Finite Group Databasebeta

← #13156037

Subgroups — #13156037

Order 1728.

708 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723677C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#3716C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7C3×#63572C211yesmaximal, characteristic, complemented
8#10591476—31nomaximal
9#10590755—31nomaximal
10#10590754—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×D3×C3⋊Dic3—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—21no—
20#88814—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C3×#3716—22no—
23C3×#3714—22no—
24D4×C3⋊Dic3—31no—
25C2×C3×#214—31no—
26C2×C3×#213—31no—
27C2×C3×D24—31no—
28C2×C3×#204—31no—
29Dic3×D12—31no—
30Dic3×D12—31no—
31Dic3×D12—61no—
32#63864—31no—
33#63863—31no—
34#63860—31no—
35C2×#3716D311yescharacteristic, complemented
36C2×#3714D311yescharacteristic
37#63648—31no—
38#63643—31no—
39C3×Dic3⋊C8—31no—
40C3×Dic3⋊C8—31no—
41#63572D311yescharacteristic, complemented
42#63566—31no—
43#63566—31no—
44#63566—61no—
45#63563—31no—
46#63563—31no—
47#63563—61no—
48C22×C3×C3⋊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×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×#634—21no—
58C3×#634—21no—
59C32×D12—21no—
60C32×D12D412yes—
61D3×C3⋊Dic3—42no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64#5165—91no—
65#5153—91no—
66#4844—91no—
67C2×C32×D4—31no—
68C22×C3⋊Dic3—61no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—61no—
80C2×D3×Dic3—61no—
81C2×D3×Dic3—61no—
82C2×D3×Dic3—62no—
83C2×C32×C8—31no—
84C2×C32×C8—31no—
85#3757—61no—
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—31no—
89C4⋊(C3⋊Dic3)—61no—
90C4⋊(C3⋊Dic3)—61no—
91C4⋊(C3⋊Dic3)—61no—
92C4⋊(C3⋊Dic3)—121no—
93C4×C3⋊Dic3C2×D311yescharacteristic
94C3×#214—62no—
95C3×#213—62no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—61no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C3×D24—62no—
106C3×#204—62no—
107#3722—61no—
108#3722—61no—
109(C2×D3)⋊Dic3—61no—
110(C2×D3)⋊Dic3—61no—
111(C2×D3)⋊Dic3—61no—
112(C2×D3)⋊Dic3—61no—
113(C2×D3)⋊Dic3—62no—
114(C2×D3)⋊Dic3—61no—
115(C2×D3)⋊Dic3—61no—
116#3716—22no—
117#3714—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C32×D3—22no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3⋊(C3⋊Dic3)—41no—
128C3×C3⋊Dic3—21no—
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3—41no—
131C2×C3×SD16—91no—
132C2×C3×D8—91no—
133#947—91no—
134#936—91no—
135D4×Dic3—91no—
136D4×Dic3—91no—
137D4×Dic3—181no—
138C2×#214—31no—
139C2×#213—31no—
140C2×D24—31no—
141C2×#204—31no—
142#875—91no—
143C4×D12—91no—
144C3×C4⋊C8—91no—
145#835—91no—
146#832—91no—
147#821—91no—
148#821—91no—
149#821—181no—
150#816—91no—
151#816—91no—
152#816—181no—
153Dic3⋊C8—31no—
154Dic3⋊C8—31no—
155C23×C32—61no—
156C22×C3⋊D3—21no—
157C22×C3×D3—22no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—121no—
163C32×D4—32no—
164C32×D4—61no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—41no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—62no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—121no—
178C2×C3⋊Dic3—121no—
179C2×C3⋊Dic3—122no—
180C2×C3⋊Dic3—122no—
181C2×C3⋊Dic3D1211yescharacteristic
182C2×C3⋊Dic3D1211yescharacteristic
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)—21no—
185#634—21no—
186#634—21no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—121no—
196C3×D12—22no—
197C3×D12—21no—
198C3×D12—21no—
199C3×D12—22no—
200C3×D12—62no—
201C3×D12—61no—
202C3×D12—61no—
203C3×D12—61no—
204C3×D12—61no—
205C3×D12—121no—
206C3×D12#6712yes—
207C3×D12#6712yes—
208C3×Dic6—62no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—121no—
214D3×Dic3—124no—
215D3×Dic3—122no—
216D3×Dic3—122no—
217C32×C8—61no—
218C32×C8—61no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—41no—
224C3×C3⋊C8—41no—
225#463—271noSylow
226C2×C33#12111yescharacteristic
227C2×C33#13511yescharacteristic
228C2×C33AGL(1,ℤ/8)11yescharacteristic
229C3×C3⋊D3—81no—
230C32×D3—42no—
231C2×C3×Q8—91no—
232C2×C3×D4—31no—
233C2×C3×D4—31no—
234C2×C3×D4—61no—
235C2×C3×D4—91no—
236C22×Dic3—181no—
237C22×Dic3—181no—
238C22×Dic3—182no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×D12D3211yescharacteristic
243C2×C4×D3—181no—
244C2×Dic6—31no—
245C2×Dic6—31no—
246C2×Dic6—61no—
247C3×SD16—182no—
248C3×D8—182no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—31no—
253C2×C3×C8—61no—
254C2×C3×C8—61no—
255C3×C42—91no—
256#217—181no—
257#217—181no—
258#217—181no—
259#217—181no—
260#217—182no—
261#214—62no—
262#213—62no—
263(C2×D3)⋊C4—181no—
264(C2×D3)⋊C4—181no—
265(C2×D3)⋊C4—181no—
266(C2×D3)⋊C4—181no—
267(C2×D3)⋊C4—181no—
268C4⋊Dic3—91no—
269C4⋊Dic3—91no—
270C4⋊Dic3—91no—
271C4⋊Dic3—181no—
272C4⋊Dic3—181no—
273C4⋊Dic3—181no—
274C4⋊Dic3—361no—
275C4×Dic3—31no—
276C4×Dic3—31no—
277C4×Dic3—61no—
278C2×C3⋊C8D3211yescharacteristic
279C2×C3⋊C8D3211yescharacteristic
280D24—62no—
281#204—62no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32—62no—
287C22×C32—121no—
288C22×C32—121no—
289C22×C32C2×D1211yescharacteristic
290C22×C32D3×D411yescharacteristic
291C22×C32D3×D411yescharacteristic
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3⋊D3—41no—
295C2×C3×D3—24no—
296C2×C3×D3—22no—
297C2×C3×D3—22no—
298C2×C3×D3—42no—
299C2×C3×D3—42no—
300C2×C3×D3—41no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—122no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—241no—
312C2×C3×D3—241no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4D3×D411yescharacteristic
322C32×C4#23711yescharacteristic
323C32×C4#23711yescharacteristic
324C32×C4#23911yescharacteristic
325C32×C4C2×#6711yescharacteristic
326C32×C4C2×#6711yescharacteristic
327C3⋊Dic3—21no—
328C3⋊Dic3—21no—
329C3⋊Dic3—41no—
330C3⋊Dic3—121no—
331C3⋊Dic3—122no—
332C3⋊Dic3—121no—
333C3⋊Dic3—121no—
334C3⋊Dic3—241no—
335C3⋊Dic3—241no—
336C3⋊Dic3—242no—
337C3×Dic3—62no—
338C3×Dic3—61no—
339C3×Dic3—61no—
340C3×Dic3—61no—
341C3×Dic3—61no—
342C3×Dic3—121no—
343C3×Dic3—122no—
344C3×Dic3—121no—
345C3×Dic3—121no—
346C2×SD16—91no—
347C2×D8—91no—
348#124—271no—
349C4×D4—271no—
350C4⋊C8—91no—
351#103—271no—
352#102—271no—
353C33#46311yescharacteristic, complemented, Sylow
354C23×C3—62no—
355C23×C3—61no—
356C23×C3—61no—
357C23×C3—181no—
358C22×D3—21no—
359C22×D3—61no—
360C22×D3—61no—
361C22×D3—121no—
362C3×Q8—181no—
363C3×Q8—181no—
364C3×D4—32no—
365C3×D4—32no—
366C3×D4—62no—
367C3×D4—62no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—181no—
371C3×D4—181no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—91no—
377C2×C3×C4—91no—
378C2×C3×C4—181no—
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—31no—
385C2×Dic3—31no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—121no—
391C2×Dic3—182no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—361no—
396C2×Dic3—361no—
397C2×Dic3—362no—
398C2×Dic3—364no—
399C2×Dic3—362no—
400C2×Dic3—362no—
401D12—21no—
402D12—62no—
403D12—61no—
404D12—61no—
405D12—61no—
406D12—61no—
407D12—121no—
408D12#62512yes—
409C4×D3—362no—
410Dic6—62no—
411Dic6—61no—
412Dic6—61no—
413Dic6—61no—
414Dic6—61no—
415Dic6—121no—
416C3×C8—61no—
417C3×C8—61no—
418C3×C8—61no—
419C3×C8—61no—
420C3×C8—121no—
421C3×C8—121no—
422C3⋊C8—21no—
423C3⋊C8—21no—
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—122no—
437C2×C32#89511yescharacteristic
438C2×C32#90411yescharacteristic
439C2×C32#90811yescharacteristic
440C2×C32#91111yescharacteristic
441C2×C32#91211yescharacteristic
442C2×C32#91411yescharacteristic
443C2×C32#91611yescharacteristic
444C2×C32#93911yescharacteristic
445C2×C32#93911yescharacteristic
446C3⋊D3—81no—
447C3×D3—44no—
448C3×D3—42no—
449C3×D3—42no—
450C3×D3—242no—
451C3×D3—241no—
452C3×D3—241no—
453C2×Q8—91no—
454C2×D4—31no—
455C2×D4—91no—
456C22×C4—541no—
457SD16—182no—
458D8—182no—
459C2×C8—31no—
460C2×C8—31no—
461C4⋊C4—271no—
462C22⋊C4—541no—
463C22⋊C4—541no—
464C22⋊C4—541no—
465C42—91no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—64no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—122no—
474C22×C3—122no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—361no—
480C22×C3—361no—
481C22×C3—361no—
482C22×C3D3×D1211yescharacteristic
483C22×C3D3×D1211yescharacteristic
484C22×C3#381111yescharacteristic
485C2×D3—22no—
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—41no—
504C3×C4—41no—
505C3×C4—181no—
506C3×C4—181no—
507C3×C4—361no—
508C3×C4#379611yescharacteristic
509C3×C4#379611yescharacteristic
510C3×C4#380511yescharacteristic
511C3×C4C2×#62511yescharacteristic
512C3×C4D3×#6711yescharacteristic
513C3×C4D3×#6711yescharacteristic
514Dic3—62no—
515Dic3—61no—
516Dic3—61no—
517Dic3—61no—
518Dic3—61no—
519Dic3—121no—
520Dic3—122no—
521Dic3—121no—
522Dic3—121no—
523Dic3—362no—
524Dic3—361no—
525Dic3—361no—
526Dic3—361no—
527Dic3—721no—
528Dic3—721no—
529Dic3—722no—
530C32—21no—
531C32—21no—
532C32—21no—
533C32—41no—
534C32#484411yescharacteristic, complemented
535C32#515311yescharacteristic, complemented
536C32#516511yescharacteristic, complemented
537C23—61no—
538C23—181no—
539Q8—181no—
540Q8—181no—
541D4—32no—
542D4—61no—
543D4—181no—
544D4—181no—
545C2×C4—91no—
546C2×C4—91no—
547C2×C4—181no—
548C2×C4—541no—
549C2×C4—1082no—
550C2×C4D3311yescharacteristic
551C8—61no—
552C8—61no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—124no—
566C2×C3—122no—
567C2×C3—122no—
568C2×C3—721no—
569C2×C3#6379411yescharacteristic
570C2×C3#6379511yescharacteristic
571C2×C3#6380611yescharacteristic
572C2×C3#6380711yescharacteristic
573C2×C3#6388611yescharacteristic
574C2×C3#6388611yescharacteristic
575C2×C3#6393611yescharacteristic
576C2×C3#6393711yescharacteristic
577C2×C3#6398011yescharacteristic
578D3—42no—
579D3—242no—
580D3—241no—
581D3—241no—
582C22—62no—
583C22—121no—
584C22—121no—
585C22—361no—
586C22—361no—
587C22—361no—
588C22#8895511yescharacteristic
589C4—181no—
590C4—181no—
591C4—361no—
592C4—1081no—
593C4D3×#62511yescharacteristic
594C4#8896411yescharacteristic
595C3—21no—
596C3—21no—
597C3—21no—
598C3—41no—
599C3#1059075411yescharacteristic, complemented
600C3#1059075511yescharacteristic, complemented
601C3#1059147611yescharacteristic, complemented
602C2—122no—
603C2—721no—
604C2#1172286411yescharacteristic
605C2#1172287911yescharacteristic
606C2#1172295511yescharacteristic
subgroupG/H
centerC22#88955
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. #11723677C2semidirect
2. D12×C3⋊Dic3C2semidirect
3. C2×C3×#3716C2semidirect
5. #11722040C2semidirect
6. #11722034C2semidirect
7. C3×#63572C2semidirect
35. C2×#3716D3semidirect
41. #63572D3semidirect
353. C33#463semidirect
534. C32#4844semidirect
535. C32#5153semidirect
536. C32#5165semidirect
599. C3#10590754semidirect
600. C3#10590755semidirect
601. C3#10591476semidirect

Maximal subgroups

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

subgroupindex
#117236772
D12×C3⋊Dic32
C2×C3×#37162
C2×C3×#37142
#117220402
#117220342
C3×#635722
#105914763
#105907553
#105907543