Finite Group Databasebeta

← #13155974

Subgroups — #13155974

Order 1728.

860 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
1#11723484C211yesmaximal, characteristic, complemented
2C2×#88791C211yesmaximal, characteristic, complemented
3C2×#88790C211yesmaximal, characteristic, complemented
4C3×#63911C211yesmaximal, characteristic, complemented
5#11722075C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591429—31nomaximal
9#10590765—31nomaximal
10#10590355—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6289—21no—
15C2×C3×#626—21no—
16C2×C3×#626—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88804—21no—
22#88791—22no—
23#88790—22no—
24C2×#3753—31no—
25C2×#3745—31no—
26C4×C4⋊(C3⋊D3)—31no—
27C3×#940—31no—
28C3×#874—31no—
29#63911D311yescharacteristic, complemented
30#63894—31no—
31#63894—31no—
32#63894—61no—
33C2×#3714—31no—
34C2×#3714—31no—
35C2×#3714—31no—
36C2×#3714—31no—
37C2×#3714—61no—
38C2×#3714—61no—
39C3×#832—31no—
40C3×#821—31no—
41C3×Dic3⋊C8—31no—
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
44#63564—31no—
45#63563—31no—
46#63563D311yescharacteristic, complemented
47#63562—31no—
48C22×C3⋊(C3⋊D3)—21no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C4⋊(C3⋊(C3⋊D3))—21no—
53C4⋊(C3⋊(C3⋊D3))D412yescomplemented
54C2×C3×C3⋊Dic3—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61#6289—42no—
62C3×#626—42no—
63C3×#626—42no—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66#5072—91no—
67#4843—91no—
68#4786—91no—
69C2×C32×D4—31no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—61no—
76C2×C4⋊(C3⋊D3)—121no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C2×C4×C3⋊D3—61no—
79C2×C3×#67—61no—
80C2×C3×#67—61no—
81C2×C3×D12—21no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—61no—
85C2×C3×D12—61no—
86C2×C3×D12—61no—
87C2×C3×D12C2×D311yescharacteristic
88C2×C3×D12C2×D311yescharacteristic
89C2×#626—21no—
90C2×#626—21no—
91C2×#624—62no—
92C2×#624—61no—
93C2×#624—61no—
94C2×C32×C8—31no—
95C32×C42—31no—
96#3753—62no—
97#3752—61no—
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
100#3745—62no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—61no—
104C3×C4×Dic3—21no—
105C3×C4×Dic3C2×D311yescharacteristic
106C3×C4×Dic3C2×D311yescharacteristic
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—61no—
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C2×D311yescharacteristic
113#3722—62no—
114#3722—61no—
115#3722—61no—
116#3714—62no—
117#3714—62no—
118#3714—64no—
119#3714—62no—
120#3714—62no—
121#3714—64no—
122C22×C33C2×D411yescharacteristic
123C2×C3⋊(C3⋊D3)—22no—
124C2×C3⋊(C3⋊D3)—41no—
125C2×C3⋊(C3⋊D3)—41no—
126C2×C3×C3⋊D3—41no—
127C2×C3×C3⋊D3—41no—
128C2×C3×C3⋊D3—41no—
129C2×C32×D3—41no—
130C2×C32×D3—41no—
131C2×C32×D3—41no—
132C33×C4C2×D411yescharacteristic
133C33×C4#4111yescharacteristic
134C3×C3⋊Dic3—41no—
135C32×Dic3—21no—
136C32×Dic3D812yescomplemented
137C3×#127—91no—
138#940—31no—
139C2×#213—91no—
140C2×#213—91no—
141C2×#213—91no—
142C2×#213—181no—
143C2×D24—91no—
144C2×D24—91no—
145C2×D24—91no—
146C2×D24—181no—
147#893—91no—
148#874—31no—
149C4×D12—91no—
150C4×D12—91no—
151C4×D12—181no—
152C3×C4⋊C8—91no—
153C3×#102—91no—
154#832—31no—
155#832—91no—
156#821—31no—
157#821—91no—
158Dic3⋊C8—31no—
159#809—91no—
160C4⋊(C3⋊C8)—31no—
161C23×C32—61no—
162C22×C3⋊D3—21no—
163C22×C3⋊D3—61no—
164C22×C3⋊D3—62no—
165C22×C3⋊D3—61no—
166C22×C3⋊D3—61no—
167C22×C3⋊D3—121no—
168C22×C3⋊D3—121no—
169C22×C3⋊D3—122no—
170C22×C3×D3—21no—
171C22×C3×D3—21no—
172C22×C3×D3—41no—
173C22×C3×D3—61no—
174C22×C3×D3—61no—
175C22×C3×D3—121no—
176C32×D4—61no—
177C32×D4—61no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—31no—
182C2×C32×C4—31no—
183C2×C32×C4—41no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—21no—
188C4⋊(C3⋊D3)—21no—
189C4⋊(C3⋊D3)—21no—
190C4⋊(C3⋊D3)—32no—
191C4⋊(C3⋊D3)—32no—
192C4⋊(C3⋊D3)—32no—
193C4⋊(C3⋊D3)—62no—
194C4⋊(C3⋊D3)—62no—
195C4⋊(C3⋊D3)—62no—
196C4⋊(C3⋊D3)—61no—
197C4⋊(C3⋊D3)—62no—
198C4⋊(C3⋊D3)—61no—
199C4⋊(C3⋊D3)—61no—
200C4⋊(C3⋊D3)—122no—
201C4⋊(C3⋊D3)—121no—
202C4⋊(C3⋊D3)—121no—
203C4⋊(C3⋊D3)—122no—
204C4×C3⋊D3—122no—
205C3×#67—122no—
206C3×#67—122no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3D1211yescharacteristic
213C2×C3×Dic3D1211yescharacteristic
214C2×C3×Dic3#6711yescharacteristic
215C2×C3×Dic3#6711yescharacteristic
216C3×D12—22no—
217C3×D12—21no—
218C3×D12—21no—
219C3×D12—21no—
220C3×D12—21no—
221C3×D12—41no—
222C3×D12—62no—
223C3×D12—61no—
224C3×D12—61no—
225C3×D12—61no—
226C3×D12—61no—
227C3×D12—121no—
228C3×D12—122no—
229C3×D12—122no—
230#626—42no—
231#626—42no—
232#624—124no—
233#624—122no—
234#624—122no—
235C32×C8—61no—
236C3⋊(C3⋊C8)—21no—
237C3×C3⋊C8—22no—
238C3×C3⋊C8—21no—
239C3×C3⋊C8—21no—
240C3×C3⋊C8—62no—
241C3×C3⋊C8—61no—
242C3×C3⋊C8—61no—
243#459—271noSylow
244C2×C33#12111yescharacteristic
245C2×C33C2×D811yescharacteristic
246C2×C33AGL(1,ℤ/8)11yescharacteristic
247C3⋊(C3⋊D3)—42no—
248C3×C3⋊D3—81no—
249C32×D3—81no—
250C2×C3×D4—31no—
251C2×C3×D4—31no—
252C2×C3×D4—61no—
253C2×C3×D4—91no—
254C2×C3×D4—181no—
255C2×C3×D4—181no—
256C2×#67—61no—
257C2×#67—61no—
258C2×D12—31no—
259C2×D12—31no—
260C2×D12—61no—
261C2×D12—61no—
262C2×D12—61no—
263C2×D12—91no—
264C2×D12—91no—
265C2×D12—91no—
266C2×D12—181no—
267C2×D12—181no—
268C2×D12—181no—
269C2×D12—361no—
270C2×D12D3211yescharacteristic
271C2×C4×D3—181no—
272C2×C4×D3—181no—
273C2×C4×D3—182no—
274C2×C4×D3—181no—
275C2×C3×C8—31no—
276C2×C3×C8—31no—
277C2×C3×C8—61no—
278C2×C3×C8—91no—
279C3×C4⋊C4—91no—
280C3×C42—31no—
281C3×C42—31no—
282C3×C42—61no—
283#213—182no—
284#213—182no—
285#213—182no—
286#213—184no—
287(C2×D3)⋊C4—181no—
288(C2×D3)⋊C4—181no—
289(C2×D3)⋊C4—182no—
290(C2×D3)⋊C4—181no—
291C4⋊Dic3—31no—
292C4⋊Dic3—31no—
293C4⋊Dic3—61no—
294C4×Dic3D3211yescharacteristic
295C2×C3⋊C8—31no—
296C2×C3⋊C8—31no—
297C2×C3⋊C8—61no—
298C2×C3⋊C8D3211yescharacteristic
299D24—182no—
300D24—182no—
301D24—184no—
302D24—182no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—41no—
307C22×C32—121no—
308C22×C32—121no—
309C22×C32—121no—
310C22×C32C2×D1211yescharacteristic
311C22×C32D3×D411yescharacteristic
312C22×C32C2×#6711yescharacteristic
313C2×C3⋊D3—41no—
314C2×C3⋊D3—41no—
315C2×C3⋊D3—41no—
316C2×C3⋊D3—62no—
317C2×C3⋊D3—64no—
318C2×C3⋊D3—62no—
319C2×C3⋊D3—62no—
320C2×C3⋊D3—122no—
321C2×C3⋊D3—122no—
322C2×C3⋊D3—124no—
323C2×C3⋊D3—121no—
324C2×C3⋊D3—121no—
325C2×C3⋊D3—122no—
326C2×C3⋊D3—122no—
327C2×C3⋊D3—121no—
328C2×C3⋊D3—121no—
329C2×C3⋊D3—121no—
330C2×C3⋊D3—121no—
331C2×C3⋊D3—241no—
332C2×C3⋊D3—241no—
333C2×C3⋊D3—241no—
334C2×C3⋊D3—241no—
335C2×C3⋊D3—242no—
336C2×C3⋊D3—242no—
337C2×C3×D3—42no—
338C2×C3×D3—41no—
339C2×C3×D3—41no—
340C2×C3×D3—41no—
341C2×C3×D3—41no—
342C2×C3×D3—41no—
343C2×C3×D3—41no—
344C2×C3×D3—81no—
345C2×C3×D3—81no—
346C2×C3×D3—122no—
347C2×C3×D3—121no—
348C2×C3×D3—121no—
349C2×C3×D3—121no—
350C2×C3×D3—121no—
351C2×C3×D3—121no—
352C2×C3×D3—121no—
353C2×C3×D3—241no—
354C2×C3×D3—241no—
355C32×C4—21no—
356C32×C4—21no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—32no—
362C32×C4—41no—
363C32×C4—41no—
364C32×C4—61no—
365C32×C4#23511yescharacteristic
366C32×C4D3×D411yescharacteristic
367C32×C4D3×D411yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4#23711yescharacteristic
370C32×C4#23911yescharacteristic
371C3⋊Dic3—41no—
372C3×Dic3—21no—
373C3×Dic3—21no—
374C3×Dic3—22no—
375C3×Dic3—41no—
376C3×Dic3—122no—
377C3×Dic3—121no—
378C3×Dic3—121no—
379C3×Dic3D2412yescomplemented
380C3×Dic3#21312yescomplemented
381C2×D8—271no—
382C2×D8—271no—
383#127—91no—
384C4×D4—271no—
385C4⋊C8—91no—
386#102—91no—
387#102—271no—
388C33#45911yescharacteristic, complemented, Sylow
389C23×C3—61no—
390C23×C3—61no—
391C23×C3—121no—
392C23×C3—181no—
393C22×D3—21no—
394C22×D3—61no—
395C22×D3—61no—
396C22×D3—121no—
397C22×D3—182no—
398C22×D3—181no—
399C22×D3—181no—
400C22×D3—181no—
401C22×D3—361no—
402C22×D3—361no—
403C22×D3—362no—
404C3×D4—62no—
405C3×D4—61no—
406C3×D4—61no—
407C3×D4—61no—
408C3×D4—61no—
409C3×D4—121no—
410C3×D4—181no—
411C3×D4—181no—
412C3×D4—362no—
413C3×D4—362no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—31no—
418C2×C3×C4—31no—
419C2×C3×C4—31no—
420C2×C3×C4—31no—
421C2×C3×C4—41no—
422C2×C3×C4—61no—
423C2×C3×C4—61no—
424C2×C3×C4—181no—
425C2×C3×C4C2×D3211yescharacteristic
426C2×C3×C4C2×D3211yescharacteristic
427C2×C3×C4C2×D3211yescharacteristic
428#67—122no—
429#67—122no—
430C2×Dic3—62no—
431C2×Dic3—61no—
432C2×Dic3—61no—
433C2×Dic3#62611yescharacteristic
434C2×Dic3#62611yescharacteristic
435D12—21no—
436D12—21no—
437D12—62no—
438D12—61no—
439D12—61no—
440D12—61no—
441D12—61no—
442D12—92no—
443D12—92no—
444D12—92no—
445D12—121no—
446D12—122no—
447D12—122no—
448D12—182no—
449D12—182no—
450D12—182no—
451D12—182no—
452D12—181no—
453D12—181no—
454D12—181no—
455D12—362no—
456D12—361no—
457D12—361no—
458D12—362no—
459C4×D3—362no—
460C4×D3—362no—
461C4×D3—364no—
462C4×D3—362no—
463C3×C8—62no—
464C3×C8—61no—
465C3×C8—61no—
466C3×C8—181no—
467C3⋊C8—21no—
468C3⋊C8—62no—
469C3⋊C8—61no—
470C3⋊C8—61no—
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—241no—
484C2×C32#88411yescharacteristic
485C2×C32#89511yescharacteristic
486C2×C32C2×D2411yescharacteristic
487C2×C32#90811yescharacteristic
488C2×C32D3×D811yescharacteristic
489C2×C32#91411yescharacteristic
490C2×C32C2×#21311yescharacteristic
491C2×C32#93211yescharacteristic
492C2×C32#93811yescharacteristic
493C3⋊D3—81no—
494C3⋊D3—122no—
495C3⋊D3—124no—
496C3⋊D3—122no—
497C3⋊D3—122no—
498C3⋊D3—242no—
499C3⋊D3—242no—
500C3⋊D3—244no—
501C3×D3—82no—
502C3×D3—81no—
503C3×D3—81no—
504C3×D3—242no—
505C3×D3—241no—
506C3×D3—241no—
507C2×D4—31no—
508C2×D4—91no—
509C2×D4—181no—
510C2×D4—181no—
511C2×D4—271no—
512C22×C4—541no—
513D8—542no—
514D8—542no—
515C2×C8—31no—
516C2×C8—91no—
517C4⋊C4—91no—
518C22⋊C4—541no—
519C42—31no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—41no—
524C22×C3—122no—
525C22×C3—121no—
526C22×C3—121no—
527C22×C3—121no—
528C22×C3—121no—
529C22×C3—121no—
530C22×C3—121no—
531C22×C3—241no—
532C22×C3—241no—
533C22×C3—361no—
534C22×C3—361no—
535C22×C3—361no—
536C22×C3D3×D1211yescharacteristic
537C22×C3C2×#62611yescharacteristic
538C22×C3D3×#6711yescharacteristic
539C2×D3—41no—
540C2×D3—41no—
541C2×D3—41no—
542C2×D3—122no—
543C2×D3—121no—
544C2×D3—121no—
545C2×D3—121no—
546C2×D3—121no—
547C2×D3—121no—
548C2×D3—121no—
549C2×D3—184no—
550C2×D3—182no—
551C2×D3—182no—
552C2×D3—182no—
553C2×D3—241no—
554C2×D3—241no—
555C2×D3—362no—
556C2×D3—362no—
557C2×D3—364no—
558C2×D3—362no—
559C2×D3—362no—
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—61no—
584C3×C4—61no—
585C3×C4—121no—
586C3×C4—361no—
587C3×C4#379611yescharacteristic
588C3×C4#379911yescharacteristic
589C3×C4#380211yescharacteristic
590C3×C4#380411yescharacteristic
591C3×C4#381111yescharacteristic
592C3×C4#381111yescharacteristic
593Dic3—21no—
594Dic3—122no—
595Dic3—121no—
596Dic3—121no—
597Dic3#371412yescomplemented
598C32—21no—
599C32—21no—
600C32—21no—
601C32—41no—
602C32#478611yescharacteristic, complemented
603C32#484311yescharacteristic, complemented
604C32#507211yescharacteristic, complemented
605C23—61no—
606C23—181no—
607C23—541no—
608D4—61no—
609D4—61no—
610D4—181no—
611D4—181no—
612D4—272no—
613D4—362no—
614D4—362no—
615D4—541no—
616C2×C4—31no—
617C2×C4—31no—
618C2×C4—181no—
619C2×C4—1082no—
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—242no—
636C2×C3—241no—
637C2×C3—241no—
638C2×C3—721no—
639C2×C3D3×D2411yescharacteristic
640C2×C3#6379411yescharacteristic
641C2×C3C2×#371411yescharacteristic
642C2×C3#6383011yescharacteristic
643C2×C3#6391011yescharacteristic
644C2×C3#6391611yescharacteristic
645C2×C3D3×#21311yescharacteristic
646C2×C3#6393411yescharacteristic
647C2×C3#6395811yescharacteristic
648D3—81no—
649D3—242no—
650D3—241no—
651D3—241no—
652D3—364no—
653D3—362no—
654D3—362no—
655D3—362no—
656D3—722no—
657D3—722no—
658D3—724no—
659C22—121no—
660C22—121no—
661C22—121no—
662C22—361no—
663C22—361no—
664C22—361no—
665C22—542no—
666C22—1081no—
667C22—1081no—
668C22D3×#62611yescharacteristic
669C4—32no—
670C4—61no—
671C4—361no—
672C4#8895411yescharacteristic
673C4#8896211yescharacteristic
674C3—21no—
675C3—21no—
676C3—21no—
677C3—41no—
678C3#1059035511yescharacteristic, complemented
679C3#1059076511yescharacteristic, complemented
680C3#1059142911yescharacteristic, complemented
681C2—241no—
682C2—721no—
683C2—1082no—
684C2D3×#371411yescharacteristic
685C2#1172286811yescharacteristic
686C2#1172296311yescharacteristic
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. #11723484C2semidirect
2. C2×#88791C2semidirect
3. C2×#88790C2semidirect
4. C3×#63911C2semidirect
5. #11722075C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×#63563C2semidirect
29. #63911D3semidirect
43. Dic3⋊(C3⋊C8)D3semidirect
46. #63563D3semidirect
53. C4⋊(C3⋊(C3⋊D3))D4semidirect
136. C32×Dic3D8semidirect
379. C3×Dic3D24semidirect
380. C3×Dic3#213semidirect
388. C33#459semidirect
597. Dic3#3714semidirect
602. C32#4786semidirect
603. C32#4843semidirect
604. C32#5072semidirect
678. C3#10590355semidirect
679. C3#10590765semidirect
680. C3#10591429semidirect

Maximal subgroups

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

subgroupindex
#117234842
C2×#887912
C2×#887902
C3×#639112
#117220752
C3×Dic3⋊(C3⋊C8)2
C3×#635632
#105914293
#105907653
#105903553