Finite Group Databasebeta

← #13155972

Subgroups — #13155972

Order 1728.

848 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88810C211yesmaximal, characteristic, complemented
2#11723517C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic, complemented
4C3×#63894C211yesmaximal, characteristic, complemented
5#11722035C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8#10590765—31nomaximal
9#10590547—31nomaximal
10#10590349—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×#6292—21no—
15C2×#6292—21no—
16C2×C3×#624—21no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88810—22no—
22#88791—22no—
23C3×#3722—21no—
24C2×#3745—31no—
25#64083—31no—
26C3×#893—31no—
27C3×C4×D12—31no—
28#63911—31no—
29#63911—31no—
30#63911—61no—
31#63894D311yescharacteristic, complemented
32C2×#3714—31no—
33C2×#3714—31no—
34C2×#3714—31no—
35C2×#3714—31no—
36C2×#3714—61no—
37C2×#3713—31no—
38#63636—31no—
39C3×#821—31no—
40C3×Dic3⋊C8—31no—
41C3×#809—31no—
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
44#63564—31no—
45#63564—31no—
46#63564—61no—
47#63564D311yescharacteristic, complemented
48C22×C3⋊(C3⋊D3)—21no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3—21no—
51C2×C33×C4C2311yescharacteristic
52C4⋊(C3⋊(C3⋊D3))—21no—
53C4⋊(C3⋊(C3⋊D3))—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)D412yescomplemented
58C2×C32×Dic3—21no—
59C2×C32×Dic3D411yescharacteristic
60C2×C32×Dic3D411yescharacteristic
61#6292—42no—
62#6292—42no—
63C3×#624—42no—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66#5015—91no—
67#4843—91no—
68#4786—91no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—61no—
76C2×C4⋊(C3⋊D3)—61no—
77C2×C4⋊(C3⋊D3)—121no—
78C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
79C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—61no—
85C2×C3×D12—61no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3—61no—
88C2×#626—62no—
89C2×#626—62no—
90C2×#626—61no—
91C2×#626—61no—
92C2×#626—61no—
93C2×#626—61no—
94C2×#624—21no—
95C2×C32×C8—31no—
96C32×C4⋊C4—31no—
97C32×C42—31no—
98C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
99#3745—62no—
100C3×(C2×D3)⋊C4—61no—
101C3×(C2×D3)⋊C4—61no—
102C3×C4⋊Dic3—21no—
103C3×C4⋊Dic3C2×D311yescharacteristic
104C3×C4⋊Dic3C2×D311yescharacteristic
105C3×C4×Dic3—21no—
106C3×C4×Dic3C2×D311yescharacteristic
107C3×C4×Dic3C2×D311yescharacteristic
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—61no—
112C2×C3×C3⋊C8C2×D311yescharacteristic
113C2×C3×C3⋊C8C2×D311yescharacteristic
114#3722—21no—
115#3714—62no—
116#3714—62no—
117#3714—64no—
118#3714—62no—
119#3714—62no—
120#3713—62no—
121C22×C33C2×D411yescharacteristic
122C2×C3⋊(C3⋊D3)—41no—
123C2×C3⋊(C3⋊D3)—41no—
124C2×C3⋊(C3⋊D3)—41no—
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—41no—
127C2×C3×C3⋊D3—41no—
128C2×C3×C3⋊D3—41no—
129C2×C3×C3⋊D3—41no—
130C2×C3×C3⋊D3—41no—
131C33×C4C2×D411yescharacteristic
132C33×C4#4111yescharacteristic
133C32×Dic3—21no—
134C32×Dic3—41no—
135C32×Dic3D812yescomplemented
136C3×C4×D4—91no—
137#940—91no—
138C2×#213—91no—
139C2×#213—91no—
140C2×#213—91no—
141C2×D24—91no—
142C2×D24—91no—
143C2×D24—91no—
144C2×D24—181no—
145#893—31no—
146#874—91no—
147#874—91no—
148#874—181no—
149C4×D12—31no—
150C3×C4⋊C8—91no—
151C3×#102—91no—
152#821—31no—
153#821—91no—
154Dic3⋊C8—31no—
155#809—31no—
156#809—91no—
157#809—91no—
158#809—181no—
159C4⋊(C3⋊C8)—31no—
160C22×C3⋊D3—21no—
161C22×C3⋊D3—21no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—62no—
164C22×C3⋊D3—61no—
165C22×C3⋊D3—61no—
166C22×C3⋊D3—121no—
167C22×C3⋊D3—121no—
168C22×C3⋊D3—122no—
169C22×C3×D3—61no—
170C22×C3×D3—61no—
171C22×C3×D3—62no—
172C22×C3×D3—61no—
173C22×C3×D3—61no—
174C22×C3×D3—121no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—31no—
179C2×C32×C4—31no—
180C2×C32×C4—41no—
181C2×C32×C4—61no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C4⋊(C3⋊D3)—21no—
186C4⋊(C3⋊D3)—21no—
187C4⋊(C3⋊D3)—21no—
188C4⋊(C3⋊D3)—61no—
189C4⋊(C3⋊D3)—61no—
190C4⋊(C3⋊D3)—62no—
191C4⋊(C3⋊D3)—62no—
192C4⋊(C3⋊D3)—61no—
193C4⋊(C3⋊D3)—61no—
194C4⋊(C3⋊D3)—61no—
195C4⋊(C3⋊D3)—61no—
196C4⋊(C3⋊D3)—121no—
197C4⋊(C3⋊D3)—121no—
198C4⋊(C3⋊D3)—121no—
199C4⋊(C3⋊D3)—121no—
200C4⋊(C3⋊D3)—122no—
201C4⋊(C3⋊D3)—122no—
202C4⋊(C3⋊D3)—122no—
203C4⋊(C3⋊D3)—122no—
204C4⋊(C3⋊D3)D1212yescomplemented
205C2×C3×Dic3—21no—
206C2×C3×Dic3—21no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—41no—
210C2×C3×Dic3D1211yescharacteristic
211C2×C3×Dic3D1211yescharacteristic
212C2×C3×Dic3#6711yescharacteristic
213C2×C3×Dic3#6711yescharacteristic
214C3×D12—32no—
215C3×D12—32no—
216C3×D12—62no—
217C3×D12—61no—
218C3×D12—61no—
219C3×D12—61no—
220C3×D12—61no—
221C3×D12—61no—
222C3×D12—61no—
223C3×D12—121no—
224C3×D12—121no—
225C3×D12—121no—
226C3×C4×D3—122no—
227C3×C4×D3—122no—
228#626—124no—
229#626—122no—
230#626—122no—
231#626—124no—
232#626—122no—
233#626—122no—
234#624—42no—
235C32×C8—61no—
236C3⋊(C3⋊C8)—21no—
237C3×C3⋊C8—22no—
238C3×C3⋊C8—21no—
239C3×C3⋊C8—21no—
240C3×C3⋊C8—61no—
241C3×C3⋊C8—61no—
242C3×C3⋊C8—121no—
243#459—271noSylow
244C2×C33#12111yescharacteristic
245C2×C33C2×D811yescharacteristic
246C2×C33AGL(1,ℤ/8)11yescharacteristic
247C3⋊(C3⋊D3)—81no—
248C3×C3⋊D3—42no—
249C3×C3⋊D3—81no—
250C2×C3×D4—91no—
251C2×C3×D4—91no—
252C22×C3×C4—181no—
253C2×#67—181no—
254C2×#67—181no—
255C2×D12—31no—
256C2×D12—31no—
257C2×D12—31no—
258C2×D12—31no—
259C2×D12—61no—
260C2×D12—61no—
261C2×D12—91no—
262C2×D12—91no—
263C2×D12—91no—
264C2×D12—181no—
265C2×D12—181no—
266C2×D12—181no—
267C2×D12—181no—
268C2×D12—181no—
269C2×D12—181no—
270C2×D12—181no—
271C2×D12—182no—
272C2×D12—182no—
273C2×D12—361no—
274C2×C4×D3—61no—
275C2×C4×D3—61no—
276C2×C3×C8—31no—
277C2×C3×C8—31no—
278C2×C3×C8—61no—
279C2×C3×C8—91no—
280C3×C4⋊C4—31no—
281C3×C4⋊C4—31no—
282C3×C4⋊C4—61no—
283C3×C22⋊C4—181no—
284C3×C42—31no—
285C3×C42—31no—
286C3×C42—61no—
287#213—182no—
288#213—182no—
289#213—182no—
290(C2×D3)⋊C4—61no—
291(C2×D3)⋊C4—61no—
292C4⋊Dic3D3211yescharacteristic
293C4×Dic3D3211yescharacteristic
294C2×C3⋊C8—31no—
295C2×C3⋊C8—31no—
296C2×C3⋊C8—61no—
297C2×C3⋊C8D3211yescharacteristic
298D24—182no—
299D24—182no—
300D24—184no—
301D24—182no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—41no—
306C22×C32C2×D1211yescharacteristic
307C22×C32D3×D411yescharacteristic
308C22×C32C2×#6711yescharacteristic
309C2×C3⋊D3—22no—
310C2×C3⋊D3—41no—
311C2×C3⋊D3—41no—
312C2×C3⋊D3—41no—
313C2×C3⋊D3—41no—
314C2×C3⋊D3—41no—
315C2×C3⋊D3—122no—
316C2×C3⋊D3—122no—
317C2×C3⋊D3—122no—
318C2×C3⋊D3—124no—
319C2×C3⋊D3—121no—
320C2×C3⋊D3—121no—
321C2×C3⋊D3—121no—
322C2×C3⋊D3—121no—
323C2×C3⋊D3—121no—
324C2×C3⋊D3—122no—
325C2×C3⋊D3—122no—
326C2×C3⋊D3—121no—
327C2×C3⋊D3—121no—
328C2×C3⋊D3—121no—
329C2×C3⋊D3—121no—
330C2×C3⋊D3—241no—
331C2×C3⋊D3—241no—
332C2×C3⋊D3—241no—
333C2×C3⋊D3—241no—
334C2×C3⋊D3—242no—
335C2×C3⋊D3—242no—
336C2×C3×D3—62no—
337C2×C3×D3—62no—
338C2×C3×D3—122no—
339C2×C3×D3—121no—
340C2×C3×D3—121no—
341C2×C3×D3—121no—
342C2×C3×D3—121no—
343C2×C3×D3—122no—
344C2×C3×D3—122no—
345C2×C3×D3—122no—
346C2×C3×D3—121no—
347C2×C3×D3—121no—
348C2×C3×D3—121no—
349C2×C3×D3—121no—
350C2×C3×D3—121no—
351C2×C3×D3—121no—
352C2×C3×D3—241no—
353C2×C3×D3—241no—
354C32×C4—21no—
355C32×C4—21no—
356C32×C4—21no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—32no—
361C32×C4—41no—
362C32×C4—41no—
363C32×C4—61no—
364C32×C4—121no—
365C32×C4C2×D1211yescharacteristic
366C32×C4#23511yescharacteristic
367C32×C4#23511yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4D3×D411yescharacteristic
370C32×C4#23911yescharacteristic
371C3×Dic3—22no—
372C3×Dic3—21no—
373C3×Dic3—21no—
374C3×Dic3—22no—
375C3×Dic3—41no—
376C3×Dic3—41no—
377C3×Dic3—81no—
378C3×Dic3D2412yescomplemented
379C3×Dic3#21312yescomplemented
380C2×D8—271no—
381C2×D8—271no—
382#127—271no—
383C4×D4—91no—
384C4⋊C8—91no—
385#102—91no—
386#102—271no—
387C33#45911yescharacteristic, complemented, Sylow
388C23×C3—181no—
389C23×C3—181no—
390C22×D3—61no—
391C22×D3—61no—
392C22×D3—62no—
393C22×D3—61no—
394C22×D3—61no—
395C22×D3—121no—
396C22×D3—182no—
397C22×D3—181no—
398C22×D3—181no—
399C22×D3—181no—
400C22×D3—361no—
401C22×D3—361no—
402C22×D3—362no—
403C3×D4—92no—
404C3×D4—181no—
405C3×D4—181no—
406C3×D4—181no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—31no—
411C2×C3×C4—31no—
412C2×C3×C4—31no—
413C2×C3×C4—31no—
414C2×C3×C4—41no—
415C2×C3×C4—61no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—61no—
419C2×C3×C4—121no—
420C2×C3×C4—362no—
421C2×C3×C4C2×D3211yescharacteristic
422C2×C3×C4C2×D3211yescharacteristic
423C2×C3×C4C2×D3211yescharacteristic
424#67—362no—
425#67—362no—
426C2×Dic3—21no—
427C2×Dic3#62611yescharacteristic
428C2×Dic3#62611yescharacteristic
429D12—32no—
430D12—32no—
431D12—62no—
432D12—61no—
433D12—61no—
434D12—61no—
435D12—61no—
436D12—61no—
437D12—61no—
438D12—121no—
439D12—121no—
440D12—121no—
441D12—182no—
442D12—182no—
443D12—181no—
444D12—181no—
445D12—181no—
446D12—181no—
447D12—181no—
448D12—181no—
449D12—361no—
450D12—361no—
451D12—361no—
452D12—361no—
453D12—362no—
454D12—362no—
455D12—362no—
456D12—362no—
457D12—364no—
458D12—364no—
459D12—362no—
460D12—362no—
461C4×D3—122no—
462C4×D3—122no—
463C3×C8—62no—
464C3×C8—61no—
465C3×C8—61no—
466C3×C8—181no—
467C3⋊C8—21no—
468C3⋊C8—61no—
469C3⋊C8—61no—
470C3⋊C8—121no—
471C2×C32—21no—
472C2×C32—21no—
473C2×C32—21no—
474C2×C32—21no—
475C2×C32—21no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—41no—
481C2×C32—41no—
482C2×C32—41no—
483C2×C32#88411yescharacteristic
484C2×C32#89511yescharacteristic
485C2×C32C2×D2411yescharacteristic
486C2×C32#90811yescharacteristic
487C2×C32D3×D811yescharacteristic
488C2×C32#91411yescharacteristic
489C2×C32#93011yescharacteristic
490C2×C32C2×#21311yescharacteristic
491C2×C32#94911yescharacteristic
492C3⋊D3—42no—
493C3⋊D3—81no—
494C3⋊D3—242no—
495C3⋊D3—242no—
496C3⋊D3—242no—
497C3⋊D3—244no—
498C3⋊D3—241no—
499C3⋊D3—241no—
500C3⋊D3—241no—
501C3×D3—122no—
502C3×D3—122no—
503C3×D3—242no—
504C3×D3—242no—
505C3×D3—241no—
506C3×D3—241no—
507C2×D4—91no—
508C2×D4—91no—
509C2×D4—271no—
510C2×D4—541no—
511C2×D4—541no—
512C22×C4—181no—
513D8—542no—
514D8—542no—
515C2×C8—31no—
516C2×C8—91no—
517C4⋊C4—31no—
518C22⋊C4—181no—
519C42—31no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—41no—
524C22×C3—182no—
525C22×C3—361no—
526C22×C3—361no—
527C22×C3—361no—
528C22×C3—361no—
529C22×C3—361no—
530C22×C3D3×D1211yescharacteristic
531C22×C3C2×#62611yescharacteristic
532C22×C3D3×#6711yescharacteristic
533C2×D3—62no—
534C2×D3—62no—
535C2×D3—122no—
536C2×D3—121no—
537C2×D3—121no—
538C2×D3—121no—
539C2×D3—121no—
540C2×D3—122no—
541C2×D3—122no—
542C2×D3—122no—
543C2×D3—121no—
544C2×D3—121no—
545C2×D3—121no—
546C2×D3—121no—
547C2×D3—121no—
548C2×D3—121no—
549C2×D3—241no—
550C2×D3—241no—
551C2×D3—362no—
552C2×D3—362no—
553C2×D3—362no—
554C2×D3—362no—
555C2×D3—362no—
556C2×D3—364no—
557C2×D3—361no—
558C2×D3—361no—
559C2×D3—361no—
560C2×D3—361no—
561C2×D3—361no—
562C2×D3—361no—
563C2×D3—361no—
564C2×D3—361no—
565C2×D3—361no—
566C2×D3—721no—
567C2×D3—721no—
568C2×D3—721no—
569C2×D3—721no—
570C2×D3—722no—
571C2×D3—722no—
572C3×C4—21no—
573C3×C4—21no—
574C3×C4—21no—
575C3×C4—21no—
576C3×C4—21no—
577C3×C4—21no—
578C3×C4—32no—
579C3×C4—32no—
580C3×C4—41no—
581C3×C4—41no—
582C3×C4—62no—
583C3×C4—62no—
584C3×C4—61no—
585C3×C4—61no—
586C3×C4—121no—
587C3×C4—121no—
588C3×C4—241no—
589C3×C4#379811yescharacteristic
590C3×C4#379911yescharacteristic
591C3×C4#379911yescharacteristic
592C3×C4D3×D1211yescharacteristic
593C3×C4D3×D1211yescharacteristic
594C3×C4#381111yescharacteristic
595Dic3—21no—
596Dic3—41no—
597Dic3#371412yescomplemented
598C32—21no—
599C32—21no—
600C32—21no—
601C32—41no—
602C32#478611yescharacteristic, complemented
603C32#484311yescharacteristic, complemented
604C32#501511yescharacteristic, complemented
605C23—181no—
606C23—181no—
607C23—541no—
608D4—92no—
609D4—181no—
610D4—181no—
611D4—181no—
612D4—541no—
613D4—541no—
614D4—1082no—
615D4—1082no—
616C2×C4—31no—
617C2×C4—31no—
618C2×C4—61no—
619C2×C4—362no—
620C2×C4D3311yescharacteristic
621C8—61no—
622C8—181no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—21no—
628C2×C3—21no—
629C2×C3—21no—
630C2×C3—21no—
631C2×C3—21no—
632C2×C3—41no—
633C2×C3—41no—
634C2×C3—41no—
635C2×C3—362no—
636C2×C3—721no—
637C2×C3D3×D2411yescharacteristic
638C2×C3#6379411yescharacteristic
639C2×C3C2×#371411yescharacteristic
640C2×C3#6382511yescharacteristic
641C2×C3#6390611yescharacteristic
642C2×C3#6390911yescharacteristic
643C2×C3#6391011yescharacteristic
644C2×C3D3×#21311yescharacteristic
645C2×C3#6393911yescharacteristic
646D3—122no—
647D3—122no—
648D3—242no—
649D3—242no—
650D3—241no—
651D3—241no—
652D3—722no—
653D3—722no—
654D3—722no—
655D3—724no—
656D3—721no—
657D3—721no—
658D3—721no—
659C22—182no—
660C22—361no—
661C22—361no—
662C22—361no—
663C22—361no—
664C22—361no—
665C22—1081no—
666C22—1081no—
667C22—1081no—
668C22D3×#62611yescharacteristic
669C4—32no—
670C4—61no—
671C4—121no—
672C4#8895511yescharacteristic
673C4#8896511yescharacteristic
674C3—21no—
675C3—21no—
676C3—21no—
677C3—41no—
678C3#1059034911yescharacteristic, complemented
679C3#1059054711yescharacteristic, complemented
680C3#1059076511yescharacteristic, complemented
681C2—362no—
682C2—721no—
683C2—2161no—
684C2D3×#371411yescharacteristic
685C2#1172287111yescharacteristic
686C2#1172296111yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#459
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×#88810C2semidirect
2. #11723517C2semidirect
3. C2×#88791C2semidirect
4. C3×#63894C2semidirect
5. #11722035C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×#63564C2semidirect
31. #63894D3semidirect
43. Dic3⋊(C3⋊C8)D3semidirect
47. #63564D3semidirect
57. C3×C4⋊(C3⋊D3)D4semidirect
135. C32×Dic3D8semidirect
204. C4⋊(C3⋊D3)D12semidirect
378. C3×Dic3D24semidirect
379. C3×Dic3#213semidirect
387. C33#459semidirect
597. Dic3#3714semidirect
602. C32#4786semidirect
603. C32#4843semidirect
604. C32#5015semidirect
678. C3#10590349semidirect
679. C3#10590547semidirect
680. C3#10590765semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
#117235172
C2×#887912
C3×#638942
#117220352
C3×Dic3⋊(C3⋊C8)2
C3×#635642
#105907653
#105905473
#105903493