Finite Group Databasebeta

← #13156771

Subgroups — #13156771

Order 1728.

1778 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6292C211yesmaximal, characteristic
2C22×C3×#624C211yesmaximal, characteristic, complemented
3C2×#88814C211yesmaximal, characteristic, complemented
4C2×#88804C212yesmaximal, complemented
5C3×#63579C212yesmaximal, complemented
6#10591382—32nomaximal
7#10591380—31nomaximal
8C23×C3⋊(C3⋊D3)C2211yescharacteristic
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×#6292—22no—
14C2×#6292—22no—
15C2×#6292C414yescomplemented
16C2×C3×#624—22no—
17C2×C3×#624—24no—
18#88814—22no—
19#88804—24no—
20C22×C4⋊(C3⋊D3)—31no—
21C22×C3×C4×D3—32no—
22C22×#626—32no—
23C22×#626—61no—
24C22×#624D311yescharacteristic, complemented
25C2×#3752—32no—
26C2×#3722—32no—
27C2×#3722—32no—
28C2×#3722—31no—
29C2×#3722—62no—
30C2×(C2×D3)⋊Dic3—32no—
31C3×C3⋊#95—32no—
32C3×C3⋊#95—32no—
33#63579D312yescomplemented
34C23×C33C2311yescharacteristic
35C22×C3⋊(C3⋊D3)—21no—
36C22×C3⋊(C3⋊D3)—21no—
37C22×C3⋊(C3⋊D3)—22no—
38C22×C3⋊(C3⋊D3)—22no—
39C22×C3⋊(C3⋊D3)C2×C412yescomplemented
40C22×C3×C3⋊D3—22no—
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3C2×C412yes—
43C22×C3×C3⋊D3D412yes—
44C22×C3×C3⋊D3D412yes—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3—22no—
47C2×C3×C3⋊Dic3—22no—
48C2×C32×Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—22no—
51C2×C32×Dic3—22no—
52C2×C32×Dic3C2×C412yescomplemented
53C2×C32×Dic3D414yescomplemented
54#6292—28no—
55C3×#624—28no—
56#5213—91no—
57#4665—92no—
58C23×C3⋊D3—32no—
59C23×C3⋊D3—31no—
60C23×C3⋊D3—61no—
61C23×C3⋊D3—62no—
62C23×C3⋊D3—121no—
63C23×C3⋊D3C2×D311yescharacteristic
64C23×C3×D3—32no—
65C23×C3×D3—61no—
66C22×C32×C4—32no—
67C22×C32×C4—31no—
68C22×C3⋊Dic3C2×D312yes—
69C2×C4⋊(C3⋊D3)—34no—
70C2×C4⋊(C3⋊D3)—62no—
71C2×C4⋊(C3⋊D3)—62no—
72C22×C3×Dic3—21no—
73C22×C3×Dic3—22no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—32no—
76C22×C3×Dic3—62no—
77C22×C3×Dic3C2×D312yes—
78C22×C3×Dic3C2×D312yes—
79C22×C3×Dic3C2×D312yes—
80C2×C3×C4×D3—64no—
81C2×C3×C4×D3—64no—
82C2×C3×C4×D3—64no—
83C2×#626—38no—
84C2×#626—64no—
85C2×#626—64no—
86C2×#626—64no—
87C2×#626—64no—
88C2×#626—64no—
89C2×#624—22no—
90C2×#624—24no—
91#3752—64no—
92#3722—68no—
93#3722—62no—
94#3722—64no—
95#3722—64no—
96(C2×D3)⋊Dic3—64no—
97C22×C33C22×C411yescharacteristic
98C22×C33C2×D411yescharacteristic
99C22×C33C2×D411yescharacteristic
100C22×C33C2×D412yes—
101C22×C33#4112yes—
102C2×C3⋊(C3⋊D3)—22no—
103C2×C3⋊(C3⋊D3)—24no—
104C2×C3⋊(C3⋊D3)—41no—
105C2×C3⋊(C3⋊D3)—42no—
106C2×C3⋊(C3⋊D3)—41no—
107C2×C3×C3⋊D3—24no—
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3—24no—
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3C22⋊C414yescomplemented
112C3×C3⋊Dic3—42no—
113C32×Dic3—22no—
114C32×Dic3—24no—
115C23×C3×C4—91no—
116C22×#67—91no—
117C22×D12—92no—
118C22×D12—181no—
119C22×C4×D3—32no—
120C2×#217—91no—
121C2×(C2×D3)⋊C4—92no—
122C2×(C2×D3)⋊C4—92no—
123C2×(C2×D3)⋊C4—92no—
124C2×(C2×D3)⋊C4—92no—
125C2×(C2×D3)⋊C4—182no—
126C3×#95—92no—
127C3⋊#95—32no—
128C3⋊#95—32no—
129C23×C32—21no—
130C23×C32—22no—
131C23×C32—41no—
132C23×C32C22×D311yescharacteristic
133C23×C32C22×D312yes—
134C22×C3⋊D3—22no—
135C22×C3⋊D3—22no—
136C22×C3⋊D3—34no—
137C22×C3⋊D3—32no—
138C22×C3⋊D3—64no—
139C22×C3⋊D3—62no—
140C22×C3⋊D3—61no—
141C22×C3⋊D3—62no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—61no—
144C22×C3⋊D3—64no—
145C22×C3⋊D3—64no—
146C22×C3⋊D3—62no—
147C22×C3⋊D3—62no—
148C22×C3⋊D3—62no—
149C22×C3⋊D3—62no—
150C22×C3⋊D3—62no—
151C22×C3⋊D3—62no—
152C22×C3⋊D3—62no—
153C22×C3⋊D3—62no—
154C22×C3⋊D3—122no—
155C22×C3⋊D3—122no—
156C22×C3⋊D3—122no—
157C22×C3⋊D3—122no—
158C22×C3⋊D3—122no—
159C22×C3⋊D3—122no—
160C22×C3⋊D3—122no—
161C22×C3⋊D3—124no—
162C22×C3⋊D3—122no—
163C22×C3⋊D3—122no—
164C22×C3⋊D3—124no—
165C22×C3⋊D3C4×D312yes—
166C22×C3⋊D3D1212yes—
167C22×C3⋊D3#6712yes—
168C22×C3×D3—34no—
169C22×C3×D3—34no—
170C22×C3×D3—34no—
171C22×C3×D3—62no—
172C22×C3×D3—62no—
173C22×C3×D3—62no—
174C22×C3×D3—62no—
175C22×C3×D3—62no—
176C22×C3×D3—62no—
177C22×C3×D3—62no—
178C22×C3×D3—122no—
179C22×C3×D3—122no—
180C2×C32×C4—32no—
181C2×C32×C4—34no—
182C2×C32×C4—61no—
183C2×C32×C4—61no—
184C2×C32×C4—62no—
185C2×C32×C4—62no—
186C2×C3⋊Dic3—22no—
187C2×C3⋊Dic3—22no—
188C2×C3⋊Dic3—22no—
189C4⋊(C3⋊D3)—68no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—22no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—22no—
198C2×C3×Dic3—22no—
199C2×C3×Dic3—24no—
200C2×C3×Dic3—42no—
201C2×C3×Dic3—42no—
202C2×C3×Dic3—64no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—122no—
210C2×C3×Dic3—122no—
211C2×C3×Dic3C4×D314yescomplemented
212C2×C3×Dic3D1214yescomplemented
213C2×C3×Dic3#6714yescomplemented
214C3×C4×D3—616no—
215#626—616no—
216#626—616no—
217#624—28no—
218#386—271noSylow
219C2×C33C2×C22⋊C411yescharacteristic
220C2×C33C4×D412yes—
221C2×C33#12011yescharacteristic
222C2×C33#12112yes—
223C2×C33#12311yescharacteristic
224C3⋊(C3⋊D3)—42no—
225C3×C3⋊D3—24no—
226C24×C3—91no—
227C23×D3—32no—
228C23×D3—61no—
229C23×D3—92no—
230C23×D3—91no—
231C23×D3—182no—
232C23×D3—181no—
233C23×D3—361no—
234C22×C3×C4—32no—
235C22×C3×C4—32no—
236C22×C3×C4—32no—
237C22×C3×C4—61no—
238C22×C3×C4—62no—
239C22×C3×C4—92no—
240C22×C3×C4—184no—
241C22×C3×C4—182no—
242C2×#67—94no—
243C2×#67—182no—
244C2×#67—182no—
245C22×Dic3—32no—
246C22×Dic3—32no—
247C22×Dic3—62no—
248C22×Dic3D3211yescharacteristic
249C22×Dic3D3212yes—
250C2×D12—98no—
251C2×D12—184no—
252C2×D12—184no—
253C2×D12—184no—
254C2×D12—184no—
255C2×D12—184no—
256C2×C4×D3—64no—
257C2×C4×D3—64no—
258C2×C4×D3—64no—
259#217—182no—
260(C2×D3)⋊C4—184no—
261(C2×D3)⋊C4—184no—
262(C2×D3)⋊C4—188no—
263(C2×D3)⋊C4—184no—
264(C2×D3)⋊C4—184no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—41no—
278C22×C32—42no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—42no—
282C22×C32C2×C4×D311yescharacteristic
283C22×C32C2×C4×D312yes—
284C22×C32C2×D1211yescharacteristic
285C22×C32C2×D1212yes—
286C22×C32#23512yes—
287C22×C32#23512yes—
288C22×C32D3×D412yes—
289C22×C32D3×D412yes—
290C22×C32D3×D412yes—
291C22×C32#23912yes—
292C22×C32C2×#6712yes—
293C22×C32C2×#6711yescharacteristic
294C2×C3⋊D3—24no—
295C2×C3⋊D3—22no—
296C2×C3⋊D3—24no—
297C2×C3⋊D3—22no—
298C2×C3⋊D3—64no—
299C2×C3⋊D3—62no—
300C2×C3⋊D3—64no—
301C2×C3⋊D3—68no—
302C2×C3⋊D3—64no—
303C2×C3⋊D3—64no—
304C2×C3⋊D3—64no—
305C2×C3⋊D3—124no—
306C2×C3⋊D3—124no—
307C2×C3⋊D3—124no—
308C2×C3⋊D3—124no—
309C2×C3⋊D3—128no—
310C2×C3⋊D3—124no—
311C2×C3⋊D3—122no—
312C2×C3⋊D3—124no—
313C2×C3⋊D3—124no—
314C2×C3⋊D3—124no—
315C2×C3⋊D3—122no—
316C2×C3⋊D3—124no—
317C2×C3⋊D3—124no—
318C2×C3⋊D3—128no—
319C2×C3⋊D3—124no—
320C2×C3⋊D3—122no—
321C2×C3⋊D3—122no—
322C2×C3⋊D3—122no—
323C2×C3⋊D3—122no—
324C2×C3⋊D3—121no—
325C2×C3⋊D3—122no—
326C2×C3⋊D3—121no—
327C2×C3⋊D3(C2×D3)⋊C414yescomplemented
328C2×C3×D3—38no—
329C2×C3×D3—64no—
330C2×C3×D3—64no—
331C2×C3×D3—64no—
332C2×C3×D3—64no—
333C2×C3×D3—64no—
334C2×C3×D3—64no—
335C2×C3×D3—64no—
336C2×C3×D3—124no—
337C2×C3×D3—122no—
338C2×C3×D3—124no—
339C2×C3×D3—122no—
340C32×C4—62no—
341C32×C4—64no—
342C3⋊Dic3—42no—
343C3×Dic3—24no—
344C3×Dic3—24no—
345C3×Dic3—24no—
346C3×Dic3—24no—
347C3×Dic3—44no—
348C3×Dic3—124no—
349C3×Dic3—122no—
350C3×Dic3—122no—
351C22×D4—271no—
352C23×C4—91no—
353C2×C22⋊C4—272no—
354C2×C22⋊C4—271no—
355#95—92no—
356C33#38611yescharacteristic, complemented, Sylow
357C23×C3—22no—
358C23×C3—21no—
359C23×C3—41no—
360C23×C3—92no—
361C23×C3—92no—
362C23×C3—92no—
363C23×C3—182no—
364C23×C3—182no—
365C23×C3C2×D3212yes—
366C23×C3C2×D3211yescharacteristic
367C22×D3—34no—
368C22×D3—34no—
369C22×D3—34no—
370C22×D3—62no—
371C22×D3—62no—
372C22×D3—62no—
373C22×D3—62no—
374C22×D3—62no—
375C22×D3—62no—
376C22×D3—62no—
377C22×D3—94no—
378C22×D3—92no—
379C22×D3—122no—
380C22×D3—122no—
381C22×D3—182no—
382C22×D3—184no—
383C22×D3—184no—
384C22×D3—182no—
385C22×D3—182no—
386C22×D3—184no—
387C22×D3—182no—
388C22×D3—182no—
389C22×D3—182no—
390C22×D3—182no—
391C22×D3—182no—
392C22×D3—182no—
393C22×D3—182no—
394C22×D3—182no—
395C22×D3—181no—
396C22×D3—181no—
397C22×D3—362no—
398C22×D3—362no—
399C22×D3—362no—
400C22×D3—362no—
401C22×D3—362no—
402C22×D3—362no—
403C22×D3—362no—
404C22×D3—364no—
405C22×D3—362no—
406C22×D3—362no—
407C22×D3—364no—
408C2×C3×C4—34no—
409C2×C3×C4—34no—
410C2×C3×C4—34no—
411C2×C3×C4—62no—
412C2×C3×C4—64no—
413C2×C3×C4—62no—
414C2×C3×C4—62no—
415C2×C3×C4—62no—
416C2×C3×C4—62no—
417C2×C3×C4—62no—
418C2×C3×C4—62no—
419C2×C3×C4—62no—
420C2×C3×C4—62no—
421C2×C3×C4—122no—
422C2×C3×C4—122no—
423C2×C3×C4—188no—
424C2×C3×C4—182no—
425C2×C3×C4—182no—
426C2×C3×C4—182no—
427#67—188no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—64no—
433C2×Dic3—62no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3—62no—
437C2×Dic3—62no—
438C2×Dic3—62no—
439C2×Dic3—122no—
440C2×Dic3—122no—
441C2×Dic3#62412yescomplemented
442C2×Dic3#62614yescomplemented
443D12—1816no—
444D12—1816no—
445C4×D3—616no—
446C2×C32—22no—
447C2×C32—22no—
448C2×C32—21no—
449C2×C32—22no—
450C2×C32—22no—
451C2×C32—22no—
452C2×C32—22no—
453C2×C32—22no—
454C2×C32—22no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—22no—
458C2×C32—41no—
459C2×C32—42no—
460C2×C32—41no—
461C2×C32—42no—
462C2×C32—41no—
463C2×C32C4×D1212yes—
464C2×C32D3×C22⋊C412yes—
465C2×C32#88212yes—
466C2×C32#88412yes—
467C2×C32#89312yes—
468C2×C32#89412yes—
469C2×C32#89512yes—
470C2×C32C2×(C2×D3)⋊C411yescharacteristic
471C2×C32C4×#6712yes—
472C2×C32#92911yescharacteristic
473C2×C32#93012yes—
474C2×C32#95311yescharacteristic
475C3⋊D3—24no—
476C3⋊D3—124no—
477C3⋊D3—128no—
478C3⋊D3—128no—
479C3⋊D3—124no—
480C3⋊D3—122no—
481C3×D3—68no—
482C3×D3—124no—
483C24—91no—
484C24—271no—
485C2×D4—274no—
486C2×D4—542no—
487C2×D4—542no—
488C22×C4—31no—
489C22×C4—32no—
490C22×C4—92no—
491C22×C4—184no—
492C22×C4—182no—
493C22⋊C4—544no—
494C22⋊C4—542no—
495C22×C3—21no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—21no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—21no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—41no—
508C22×C3—42no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—42no—
512C22×C3—94no—
513C22×C3—184no—
514C22×C3—182no—
515C22×C3—184no—
516C22×C3—182no—
517C22×C3#379812yes—
518C22×C3#379912yes—
519C22×C3#379912yes—
520C22×C3C4×D3212yes—
521C22×C3D3×D1212yes—
522C22×C3D3×D1212yes—
523C22×C3C2×#62411yescharacteristic
524C22×C3C2×#62612yes—
525C22×C3D3×#6712yes—
526C22×C3D3×#6712yes—
527C22×C3#381111yescharacteristic
528C22×C3#381111yescharacteristic
529C2×D3—38no—
530C2×D3—64no—
531C2×D3—64no—
532C2×D3—64no—
533C2×D3—64no—
534C2×D3—64no—
535C2×D3—64no—
536C2×D3—64no—
537C2×D3—124no—
538C2×D3—122no—
539C2×D3—124no—
540C2×D3—122no—
541C2×D3—184no—
542C2×D3—188no—
543C2×D3—184no—
544C2×D3—184no—
545C2×D3—184no—
546C2×D3—184no—
547C2×D3—182no—
548C2×D3—364no—
549C2×D3—364no—
550C2×D3—364no—
551C2×D3—364no—
552C2×D3—368no—
553C2×D3—364no—
554C2×D3—362no—
555C2×D3—364no—
556C2×D3—364no—
557C2×D3—364no—
558C2×D3—362no—
559C2×D3—364no—
560C2×D3—364no—
561C2×D3—368no—
562C2×D3—364no—
563C2×D3—362no—
564C2×D3—361no—
565C2×D3—362no—
566C2×D3—362no—
567C2×D3—362no—
568C2×D3—361no—
569C2×D3—362no—
570C3×C4—64no—
571C3×C4—64no—
572C3×C4—64no—
573C3×C4—64no—
574C3×C4—124no—
575C3×C4—362no—
576Dic3—22no—
577Dic3—24no—
578Dic3—124no—
579Dic3—122no—
580Dic3—122no—
581C32—21no—
582C32—22no—
583C32—41no—
584C32#466512yescomplemented
585C32#521311yescharacteristic, complemented
586C23—92no—
587C23—92no—
588C23—92no—
589C23—182no—
590C23—182no—
591C23—272no—
592C23—542no—
593C23—541no—
594C23—541no—
595C23—542no—
596C23D3311yescharacteristic
597D4—548no—
598C2×C4—32no—
599C2×C4—34no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—188no—
605C2×C4—182no—
606C2×C4—182no—
607C2×C4—182no—
608C2×C3—21no—
609C2×C3—22no—
610C2×C3—22no—
611C2×C3—22no—
612C2×C3—22no—
613C2×C3—22no—
614C2×C3—21no—
615C2×C3—22no—
616C2×C3—22no—
617C2×C3—22no—
618C2×C3—21no—
619C2×C3—22no—
620C2×C3—41no—
621C2×C3—42no—
622C2×C3—41no—
623C2×C3—42no—
624C2×C3—41no—
625C2×C3—184no—
626C2×C3#6385012yes—
627C2×C3#6389412yes—
628C2×C3#6389711yescharacteristic
629C2×C3C4×#62612yes—
630C2×C3#6390412yes—
631C2×C3#6390612yes—
632C2×C3#6390912yes—
633C2×C3#6391012yes—
634C2×C3D3×(C2×D3)⋊C412yes—
635C2×C3#6392311yescharacteristic
636C2×C3#6397411yescharacteristic
637C2×C3#6397712yes—
638D3—68no—
639D3—124no—
640D3—364no—
641D3—368no—
642D3—368no—
643D3—362no—
644D3—364no—
645C22—94no—
646C22—184no—
647C22—182no—
648C22—184no—
649C22—182no—
650C22—542no—
651C22—544no—
652C22—1081no—
653C22—1082no—
654C22—1081no—
655C22D3×#62411yescharacteristic
656C22D3×#62612yes—
657C22#8895211yescharacteristic
658C22#8895511yescharacteristic
659C22#8896512yes—
660C4—62no—
661C4—64no—
662C4—362no—
663C3—21no—
664C3—22no—
665C3—41no—
666C3#1059138011yescharacteristic, complemented
667C3#1059138212yescomplemented
668C2—184no—
669C2—1082no—
670C2#1172290411yescharacteristic
671C2#1172293312yes—
672C2#1172294811yescharacteristic
673C2#1172296112yes—
674C2#1172299311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC22×#6292C2
hypocenterC33#386
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×#624C2semidirect
3. C2×#88814C2semidirect
4. C2×#88804C2semidirect
5. C3×#63579C2semidirect
15. C2×#6292C4semidirect
24. C22×#624D3semidirect
33. #63579D3semidirect
39. C22×C3⋊(C3⋊D3)C2×C4semidirect
52. C2×C32×Dic3C2×C4semidirect
53. C2×C32×Dic3D4semidirect
111. C2×C3×C3⋊D3C22⋊C4semidirect
211. C2×C3×Dic3C4×D3semidirect
212. C2×C3×Dic3D12semidirect
213. C2×C3×Dic3#67semidirect
327. C2×C3⋊D3(C2×D3)⋊C4semidirect
356. C33#386semidirect
441. C2×Dic3#624semidirect
442. C2×Dic3#626semidirect
584. C32#4665semidirect
585. C32#5213semidirect
666. C3#10591380semidirect
667. C3#10591382semidirect

Maximal subgroups

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

subgroupindex
C22×#62922
C22×C3×#6242
C2×#888142
C2×#888042
C3×#635792
#105913823
#105913803