Finite Group Databasebeta

← #13176551

Subgroups — #13176551

Order 1728.

2284 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89017C211yesmaximal, characteristic, complemented
3#11723933C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic, complemented
5C32×#897C211yesmaximal, characteristic, complemented
6#11723541C212yesmaximal, complemented
7#11723515C211yesmaximal, characteristic, complemented
8#11723508C212yesmaximal, complemented
9#11723504C212yesmaximal, complemented
10#11723501C212yesmaximal, complemented
11#11723435C211yesmaximal, characteristic, complemented
12#10596098—31nomaximal
13#10594632—34nomaximal
14C22×D3×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2212yes—
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C32×C4×D3C2211yescharacteristic
20C2×C32×Dic6C2211yescharacteristic, complemented
21C2×#6292C2212yes—
22C2×#6290C2212yes—
23C2×#6289C2211yescharacteristic
24C2×Dic3×C3⋊D3C2211yescharacteristic
25C2×Dic3×C3⋊D3C2212yescomplemented
26C2×D3×C3⋊Dic3C2211yescharacteristic
27C4×D3×C3⋊D3—24no—
28#89017—24no—
29C33×C4⋊C4C2211yescharacteristic, complemented
30#88859C2212yes—
31C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
32C4×C3⋊(C3⋊Dic3)C2212yescomplemented
33C3×#3752C2212yescomplemented
34C32×(C2×D3)⋊C4C2212yes—
35C32×C4⋊Dic3C2212yescomplemented
36#88806C2212yescomplemented
37#88803C2212yes—
38#88803C2212yescomplemented
39C2×#3832—31no—
40C22×C4×C3⋊D3—31no—
41C2×C4×D32—34no—
42C2×#3796—34no—
43C32×C22⋊Q8—31no—
44#64155—31no—
45#64139—31no—
46#64104D311yescharacteristic, complemented
47#64103—32no—
48#64102—32no—
49#64101—31no—
50#64101—34no—
51#64101—64no—
52#64091—32no—
53#64090—32no—
54C3×#897D314yescomplemented
55C3×#895—34no—
56#63922—38no—
57#63908—34no—
58#63902—38no—
59#63898—38no—
60#63856—34no—
61#63854—38no—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C22×C3×C3⋊D3C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3C2311yescharacteristic
66C2×D3×C3⋊D3—22no—
67C2×D3×C3⋊D3—22no—
68C2×D3×C3⋊D3—22no—
69C2×C33×C4C2312yes—
70C2×C33×C4C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2312yes—
72C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
73C4×C3⋊(C3⋊D3)D414yescomplemented
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C3×C4⋊(C3⋊D3)—24no—
76C3×C4⋊(C3⋊D3)—22no—
77C3×C4×C3⋊D3—22no—
78C2×C32×Dic3C2311yescharacteristic
79C2×C32×Dic3C2312yes—
80C32×C4×D3—22no—
81C32×Dic6—22no—
82#6292—24no—
83#6290—24no—
84#6289—22no—
85Dic3×C3⋊D3—24no—
86Dic3×C3⋊D3—22no—
87D3×C3⋊Dic3—22no—
88#5627—94no—
89#5606—91no—
90C23×C3⋊D3—31no—
91C22×D32—34no—
92C2×C32×Q8—31no—
93C22×C32×C4—31no—
94C2×#638—32no—
95C22×C3⋊Dic3—31no—
96#3832—64no—
97C2×C4⋊(C3⋊D3)—32no—
98C2×C4⋊(C3⋊D3)C2×D312yes—
99C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
100C2×C4×C3⋊D3—31no—
101C2×C4×C3⋊D3—34no—
102C2×C4×C3⋊D3—32no—
103C2×C4×C3⋊D3—31no—
104C2×C4×C3⋊D3—31no—
105C2×C4×C3⋊D3—64no—
106C2×C4×C3⋊D3—64no—
107C2×C4×C3⋊D3C2×D311yescharacteristic
108C2×C3×D12—34no—
109C2×C3×D12—38no—
110C2×C3×C4×D3—34no—
111C2×C3×C4×D3C2×D314yes—
112C2×C3×Dic6C2×D314yescomplemented
113C2×#626—38no—
114C2×#625—38no—
115C2×#624—34no—
116C2×D3×Dic3—34no—
117C2×D3×Dic3—38no—
118C2×D3×Dic3—34no—
119C4×D32—616no—
120#3796—616no—
121C32×C4⋊C4—24no—
122C32×C4⋊C4—32no—
123C32×C4⋊C4C2×D314yescomplemented
124C32×C4⋊C4C2×D311yescharacteristic, complemented
125C32×C22⋊C4—32no—
126#3752—32no—
127#3752—32no—
128#3752—32no—
129#3752—38no—
130#3752—68no—
131#3752C2×D312yescomplemented
132C4⋊(C3⋊Dic3)—31no—
133C4⋊(C3⋊Dic3)—34no—
134C4⋊(C3⋊Dic3)—64no—
135#3750—32no—
136C4×C3⋊Dic3—32no—
137C4×C3⋊Dic3—38no—
138C4×C3⋊Dic3—68no—
139C3×(C2×D3)⋊C4—38no—
140C3×(C2×D3)⋊C4C2×D318yes—
141C3×C4⋊Dic3C2×D318yescomplemented
142#3724—38no—
143(C2×D3)⋊Dic3—38no—
144(C2×D3)⋊Dic3—38no—
145C22×C33C2411yescharacteristic
146C2×C3⋊(C3⋊D3)—21no—
147C2×C3⋊(C3⋊D3)—21no—
148C2×C3⋊(C3⋊D3)C2×D412yes—
149C2×C3×C3⋊D3—21no—
150C2×C3×C3⋊D3—21no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—22no—
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3#4112yescomplemented
155C2×C32×D3—21no—
156C2×C32×D3—21no—
157C2×C32×D3#4112yescomplemented
158D3×C3⋊D3—28no—
159C33×C4—22no—
160C33×C4C2×D412yes—
161C3⋊(C3⋊Dic3)—22no—
162C3⋊(C3⋊Dic3)C2×D412yes—
163C3×C3⋊Dic3—21no—
164C32×Dic3—21no—
165C32×Dic3—22no—
166C2×#239—94no—
167C2×#237—91no—
168C22×C4×D3—94no—
169C22×C4×D3—91no—
170C3×C22⋊Q8—34no—
171C3×#121—91no—
172#946—94no—
173#938—91no—
174#937—92no—
175D4×Dic3—92no—
176#930—94no—
177#924—91no—
178#897C3⋊D311yescharacteristic, complemented
179#895—34no—
180#894—98no—
181#893—98no—
182#892—94no—
183#892—91no—
184#892—184no—
185#883—92no—
186#882—98no—
187#67⋊C4—98no—
188#67⋊C4—92no—
189C23×C32—31no—
190C22×C3⋊D3—31no—
191C22×C3⋊D3—34no—
192C22×C3⋊D3—64no—
193C22×C3⋊D3—62no—
194C22×C3⋊D3—62no—
195C22×C3⋊D3—62no—
196C22×C3⋊D3C22×D312yes—
197C22×C3⋊D3C22×D311yescharacteristic
198C22×C3×D3—34no—
199C22×C3×D3—38no—
200C22×C3×D3C22×D314yes—
201C2×D32—68no—
202C2×D32—68no—
203C2×D32—68no—
204C32×Q8—62no—
205C2×C32×C4—24no—
206C2×C32×C4—28no—
207C2×C32×C4—32no—
208C2×C32×C4—31no—
209C2×C32×C4—62no—
210C2×C32×C4C22×D318yes—
211C2×C32×C4C22×D314yes—
212C2×C32×C4C22×D312yes—
213C2×C32×C4C22×D311yescharacteristic
214#638—64no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—32no—
217C2×C3⋊Dic3—34no—
218C2×C3⋊Dic3—38no—
219C2×C3⋊Dic3—64no—
220C2×C3⋊Dic3—68no—
221C2×C3⋊Dic3—62no—
222C2×C3⋊Dic3C22×D311yescharacteristic
223C4⋊(C3⋊D3)—24no—
224C4⋊(C3⋊D3)—22no—
225C4⋊(C3⋊D3)—64no—
226C4×C3⋊D3—22no—
227C4×C3⋊D3—34no—
228C4×C3⋊D3—316no—
229C4×C3⋊D3—616no—
230C4×C3⋊D3—62no—
231C4×C3⋊D3—62no—
232C4×C3⋊D3—64no—
233C2×C3×Dic3—34no—
234C2×C3×Dic3C22×D314yes—
235C2×C3×Dic3C22×D318yes—
236C3×D12—616no—
237C3×D12—68no—
238C3×C4×D3—28no—
239C3×C4×D3—68no—
240C3×Dic6—28no—
241#626—616no—
242#625—616no—
243#624—68no—
244D3×Dic3—616no—
245D3×Dic3—68no—
246D3×Dic3—68no—
247#525—271noSylow
248C2×C33C22×D411yescharacteristic
249C2×C33C2×#4111yescharacteristic
250C2×C33C2×#4111yescharacteristic
251C3⋊(C3⋊D3)—22no—
252C3×C3⋊D3—22no—
253C3×C3⋊D3—42no—
254C32×D3—22no—
255C23×D3—94no—
256C23×D3—91no—
257C2×C3×Q8—34no—
258C2×C3×D4—92no—
259C2×C3×D4—91no—
260C22×C3×C4—34no—
261C22×C3×C4—91no—
262C2×#67—98no—
263C2×#67—92no—
264C2×#67—92no—
265C22×Dic3—94no—
266C22×Dic3—92no—
267C22×Dic3—91no—
268#239—1816no—
269#237—184no—
270C2×D12—38no—
271C2×D12—34no—
272C2×D12—98no—
273C2×C4×D3—34no—
274C2×C4×D3—94no—
275C2×C4×D3—91no—
276C2×C4×D3—94no—
277C2×C4×D3—98no—
278C2×C4×D3—94no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—184no—
282C2×C4×D3—184no—
283C2×C4×D3—1816no—
284C2×C4×D3C2×C3⋊D311yescharacteristic
285C2×Dic6C2×C3⋊D311yescharacteristic, complemented
286C3×C4⋊C4—24no—
287C3×C4⋊C4—38no—
288C3×C4⋊C4D3214yescomplemented
289C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
290C3×C22⋊C4—38no—
291C3×C22⋊C4—92no—
292#217—92no—
293#217—92no—
294(C2×D3)⋊C4—38no—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—98no—
297(C2×D3)⋊C4—98no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—188no—
300(C2×D3)⋊C4C2×C3⋊D312yes—
301C4⋊Dic3—94no—
302C4⋊Dic3—91no—
303C4⋊Dic3—184no—
304C4⋊Dic3C2×C3⋊D312yescomplemented
305Dic3⋊C4—98no—
306Dic3⋊C4—92no—
307C4×Dic3—98no—
308C4×Dic3—92no—
309C4×Dic3—188no—
310C22×C32—24no—
311C22×C32—32no—
312C22×C32—61no—
313C22×C32—61no—
314C22×C32C23×D314yes—
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3—21no—
317C2×C3⋊D3—21no—
318C2×C3⋊D3—22no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3—22no—
321C2×C3⋊D3—32no—
322C2×C3⋊D3—38no—
323C2×C3⋊D3—68no—
324C2×C3⋊D3—68no—
325C2×C3⋊D3—61no—
326C2×C3⋊D3—61no—
327C2×C3⋊D3—64no—
328C2×C3⋊D3—64no—
329C2×C3⋊D3—124no—
330C2×C3⋊D3—124no—
331C2×C3⋊D3#23712yescomplemented
332C2×C3×D3—24no—
333C2×C3×D3—24no—
334C2×C3×D3—38no—
335C2×C3×D3—64no—
336C2×C3×D3—64no—
337C2×C3×D3—68no—
338C2×C3×D3—68no—
339C2×C3×D3—68no—
340C2×C3×D3#23918yescomplemented
341D32—632no—
342C32×C4—22no—
343C32×C4—28no—
344C32×C4—28no—
345C32×C4—48no—
346C32×C4—61no—
347C32×C4—62no—
348C32×C4D3×D418yes—
349C32×C4D3×D412yes—
350C3⋊Dic3—21no—
351C3⋊Dic3—32no—
352C3⋊Dic3—38no—
353C3⋊Dic3—68no—
354C3⋊Dic3—62no—
355C3⋊Dic3—68no—
356C3⋊Dic3—128no—
357C3×Dic3—24no—
358C3×Dic3—28no—
359C3×Dic3—64no—
360C2×#41—271no—
361C23×C4—271no—
362#123—272no—
363C22⋊Q8—31no—
364C22⋊Q8—271no—
365#121—91no—
366#121—271no—
367#120—272no—
368C4×D4—272no—
369C4×D4—272no—
370#117—271no—
371C33#52511yescharacteristic, complemented, Sylow
372C23×C3—34no—
373C23×C3—91no—
374C23×C3—92no—
375C22×D3—34no—
376C22×D3—38no—
377C22×D3—94no—
378C22×D3—91no—
379C22×D3—184no—
380C22×D3—182no—
381C22×D3—182no—
382C22×D3—188no—
383C22×D3—188no—
384C22×D3—188no—
385C22×D3—182no—
386C22×D3C22×C3⋊D311yescharacteristic
387C3×Q8—68no—
388C3×D4—184no—
389C3×D4—182no—
390C2×C3×C4—24no—
391C2×C3×C4—28no—
392C2×C3×C4—38no—
393C2×C3×C4—34no—
394C2×C3×C4—68no—
395C2×C3×C4—91no—
396C2×C3×C4—182no—
397C2×C3×C4C2×D3218yes—
398C2×C3×C4C2×D3214yes—
399C2×C3×C4C22×C3⋊D312yes—
400C2×C3×C4C22×C3⋊D311yescharacteristic
401#67—184no—
402#67—1816no—
403#67—184no—
404C2×Dic3—34no—
405C2×Dic3—94no—
406C2×Dic3—91no—
407C2×Dic3—98no—
408C2×Dic3—92no—
409C2×Dic3—184no—
410C2×Dic3—188no—
411C2×Dic3—182no—
412C2×Dic3—184no—
413C2×Dic3—188no—
414C2×Dic3C22×C3⋊D312yes—
415C2×Dic3C22×C3⋊D311yescharacteristic
416D12—616no—
417D12—68no—
418D12—1816no—
419C4×D3—22no—
420C4×D3—68no—
421C4×D3—916no—
422C4×D3—94no—
423C4×D3—1816no—
424C4×D3—182no—
425C4×D3—188no—
426C4×D3—188no—
427C4×D3—1816no—
428C4×D3—182no—
429Dic6—22no—
430C2×C32—24no—
431C2×C32—24no—
432C2×C32—24no—
433C2×C32—62no—
434C2×C32C2×D3×D414yes—
435C2×C32C2×D3×D411yescharacteristic
436C2×C32C2×#23711yescharacteristic
437C2×C32C2×#23914yes—
438C2×C32D3×#4114yes—
439C2×C32D3×#4111yescharacteristic
440C3⋊D3—22no—
441C3⋊D3—42no—
442C3⋊D3—62no—
443C3⋊D3—68no—
444C3⋊D3—128no—
445C3×D3—28no—
446C3×D3—68no—
447C3×D3—128no—
448C24—271no—
449#41—544no—
450C2×Q8—31no—
451C2×D4—92no—
452C2×D4—91no—
453C2×D4—272no—
454C2×D4—272no—
455C22×C4—31no—
456C22×C4—91no—
457C22×C4—271no—
458C22×C4—271no—
459C22×C4—272no—
460C22×C4—271no—
461C22×C4—271no—
462C22×C4—544no—
463C4⋊C4—32no—
464C4⋊C4—272no—
465C4⋊C4—271no—
466C4⋊C4D3×C3⋊D311yescharacteristic, complemented
467C22⋊C4—32no—
468C22⋊C4—92no—
469C22⋊C4—272no—
470C22⋊C4—272no—
471C22⋊C4—272no—
472C42—272no—
473C22×C3—24no—
474C22×C3—38no—
475C22×C3—64no—
476C22×C3—64no—
477C22×C3—92no—
478C22×C3—181no—
479C22×C3—181no—
480C22×C3—182no—
481C22×C3—182no—
482C22×C3—182no—
483C22×C3C22×D3214yes—
484C22×C3C23×C3⋊D311yescharacteristic
485C2×D3—21no—
486C2×D3—21no—
487C2×D3—38no—
488C2×D3—64no—
489C2×D3—64no—
490C2×D3—68no—
491C2×D3—68no—
492C2×D3—68no—
493C2×D3—98no—
494C2×D3—92no—
495C2×D3—188no—
496C2×D3—188no—
497C2×D3—1832no—
498C2×D3—184no—
499C2×D3—184no—
500C2×D3—181no—
501C2×D3—181no—
502C2×D3—364no—
503C2×D3—364no—
504C2×D3#383212yescomplemented
505C3×C4—28no—
506C3×C4—22no—
507C3×C4—28no—
508C3×C4—48no—
509C3×C4—64no—
510C3×C4—68no—
511C3×C4—181no—
512C3×C4#380218yes—
513C3×C4D4×C3⋊D312yes—
514Dic3—22no—
515Dic3—21no—
516Dic3—64no—
517Dic3—98no—
518Dic3—92no—
519Dic3—188no—
520Dic3—188no—
521Dic3—182no—
522Dic3—368no—
523C32—24no—
524C32#560611yescharacteristic, complemented
525C32#562714yescomplemented
526C23—31no—
527C23—91no—
528C23—92no—
529C23—271no—
530C23—542no—
531C23—542no—
532C23—542no—
533Q8—62no—
534D4—184no—
535D4—182no—
536D4—544no—
537D4—544no—
538C2×C4—32no—
539C2×C4—31no—
540C2×C4—62no—
541C2×C4—91no—
542C2×C4—182no—
543C2×C4—271no—
544C2×C4—274no—
545C2×C4—272no—
546C2×C4—542no—
547C2×C4—542no—
548C2×C4—544no—
549C2×C4—542no—
550C2×C4C2×D3×C3⋊D312yes—
551C2×C4C2×D3×C3⋊D311yescharacteristic
552C2×C3—24no—
553C2×C3—24no—
554C2×C3—24no—
555C2×C3—68no—
556C2×C3—182no—
557C2×C3—362no—
558C2×C3C2×#380214yes—
559C2×C3D3×#23714yes—
560C2×C3D3×#23914yes—
561C2×C3C2×D4×C3⋊D311yescharacteristic
562C2×C3C2×#383211yescharacteristic
563C2×C3#41×C3⋊D311yescharacteristic
564D3—22no—
565D3—68no—
566D3—128no—
567D3—182no—
568D3—188no—
569D3—368no—
570C22—32no—
571C22—61no—
572C22—61no—
573C22—92no—
574C22—181no—
575C22—181no—
576C22—182no—
577C22—182no—
578C22—182no—
579C22—272no—
580C22—548no—
581C22—541no—
582C22—541no—
583C22C22×D3×C3⋊D311yescharacteristic
584C4—22no—
585C4—61no—
586C4—62no—
587C4—181no—
588C4—272no—
589C4—542no—
590C4#8902612yes—
591C3—24no—
592C3#1059463214yescomplemented
593C3#1059609811yescharacteristic, complemented
594C2—62no—
595C2—182no—
596C2—362no—
597C2—542no—
598C2C2×#8902611yescharacteristic
599C2C3⋊D3×#23711yescharacteristic
600C2D3×#383211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D3×C3⋊D3C2semidirect
2. C2×#89017C2semidirect
3. #11723933C2semidirect
4. C3×#64104C2semidirect
5. C32×#897C2semidirect
6. #11723541C2semidirect
7. #11723515C2semidirect
8. #11723508C2semidirect
9. #11723504C2semidirect
10. #11723501C2semidirect
11. #11723435C2semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C2×C32×Dic6C22semidirect
25. C2×Dic3×C3⋊D3C22semidirect
29. C33×C4⋊C4C22semidirect
31. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
32. C4×C3⋊(C3⋊Dic3)C22semidirect
33. C3×#3752C22semidirect
35. C32×C4⋊Dic3C22semidirect
36. #88806C22semidirect
38. #88803C22semidirect
46. #64104D3semidirect
54. C3×#897D3semidirect
73. C4×C3⋊(C3⋊D3)D4semidirect
99. C2×C4⋊(C3⋊D3)C2×D3semidirect
112. C2×C3×Dic6C2×D3semidirect
123. C32×C4⋊C4C2×D3semidirect
124. C32×C4⋊C4C2×D3semidirect
131. #3752C2×D3semidirect
141. C3×C4⋊Dic3C2×D3semidirect
154. C2×C3×C3⋊D3#41semidirect
157. C2×C32×D3#41semidirect
178. #897C3⋊D3semidirect
285. C2×Dic6C2×C3⋊D3semidirect
288. C3×C4⋊C4D32semidirect
289. C3×C4⋊C4C2×C3⋊D3semidirect
304. C4⋊Dic3C2×C3⋊D3semidirect
331. C2×C3⋊D3#237semidirect
340. C2×C3×D3#239semidirect
371. C33#525semidirect
466. C4⋊C4D3×C3⋊D3semidirect
504. C2×D3#3832semidirect
524. C32#5606semidirect
525. C32#5627semidirect
592. C3#10594632semidirect
593. C3#10596098semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
C2×#890172
#117239332
C3×#641042
C32×#8972
#117235412
#117235152
#117235082
#117235042
#117235012
#117234352
#105960983
#105946323