Finite Group Databasebeta

← #13156856

Subgroups — #13156856

Order 1728.

1666 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6291C211yesmaximal, characteristic
2C2×#88815C211yesmaximal, characteristic, complemented
3C2×#88804C212yesmaximal, complemented
4C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
5C2×C3×Dic32C211yesmaximal, characteristic, complemented
6#10591704—31nomaximal
7#10591383—32nomaximal
8C23×C3⋊(C3⋊D3)C2211yescharacteristic
9C23×C32×D3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×#6291—22no—
14C2×#6291—22no—
15C2×#6291C414yescomplemented
16#88815—22no—
17#88804—24no—
18C3×(C2×D3)⋊Dic3—24no—
19C3×Dic32—22no—
20C22×#638—31no—
21C22×#626—32no—
22C22×#626—61no—
23C2×#3752—32no—
24C2×C3×#217—32no—
25C2×C3×(C2×D3)⋊C4—32no—
26C2×C3×C4×Dic3—32no—
27C2×#3724—31no—
28C2×Dic3⋊Dic3—32no—
29C2×#3722—32no—
30C2×#3722—32no—
31C2×#3722—62no—
32C2×(C2×D3)⋊Dic3D312yescomplemented
33C2×Dic32D311yescharacteristic, complemented
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×C412yes—
40C22×C32×D3—21no—
41C22×C32×D3—21no—
42C22×C32×D3—22no—
43C22×C32×D3—22no—
44C22×C32×D3C2×C412yes—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3—22no—
47C2×C3×C3⋊Dic3—22no—
48C2×C3×C3⋊Dic3C2×C412yes—
49C2×C3×C3⋊Dic3D412yes—
50C2×C3×C3⋊Dic3D412yes—
51C2×C32×Dic3—22no—
52C2×C32×Dic3—22no—
53C2×C32×Dic3D414yescomplemented
54#6291—28no—
55#4957—92no—
56#4939—91no—
57C24×C32—31no—
58C23×C3⋊D3—32no—
59C23×C3⋊D3—31no—
60C23×C3⋊D3—61no—
61C23×C3⋊D3—62no—
62C23×C3⋊D3—121no—
63C23×C3×D3—21no—
64C23×C3×D3C2×D312yes—
65C22×C32×C4—32no—
66C2×#638—34no—
67C2×#638—62no—
68C2×#638—62no—
69C22×C3⋊Dic3C2×D312yes—
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3×Dic3—22no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—61no—
76C22×C3×Dic3—62no—
77C22×C3×Dic3C2×D312yes—
78C22×C3×Dic3C2×D312yes—
79C2×#626—38no—
80C2×#626—64no—
81C2×#626—64no—
82C2×#626—64no—
83C2×#626—64no—
84C2×#626—64no—
85#3752—64no—
86C3×#217—64no—
87C3×(C2×D3)⋊C4—64no—
88C3×C4×Dic3—64no—
89#3724—62no—
90Dic3⋊Dic3—64no—
91#3722—68no—
92#3722—64no—
93#3722—64no—
94(C2×D3)⋊Dic3—24no—
95Dic32—22no—
96C22×C33C22×C411yescharacteristic
97C22×C33C2×D411yescharacteristic
98C22×C33C2×D411yescharacteristic
99C22×C33C2×D412yes—
100C22×C33#4112yes—
101C2×C3⋊(C3⋊D3)—22no—
102C2×C3⋊(C3⋊D3)—24no—
103C2×C3⋊(C3⋊D3)—42no—
104C2×C3⋊(C3⋊D3)—41no—
105C2×C3⋊(C3⋊D3)—41no—
106C2×C32×D3—22no—
107C2×C32×D3—24no—
108C2×C32×D3—42no—
109C2×C32×D3—41no—
110C2×C32×D3—41no—
111C3×C3⋊Dic3—42no—
112C3×C3⋊Dic3C22⋊C414yescomplemented
113C32×Dic3—24no—
114C22×#67—92no—
115C22×#67—181no—
116C22×D12—91no—
117C2×C3×C22⋊C4—92no—
118C2×C3×C42—91no—
119C2×#217—32no—
120C2×(C2×D3)⋊C4—32no—
121C2×(C2×D3)⋊C4—92no—
122C2×(C2×D3)⋊C4—92no—
123C2×(C2×D3)⋊C4—92no—
124C2×(C2×D3)⋊C4—182no—
125C2×C4⋊Dic3—91no—
126C2×Dic3⋊C4—92no—
127C2×C4×Dic3—32no—
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—32no—
131C23×C32—41no—
132C23×C32—61no—
133C23×C32—61no—
134C23×C32—62no—
135C23×C32—62no—
136C23×C32C22×D311yescharacteristic
137C23×C32C22×D312yes—
138C22×C3⋊D3—34no—
139C22×C3⋊D3—32no—
140C22×C3⋊D3—64no—
141C22×C3⋊D3—62no—
142C22×C3⋊D3—61no—
143C22×C3⋊D3—62no—
144C22×C3⋊D3—62no—
145C22×C3⋊D3—61no—
146C22×C3⋊D3—64no—
147C22×C3⋊D3—64no—
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—62no—
155C22×C3⋊D3—62no—
156C22×C3⋊D3—122no—
157C22×C3⋊D3—122no—
158C22×C3⋊D3—122no—
159C22×C3⋊D3—122no—
160C22×C3⋊D3—122no—
161C22×C3⋊D3—122no—
162C22×C3⋊D3—122no—
163C22×C3⋊D3—124no—
164C22×C3⋊D3—122no—
165C22×C3⋊D3—122no—
166C22×C3⋊D3—124no—
167C22×C3×D3—22no—
168C22×C3×D3—22no—
169C22×C3×D3—22no—
170C22×C3×D3—22no—
171C22×C3×D3—22no—
172C22×C3×D3—22no—
173C22×C3×D3—22no—
174C22×C3×D3—22no—
175C22×C3×D3—22no—
176C22×C3×D3—42no—
177C22×C3×D3—42no—
178C22×C3×D3C4×D314yes—
179C2×C32×C4—34no—
180C2×C32×C4—62no—
181C2×C32×C4—62no—
182#638—68no—
183C2×C3⋊Dic3—22no—
184C2×C3⋊Dic3—22no—
185C2×C3⋊Dic3—22no—
186C2×C3⋊Dic3C4×D312yes—
187C2×C3⋊Dic3D1212yes—
188C2×C3⋊Dic3#6712yes—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3—34no—
197C2×C3×Dic3—42no—
198C2×C3×Dic3—42no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—62no—
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×Dic3D1214yescomplemented
212C2×C3×Dic3#6714yescomplemented
213#626—616no—
214#626—616no—
215#390—271noSylow
216C2×C33C2×C22⋊C411yescharacteristic
217C2×C33C4×D412yes—
218C2×C33#12112yes—
219C2×C33#12411yescharacteristic
220C2×C33#12711yescharacteristic
221C3⋊(C3⋊D3)—42no—
222C32×D3—42no—
223C24×C3—32no—
224C24×C3—61no—
225C23×D3—91no—
226C23×D3—92no—
227C23×D3—182no—
228C23×D3—181no—
229C23×D3—361no—
230C23×D3D3211yescharacteristic
231C22×C3×C4—32no—
232C22×C3×C4—32no—
233C22×C3×C4—62no—
234C22×C3×C4—91no—
235C22×C3×C4—92no—
236C2×#67—98no—
237C2×#67—184no—
238C2×#67—184no—
239C2×#67—184no—
240C2×#67—184no—
241C2×#67—184no—
242C22×Dic3—32no—
243C22×Dic3—32no—
244C22×Dic3—32no—
245C22×Dic3—61no—
246C22×Dic3—62no—
247C22×Dic3D3212yes—
248C2×D12—94no—
249C2×D12—182no—
250C2×D12—182no—
251C3×C22⋊C4—184no—
252C3×C42—182no—
253#217—64no—
254(C2×D3)⋊C4—64no—
255(C2×D3)⋊C4—184no—
256(C2×D3)⋊C4—184no—
257(C2×D3)⋊C4—188no—
258(C2×D3)⋊C4—184no—
259C4⋊Dic3—182no—
260Dic3⋊C4—184no—
261C4×Dic3—64no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—21no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—41no—
275C22×C32—42no—
276C22×C32—41no—
277C22×C32—41no—
278C22×C32—42no—
279C22×C32—62no—
280C22×C32—64no—
281C22×C32—121no—
282C22×C32—121no—
283C22×C32—122no—
284C22×C32C2×C4×D311yescharacteristic
285C22×C32C2×C4×D312yes—
286C22×C32C2×D1211yescharacteristic
287C22×C32C2×D1212yes—
288C22×C32#23512yes—
289C22×C32#23512yes—
290C22×C32D3×D412yes—
291C22×C32D3×D412yes—
292C22×C32D3×D412yes—
293C22×C32#23712yes—
294C22×C32C2×#6712yes—
295C22×C32C2×#6711yescharacteristic
296C2×C3⋊D3—64no—
297C2×C3⋊D3—62no—
298C2×C3⋊D3—64no—
299C2×C3⋊D3—68no—
300C2×C3⋊D3—64no—
301C2×C3⋊D3—64no—
302C2×C3⋊D3—64no—
303C2×C3⋊D3—124no—
304C2×C3⋊D3—124no—
305C2×C3⋊D3—124no—
306C2×C3⋊D3—124no—
307C2×C3⋊D3—128no—
308C2×C3⋊D3—124no—
309C2×C3⋊D3—124no—
310C2×C3⋊D3—124no—
311C2×C3⋊D3—122no—
312C2×C3⋊D3—124no—
313C2×C3⋊D3—124no—
314C2×C3⋊D3—122no—
315C2×C3⋊D3—128no—
316C2×C3⋊D3—124no—
317C2×C3⋊D3—124no—
318C2×C3⋊D3—121no—
319C2×C3⋊D3—121no—
320C2×C3⋊D3—122no—
321C2×C3⋊D3—122no—
322C2×C3⋊D3—122no—
323C2×C3⋊D3—122no—
324C2×C3⋊D3—122no—
325C2×C3×D3—24no—
326C2×C3×D3—24no—
327C2×C3×D3—24no—
328C2×C3×D3—24no—
329C2×C3×D3—44no—
330C2×C3×D3—42no—
331C2×C3×D3—44no—
332C2×C3×D3—42no—
333C2×C3×D3—42no—
334C2×C3×D3—42no—
335C2×C3×D3—42no—
336C2×C3×D3—42no—
337C32×C4—64no—
338C3⋊Dic3—42no—
339C3⋊Dic3(C2×D3)⋊C414yescomplemented
340C3×Dic3—24no—
341C3×Dic3—24no—
342C3×Dic3—38no—
343C3×Dic3—44no—
344C3×Dic3—64no—
345C3×Dic3—124no—
346C3×Dic3—122no—
347C3×Dic3—122no—
348C22×D4—271no—
349C2×C4⋊C4—271no—
350C2×C22⋊C4—92no—
351C2×C22⋊C4—272no—
352C2×C42—91no—
353C33#39011yescharacteristic, complemented, Sylow
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—34no—
357C23×C3—41no—
358C23×C3—62no—
359C23×C3—62no—
360C23×C3—62no—
361C23×C3—62no—
362C23×C3—62no—
363C23×C3—62no—
364C23×C3—62no—
365C23×C3—62no—
366C23×C3—62no—
367C23×C3—122no—
368C23×C3—122no—
369C23×C3C2×D3212yes—
370C23×C3C2×D3211yescharacteristic
371C22×D3—22no—
372C22×D3—22no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—94no—
376C22×D3—92no—
377C22×D3—182no—
378C22×D3—184no—
379C22×D3—184no—
380C22×D3—182no—
381C22×D3—182no—
382C22×D3—184no—
383C22×D3—182no—
384C22×D3—182no—
385C22×D3—182no—
386C22×D3—182no—
387C22×D3—182no—
388C22×D3—182no—
389C22×D3—182no—
390C22×D3—182no—
391C22×D3—181no—
392C22×D3—181no—
393C22×D3—362no—
394C22×D3—362no—
395C22×D3—362no—
396C22×D3—362no—
397C22×D3—362no—
398C22×D3—362no—
399C22×D3—362no—
400C22×D3—364no—
401C22×D3—362no—
402C22×D3—362no—
403C22×D3—364no—
404C22×D3#62412yes—
405C2×C3×C4—34no—
406C2×C3×C4—34no—
407C2×C3×C4—64no—
408C2×C3×C4—62no—
409C2×C3×C4—62no—
410C2×C3×C4—62no—
411C2×C3×C4—62no—
412C2×C3×C4—92no—
413C2×C3×C4—92no—
414C2×C3×C4—92no—
415C2×C3×C4—122no—
416C2×C3×C4—122no—
417C2×C3×C4—182no—
418C2×C3×C4—182no—
419C2×C3×C4—182no—
420#67—1816no—
421#67—1816no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—34no—
425C2×Dic3—34no—
426C2×Dic3—34no—
427C2×Dic3—62no—
428C2×Dic3—62no—
429C2×Dic3—62no—
430C2×Dic3—64no—
431C2×Dic3—62no—
432C2×Dic3—62no—
433C2×Dic3—62no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3—62no—
437C2×Dic3—122no—
438C2×Dic3—122no—
439C2×Dic3#62614yescomplemented
440D12—188no—
441C2×C32—22no—
442C2×C32—22no—
443C2×C32—22no—
444C2×C32—22no—
445C2×C32—22no—
446C2×C32—22no—
447C2×C32—22no—
448C2×C32—21no—
449C2×C32—22no—
450C2×C32—21no—
451C2×C32—22no—
452C2×C32—21no—
453C2×C32—41no—
454C2×C32—42no—
455C2×C32—42no—
456C2×C32—41no—
457C2×C32—41no—
458C2×C32—122no—
459C2×C32C4×D1212yes—
460C2×C32#87411yescharacteristic
461C2×C32#87511yescharacteristic
462C2×C32D3×C22⋊C412yes—
463C2×C32#67⋊C412yes—
464C2×C32#88412yes—
465C2×C32#88512yes—
466C2×C32C2×(C2×D3)⋊C411yescharacteristic
467C2×C32C4×#6712yes—
468C2×C32#93012yes—
469C2×C32#93912yes—
470C2×C32#94012yes—
471C3⋊D3—128no—
472C3⋊D3—124no—
473C3⋊D3—128no—
474C3⋊D3—122no—
475C3⋊D3—124no—
476C3×D3—44no—
477C3×D3—44no—
478C24—31no—
479C24—271no—
480C2×D4—274no—
481C2×D4—542no—
482C2×D4—542no—
483C22×C4—32no—
484C22×C4—91no—
485C22×C4—92no—
486C4⋊C4—542no—
487C22⋊C4—184no—
488C22⋊C4—544no—
489C42—182no—
490C22×C3—21no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—22no—
497C22×C3—21no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—41no—
503C22×C3—42no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—42no—
507C22×C3—64no—
508C22×C3—64no—
509C22×C3—64no—
510C22×C3—64no—
511C22×C3—124no—
512C22×C3—122no—
513C22×C3—122no—
514C22×C3—124no—
515C22×C3—122no—
516C22×C3—122no—
517C22×C3—122no—
518C22×C3—122no—
519C22×C3#379812yes—
520C22×C3C4×D3212yes—
521C22×C3D3×D1212yes—
522C22×C3D3×D1212yes—
523C22×C3#380412yes—
524C22×C3#380412yes—
525C22×C3C2×#62411yescharacteristic
526C22×C3C2×#62612yes—
527C22×C3D3×#6712yes—
528C22×C3D3×#6712yes—
529C22×C3#381111yescharacteristic
530C22×C3#381111yescharacteristic
531C2×D3—24no—
532C2×D3—22no—
533C2×D3—41no—
534C2×D3—42no—
535C2×D3—41no—
536C2×D3—184no—
537C2×D3—188no—
538C2×D3—184no—
539C2×D3—184no—
540C2×D3—184no—
541C2×D3—184no—
542C2×D3—182no—
543C2×D3—364no—
544C2×D3—364no—
545C2×D3—364no—
546C2×D3—364no—
547C2×D3—368no—
548C2×D3—362no—
549C2×D3—364no—
550C2×D3—364no—
551C2×D3—362no—
552C2×D3—364no—
553C2×D3—364no—
554C2×D3—364no—
555C2×D3—364no—
556C2×D3—364no—
557C2×D3—368no—
558C2×D3—362no—
559C2×D3—362no—
560C2×D3—362no—
561C2×D3—361no—
562C2×D3—362no—
563C2×D3—362no—
564C2×D3—361no—
565C3×C4—64no—
566C3×C4—64no—
567C3×C4—94no—
568C3×C4—124no—
569C3×C4—362no—
570Dic3—24no—
571Dic3—38no—
572Dic3—64no—
573Dic3—124no—
574Dic3—122no—
575Dic3—122no—
576C32—22no—
577C32—21no—
578C32—41no—
579C32#493911yescharacteristic, complemented
580C32#495712yescomplemented
581C23—32no—
582C23—62no—
583C23—61no—
584C23—61no—
585C23—62no—
586C23—272no—
587C23—542no—
588C23—541no—
589C23—541no—
590C23—542no—
591C23D3311yescharacteristic
592D4—548no—
593C2×C4—34no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—92no—
597C2×C4—92no—
598C2×C4—92no—
599C2×C4—182no—
600C2×C4—182no—
601C2×C4—182no—
602C2×C3—21no—
603C2×C3—22no—
604C2×C3—22no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—22no—
608C2×C3—22no—
609C2×C3—22no—
610C2×C3—22no—
611C2×C3—22no—
612C2×C3—22no—
613C2×C3—22no—
614C2×C3—41no—
615C2×C3—41no—
616C2×C3—41no—
617C2×C3—42no—
618C2×C3—42no—
619C2×C3—124no—
620C2×C3—124no—
621C2×C3#6385212yes—
622C2×C3#6388012yes—
623C2×C3C4×#62612yes—
624C2×C3#6390612yes—
625C2×C3#6391112yes—
626C2×C3D3×(C2×D3)⋊C412yes—
627C2×C3#6395212yes—
628C2×C3#6395311yescharacteristic
629C2×C3#6397112yes—
630C2×C3#6397411yescharacteristic
631C2×C3#6397812yes—
632C2×C3#6397911yescharacteristic
633D3—42no—
634D3—364no—
635D3—368no—
636D3—368no—
637D3—364no—
638D3—362no—
639C22—62no—
640C22—64no—
641C22—121no—
642C22—121no—
643C22—122no—
644C22—542no—
645C22—544no—
646C22—1081no—
647C22—1081no—
648C22—1082no—
649C22D3×#62411yescharacteristic
650C22D3×#62612yes—
651C22#8895211yescharacteristic
652C22#8895511yescharacteristic
653C22#8895912yes—
654C4—64no—
655C4—94no—
656C4—362no—
657C3—21no—
658C3—22no—
659C3—41no—
660C3#1059138312yescomplemented
661C3#1059170411yescharacteristic, complemented
662C2—122no—
663C2—1082no—
664C2#1172290411yescharacteristic
665C2#1172293212yes—
666C2#1172295812yes—
667C2#1172300811yescharacteristic
668C2#1172303011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC22×#6291C2
hypocenterC33#390
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88815C2semidirect
3. C2×#88804C2semidirect
4. C2×C3×(C2×D3)⋊Dic3C2semidirect
5. C2×C3×Dic32C2semidirect
15. C2×#6291C4semidirect
32. C2×(C2×D3)⋊Dic3D3semidirect
33. C2×Dic32D3semidirect
53. C2×C32×Dic3D4semidirect
112. C3×C3⋊Dic3C22⋊C4semidirect
211. C2×C3×Dic3D12semidirect
212. C2×C3×Dic3#67semidirect
339. C3⋊Dic3(C2×D3)⋊C4semidirect
353. C33#390semidirect
439. C2×Dic3#626semidirect
579. C32#4939semidirect
580. C32#4957semidirect
660. C3#10591383semidirect
661. C3#10591704semidirect

Maximal subgroups

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

subgroupindex
C22×#62912
C2×#888152
C2×#888042
C2×C3×(C2×D3)⋊Dic32
C2×C3×Dic322
#105917043
#105913833