Finite Group Databasebeta

← #13155717

Subgroups — #13155717

Order 1728.

786 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#11723501C211yesmaximal, characteristic, complemented
2#11723499C211yesmaximal, characteristic, complemented
3C2×C3×#3714C211yesmaximal, characteristic
4#11722075C211yesmaximal, characteristic, complemented
5#11722021C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591436—31nomaximal
9#10590781—31nomaximal
10#10590462—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C2×D3×C3⋊Dic3—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88803—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C3×#3714C414yescomplemented
23D4×C3⋊Dic3—31no—
24#64102—31no—
25C2×C3×#213—31no—
26C2×C3×D24—31no—
27#63898—31no—
28#63898—31no—
29#63898—31no—
30#63898—31no—
31#63898—61no—
32#63898—61no—
33C2×#3714D311yescharacteristic
34#63642—31no—
35C3×#832—31no—
36C3×#817—31no—
37C3×#809—31no—
38C3×#808—31no—
39#63573D311yescharacteristic, complemented
40#63564—31no—
41#63563—31no—
42#63562—31no—
43#63562D311yescharacteristic, complemented
44#63555—31no—
45#63555—31no—
46#63555—61no—
47C22×C3×C3⋊D3—21no—
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3⋊(C3⋊Dic3)D411yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12C2×C412yescomplemented
58Dic3×C3⋊D3—42no—
59D3×C3⋊Dic3—42no—
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8C2×C412yescomplemented
62#5152—91no—
63#4869—91no—
64#4785—91no—
65C2×C32×D4—31no—
66C22×C3⋊Dic3—61no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C4×C3⋊D3—61no—
69C2×C3×D12—21no—
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—61no—
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×D12C2×D311yescharacteristic
75C2×D3×Dic3—61no—
76C2×D3×Dic3—61no—
77C2×D3×Dic3—62no—
78C2×D3×Dic3—61no—
79C2×D3×Dic3—61no—
80C2×D3×Dic3—62no—
81C2×C32×C8—31no—
82C32×C4⋊C4—31no—
83#3757—61no—
84#3752—61no—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
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—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94C3×#213—34no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—61no—
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—34no—
108(C2×D3)⋊Dic3—61no—
109(C2×D3)⋊Dic3—61no—
110(C2×D3)⋊Dic3—62no—
111(C2×D3)⋊Dic3—61no—
112(C2×D3)⋊Dic3—61no—
113(C2×D3)⋊Dic3—62no—
114#3714Dic314yescomplemented
115C22×C33C2×D411yescharacteristic
116C2×C3×C3⋊D3—22no—
117C2×C3×C3⋊D3—41no—
118C2×C3×C3⋊D3—41no—
119C2×C32×D3—22no—
120C2×C32×D3—41no—
121C2×C32×D3—41no—
122C33×C4C22×C411yescharacteristic
123C33×C4#4111yescharacteristic
124C3⋊(C3⋊Dic3)—21no—
125C3⋊(C3⋊Dic3)—21no—
126C3×C3⋊Dic3—41no—
127C32×Dic3—41no—
128C2×C3×D8—91no—
129D4×Dic3—91no—
130D4×Dic3—91no—
131D4×Dic3—91no—
132D4×Dic3—181no—
133C2×#213—31no—
134C2×D24—31no—
135#893—91no—
136#893—91no—
137#893—91no—
138#893—181no—
139C3×#106—91no—
140C3×#102—91no—
141#832—31no—
142#832—91no—
143#821—91no—
144#817—31no—
145#815—91no—
146#815—91no—
147#815—91no—
148#815—181no—
149#809—31no—
150#809—91no—
151#808—31no—
152C23×C32—61no—
153C22×C3⋊D3—21no—
154C22×C3×D3—22no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—62no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C32×D4—32no—
161C32×D4—61no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—41no—
166C2×C32×C4—61no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—61no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—61no—
183C2×C3⋊Dic3—121no—
184C2×C3⋊Dic3—121no—
185C2×C3⋊Dic3—122no—
186C4⋊(C3⋊D3)—21no—
187C4⋊(C3⋊D3)C2×Dic312yescomplemented
188C4×C3⋊D3—122no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C3×D12—22no—
196C3×D12—21no—
197C3×D12—21no—
198C3×D12—22no—
199C3×D12—32no—
200C3×D12—32no—
201C3×D12—62no—
202C3×D12—62no—
203C3×D12—61no—
204C3×D12—61no—
205C3×D12C4×D312yescomplemented
206C3×D12C2×Dic312yescomplemented
207D3×Dic3—124no—
208D3×Dic3—122no—
209D3×Dic3—122no—
210D3×Dic3—124no—
211D3×Dic3—122no—
212D3×Dic3—122no—
213C32×C8—32no—
214C3⋊(C3⋊C8)—21no—
215C3×C3⋊C8—22no—
216C3×C3⋊C8—62no—
217C3×C3⋊C8—61no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8C4×D312yescomplemented
220C3×C3⋊C8C2×Dic312yescomplemented
221#442—271noSylow
222C2×C33C4×D411yescharacteristic
223C2×C33AGL(1,ℤ/8)11yescharacteristic
224C2×C33AGL(1,ℤ/8)11yescharacteristic
225C3×C3⋊D3—42no—
226C32×D3—42no—
227C2×C3×D4—31no—
228C2×C3×D4—31no—
229C2×C3×D4—61no—
230C2×C3×D4—91no—
231C22×Dic3—181no—
232C22×Dic3—181no—
233C22×Dic3—181no—
234C22×Dic3—182no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×D12D3211yescharacteristic
239C2×C4×D3—181no—
240C2×C4×D3—181no—
241C2×C4×D3—181no—
242C2×C4×D3—182no—
243C3×D8—94no—
244C2×C3×C8—31no—
245C2×C3×C8—31no—
246C2×C3×C8—61no—
247C2×C3×C8—91no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—31no—
250C3×C4⋊C4—61no—
251C3×C4⋊C4—91no—
252#217—181no—
253#217—181no—
254#217—181no—
255#217—182no—
256#213—34no—
257(C2×D3)⋊C4—181no—
258(C2×D3)⋊C4—181no—
259(C2×D3)⋊C4—181no—
260(C2×D3)⋊C4—182no—
261C4⋊Dic3—31no—
262C4⋊Dic3—31no—
263C4⋊Dic3—61no—
264C4⋊Dic3D3211yescharacteristic
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—91no—
268C4×Dic3—181no—
269C4×Dic3—181no—
270C4×Dic3—181no—
271C4×Dic3—361no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—61no—
275C2×C3⋊C8D3211yescharacteristic
276D24—34no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—41no—
281C22×C32—62no—
282C22×C32—121no—
283C22×C32—121no—
284C22×C32D3×D411yescharacteristic
285C22×C32D3×D411yescharacteristic
286C22×C32D3×D411yescharacteristic
287C2×C3⋊D3—22no—
288C2×C3⋊D3—41no—
289C2×C3⋊D3—41no—
290C2×C3×D3—24no—
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C2×C3×D3—41no—
299C2×C3×D3—64no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—122no—
303C2×C3×D3—122no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—41no—
315C32×C4—41no—
316C32×C4—121no—
317C32×C4C2×C4×D311yescharacteristic
318C32×C4C2×C4×D311yescharacteristic
319C32×C4#23711yescharacteristic
320C32×C4#23711yescharacteristic
321C32×C4#23911yescharacteristic
322C32×C4C22×Dic311yescharacteristic
323C3⋊Dic3—41no—
324C3⋊Dic3—61no—
325C3⋊Dic3—61no—
326C3⋊Dic3—62no—
327C3⋊Dic3—62no—
328C3⋊Dic3—61no—
329C3⋊Dic3—61no—
330C3⋊Dic3—61no—
331C3⋊Dic3—61no—
332C3⋊Dic3—121no—
333C3⋊Dic3—121no—
334C3⋊Dic3—121no—
335C3⋊Dic3—121no—
336C3⋊Dic3—122no—
337C3⋊Dic3—122no—
338C3×Dic3—42no—
339C3×Dic3—41no—
340C3×Dic3—41no—
341C3×Dic3—122no—
342C3×Dic3—121no—
343C3×Dic3—121no—
344C2×D8—91no—
345C4×D4—271no—
346C4×D4—271no—
347#106—91no—
348#102—91no—
349#102—271no—
350#97—271no—
351C33#44211yescharacteristic, complemented, Sylow
352C23×C3—62no—
353C23×C3—61no—
354C23×C3—61no—
355C23×C3—181no—
356C22×D3—21no—
357C22×D3—62no—
358C22×D3—61no—
359C22×D3—61no—
360C3×D4—32no—
361C3×D4—32no—
362C3×D4—62no—
363C3×D4—62no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—92no—
367C3×D4—181no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—62no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—181no—
376C2×C3×C4C2×D3211yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C2×D3211yescharacteristic
379C2×Dic3—21no—
380C2×Dic3—62no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—91no—
387C2×Dic3—91no—
388C2×Dic3—91no—
389C2×Dic3—181no—
390C2×Dic3—181no—
391C2×Dic3—181no—
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—32no—
403D12—32no—
404D12—62no—
405D12—62no—
406D12—61no—
407D12—61no—
408D12D3×Dic312yescomplemented
409C4×D3—364no—
410C4×D3—362no—
411C4×D3—362no—
412C4×D3—362no—
413C3×C8—32no—
414C3×C8—32no—
415C3×C8—62no—
416C3×C8—181no—
417C3⋊C8—62no—
418C3⋊C8—61no—
419C3⋊C8—61no—
420C3⋊C8D3×Dic312yescomplemented
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—41no—
431C2×C32—41no—
432C2×C32—41no—
433C2×C32—122no—
434C2×C32#67⋊C411yescharacteristic
435C2×C32#89311yescharacteristic
436C2×C32#91111yescharacteristic
437C2×C32#91111yescharacteristic
438C2×C32#91111yescharacteristic
439C2×C32#91411yescharacteristic
440C2×C32#91411yescharacteristic
441C2×C32#91411yescharacteristic
442C2×C32D4×Dic311yescharacteristic
443C3⋊D3—42no—
444C3×D3—44no—
445C3×D3—42no—
446C3×D3—42no—
447C3×D3—124no—
448C3×D3—122no—
449C3×D3—122no—
450C2×D4—31no—
451C2×D4—91no—
452C22×C4—541no—
453C22×C4—541no—
454D8—94no—
455C2×C8—31no—
456C2×C8—91no—
457C4⋊C4—31no—
458C4⋊C4—91no—
459C22⋊C4—541no—
460C22⋊C4—541no—
461C42—271no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—41no—
466C22×C3—64no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3—122no—
470C22×C3—122no—
471C22×C3—121no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—182no—
476C22×C3—361no—
477C22×C3—361no—
478C22×C3#380211yescharacteristic
479C22×C3#381111yescharacteristic
480C22×C3#381111yescharacteristic
481C2×D3—22no—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—64no—
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—122no—
488C2×D3—122no—
489C2×D3—121no—
490C2×D3—121no—
491C2×D3—121no—
492C2×D3—121no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—122no—
502C3×C4—121no—
503C3×C4—121no—
504C3×C4—361no—
505C3×C4#379611yescharacteristic
506C3×C4#379611yescharacteristic
507C3×C4C4×D3211yescharacteristic
508C3×C4C2×D3×Dic311yescharacteristic
509C3×C4C2×D3×Dic311yescharacteristic
510C3×C4#380511yescharacteristic
511Dic3—41no—
512Dic3—122no—
513Dic3—121no—
514Dic3—121no—
515Dic3—182no—
516Dic3—182no—
517Dic3—181no—
518Dic3—181no—
519Dic3—181no—
520Dic3—181no—
521Dic3—181no—
522Dic3—181no—
523Dic3—361no—
524Dic3—361no—
525Dic3—361no—
526Dic3—361no—
527Dic3—362no—
528Dic3—362no—
529C32—21no—
530C32—21no—
531C32—21no—
532C32—41no—
533C32#478511yescharacteristic, complemented
534C32#486911yescharacteristic, complemented
535C32#515211yescharacteristic, complemented
536C23—61no—
537C23—181no—
538D4—32no—
539D4—61no—
540D4—92no—
541D4—181no—
542C2×C4—61no—
543C2×C4—181no—
544C2×C4—271no—
545C2×C4—271no—
546C2×C4—1082no—
547C2×C4—1082no—
548C2×C4D3311yescharacteristic
549C8—32no—
550C8—181no—
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—124no—
564C2×C3—122no—
565C2×C3—122no—
566C2×C3—362no—
567C2×C3#6379811yescharacteristic
568C2×C3#6379811yescharacteristic
569C2×C3#6388111yescharacteristic
570C2×C3#6389811yescharacteristic
571C2×C3#6392711yescharacteristic
572C2×C3#6393711yescharacteristic
573C2×C3#6393711yescharacteristic
574C2×C3#6394111yescharacteristic
575C2×C3#67⋊Dic311yescharacteristic
576D3—42no—
577D3—124no—
578D3—122no—
579D3—122no—
580C22—62no—
581C22—121no—
582C22—121no—
583C22—182no—
584C22—361no—
585C22—361no—
586C22#8895411yescharacteristic
587C4—121no—
588C4—361no—
589C4—541no—
590C4—541no—
591C4D32×Dic311yescharacteristic
592C4#8896411yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059046211yescharacteristic, complemented
598C3#1059078111yescharacteristic, complemented
599C3#1059143611yescharacteristic, complemented
600C2—122no—
601C2—362no—
602C2#1172287711yescharacteristic
603C2#1172287811yescharacteristic
604C2#1172292911yescharacteristic
subgroupG/H
centerC22#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723501C2semidirect
2. #11723499C2semidirect
4. #11722075C2semidirect
5. #11722021C2semidirect
6. C3×#63573C2semidirect
7. C3×#63562C2semidirect
22. C3×#3714C4semidirect
39. #63573D3semidirect
43. #63562D3semidirect
54. C3×C4⋊(C3⋊D3)C2×C4semidirect
57. C32×D12C2×C4semidirect
61. C32×C3⋊C8C2×C4semidirect
114. #3714Dic3semidirect
187. C4⋊(C3⋊D3)C2×Dic3semidirect
205. C3×D12C4×D3semidirect
206. C3×D12C2×Dic3semidirect
219. C3×C3⋊C8C4×D3semidirect
220. C3×C3⋊C8C2×Dic3semidirect
351. C33#442semidirect
408. D12D3×Dic3semidirect
420. C3⋊C8D3×Dic3semidirect
533. C32#4785semidirect
534. C32#4869semidirect
535. C32#5152semidirect
597. C3#10590462semidirect
598. C3#10590781semidirect
599. C3#10591436semidirect

Maximal subgroups

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

subgroupindex
#117235012
#117234992
C2×C3×#37142
#117220752
#117220212
C3×#635732
C3×#635622
#105914363
#105907813
#105904623