Finite Group Databasebeta

← #13155994

Subgroups — #13155994

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
1#11723668C211yesmaximal, characteristic, complemented
2C2×#88792C211yesmaximal, characteristic, complemented
3C3×#63914C211yesmaximal, characteristic
4C2×C3×#3716C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591439—31nomaximal
9#10591438—31nomaximal
10#10590876—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3×#625—21no—
15C2×C3×#625—21no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88815—21no—
21#88813—21no—
22#88792—22no—
23C3×#3716—22no—
24C2×#3754—31no—
25C3×#940—31no—
26C3×#940—31no—
27C2×C3×#214—31no—
28C2×C3×#204—31no—
29#63914D311yescharacteristic
30#63844—31no—
31#63843—31no—
32Dic3×Dic6—31no—
33C2×#3716D311yescharacteristic, complemented
34C2×#3715—31no—
35C2×#3715—31no—
36C2×#3715—61no—
37#63659—31no—
38#63641—31no—
39C3×#832—31no—
40C3×#821—31no—
41Dic3⋊(C3⋊C8)—31no—
42Dic3⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)—61no—
44#63563D311yescharacteristic, complemented
45#63562—31no—
46#63562—31no—
47#63562—61no—
48C22×C32×D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×#634—21no—
56C3×#634D412yescomplemented
57C32×D12—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×D12—21no—
61C3×#625—42no—
62C3×#625—42no—
63C3⋊(C3⋊(C3⋊C8))—21no—
64C32×C3⋊C8—21no—
65#5163—91no—
66#5075—91no—
67#4762—91no—
68C2×C32×D4—31no—
69C2×C32×D4—31no—
70C2×#634C2×D311yescharacteristic
71C2×C3×#67—61no—
72C2×C3×#67—61no—
73C2×C3×#67—61no—
74C2×C3×#67—61no—
75C2×C3×D12—21no—
76C2×C3×D12—21no—
77C2×C3×D12C2×D311yescharacteristic
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×#625—21no—
85C2×#625—21no—
86C2×C32×C8—31no—
87#3754—62no—
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89C4×C3⋊Dic3C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)—31no—
91C2×C3⋊(C3⋊C8)—31no—
92C2×C3⋊(C3⋊C8)—31no—
93C2×C3⋊(C3⋊C8)—61no—
94C2×C3⋊(C3⋊C8)—61no—
95C2×C3⋊(C3⋊C8)—61no—
96C2×C3⋊(C3⋊C8)—121no—
97C3×#214—62no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—61no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—61no—
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C3×#204—62no—
108#3724—61no—
109Dic3⋊Dic3—61no—
110Dic3⋊Dic3—61no—
111Dic32—61no—
112Dic32—61no—
113Dic32—61no—
114#3716—22no—
115#3715—62no—
116#3715—62no—
117#3715—64no—
118C22×C33C2×D411yescharacteristic
119C2×C32×D3—41no—
120C2×C32×D3—41no—
121C2×C32×D3—41no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3×C3⋊Dic3—22no—
128C3×C3⋊Dic3—21no—
129C3×C3⋊Dic3—41no—
130C3×C3⋊Dic3SD1612yescomplemented
131C2×C3×SD16—91no—
132C3×#127—91no—
133Q8×Dic3—91no—
134C2×#215—91no—
135#940—31no—
136#940—31no—
137C2×#214—31no—
138C2×#214—91no—
139C2×#214—91no—
140C2×#214—181no—
141C2×#204—31no—
142#887—91no—
143C4×Dic6—91no—
144C3×#102—91no—
145#832—31no—
146#832—91no—
147#832—91no—
148#832—181no—
149#821—31no—
150Dic3⋊C8—91no—
151Dic3⋊C8—91no—
152Dic3⋊C8—181no—
153#809—91no—
154C4⋊(C3⋊C8)—91no—
155C23×C32—61no—
156C23×C32—61no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—21no—
161C22×C3×D3—41no—
162C22×C3×D3—41no—
163C32×D4—61no—
164C32×D4—61no—
165C32×D4—61no—
166C32×D4—61no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—41no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3#6711yescharacteristic
177C2×C3⋊Dic3#6711yescharacteristic
178#634—21no—
179#634#6712yescomplemented
180C3×#67—122no—
181C3×#67—122no—
182C3×#67—122no—
183C3×#67—122no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—121no—
195C2×C3×Dic3—121no—
196C3×D12—22no—
197C3×D12—22no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—21no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—41no—
207C3×D12—41no—
208C3×Dic6—32no—
209C3×Dic6—32no—
210C3×Dic6—62no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—121no—
214#625—42no—
215#625—42no—
216C32×C8—61no—
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—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—41no—
227#460—271noSylow
228C2×C33#12111yescharacteristic
229C2×C33C2×SD1611yescharacteristic
230C2×C33AGL(1,ℤ/8)11yescharacteristic
231C32×D3—81no—
232C32×D3—81no—
233C2×C3×Q8—91no—
234C2×C3×D4—31no—
235C2×C3×D4—31no—
236C2×C3×D4—31no—
237C2×C3×D4—31no—
238C2×C3×D4—61no—
239C2×C3×D4—61no—
240C2×C3×D4—181no—
241C2×C3×D4—181no—
242C2×#67—61no—
243C2×#67—61no—
244C2×#67—61no—
245C2×#67—61no—
246C2×D12D3211yescharacteristic
247C2×D12D3211yescharacteristic
248C2×Dic6—31no—
249C2×Dic6—31no—
250C2×Dic6—61no—
251C3×SD16—182no—
252C2×C3×C8—31no—
253C2×C3×C8—31no—
254C2×C3×C8—61no—
255C3×C4⋊C4—91no—
256C3×C42—91no—
257#215—182no—
258#214—62no—
259#214—182no—
260#214—182no—
261#214—184no—
262C4⋊Dic3—31no—
263C4⋊Dic3—31no—
264C4⋊Dic3—61no—
265C4⋊Dic3—181no—
266Dic3⋊C4—181no—
267Dic3⋊C4—181no—
268C4×Dic3—31no—
269C4×Dic3—31no—
270C4×Dic3—61no—
271C4×Dic3—181no—
272C4×Dic3—181no—
273C4×Dic3—181no—
274C2×C3⋊C8—91no—
275C2×C3⋊C8—91no—
276C2×C3⋊C8—91no—
277C2×C3⋊C8—181no—
278C2×C3⋊C8—181no—
279C2×C3⋊C8—181no—
280C2×C3⋊C8—361no—
281C2×C3⋊C8D3211yescharacteristic
282#204—62no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—41no—
287C22×C32—121no—
288C22×C32—121no—
289C22×C32—121no—
290C22×C32—121no—
291C22×C32—121no—
292C22×C32—121no—
293C22×C32D3×D411yescharacteristic
294C22×C32D3×D411yescharacteristic
295C22×C32C2×#6711yescharacteristic
296C2×C3×D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—41no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—81no—
311C2×C3×D3—81no—
312C2×C3×D3—81no—
313C2×C3×D3—81no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—41no—
321C32×C4—41no—
322C32×C4#23511yescharacteristic
323C32×C4D3×D411yescharacteristic
324C32×C4D3×D411yescharacteristic
325C32×C4#23711yescharacteristic
326C32×C4#23711yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C3⋊Dic3—21no—
329C3⋊Dic3—22no—
330C3⋊Dic3—41no—
331C3⋊Dic3#21412yescomplemented
332C3×Dic3—32no—
333C3×Dic3—32no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—61no—
337C3×Dic3—61no—
338C3×Dic3—62no—
339C3×Dic3—122no—
340C3×Dic3—121no—
341C3×Dic3—121no—
342C3×Dic3—121no—
343C3×Dic3—241no—
344C2×SD16—91no—
345C2×SD16—271no—
346#127—91no—
347C4×Q8—271no—
348C4⋊C8—271no—
349#102—91no—
350#102—271no—
351C33#46011yescharacteristic, complemented, Sylow
352C23×C3—61no—
353C23×C3—61no—
354C23×C3—61no—
355C23×C3—61no—
356C23×C3—121no—
357C23×C3—121no—
358C22×D3—21no—
359C22×D3—21no—
360C3×Q8—92no—
361C3×Q8—181no—
362C3×D4—62no—
363C3×D4—62no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—61no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—61no—
372C3×D4—121no—
373C3×D4—121no—
374C3×D4—362no—
375C3×D4—362no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—91no—
381C2×C3×C4—91no—
382C2×C3×C4—181no—
383C2×C3×C4—181no—
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C2×D3211yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387#67—122no—
388#67—122no—
389#67—122no—
390#67—122no—
391C2×Dic3—31no—
392C2×Dic3—31no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—61no—
400C2×Dic3—61no—
401C2×Dic3—121no—
402C2×Dic3—121no—
403D12—21no—
404D12—21no—
405D12—21no—
406D12—21no—
407Dic6—32no—
408Dic6—32no—
409Dic6—62no—
410Dic6—61no—
411Dic6—61no—
412Dic6—121no—
413C3×C8—61no—
414C3×C8—61no—
415C3×C8—121no—
416C3⋊C8—21no—
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×C32—241no—
438C2×C32#88411yescharacteristic
439C2×C32#91111yescharacteristic
440C2×C32D3×SD1611yescharacteristic
441C2×C32D3×SD1611yescharacteristic
442C2×C32#91411yescharacteristic
443C2×C32#93211yescharacteristic
444C2×C32C2×#21411yescharacteristic
445C2×C32#93811yescharacteristic
446C2×C32#93911yescharacteristic
447C3×D3—82no—
448C3×D3—82no—
449C3×D3—81no—
450C3×D3—81no—
451C3×D3—81no—
452C3×D3—81no—
453C2×Q8—91no—
454C2×D4—31no—
455C2×D4—31no—
456C2×D4—181no—
457C2×D4—181no—
458SD16—182no—
459SD16—542no—
460C2×C8—31no—
461C2×C8—271no—
462C4⋊C4—91no—
463C4⋊C4—541no—
464C42—91no—
465C42—541no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—122no—
471C22×C3—122no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—121no—
480C22×C3—121no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3—121no—
484C22×C3—241no—
485C22×C3—241no—
486C22×C3—241no—
487C22×C3—241no—
488C22×C3#380211yescharacteristic
489C22×C3D3×#6711yescharacteristic
490C22×C3D3×#6711yescharacteristic
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—41no—
496C2×D3—41no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—92no—
506C3×C4—181no—
507C3×C4—182no—
508C3×C4—361no—
509C3×C4#379511yescharacteristic
510C3×C4#380411yescharacteristic
511C3×C4#380511yescharacteristic
512C3×C4D3×#6711yescharacteristic
513C3×C4D3×#6711yescharacteristic
514C3×C4#381111yescharacteristic
515Dic3—32no—
516Dic3—32no—
517Dic3—62no—
518Dic3—62no—
519Dic3—61no—
520Dic3—61no—
521Dic3—62no—
522Dic3—122no—
523Dic3—121no—
524Dic3—121no—
525Dic3—121no—
526Dic3—241no—
527C32—21no—
528C32—21no—
529C32—21no—
530C32—41no—
531C32#476211yescharacteristic, complemented
532C32#507511yescharacteristic, complemented
533C32#516311yescharacteristic, complemented
534C23—61no—
535C23—61no—
536Q8—92no—
537Q8—181no—
538D4—61no—
539D4—61no—
540D4—61no—
541D4—61no—
542D4—362no—
543D4—362no—
544C2×C4—91no—
545C2×C4—91no—
546C2×C4—181no—
547C2×C4—181no—
548C2×C4D3311yescharacteristic
549C8—61no—
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—242no—
565C2×C3—241no—
566C2×C3—241no—
567C2×C3—241no—
568C2×C3—241no—
569C2×C3#6379611yescharacteristic
570C2×C3#6379811yescharacteristic
571C2×C3#6388511yescharacteristic
572C2×C3D3×#21411yescharacteristic
573C2×C3D3×#21411yescharacteristic
574C2×C3#6393211yescharacteristic
575C2×C3#6393411yescharacteristic
576C2×C3#6395811yescharacteristic
577C2×C3#6397011yescharacteristic
578D3—81no—
579D3—81no—
580C22—121no—
581C22—121no—
582C22—121no—
583C22—121no—
584C22—121no—
585C22—121no—
586C22#8895311yescharacteristic
587C4—92no—
588C4—181no—
589C4—182no—
590C4—361no—
591C4#8895211yescharacteristic
592C4#8896111yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059087611yescharacteristic, complemented
598C3#1059143811yescharacteristic, complemented
599C3#1059143911yescharacteristic, complemented
600C2—241no—
601C2—241no—
602C2#1172283911yescharacteristic
603C2#1172287511yescharacteristic
604C2#1172296511yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63914C2
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. #11723668C2semidirect
2. C2×#88792C2semidirect
4. C2×C3×#3716C2semidirect
5. #11722051C2semidirect
6. #11722031C2semidirect
7. C3×#63563C2semidirect
33. C2×#3716D3semidirect
44. #63563D3semidirect
56. C3×#634D4semidirect
130. C3×C3⋊Dic3SD16semidirect
179. #634#67semidirect
331. C3⋊Dic3#214semidirect
351. C33#460semidirect
531. C32#4762semidirect
532. C32#5075semidirect
533. C32#5163semidirect
597. C3#10590876semidirect
598. C3#10591438semidirect
599. C3#10591439semidirect

Maximal subgroups

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

subgroupindex
#117236682
C2×#887922
C3×#639142
C2×C3×#37162
#117220512
#117220312
C3×#635632
#105914393
#105914383
#105908763