Finite Group Databasebeta

← #13155723

Subgroups — #13155723

Order 1728.

802 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#11723494C211yesmaximal, characteristic, complemented
2C2×#88790C211yesmaximal, characteristic
3C3×#63898C211yesmaximal, characteristic, complemented
4#11722071C211yesmaximal, characteristic, complemented
5#11722035C211yesmaximal, characteristic, complemented
6C3×#63574C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591404—31nomaximal
9#10590589—31nomaximal
10#10590462—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6289—21no—
14C2×C3×D3×Dic3—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20#88804—21no—
21#88790C414yescomplemented
22C3×(C2×D3)⋊Dic3—21no—
23C2×#3753—31no—
24#64102—31no—
25C3×D4×Dic3—31no—
26C3×#893—31no—
27#63898D311yescharacteristic, complemented
28#63894—31no—
29#63894—31no—
30#63894—61no—
31C2×#3714—31no—
32C2×#3714—31no—
33C2×#3714—61no—
34#63636—31no—
35C3×#832—31no—
36C3×#809—31no—
37C3×#808—31no—
38C3×#808—31no—
39#63574D311yescharacteristic, complemented
40#63564—31no—
41#63564—31no—
42#63564—61no—
43#63562D311yescharacteristic, complemented
44#63555—31no—
45#63554—31no—
46#63554—31no—
47C22×C3⋊(C3⋊D3)—21no—
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
52C2×C3×C3⋊Dic3D411yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12C2×C412yescomplemented
58#6289—42no—
59C3×D3×Dic3—42no—
60C3×C3⋊(C3⋊C8)—21no—
61C3×C3⋊(C3⋊C8)C2×C412yescomplemented
62#5014—91no—
63#4869—91no—
64#4785—91no—
65C2×C32×D4—31no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—121no—
73C2×C4×C3⋊D3—61no—
74C22×C3×Dic3—61no—
75C2×C3×D12—21no—
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×D12C2×D311yescharacteristic
78C2×C3×C4×D3—61no—
79C2×#624—61no—
80C2×#624—61no—
81C2×#624—62no—
82C2×D3×Dic3—21no—
83C32×C4⋊C4—31no—
84C32×C4⋊C4—31no—
85#3753—34no—
86#3752—61no—
87C4×C3⋊Dic3C2×D311yescharacteristic
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×#217—61no—
91C3×(C2×D3)⋊C4—61no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4×Dic3—31no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—61no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8—61no—
107#3722—61no—
108#3722—61no—
109#3722—62no—
110(C2×D3)⋊Dic3—21no—
111#3714—34no—
112#3714—34no—
113#3714—64no—
114C22×C33C2×D411yescharacteristic
115C2×C3⋊(C3⋊D3)—22no—
116C2×C3⋊(C3⋊D3)—41no—
117C2×C3⋊(C3⋊D3)—41no—
118C2×C32×D3—22no—
119C2×C32×D3—41no—
120C2×C32×D3—41no—
121C33×C4C22×C411yescharacteristic
122C33×C4#4111yescharacteristic
123C3×C3⋊Dic3—21no—
124C3×C3⋊Dic3—21no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C3×C4×D4—91no—
128D4×Dic3—31no—
129C2×#213—91no—
130C2×#213—91no—
131C2×#213—181no—
132C2×D24—91no—
133#893—31no—
134#893—91no—
135#893—91no—
136#893—181no—
137C4×D12—91no—
138C3×#106—91no—
139C3×#102—91no—
140#832—31no—
141#821—91no—
142#815—91no—
143#815—91no—
144#809—31no—
145#809—91no—
146#809—91no—
147#809—181no—
148#808—31no—
149#808—31no—
150#803—91no—
151C23×C32—61no—
152C22×C3⋊D3—61no—
153C22×C3⋊D3—62no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—121no—
157C22×C3⋊D3—121no—
158C22×C3⋊D3—122no—
159C22×C3×D3—21no—
160C22×C3×D3—21no—
161C22×C3×D3—41no—
162C32×D4—32no—
163C32×D4—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3#6711yescharacteristic
174C2×C3⋊Dic3#6711yescharacteristic
175C4⋊(C3⋊D3)—32no—
176C4⋊(C3⋊D3)—32no—
177C4⋊(C3⋊D3)—32no—
178C4⋊(C3⋊D3)—62no—
179C4⋊(C3⋊D3)—62no—
180C4⋊(C3⋊D3)—62no—
181C4⋊(C3⋊D3)—61no—
182C4⋊(C3⋊D3)—62no—
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)—61no—
185C4⋊(C3⋊D3)—122no—
186C4⋊(C3⋊D3)—121no—
187C4⋊(C3⋊D3)—121no—
188C4⋊(C3⋊D3)—122no—
189C4×C3⋊D3—122no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—41no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—122no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—22no—
206C3×D12—41no—
207C3×D12C4×D312yescomplemented
208C3×D12C4×D312yescomplemented
209C3×C4×D3—122no—
210#624—124no—
211#624—122no—
212#624—122no—
213D3×Dic3—42no—
214C3⋊(C3⋊C8)—21no—
215C3⋊(C3⋊C8)C4×D312yescomplemented
216C3×C3⋊C8—32no—
217C3×C3⋊C8—32no—
218C3×C3⋊C8—62no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—121no—
222#442—271noSylow
223C2×C33C4×D411yescharacteristic
224C2×C33AGL(1,ℤ/8)11yescharacteristic
225C2×C33AGL(1,ℤ/8)11yescharacteristic
226C3⋊(C3⋊D3)—42no—
227C32×D3—42no—
228C2×C3×D4—31no—
229C2×C3×D4—31no—
230C2×C3×D4—61no—
231C22×C3×C4—181no—
232C22×Dic3—61no—
233C2×D12—91no—
234C2×D12—91no—
235C2×D12—91no—
236C2×D12—181no—
237C2×D12—181no—
238C2×D12—181no—
239C2×D12—361no—
240C2×D12D3211yescharacteristic
241C2×C4×D3—61no—
242C2×C4×D3—181no—
243C2×C4×D3—181no—
244C2×C4×D3—181no—
245C2×C4×D3—182no—
246C2×C3×C8—91no—
247C2×C3×C8—91no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—31no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—31no—
252C3×C4⋊C4—61no—
253C3×C4⋊C4—61no—
254C3×C22⋊C4—181no—
255C3×C42—91no—
256#217—61no—
257#213—94no—
258#213—94no—
259#213—184no—
260(C2×D3)⋊C4—61no—
261(C2×D3)⋊C4—181no—
262(C2×D3)⋊C4—181no—
263(C2×D3)⋊C4—181no—
264(C2×D3)⋊C4—182no—
265C4⋊Dic3D3211yescharacteristic
266C4⋊Dic3D3211yescharacteristic
267C4×Dic3—31no—
268C4×Dic3—31no—
269C4×Dic3—61no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—61no—
275C2×C3⋊C8—61no—
276D24—94no—
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×C32C2×#6711yescharacteristic
287C2×C3⋊D3—62no—
288C2×C3⋊D3—64no—
289C2×C3⋊D3—62no—
290C2×C3⋊D3—62no—
291C2×C3⋊D3—122no—
292C2×C3⋊D3—122no—
293C2×C3⋊D3—124no—
294C2×C3⋊D3—121no—
295C2×C3⋊D3—121no—
296C2×C3⋊D3—122no—
297C2×C3⋊D3—122no—
298C2×C3⋊D3—121no—
299C2×C3⋊D3—121no—
300C2×C3⋊D3—121no—
301C2×C3⋊D3—121no—
302C2×C3⋊D3—241no—
303C2×C3⋊D3—241no—
304C2×C3⋊D3—241no—
305C2×C3⋊D3—241no—
306C2×C3⋊D3—242no—
307C2×C3⋊D3—242no—
308C2×C3×D3—22no—
309C2×C3×D3—22no—
310C2×C3×D3—42no—
311C2×C3×D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C2×C3×D3—81no—
316C2×C3×D3—81no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—121no—
326C32×C4—121no—
327C32×C4C2×C4×D311yescharacteristic
328C32×C4C2×C4×D311yescharacteristic
329C32×C4C2×C4×D311yescharacteristic
330C32×C4#23511yescharacteristic
331C32×C4#23711yescharacteristic
332C32×C4#23911yescharacteristic
333C3⋊Dic3—21no—
334C3⋊Dic3—21no—
335C3×Dic3—42no—
336C3×Dic3—41no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C3×Dic3—62no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3—61no—
344C3×Dic3—61no—
345C3×Dic3—81no—
346C3×Dic3—121no—
347C2×D8—271no—
348C4×D4—91no—
349C4×D4—271no—
350#106—91no—
351#102—91no—
352#102—271no—
353#97—271no—
354C33#44211yescharacteristic, complemented, Sylow
355C23×C3—61no—
356C23×C3—61no—
357C23×C3—121no—
358C22×D3—21no—
359C22×D3—182no—
360C22×D3—181no—
361C22×D3—181no—
362C22×D3—181no—
363C22×D3—361no—
364C22×D3—361no—
365C22×D3—362no—
366C3×D4—32no—
367C3×D4—32no—
368C3×D4—62no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—121no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—62no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—91no—
382C2×C3×C4—91no—
383C2×C3×C4—121no—
384C2×C3×C4—362no—
385C2×C3×C4C2×D3211yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—31no—
393C2×Dic3—31no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—122no—
397D12—21no—
398D12—92no—
399D12—92no—
400D12—92no—
401D12—182no—
402D12—182no—
403D12—182no—
404D12—182no—
405D12—181no—
406D12—181no—
407D12—181no—
408D12—362no—
409D12—361no—
410D12—361no—
411D12—362no—
412D12#62412yescomplemented
413C4×D3—122no—
414C4×D3—362no—
415C4×D3—364no—
416C4×D3—362no—
417C4×D3—362no—
418C3×C8—92no—
419C3×C8—181no—
420C3⋊C8—32no—
421C3⋊C8—32no—
422C3⋊C8—62no—
423C3⋊C8—61no—
424C3⋊C8—61no—
425C3⋊C8—121no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—41no—
438C2×C32—122no—
439C2×C32#67⋊C411yescharacteristic
440C2×C32#89311yescharacteristic
441C2×C32#91111yescharacteristic
442C2×C32#91411yescharacteristic
443C2×C32#91411yescharacteristic
444C2×C32#91411yescharacteristic
445C2×C32C4×#6711yescharacteristic
446C2×C32#93211yescharacteristic
447C2×C32#94911yescharacteristic
448C3⋊D3—122no—
449C3⋊D3—124no—
450C3⋊D3—122no—
451C3⋊D3—122no—
452C3⋊D3—242no—
453C3⋊D3—242no—
454C3⋊D3—244no—
455C3×D3—42no—
456C3×D3—42no—
457C3×D3—82no—
458C2×D4—31no—
459C2×D4—271no—
460C22×C4—181no—
461C22×C4—541no—
462D8—274no—
463C2×C8—91no—
464C2×C8—91no—
465C4⋊C4—31no—
466C4⋊C4—31no—
467C22⋊C4—181no—
468C22⋊C4—541no—
469C42—91no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—41no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—122no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—121no—
480C22×C3—121no—
481C22×C3—241no—
482C22×C3—241no—
483C22×C3D3×#6711yescharacteristic
484C22×C3D3×#6711yescharacteristic
485C22×C3#381111yescharacteristic
486C2×D3—22no—
487C2×D3—41no—
488C2×D3—41no—
489C2×D3—184no—
490C2×D3—182no—
491C2×D3—182no—
492C2×D3—182no—
493C2×D3—362no—
494C2×D3—362no—
495C2×D3—364no—
496C2×D3—362no—
497C2×D3—362no—
498C2×D3—361no—
499C2×D3—361no—
500C2×D3—361no—
501C2×D3—361no—
502C2×D3—361no—
503C2×D3—361no—
504C2×D3—721no—
505C2×D3—721no—
506C2×D3—721no—
507C2×D3—721no—
508C2×D3—722no—
509C2×D3—722no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—122no—
519C3×C4—121no—
520C3×C4—121no—
521C3×C4—121no—
522C3×C4—121no—
523C3×C4—181no—
524C3×C4—181no—
525C3×C4—241no—
526C3×C4#379611yescharacteristic
527C3×C4#379911yescharacteristic
528C3×C4C4×D3211yescharacteristic
529C3×C4C4×D3211yescharacteristic
530C3×C4#380411yescharacteristic
531C3×C4C2×#62411yescharacteristic
532Dic3—41no—
533Dic3—41no—
534Dic3—62no—
535Dic3—61no—
536Dic3—61no—
537Dic3—61no—
538Dic3—61no—
539Dic3—121no—
540C32—21no—
541C32—21no—
542C32—21no—
543C32—41no—
544C32#478511yescharacteristic, complemented
545C32#486911yescharacteristic, complemented
546C32#501411yescharacteristic, complemented
547C23—61no—
548C23—541no—
549D4—32no—
550D4—61no—
551D4—272no—
552D4—541no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—91no—
556C2×C4—91no—
557C2×C4—362no—
558C2×C4—1082no—
559C2×C4D3311yescharacteristic
560C8—92no—
561C8—181no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—41no—
572C2×C3—41no—
573C2×C3—41no—
574C2×C3—122no—
575C2×C3—122no—
576C2×C3—242no—
577C2×C3#6388111yescharacteristic
578C2×C3#6390411yescharacteristic
579C2×C3#6392511yescharacteristic
580C2×C3#6393411yescharacteristic
581C2×C3#6393411yescharacteristic
582C2×C3#6393711yescharacteristic
583C2×C3#6393911yescharacteristic
584C2×C3#6394111yescharacteristic
585C2×C3#6395211yescharacteristic
586D3—42no—
587D3—364no—
588D3—362no—
589D3—362no—
590D3—362no—
591D3—722no—
592D3—722no—
593D3—724no—
594C22—62no—
595C22—121no—
596C22—121no—
597C22—542no—
598C22—1081no—
599C22—1081no—
600C22#8895211yescharacteristic
601C4—121no—
602C4—121no—
603C4—181no—
604C4—181no—
605C4D3×#62411yescharacteristic
606C4#8896211yescharacteristic
607C3—21no—
608C3—21no—
609C3—21no—
610C3—41no—
611C3#1059046211yescharacteristic, complemented
612C3#1059058911yescharacteristic, complemented
613C3#1059140411yescharacteristic, complemented
614C2—122no—
615C2—1082no—
616C2#1172287311yescharacteristic
617C2#1172287411yescharacteristic
618C2#1172293511yescharacteristic
subgroupG/H
centerC22#88952
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#88790C2
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. #11723494C2semidirect
3. C3×#63898C2semidirect
4. #11722071C2semidirect
5. #11722035C2semidirect
6. C3×#63574C2semidirect
7. C3×#63562C2semidirect
21. #88790C4semidirect
27. #63898D3semidirect
39. #63574D3semidirect
43. #63562D3semidirect
51. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
57. C32×D12C2×C4semidirect
61. C3×C3⋊(C3⋊C8)C2×C4semidirect
207. C3×D12C4×D3semidirect
208. C3×D12C4×D3semidirect
215. C3⋊(C3⋊C8)C4×D3semidirect
354. C33#442semidirect
412. D12#624semidirect
544. C32#4785semidirect
545. C32#4869semidirect
546. C32#5014semidirect
611. C3#10590462semidirect
612. C3#10590589semidirect
613. C3#10591404semidirect

Maximal subgroups

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

subgroupindex
#117234942
C2×#887902
C3×#638982
#117220712
#117220352
C3×#635742
C3×#635622
#105914043
#105905893
#105904623