Finite Group Databasebeta

← #13176545

Subgroups — #13176545

Order 1728.

2276 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3#11723938C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic, complemented
5C32×#892C211yesmaximal, characteristic, complemented
6#11723541C212yesmaximal, complemented
7#11723532C211yesmaximal, characteristic, complemented
8Dic3×C4⋊(C3⋊D3)C212yesmaximal, complemented
9#11723480C212yesmaximal, complemented
10#11723471C212yesmaximal, complemented
11#11723431C211yesmaximal, characteristic, complemented
12#10595995—31nomaximal
13#10594626—34nomaximal
14C22×D3×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×#6308C2211yescharacteristic, complemented
17C2×C3×C4⋊(C3⋊D3)C2212yes—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C32×C4×D3C2211yescharacteristic
21C2×#6292C2212yes—
22C2×#6290C2212yes—
23C2×#6289C2211yescharacteristic
24C2×Dic3×C3⋊D3C2211yescharacteristic
25C2×Dic3×C3⋊D3C2212yescomplemented
26C2×D3×C3⋊Dic3C2211yescharacteristic
27C4×D3×C3⋊D3—24no—
28#89021—24no—
29C33×C4⋊C4C2211yescharacteristic, complemented
30#88859C2212yes—
31C4⋊(C3⋊(C3⋊Dic3))C2212yescomplemented
32C3×#3752C2212yescomplemented
33C32×(C2×D3)⋊C4C2212yes—
34C32×C4⋊Dic3C2211yescharacteristic, complemented
35C32×C4×Dic3C2212yescomplemented
36Dic3⋊(C3⋊Dic3)C2212yescomplemented
37#88803C2212yes—
38#88803C2212yescomplemented
39C2×#3828—31no—
40C22×C4×C3⋊D3—31no—
41C2×C4×D32—34no—
42C2×#3795—34no—
43C32×#117—31no—
44#64106—31no—
45#64106—31no—
46#64106—34no—
47#64106—64no—
48#64104—31no—
49#64104D311yescharacteristic, complemented
50#64095—32no—
51#64091—32no—
52C4×C4⋊(C3⋊D3)—32no—
53C4×C4⋊(C3⋊D3)—32no—
54C3×#895—34no—
55C3×#892D314yescomplemented
56#63922—38no—
57#63916—34no—
58Dic3×D12—38no—
59#63891—38no—
60#63880—38no—
61#63857—34no—
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)C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C4×C3⋊(C3⋊D3)—22no—
74#6308—22no—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C3×C4⋊(C3⋊D3)—24no—
77C3×C4⋊(C3⋊D3)—22no—
78C3×C4×C3⋊D3—22no—
79C2×C32×Dic3C2312yes—
80C2×C32×Dic3C2311yescharacteristic
81C32×C4×D3D414yescomplemented
82#6292—24no—
83#6290—24no—
84#6289—22no—
85Dic3×C3⋊D3—24no—
86Dic3×C3⋊D3—22no—
87D3×C3⋊Dic3—22no—
88#5606—91no—
89#5524—94no—
90C23×C3⋊D3—31no—
91C22×D32—34no—
92C22×C32×C4—31no—
93C2×#638—32no—
94C22×C3⋊Dic3—31no—
95#3828—64no—
96C2×C4⋊(C3⋊D3)—32no—
97C2×C4⋊(C3⋊D3)C2×D312yes—
98C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
99C2×C4×C3⋊D3—31no—
100C2×C4×C3⋊D3—32no—
101C2×C4×C3⋊D3—31no—
102C2×C4×C3⋊D3—34no—
103C2×C4×C3⋊D3—31no—
104C2×C4×C3⋊D3—64no—
105C2×C4×C3⋊D3—64no—
106C2×C4×C3⋊D3C2×D311yescharacteristic
107C2×#634—31no—
108C2×#634—34no—
109C2×#634—64no—
110C2×C3×D12—34no—
111C2×C3×D12—38no—
112C2×C3×C4×D3—34no—
113C2×C3×C4×D3C2×D314yes—
114C2×#626—38no—
115C2×#625—38no—
116C2×#624—34no—
117C2×D3×Dic3—34no—
118C2×D3×Dic3—38no—
119C2×D3×Dic3—34no—
120C4×D32—616no—
121#3795—616no—
122C32×C4⋊C4—24no—
123C32×C4⋊C4—31no—
124C32×C4⋊C4C2×D314yescomplemented
125C32×C4⋊C4C2×D311yescharacteristic, complemented
126C32×C22⋊C4—32no—
127C32×C42—32no—
128#3752—32no—
129#3752—32no—
130#3752—32no—
131#3752—38no—
132#3752—68no—
133#3752C2×D312yescomplemented
134C4⋊(C3⋊Dic3)—32no—
135C4⋊(C3⋊Dic3)—38no—
136C4⋊(C3⋊Dic3)—32no—
137C4⋊(C3⋊Dic3)—68no—
138C3×(C2×D3)⋊C4—38no—
139C3×(C2×D3)⋊C4C2×D318yes—
140C3×C4⋊Dic3C2×D314yescomplemented
141C3×C4×Dic3C2×D318yescomplemented
142Dic3⋊Dic3—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)#4112yescomplemented
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×D3C2×D412yes—
158D3×C3⋊D3—28no—
159C33×C4—22no—
160C33×C4C2×D412yes—
161C3⋊(C3⋊Dic3)—21no—
162C3⋊(C3⋊Dic3)—22no—
163C3×C3⋊Dic3—21no—
164C32×Dic3—22no—
165C32×Dic3C2×D412yes—
166C2×#237—91no—
167C2×#235—94no—
168C22×C4×D3—94no—
169C22×C4×D3—91no—
170C3×#121—91no—
171C3×#117—34no—
172#938—91no—
173#937—92no—
174D4×Dic3—92no—
175#924—91no—
176#897—94no—
177#897—94no—
178#897—91no—
179#897—184no—
180#895—34no—
181#895—94no—
182#892C3⋊D311yescharacteristic, complemented
183#886—98no—
184#883—92no—
185#882—98no—
186#67⋊C4—92no—
187C4×D12—98no—
188C4×D12—98no—
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—
204C2×C32×C4—24no—
205C2×C32×C4—28no—
206C2×C32×C4—31no—
207C2×C32×C4—34no—
208C2×C32×C4—32no—
209C2×C32×C4C22×D318yes—
210C2×C32×C4C22×D314yes—
211C2×C32×C4C22×D312yes—
212C2×C32×C4C22×D311yescharacteristic
213#638—64no—
214C2×C3⋊Dic3—32no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—34no—
217C2×C3⋊Dic3—38no—
218C2×C3⋊Dic3—64no—
219C2×C3⋊Dic3—68no—
220C2×C3⋊Dic3—62no—
221C2×C3⋊Dic3C22×D311yescharacteristic
222C4⋊(C3⋊D3)—24no—
223C4⋊(C3⋊D3)—22no—
224C4⋊(C3⋊D3)—64no—
225C4×C3⋊D3—22no—
226C4×C3⋊D3—62no—
227C4×C3⋊D3—62no—
228C4×C3⋊D3—64no—
229C4×C3⋊D3—62no—
230C4×C3⋊D3—68no—
231C4×C3⋊D3—128no—
232#634—62no—
233#634—68no—
234#634—128no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3C22×D318yes—
237C2×C3×Dic3C22×D314yes—
238C3×D12—616no—
239C3×D12—68no—
240C3×C4×D3—68no—
241C3×C4×D3D12116yescomplemented
242#626—616no—
243#625—616no—
244#624—68no—
245D3×Dic3—68no—
246D3×Dic3—616no—
247D3×Dic3—68no—
248#525—271noSylow
249C2×C33C22×D411yescharacteristic
250C2×C33C2×#4111yescharacteristic
251C2×C33C2×#4111yescharacteristic
252C3⋊(C3⋊D3)—22no—
253C3×C3⋊D3—22no—
254C3×C3⋊D3—42no—
255C32×D3—22no—
256C23×D3—94no—
257C23×D3—91no—
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#237—184no—
269#235—1816no—
270C2×D12—38no—
271C2×D12—34no—
272C2×D12—98no—
273C2×C4×D3—34no—
274C2×C4×D3—94no—
275C2×C4×D3—94no—
276C2×C4×D3—98no—
277C2×C4×D3—91no—
278C2×C4×D3—94no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—184no—
282C2×C4×D3—1816no—
283C2×C4×D3—184no—
284C2×C4×D3C2×C3⋊D311yescharacteristic
285C2×Dic6—94no—
286C2×Dic6—91no—
287C2×Dic6—184no—
288C3×C4⋊C4—24no—
289C3×C4⋊C4—34no—
290C3×C4⋊C4D3214yescomplemented
291C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
292C3×C22⋊C4—38no—
293C3×C22⋊C4—92no—
294C3×C42—38no—
295#217—92no—
296#217—92no—
297(C2×D3)⋊C4—38no—
298(C2×D3)⋊C4—98no—
299(C2×D3)⋊C4—98no—
300(C2×D3)⋊C4—98no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—188no—
303(C2×D3)⋊C4C2×C3⋊D312yes—
304C4⋊Dic3—98no—
305C4⋊Dic3—98no—
306C4⋊Dic3—92no—
307C4⋊Dic3—188no—
308C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
309Dic3⋊C4—92no—
310C4×Dic3C2×C3⋊D312yescomplemented
311C22×C32—24no—
312C22×C32—32no—
313C22×C32—61no—
314C22×C32—61no—
315C22×C32C23×D314yes—
316C22×C32C23×D311yescharacteristic
317C2×C3⋊D3—21no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3—22no—
321C2×C3⋊D3—22no—
322C2×C3⋊D3—32no—
323C2×C3⋊D3—38no—
324C2×C3⋊D3—68no—
325C2×C3⋊D3—68no—
326C2×C3⋊D3—61no—
327C2×C3⋊D3—61no—
328C2×C3⋊D3—64no—
329C2×C3⋊D3—64no—
330C2×C3⋊D3—124no—
331C2×C3⋊D3—124no—
332C2×C3⋊D3#23712yescomplemented
333C2×C3×D3—24no—
334C2×C3×D3—24no—
335C2×C3×D3—38no—
336C2×C3×D3—64no—
337C2×C3×D3—64no—
338C2×C3×D3—68no—
339C2×C3×D3—68no—
340C2×C3×D3—68no—
341C2×C3×D3C2×D1218yes—
342D32—632no—
343C32×C4—22no—
344C32×C4—28no—
345C32×C4—28no—
346C32×C4—32no—
347C32×C4—48no—
348C32×C4—62no—
349C32×C4C2×D1218yes—
350C32×C4D3×D412yes—
351C3⋊Dic3—21no—
352C3⋊Dic3—62no—
353C3⋊Dic3—61no—
354C3⋊Dic3—64no—
355C3⋊Dic3—68no—
356C3⋊Dic3—128no—
357C3⋊Dic3—124no—
358C3×Dic3—28no—
359C3×Dic3—64no—
360C3×Dic3C2×D1218yes—
361C2×#41—271no—
362C23×C4—271no—
363#123—272no—
364C22⋊Q8—271no—
365C22⋊Q8—271no—
366#121—91no—
367#121—271no—
368#120—272no—
369C4×D4—272no—
370C4×D4—272no—
371#117—31no—
372C33#52511yescharacteristic, complemented, Sylow
373C23×C3—34no—
374C23×C3—91no—
375C23×C3—92no—
376C22×D3—34no—
377C22×D3—38no—
378C22×D3—94no—
379C22×D3—91no—
380C22×D3—184no—
381C22×D3—182no—
382C22×D3—182no—
383C22×D3—188no—
384C22×D3—188no—
385C22×D3—188no—
386C22×D3—182no—
387C22×D3C22×C3⋊D311yescharacteristic
388C3×D4—184no—
389C3×D4—182no—
390C2×C3×C4—24no—
391C2×C3×C4—28no—
392C2×C3×C4—34no—
393C2×C3×C4—316no—
394C2×C3×C4—38no—
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—98no—
406C2×Dic3—94no—
407C2×Dic3—92no—
408C2×Dic3—91no—
409C2×Dic3—184no—
410C2×Dic3—188no—
411C2×Dic3—188no—
412C2×Dic3—182no—
413C2×Dic3—184no—
414C2×Dic3C22×C3⋊D312yes—
415C2×Dic3C22×C3⋊D311yescharacteristic
416D12—616no—
417D12—68no—
418D12—1816no—
419C4×D3—68no—
420C4×D3—188no—
421C4×D3—188no—
422C4×D3—1816no—
423C4×D3—182no—
424C4×D3—188no—
425C4×D3—182no—
426C4×D3—182no—
427C4×D3—368no—
428C4×D3C4⋊(C3⋊D3)14yescomplemented
429Dic6—188no—
430Dic6—182no—
431Dic6—368no—
432C2×C32—24no—
433C2×C32—24no—
434C2×C32—24no—
435C2×C32—62no—
436C2×C32C22×D1214yes—
437C2×C32C2×D3×D411yescharacteristic
438C2×C32C2×#23711yescharacteristic
439C2×C32D3×#4114yes—
440C2×C32D3×#4114yes—
441C2×C32D3×#4111yescharacteristic
442C3⋊D3—22no—
443C3⋊D3—42no—
444C3⋊D3—62no—
445C3⋊D3—68no—
446C3⋊D3—128no—
447C3×D3—28no—
448C3×D3—68no—
449C3×D3—128no—
450C24—271no—
451#41—544no—
452C2×Q8—271no—
453C2×D4—92no—
454C2×D4—91no—
455C2×D4—272no—
456C2×D4—272no—
457C22×C4—31no—
458C22×C4—91no—
459C22×C4—271no—
460C22×C4—271no—
461C22×C4—272no—
462C22×C4—271no—
463C22×C4—271no—
464C22×C4—544no—
465C4⋊C4—31no—
466C4⋊C4—272no—
467C4⋊C4—272no—
468C4⋊C4D3×C3⋊D311yescharacteristic, complemented
469C22⋊C4—32no—
470C22⋊C4—92no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—32no—
475C22×C3—24no—
476C22×C3—38no—
477C22×C3—64no—
478C22×C3—64no—
479C22×C3—92no—
480C22×C3—181no—
481C22×C3—181no—
482C22×C3—182no—
483C22×C3—182no—
484C22×C3—182no—
485C22×C3C22×D3214yes—
486C22×C3C23×C3⋊D311yescharacteristic
487C2×D3—21no—
488C2×D3—21no—
489C2×D3—38no—
490C2×D3—64no—
491C2×D3—64no—
492C2×D3—68no—
493C2×D3—68no—
494C2×D3—68no—
495C2×D3—98no—
496C2×D3—92no—
497C2×D3—188no—
498C2×D3—188no—
499C2×D3—1832no—
500C2×D3—184no—
501C2×D3—184no—
502C2×D3—181no—
503C2×D3—181no—
504C2×D3—364no—
505C2×D3—364no—
506C2×D3C2×C4⋊(C3⋊D3)12yes—
507C3×C4—28no—
508C3×C4—22no—
509C3×C4—28no—
510C3×C4—38no—
511C3×C4—48no—
512C3×C4—68no—
513C3×C4—181no—
514C3×C4D3×D1218yes—
515C3×C4C2×C4⋊(C3⋊D3)12yes—
516Dic3—22no—
517Dic3—64no—
518Dic3—188no—
519Dic3—184no—
520Dic3—182no—
521Dic3—181no—
522Dic3—368no—
523Dic3—364no—
524Dic3C2×C4⋊(C3⋊D3)12yes—
525C32—24no—
526C32#552414yescomplemented
527C32#560611yescharacteristic, complemented
528C23—31no—
529C23—91no—
530C23—92no—
531C23—271no—
532C23—542no—
533C23—542no—
534C23—542no—
535Q8—542no—
536D4—184no—
537D4—182no—
538D4—544no—
539D4—544no—
540C2×C4—31no—
541C2×C4—34no—
542C2×C4—32no—
543C2×C4—91no—
544C2×C4—182no—
545C2×C4—271no—
546C2×C4—272no—
547C2×C4—542no—
548C2×C4—542no—
549C2×C4—544no—
550C2×C4—542no—
551C2×C4—542no—
552C2×C4C2×D3×C3⋊D312yes—
553C2×C4C2×D3×C3⋊D311yescharacteristic
554C2×C3—24no—
555C2×C3—24no—
556C2×C3—24no—
557C2×C3—68no—
558C2×C3—182no—
559C2×C3—362no—
560C2×C3C2×D3×D1214yes—
561C2×C3D3×#23714yes—
562C2×C3#6431914yes—
563C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
564C2×C3#41×C3⋊D311yescharacteristic
565C2×C3#41×C3⋊D311yescharacteristic
566D3—22no—
567D3—68no—
568D3—128no—
569D3—188no—
570D3—182no—
571D3—368no—
572C22—32no—
573C22—61no—
574C22—61no—
575C22—92no—
576C22—181no—
577C22—181no—
578C22—182no—
579C22—182no—
580C22—182no—
581C22—272no—
582C22—548no—
583C22—541no—
584C22—541no—
585C22C22×D3×C3⋊D311yescharacteristic
586C4—22no—
587C4—32no—
588C4—62no—
589C4—181no—
590C4—541no—
591C4—542no—
592C4D3×C4⋊(C3⋊D3)12yes—
593C3—24no—
594C3#1059462614yescomplemented
595C3#1059599511yescharacteristic, complemented
596C2—62no—
597C2—182no—
598C2—362no—
599C2—542no—
600C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
601C2C3⋊D3×#23711yescharacteristic
602C2#1172459611yescharacteristic
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×#89021C2semidirect
3. #11723938C2semidirect
4. C3×#64104C2semidirect
5. C32×#892C2semidirect
6. #11723541C2semidirect
7. #11723532C2semidirect
8. Dic3×C4⋊(C3⋊D3)C2semidirect
9. #11723480C2semidirect
10. #11723471C2semidirect
11. #11723431C2semidirect
16. C2×#6308C22semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
25. C2×Dic3×C3⋊D3C22semidirect
29. C33×C4⋊C4C22semidirect
31. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
32. C3×#3752C22semidirect
34. C32×C4⋊Dic3C22semidirect
35. C32×C4×Dic3C22semidirect
36. Dic3⋊(C3⋊Dic3)C22semidirect
38. #88803C22semidirect
49. #64104D3semidirect
55. C3×#892D3semidirect
81. C32×C4×D3D4semidirect
98. C2×C4⋊(C3⋊D3)C2×D3semidirect
124. C32×C4⋊C4C2×D3semidirect
125. C32×C4⋊C4C2×D3semidirect
133. #3752C2×D3semidirect
140. C3×C4⋊Dic3C2×D3semidirect
141. C3×C4×Dic3C2×D3semidirect
148. C2×C3⋊(C3⋊D3)#41semidirect
154. C2×C3×C3⋊D3#41semidirect
182. #892C3⋊D3semidirect
241. C3×C4×D3D12semidirect
290. C3×C4⋊C4D32semidirect
291. C3×C4⋊C4C2×C3⋊D3semidirect
308. C4⋊Dic3C2×C3⋊D3semidirect
310. C4×Dic3C2×C3⋊D3semidirect
332. C2×C3⋊D3#237semidirect
372. C33#525semidirect
428. C4×D3C4⋊(C3⋊D3)semidirect
468. C4⋊C4D3×C3⋊D3semidirect
526. C32#5524semidirect
527. C32#5606semidirect
594. C3#10594626semidirect
595. C3#10595995semidirect

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×#890212
#117239382
C3×#641042
C32×#8922
#117235412
#117235322
Dic3×C4⋊(C3⋊D3)2
#117234802
#117234712
#117234312
#105959953
#105946263