Finite Group Databasebeta

← #13155982

Subgroups — #13155982

Order 1728.

690 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88793C211yesmaximal, characteristic, complemented
2C3×#63911C211yesmaximal, characteristic
3C3×Dic3×Dic6C211yesmaximal, characteristic, complemented
4C2×C3×#3715C211yesmaximal, characteristic, complemented
5#11722075C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10591617—31nomaximal
9#10591438—31nomaximal
10#10590364—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#626—21no—
15C2×C3×#626—21no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20#88793—22no—
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3715—22no—
24C2×#3755—31no—
25C3×Q8×Dic3—31no—
26C2×C3×#215—31no—
27C3×#940—31no—
28C2×C3×#214—31no—
29C3×#874—31no—
30C3×C4×Dic6—31no—
31#63911D311yescharacteristic
32Dic3×Dic6D311yescharacteristic, complemented
33C2×#3715—31no—
34C2×#3715—31no—
35C2×#3715—61no—
36C2×#3715D311yescharacteristic, complemented
37#63659—31no—
38C4⋊(C3⋊(C3⋊C8))—31no—
39Dic3⋊(C3⋊C8)—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—61no—
42#63564—31no—
43#63563—31no—
44#63562—31no—
45#63562—31no—
46#63562—31no—
47#63562—61no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×D12—21no—
58C32×D12—21no—
59C32×Dic6—21no—
60C32×Dic6D412yescomplemented
61C3×#626—42no—
62C3×#626—42no—
63C3⋊(C3⋊(C3⋊C8))—21no—
64C3×C3⋊(C3⋊C8)—21no—
65#5163—91no—
66#5075—91no—
67#5027—91no—
68C2×C32×Q8—31no—
69C2×C32×D4—31no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C3×#67—61no—
72C2×C3×#67—61no—
73C2×C3×D12—21no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×D12—61no—
78C2×C3×D12—61no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×Dic6—21no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C2×D311yescharacteristic
84C2×#626—21no—
85C2×#626—21no—
86C32×C42—31no—
87#3755—62no—
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—31no—
91C2×C3⋊(C3⋊C8)—31no—
92C2×C3⋊(C3⋊C8)—61no—
93C2×C3⋊(C3⋊C8)—61no—
94C2×C3⋊(C3⋊C8)—61no—
95C2×C3⋊(C3⋊C8)—121no—
96C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
97C3×#215—62no—
98C3×#214—62no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—61no—
102C3×C4⋊Dic3—61no—
103C3×Dic3⋊C4—61no—
104C3×C4×Dic3—21no—
105C3×C4×Dic3—61no—
106C3×C4×Dic3—61no—
107C3×C4×Dic3C2×D311yescharacteristic
108C3×C4×Dic3C2×D311yescharacteristic
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—61no—
112Dic3⋊Dic3—21no—
113Dic32—21no—
114#3715—22no—
115#3715—64no—
116#3715—62no—
117#3715—62no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3×C3⋊Dic3—41no—
128C32×Dic3—22no—
129C32×Dic3—21no—
130C32×Dic3SD1612yescomplemented
131C2×C3×SD16—91no—
132C3×#127—91no—
133C3×C4×Q8—91no—
134Q8×Dic3—31no—
135C2×#215—31no—
136C2×#215—91no—
137C2×#215—91no—
138C2×#215—181no—
139#940—31no—
140C2×#214—31no—
141C2×#214—91no—
142#874—31no—
143C4×Dic6—31no—
144#832—91no—
145#832—91no—
146#832—91no—
147#832—181no—
148#821—91no—
149Dic3⋊C8—91no—
150#809—91no—
151#809—91no—
152C4⋊(C3⋊C8)—91no—
153C4⋊(C3⋊C8)—91no—
154C4⋊(C3⋊C8)—181no—
155C23×C32—61no—
156C22×C3⋊D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—41no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—121no—
163C32×Q8—32no—
164C32×Q8—61no—
165C32×D4—61no—
166C32×D4—61no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—31no—
171C2×C32×C4—31no—
172C2×C32×C4—41no—
173C2×C32×C4—61no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—21no—
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)—21no—
180C3×#67—122no—
181C3×#67—122no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—121no—
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3#6711yescharacteristic
192C2×C3×Dic3#6711yescharacteristic
193C2×C3×Dic3#6711yescharacteristic
194C3×D12—22no—
195C3×D12—21no—
196C3×D12—21no—
197C3×D12—21no—
198C3×D12—21no—
199C3×D12—41no—
200C3×D12—62no—
201C3×D12—61no—
202C3×D12—61no—
203C3×D12—61no—
204C3×D12—61no—
205C3×D12—121no—
206C3×D12—122no—
207C3×D12—122no—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—22no—
211C3×Dic6—41no—
212C3×Dic6D1212yescomplemented
213C3×Dic6#6712yescomplemented
214#626—42no—
215#626—42no—
216C3⋊(C3⋊C8)—21no—
217C3⋊(C3⋊C8)—61no—
218C3⋊(C3⋊C8)—62no—
219C3⋊(C3⋊C8)—61no—
220C3⋊(C3⋊C8)—61no—
221C3⋊(C3⋊C8)—121no—
222C3⋊(C3⋊C8)—121no—
223C3⋊(C3⋊C8)—122no—
224C3×C3⋊C8—61no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—121no—
227#460—271noSylow
228C2×C33#12111yescharacteristic
229C2×C33C2×SD1611yescharacteristic
230C2×C33AGL(1,ℤ/8)11yescharacteristic
231C3×C3⋊D3—81no—
232C32×D3—81no—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—61no—
236C2×C3×D4—31no—
237C2×C3×D4—31no—
238C2×C3×D4—61no—
239C2×C3×D4—91no—
240C2×C3×D4—181no—
241C2×C3×D4—181no—
242C2×#67—61no—
243C2×#67—61no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—61no—
247C2×D12—61no—
248C2×D12—61no—
249C2×D12D3211yescharacteristic
250C2×Dic6D3211yescharacteristic
251C3×SD16—182no—
252C2×C3×C8—91no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—181no—
255C3×C42—31no—
256C3×C42—31no—
257C3×C42—61no—
258C3×C42—181no—
259#215—62no—
260#215—182no—
261#215—182no—
262#215—184no—
263#214—62no—
264#214—182no—
265C4⋊Dic3—31no—
266C4⋊Dic3—31no—
267C4⋊Dic3—61no—
268C4⋊Dic3—61no—
269Dic3⋊C4—61no—
270C4×Dic3—61no—
271C4×Dic3—61no—
272C4×Dic3D3211yescharacteristic
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—61no—
276C2×C3⋊C8—91no—
277C2×C3⋊C8—91no—
278C2×C3⋊C8—91no—
279C2×C3⋊C8—181no—
280C2×C3⋊C8—181no—
281C2×C3⋊C8—181no—
282C2×C3⋊C8—361no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—41no—
287C22×C32—121no—
288C22×C32—121no—
289C22×C32—121no—
290C22×C32D3×D411yescharacteristic
291C22×C32C2×#6711yescharacteristic
292C22×C32C2×#6711yescharacteristic
293C2×C3⋊D3—41no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—41no—
296C2×C3×D3—42no—
297C2×C3×D3—41no—
298C2×C3×D3—41no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—81no—
304C2×C3×D3—81no—
305C2×C3×D3—122no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C2×C3×D3—241no—
313C2×C3×D3—241no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—32no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—61no—
324C32×C4—62no—
325C32×C4C2×D1211yescharacteristic
326C32×C4#23511yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4#23711yescharacteristic
329C32×C4#23711yescharacteristic
330C32×C4C2×#6711yescharacteristic
331C3⋊Dic3—41no—
332C3×Dic3—22no—
333C3×Dic3—21no—
334C3×Dic3—22no—
335C3×Dic3—21no—
336C3×Dic3—22no—
337C3×Dic3—42no—
338C3×Dic3—41no—
339C3×Dic3—121no—
340C3×Dic3—121no—
341C3×Dic3—241no—
342C3×Dic3#21412yescomplemented
343C3×Dic3#21512yescomplemented
344C2×SD16—91no—
345C2×SD16—271no—
346#127—91no—
347C4×Q8—91no—
348C4⋊C8—271no—
349#102—271no—
350#102—271no—
351C33#46011yescharacteristic, complemented, Sylow
352C23×C3—61no—
353C23×C3—61no—
354C23×C3—121no—
355C23×C3—181no—
356C22×D3—21no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—121no—
360C3×Q8—32no—
361C3×Q8—32no—
362C3×Q8—62no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—121no—
366C3×D4—62no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—121no—
372C3×D4—181no—
373C3×D4—181no—
374C3×D4—362no—
375C3×D4—362no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—41no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—121no—
389C2×C3×C4—181no—
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C2×D3211yescharacteristic
393#67—122no—
394#67—122no—
395C2×Dic3—21no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—121no—
399C2×Dic3#62511yescharacteristic
400C2×Dic3#62511yescharacteristic
401D12—21no—
402D12—21no—
403D12—62no—
404D12—61no—
405D12—61no—
406D12—61no—
407D12—61no—
408D12—121no—
409D12—122no—
410D12—122no—
411Dic6—21no—
412Dic6#62612yescomplemented
413C3×C8—181no—
414C3⋊C8—61no—
415C3⋊C8—61no—
416C3⋊C8—121no—
417C3⋊C8—182no—
418C3⋊C8—181no—
419C3⋊C8—181no—
420C3⋊C8—181no—
421C3⋊C8—361no—
422C3⋊C8—361no—
423C3⋊C8—362no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—241no—
437C2×C32D3×SD1611yescharacteristic
438C2×C32#91411yescharacteristic
439C2×C32#93011yescharacteristic
440C2×C32#93211yescharacteristic
441C2×C32C2×#21411yescharacteristic
442C2×C32#93811yescharacteristic
443C2×C32#93911yescharacteristic
444C2×C32C2×#21511yescharacteristic
445C2×C32#94911yescharacteristic
446C3⋊D3—81no—
447C3×D3—82no—
448C3×D3—81no—
449C3×D3—81no—
450C3×D3—242no—
451C3×D3—241no—
452C3×D3—241no—
453C2×Q8—31no—
454C2×D4—31no—
455C2×D4—91no—
456C2×D4—181no—
457C2×D4—181no—
458SD16—182no—
459SD16—542no—
460C2×C8—91no—
461C2×C8—271no—
462C4⋊C4—91no—
463C4⋊C4—181no—
464C42—31no—
465C42—181no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—122no—
471C22×C3—121no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—241no—
478C22×C3—241no—
479C22×C3—361no—
480C22×C3—361no—
481C22×C3—361no—
482C22×C3C2×#62511yescharacteristic
483C22×C3D3×#6711yescharacteristic
484C22×C3D3×#6711yescharacteristic
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—122no—
489C2×D3—121no—
490C2×D3—121no—
491C2×D3—121no—
492C2×D3—121no—
493C2×D3—121no—
494C2×D3—121no—
495C2×D3—241no—
496C2×D3—241no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—32no—
504C3×C4—32no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—62no—
508C3×C4—62no—
509C3×C4—61no—
510C3×C4—61no—
511C3×C4—62no—
512C3×C4—121no—
513C3×C4—122no—
514C3×C4—361no—
515C3×C4#379511yescharacteristic
516C3×C4D3×D1211yescharacteristic
517C3×C4#380411yescharacteristic
518C3×C4#380511yescharacteristic
519C3×C4C2×#62611yescharacteristic
520C3×C4D3×#6711yescharacteristic
521Dic3—21no—
522Dic3—22no—
523Dic3—121no—
524Dic3—121no—
525Dic3—241no—
526Dic3#371512yescomplemented
527C32—21no—
528C32—21no—
529C32—21no—
530C32—41no—
531C32#502711yescharacteristic, complemented
532C32#507511yescharacteristic, complemented
533C32#516311yescharacteristic, complemented
534C23—61no—
535C23—181no—
536Q8—32no—
537Q8—61no—
538D4—61no—
539D4—61no—
540D4—181no—
541D4—181no—
542D4—362no—
543D4—362no—
544C2×C4—31no—
545C2×C4—31no—
546C2×C4—61no—
547C2×C4—181no—
548C2×C4D3311yescharacteristic
549C8—181no—
550C8—541no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—41no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—242no—
564C2×C3—241no—
565C2×C3—241no—
566C2×C3—721no—
567C2×C3#6382311yescharacteristic
568C2×C3C2×#371511yescharacteristic
569C2×C3#6390811yescharacteristic
570C2×C3D3×#21411yescharacteristic
571C2×C3#6393411yescharacteristic
572C2×C3D3×#21511yescharacteristic
573C2×C3#6393911yescharacteristic
574C2×C3#6397011yescharacteristic
575C2×C3#6397811yescharacteristic
576D3—81no—
577D3—242no—
578D3—241no—
579D3—241no—
580C22—121no—
581C22—121no—
582C22—121no—
583C22—361no—
584C22—361no—
585C22—361no—
586C22D3×#62511yescharacteristic
587C4—32no—
588C4—61no—
589C4—62no—
590C4—361no—
591C4D3×#62611yescharacteristic
592C4#8896111yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059036411yescharacteristic, complemented
598C3#1059143811yescharacteristic, complemented
599C3#1059161711yescharacteristic, complemented
600C2—241no—
601C2—721no—
602C2D3×#371511yescharacteristic
603C2#1172286711yescharacteristic
604C2#1172295611yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypocenterC33#460
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. C2×#88793C2semidirect
3. C3×Dic3×Dic6C2semidirect
4. C2×C3×#3715C2semidirect
5. #11722075C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
7. #11722031C2semidirect
32. Dic3×Dic6D3semidirect
36. C2×#3715D3semidirect
60. C32×Dic6D4semidirect
130. C32×Dic3SD16semidirect
212. C3×Dic6D12semidirect
213. C3×Dic6#67semidirect
342. C3×Dic3#214semidirect
343. C3×Dic3#215semidirect
351. C33#460semidirect
412. Dic6#626semidirect
526. Dic3#3715semidirect
531. C32#5027semidirect
532. C32#5075semidirect
533. C32#5163semidirect
597. C3#10590364semidirect
598. C3#10591438semidirect
599. C3#10591617semidirect

Maximal subgroups

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

subgroupindex
C2×#887932
C3×#639112
C3×Dic3×Dic62
C2×C3×#37152
#117220752
Dic3⋊(C3⋊(C3⋊C8))2
#117220312
#105916173
#105914383
#105903643