteristic
234C2×C33AGL(1,ℤ/8)11yescharacteristic
235C3×C3⋊D3—81no—
236C32×D3—42no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—61no—
240C2×C3×D4—91no—
241C22×C3×C4—181no—
242C22×Dic3—61no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—61no—
246C2×D12D3211yescharacteristic
247C2×C4×D3—61no—
248C2×Dic6—91no—
249C2×Dic6—91no—
250C2×Dic6—91no—
251C2×Dic6—181no—
252C2×Dic6—181no—
253C2×Dic6—181no—
254C2×Dic6—361no—
255C3×D8—182no—
256C2×C3×C8—31no—
257C2×C3×C8—31no—
258C2×C3×C8—61no—
259C2×C3×C8—91no—
260C3×C4⋊C4—91no—
261C3×C22⋊C4—181no—
262C3×C42—31no—
263C3×C42—31no—
264C3×C42—61no—
265#217—61no—
266#217—181no—
267#217—181no—
268#214—182no—
269#214—182no—
270#214—184no—
271#213—62no—
272(C2×D3)⋊C4—61no—
273(C2×D3)⋊C4—181no—
274(C2×D3)⋊C4—181no—
275(C2×D3)⋊C4—181no—
276(C2×D3)⋊C4—181no—
277(C2×D3)⋊C4—182no—
278(C2×D3)⋊C4—182no—
279C4⋊Dic3—31no—
280C4⋊Dic3—31no—
281C4⋊Dic3—61no—
282C4×Dic3D3211yescharacteristic
283C2×C3⋊C8—31no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—61no—
286C2×C3⋊C8D3211yescharacteristic
287D24—62no—
288#204—182no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—62no—
294C22×C32—121no—
295C22×C32—121no—
296C22×C32C2×D1211yescharacteristic
297C22×C32D3×D411yescharacteristic
298C22×C32C2×#6711yescharacteristic
299C2×C3⋊D3—41no—
300C2×C3⋊D3—41no—
301C2×C3⋊D3—41no—
302C2×C3×D3—22no—
303C2×C3×D3—22no—
304C2×C3×D3—42no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—81no—
310C2×C3×D3—81no—
311C2×C3×D3—122no—
312C2×C3×D3—122no—
313C2×C3×D3—122no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C2×C3×D3—121no—
317C2×C3×D3—121no—
318C2×C3×D3—121no—
319C2×C3×D3—121no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—41no—
327C32×C4—41no—
328C32×C4—61no—
329C32×C4—61no—
330C32×C4C2×D1211yescharacteristic
331C32×C4#23511yescharacteristic
332C32×C4D3×D411yescharacteristic
333C32×C4#23711yescharacteristic
334C32×C4#23911yescharacteristic
335C32×C4C2×#6711yescharacteristic
336C3⋊Dic3—41no—
337C3⋊Dic3—122no—
338C3⋊Dic3—122no—
339C3⋊Dic3—122no—
340C3⋊Dic3—124no—
341C3⋊Dic3—121no—
342C3⋊Dic3—121no—
343C3⋊Dic3—121no—
344C3×Dic3—21no—
345C3×Dic3—21no—
346C3×Dic3—21no—
347C3×Dic3—21no—
348C3×Dic3—41no—
349C3×Dic3—41no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C3×Dic3—241no—
353C2×SD16—271no—
354C2×D8—91no—
355#124—271no—
356C4×D4—91no—
357C4⋊C8—91no—
358#103—271no—
359#102—271no—
360C33#46311yescharacteristic, complemented, Sylow
361C23×C3—61no—
362C23×C3—61no—
363C23×C3—121no—
364C23×C3—181no—
365C22×D3—21no—
366C22×D3—62no—
367C22×D3—61no—
368C22×D3—61no—
369C3×D4—32no—
370C3×D4—32no—
371C3×D4—62no—
372C3×D4—61no—
373C3×D4—61no—
374C3×D4—121no—
375C3×D4—181no—
376C3×D4—181no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—31no—
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—41no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—181no—
388C2×C3×C4—362no—
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C2×D3211yescharacteristic
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3—121no—
395C2×Dic3—122no—
396C2×Dic3—182no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—361no—
401C2×Dic3—361no—
402C2×Dic3—362no—
403C2×Dic3#62611yescharacteristic
404C2×Dic3#62611yescharacteristic
405D12—21no—
406D12—61no—
407D12—61no—
408D12—61no—
409D12—61no—
410D12—121no—
411D12—121no—
412D12#62612yes—
413C4×D3—122no—
414Dic6—182no—
415Dic6—182no—
416Dic6—181no—
417Dic6—181no—
418Dic6—181no—
419Dic6—181no—
420Dic6—181no—
421Dic6—181no—
422Dic6—361no—
423Dic6—361no—
424Dic6—361no—
425Dic6—361no—
426Dic6—362no—
427Dic6—362no—
428C3×C8—61no—
429C3×C8—61no—
430C3×C8—121no—
431C3×C8—181no—
432C3⋊C8—21no—
433C3⋊C8—62no—
434C3⋊C8—61no—
435C3⋊C8—61no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—41no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32—122no—
449C2×C32#89511yescharacteristic
450C2×C32#90411yescharacteristic
451C2×C32#90811yescharacteristic
452C2×C32#91111yescharacteristic
453C2×C32#91211yescharacteristic
454C2×C32#93011yescharacteristic
455C2×C32#93211yescharacteristic
456C2×C32#93911yescharacteristic
457C2×C32#95011yescharacteristic
458C3⋊D3—81no—
459C3×D3—42no—
460C3×D3—42no—
461C3×D3—82no—
462C3×D3—242no—
463C3×D3—241no—
464C3×D3—241no—
465C2×Q8—271no—
466C2×D4—31no—
467C2×D4—91no—
468C22×C4—181no—
469SD16—542no—
470D8—182no—
471C2×C8—31no—
472C2×C8—91no—
473C4⋊C4—91no—
474C22⋊C4—181no—
475C22⋊C4—541no—
476C22⋊C4—541no—
477C42—31no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—41no—
482C22×C3—62no—
483C22×C3—62no—
484C22×C3—122no—
485C22×C3—121no—
486C22×C3—121no—
487C22×C3—121no—
488C22×C3—121no—
489C22×C3—241no—
490C22×C3—241no—
491C22×C3—361no—
492C22×C3—361no—
493C22×C3—361no—
494C22×C3D3×D1211yescharacteristic
495C22×C3C2×#62611yescharacteristic
496C22×C3D3×#6711yescharacteristic
497C2×D3—22no—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—122no—
501C2×D3—122no—
502C2×D3—122no—
503C2×D3—121no—
504C2×D3—121no—
505C2×D3—121no—
506C2×D3—121no—
507C2×D3—121no—
508C2×D3—121no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4—61no—
518C3×C4—61no—
519C3×C4—61no—
520C3×C4—61no—
521C3×C4—121no—
522C3×C4—121no—
523C3×C4—361no—
524C3×C4#379611yescharacteristic
525C3×C4#379911yescharacteristic
526C3×C4D3×D1211yescharacteristic
527C3×C4#380411yescharacteristic
528C3×C4C2×#62611yescharacteristic
529C3×C4D3×#6711yescharacteristic
530Dic3—21no—
531Dic3—21no—
532Dic3—121no—
533Dic3—121no—
534Dic3—241no—
535Dic3—362no—
536Dic3—362no—
537Dic3—362no—
538Dic3—364no—
539Dic3—361no—
540Dic3—361no—
541Dic3—361no—
542C32—21no—
543C32—21no—
544C32—21no—
545C32—41no—
546C32#484411yescharacteristic, complemented
547C32#501611yescharacteristic, complemented
548C32#515311yescharacteristic, complemented
549C23—61no—
550C23—181no—
551Q8—541no—
552Q8—541no—
553D4—32no—
554D4—61no—
555D4—181no—
556D4—181no—
557C2×C4—31no—
558C2×C4—31no—
559C2×C4—181no—
560C2×C4—362no—
561C2×C4—541no—
562C2×C4D3311yescharacteristic
563C8—61no—
564C8—181no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—122no—
578C2×C3—122no—
579C2×C3—242no—
580C2×C3—721no—
581C2×C3#6379511yescharacteristic
582C2×C3#6380711yescharacteristic
583C2×C3#6382911yescharacteristic
584C2×C3#6383011yescharacteristic
585C2×C3#6388611yescharacteristic
586C2×C3#6390911yescharacteristic
587C2×C3#6393211yescharacteristic
588C2×C3#6393311yescharacteristic
589C2×C3#6397811yescharacteristic
590D3—42no—
591D3—242no—
592D3—241no—
593D3—241no—
594C22—62no—
595C22—121no—
596C22—121no—
597C22—361no—
598C22—361no—
599C22—361no—
600C22D3×#62611yescharacteristic
601C4—61no—
602C4—61no—
603C4—361no—
604C4—1081no—
605C4D3×#62611yescharacteristic
606C4#8896211yescharacteristic
607C3—21no—
608C3—21no—
609C3—21no—
610C3—41no—
611C3#1059036611yescharacteristic, complemented
612C3#1059075511yescharacteristic, complemented
613C3#1059146811yescharacteristic, complemented
614C2—122no—
615C2—721no—
616C2#1172285411yescharacteristic
617C2#1172286911yescharacteristic
618C2#1172295911yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723439C2semidirect
2. C2×#88794C2semidirect
3. C3×Dic3×D12C2semidirect
5. #11722075C2semidirect
6. #11722042C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
30. Dic3×D12D3semidirect
41. Dic3⋊(C3⋊C8)D3semidirect
360. C33#463semidirect
546. C32#4844semidirect
547. C32#5016semidirect
548. C32#5153semidirect
611. C3#10590366semidirect
612. C3#10590755semidirect
613. C3#10591468semidirect

Maximal subgroups

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

subgroupindex
#117234392
C2×#887942
C3×Dic3×D122
C2×C3×#37142
#117220752
#117220422
C3×Dic3⋊(C3⋊C8)2
#105914683
#105907553
#105903663