Finite Group Databasebeta

← #13175335

Subgroups — #13175335

Order 1728.

1564 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6308C211yesmaximal, characteristic, complemented
2C3×#64085C211yesmaximal, characteristic, complemented
3C32×#872C211yesmaximal, characteristic, complemented
4#11723513C211yesmaximal, characteristic
5#11723508C211yesmaximal, characteristic
6#11723497C211yesmaximal, characteristic, complemented
7#11723496C211yesmaximal, characteristic, complemented
8#11723480C211yesmaximal, characteristic
9#11723464C211yesmaximal, characteristic, complemented
10#11723460C211yesmaximal, characteristic, complemented
11D3×#3750C211yesmaximal, characteristic, complemented
12#11723448C211yesmaximal, characteristic, complemented
13#11723444C211yesmaximal, characteristic, complemented
14#11723433C211yesmaximal, characteristic, complemented
15#11723409C211yesmaximal, characteristic, complemented
16#10596002—31nomaximal
17#10593919—34nomaximal
18C2×#6308C2211yescharacteristic
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6290C2211yescharacteristic
21C2×Dic3×C3⋊D3C2211yescharacteristic
22C2×D3×C3⋊Dic3C2211yescharacteristic
23C2×D3×C3⋊Dic3C2211yescharacteristic
24C33×C42C2211yescharacteristic
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
26#88857C2211yescharacteristic
27#88857C2211yescharacteristic
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
29C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3752C2211yescharacteristic
31C3×#3752C2211yescharacteristic
32C3×#3752C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×#3750C2211yescharacteristic
35C3×#3750C2211yescharacteristic
36C32×(C2×D3)⋊C4C2211yescharacteristic
37C32×(C2×D3)⋊C4C2211yescharacteristic
38C32×Dic3⋊C4C2211yescharacteristic
39C32×Dic3⋊C4C2211yescharacteristic
40C32×C4×Dic3C2211yescharacteristic
41#88807C2211yescharacteristic
42#88807C2211yescharacteristic
43#88807C2211yescharacteristic
44#88806C2211yescharacteristic
45#88806C2211yescharacteristic
46#88806C2211yescharacteristic
47Dic3⋊(C3⋊Dic3)C2211yescharacteristic
48Dic3⋊(C3⋊Dic3)C2211yescharacteristic
49#88803C2211yescharacteristic
50#88803C2211yescharacteristic
51(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×#117—31no—
54#64138—31no—
55C2×#3750—31no—
56#64107—31no—
57#64105—31no—
58#64098—31no—
59#64098—31no—
60#64095—31no—
61#64090—31no—
62#64087—31no—
63#64086—31no—
64#64085D311yescharacteristic, complemented
65#64081—31no—
66#64079—31no—
67C4×#634—31no—
68C4×#634—34no—
69C4×#634—64no—
70C3×#876—34no—
71C3×#872D314yescomplemented
72#63905—34no—
73#63902—34no—
74#63896—34no—
75#63896—34no—
76#63891—34no—
77#63878—34no—
78#63877—34no—
79D3×Dic3⋊C4—34no—
80#63870—34no—
81#63867—34no—
82#63862—34no—
83#63845—34no—
84C22×C3×C3⋊D3C2311yescharacteristic
85C22×C32×D3C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C33×C4C2311yescharacteristic
88C2×C33×C4C2311yescharacteristic
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
93#6308—22no—
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C32×Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C32×C4×D3—22no—
101#6290—22no—
102Dic3×C3⋊D3—22no—
103D3×C3⋊Dic3—22no—
104D3×C3⋊Dic3—22no—
105#5708—91no—
106#5531—94no—
107C22×C32×C4—31no—
108C2×#638—31no—
109C22×C3⋊Dic3—31no—
110C22×C3⋊Dic3—31no—
111C2×C4×C3⋊D3—31no—
112C2×#634—31no—
113C2×#634—34no—
114C2×#634—64no—
115C2×C3×C4×D3C2×D314yes—
116C2×#625—34no—
117C2×D3×Dic3—34no—
118C2×D3×Dic3—34no—
119C2×D3×Dic3—34no—
120C32×C4⋊C4—31no—
121C32×C4⋊C4—31no—
122C32×C22⋊C4—31no—
123C32×C22⋊C4—31no—
124C32×C42—24no—
125C32×C42—31no—
126C32×C42C2×D314yes—
127C32×C42C2×D311yescharacteristic
128#3757—31no—
129#3752—31no—
130#3752—31no—
131#3752C2×D311yescharacteristic
132#3752C2×D311yescharacteristic
133#3752C2×D311yescharacteristic
134C4⋊(C3⋊Dic3)—31no—
135C4⋊(C3⋊Dic3)—34no—
136C4⋊(C3⋊Dic3)—31no—
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)—64no—
139#3750—31no—
140#3750—31no—
141#3750—34no—
142#3750—34no—
143#3750—31no—
144#3750—31no—
145#3750—31no—
146#3750—31no—
147#3750—31no—
148#3750—31no—
149#3750—64no—
150#3750—64no—
151#3750C2×D311yescharacteristic
152#3750C2×D311yescharacteristic
153#3750C2×D311yescharacteristic
154C4×C3⋊Dic3—31no—
155C4×C3⋊Dic3—31no—
156C4×C3⋊Dic3—34no—
157C4×C3⋊Dic3—34no—
158C4×C3⋊Dic3—31no—
159C4×C3⋊Dic3—64no—
160C4×C3⋊Dic3—64no—
161C3×(C2×D3)⋊C4—34no—
162C3×(C2×D3)⋊C4—34no—
163C3×(C2×D3)⋊C4—34no—
164C3×(C2×D3)⋊C4C2×D314yes—
165C3×(C2×D3)⋊C4C2×D314yes—
166C3×Dic3⋊C4—34no—
167C3×Dic3⋊C4—34no—
168C3×Dic3⋊C4—34no—
169C3×Dic3⋊C4C2×D314yes—
170C3×Dic3⋊C4C2×D314yes—
171C3×C4×Dic3C2×D314yes—
172#3724—34no—
173#3724—34no—
174#3724—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178Dic3⋊Dic3—34no—
179Dic3⋊Dic3—34no—
180(C2×D3)⋊Dic3—34no—
181(C2×D3)⋊Dic3—34no—
182(C2×D3)⋊Dic3—34no—
183Dic32—34no—
184C22×C33C2411yescharacteristic
185C2×C3×C3⋊D3—21no—
186C2×C3×C3⋊D3—21no—
187C2×C3×C3⋊D3—21no—
188C2×C32×D3—21no—
189C2×C32×D3—21no—
190C2×C32×D3#4112yes—
191C33×C4—21no—
192C33×C4—21no—
193C33×C4—21no—
194C3⋊(C3⋊Dic3)—21no—
195C3⋊(C3⋊Dic3)—21no—
196C3⋊(C3⋊Dic3)—21no—
197C3⋊(C3⋊Dic3)#4112yes—
198C3×C3⋊Dic3—21no—
199C3×C3⋊Dic3—21no—
200C3×C3⋊Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3—21no—
204C3×#126—91no—
205C3×#117—34no—
206#929—94no—
207C2×Dic3⋊C4—94no—
208#898—94no—
209#896—94no—
210#896—91no—
211#892—91no—
212D3×C4⋊C4—91no—
213#890—91no—
214#889—94no—
215#889—94no—
216#889—91no—
217#889—91no—
218#886—94no—
219#883—91no—
220#883—91no—
221#67⋊C4—94no—
222#67⋊C4—91no—
223#879—91no—
224#878—94no—
225#878—91no—
226#877—94no—
227#877—91no—
228#876—34no—
229#872—94no—
230#872C3⋊D311yescharacteristic, complemented
231#870—94no—
232C4×Dic6—94no—
233C4×Dic6—91no—
234C4×Dic6—184no—
235C23×C32—31no—
236C22×C3⋊D3C22×D311yescharacteristic
237C22×C3×D3—34no—
238C22×C3×D3C22×D314yes—
239C2×C32×C4—24no—
240C2×C32×C4—24no—
241C2×C32×C4—24no—
242C2×C32×C4—31no—
243C2×C32×C4—31no—
244C2×C32×C4—31no—
245C2×C32×C4—62no—
246C2×C32×C4C22×D314yes—
247C2×C32×C4C22×D314yes—
248C2×C32×C4C22×D314yes—
249C2×C32×C4C22×D311yescharacteristic
250C2×C32×C4C22×D311yescharacteristic
251C2×C32×C4C22×D311yescharacteristic
252#638—62no—
253C2×C3⋊Dic3—31no—
254C2×C3⋊Dic3—31no—
255C2×C3⋊Dic3—31no—
256C2×C3⋊Dic3—31no—
257C2×C3⋊Dic3—34no—
258C2×C3⋊Dic3—34no—
259C2×C3⋊Dic3—34no—
260C2×C3⋊Dic3—34no—
261C2×C3⋊Dic3—64no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3—64no—
264C2×C3⋊Dic3—64no—
265C2×C3⋊Dic3—62no—
266C2×C3⋊Dic3—62no—
267C2×C3⋊Dic3C22×D311yescharacteristic
268C2×C3⋊Dic3C22×D311yescharacteristic
269C2×C3⋊Dic3C22×D311yescharacteristic
270C4×C3⋊D3—62no—
271#634—62no—
272#634—68no—
273#634—616no—
274C2×C3×Dic3—34no—
275C2×C3×Dic3—34no—
276C2×C3×Dic3—34no—
277C2×C3×Dic3C22×D314yes—
278C2×C3×Dic3C22×D314yes—
279C2×C3×Dic3C22×D314yes—
280C3×C4×D3—28no—
281#625—68no—
282D3×Dic3—68no—
283D3×Dic3—68no—
284D3×Dic3—68no—
285#552—271noSylow
286C2×C33C2×#4111yescharacteristic
287C2×C33C2×#4111yescharacteristic
288C2×C332−1+411yescharacteristic
289C3×C3⋊D3—41no—
290C32×D3—22no—
291C22×C3×C4—34no—
292C2×#67—94no—
293C2×#67—91no—
294C22×Dic3—94no—
295C22×Dic3—94no—
296C22×Dic3—91no—
297C2×C4×D3—94no—
298C2×C4×D3—91no—
299C2×C4×D3—91no—
300C2×C4×D3C2×C3⋊D311yescharacteristic
301C2×Dic6—94no—
302C2×Dic6—91no—
303C2×Dic6—184no—
304C3×C4⋊C4—34no—
305C3×C4⋊C4—34no—
306C3×C4⋊C4—91no—
307C3×C4⋊C4—91no—
308C3×C4⋊C4—91no—
309C3×C22⋊C4—34no—
310C3×C22⋊C4—34no—
311C3×C22⋊C4—91no—
312C3×C22⋊C4—91no—
313C3×C22⋊C4—91no—
314C3×C42—24no—
315C3×C42—34no—
316C3×C42D3214yes—
317C3×C42C2×C3⋊D311yescharacteristic
318#217—94no—
319#217—91no—
320#217—91no—
321(C2×D3)⋊C4—34no—
322(C2×D3)⋊C4—34no—
323(C2×D3)⋊C4—34no—
324(C2×D3)⋊C4—94no—
325(C2×D3)⋊C4—94no—
326(C2×D3)⋊C4—91no—
327(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
328(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
329C4⋊Dic3—94no—
330C4⋊Dic3—94no—
331C4⋊Dic3—94no—
332C4⋊Dic3—91no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—184no—
337Dic3⋊C4—34no—
338Dic3⋊C4—34no—
339Dic3⋊C4—34no—
340Dic3⋊C4—94no—
341Dic3⋊C4—94no—
342Dic3⋊C4—94no—
343Dic3⋊C4—94no—
344Dic3⋊C4—94no—
345Dic3⋊C4—94no—
346Dic3⋊C4—94no—
347Dic3⋊C4—94no—
348Dic3⋊C4—91no—
349Dic3⋊C4—91no—
350Dic3⋊C4—91no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—184no—
356Dic3⋊C4—184no—
357Dic3⋊C4C2×C3⋊D311yescharacteristic
358Dic3⋊C4C2×C3⋊D311yescharacteristic
359C4×Dic3—94no—
360C4×Dic3—94no—
361C4×Dic3—94no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—184no—
366C4×Dic3—184no—
367C4×Dic3C2×C3⋊D311yescharacteristic
368C22×C32—24no—
369C22×C32—32no—
370C22×C32—61no—
371C22×C32—61no—
372C22×C32C23×D314yes—
373C22×C32C23×D311yescharacteristic
374C2×C3⋊D3—21no—
375C2×C3⋊D3—21no—
376C2×C3⋊D3—21no—
377C2×C3×D3—24no—
378C2×C3×D3—24no—
379C2×C3×D3—64no—
380C2×C3×D3—64no—
381C2×C3×D3—64no—
382C2×C3×D3#23518yes—
383C32×C4—28no—
384C32×C4—28no—
385C32×C4—24no—
386C32×C4—21no—
387C32×C4—21no—
388C32×C4—24no—
389C32×C4—21no—
390C32×C4—24no—
391C32×C4—44no—
392C32×C4—61no—
393C32×C4—61no—
394C32×C4—61no—
395C3⋊Dic3—21no—
396C3⋊Dic3—21no—
397C3⋊Dic3—21no—
398C3⋊Dic3—32no—
399C3⋊Dic3—38no—
400C3⋊Dic3—68no—
401C3⋊Dic3—68no—
402C3⋊Dic3—64no—
403C3⋊Dic3—61no—
404C3⋊Dic3—61no—
405C3⋊Dic3—61no—
406C3⋊Dic3—64no—
407C3⋊Dic3—64no—
408C3⋊Dic3—124no—
409C3⋊Dic3—124no—
410C3×Dic3—24no—
411C3×Dic3—24no—
412C3×Dic3—24no—
413C3×Dic3—64no—
414C3×Dic3—64no—
415C3×Dic3—64no—
416#126—91no—
417#126—271no—
418#125—271no—
419#125—271no—
420#125—271no—
421#123—271no—
422#123—271no—
423C22⋊Q8—271no—
424C22⋊Q8—271no—
425C4×Q8—271no—
426C4×D4—271no—
427#117—31no—
428#117—271no—
429#117—271no—
430C2×C4⋊C4—271no—
431C33#55211yescharacteristic, complemented, Sylow
432C23×C3—34no—
433C23×C3—91no—
434C22×D3—34no—
435C22×D3C22×C3⋊D311yescharacteristic
436C2×C3×C4—24no—
437C2×C3×C4—24no—
438C2×C3×C4—24no—
439C2×C3×C4—34no—
440C2×C3×C4—34no—
441C2×C3×C4—34no—
442C2×C3×C4—68no—
443C2×C3×C4—91no—
444C2×C3×C4—91no—
445C2×C3×C4—91no—
446C2×C3×C4C2×D3214yes—
447C2×C3×C4C2×D3214yes—
448C2×C3×C4C2×D3214yes—
449C2×C3×C4C22×C3⋊D311yescharacteristic
450C2×C3×C4C22×C3⋊D311yescharacteristic
451C2×C3×C4C22×C3⋊D311yescharacteristic
452#67—188no—
453#67—182no—
454C2×Dic3—34no—
455C2×Dic3—34no—
456C2×Dic3—34no—
457C2×Dic3—94no—
458C2×Dic3—91no—
459C2×Dic3—94no—
460C2×Dic3—94no—
461C2×Dic3—94no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—184no—
466C2×Dic3—184no—
467C2×Dic3—184no—
468C2×Dic3—184no—
469C2×Dic3—182no—
470C2×Dic3—188no—
471C2×Dic3—188no—
472C2×Dic3C22×C3⋊D311yescharacteristic
473C2×Dic3C22×C3⋊D311yescharacteristic
474C2×Dic3C22×C3⋊D311yescharacteristic
475C4×D3—22no—
476C4×D3—188no—
477C4×D3—182no—
478C4×D3—182no—
479Dic6—188no—
480Dic6—182no—
481Dic6—1816no—
482C2×C32—24no—
483C2×C32—24no—
484C2×C32—24no—
485C2×C32—62no—
486C2×C32C2×#23514yes—
487C2×C32D3×#4114yes—
488C2×C32D3×#4111yescharacteristic
489C2×C32D3×#4111yescharacteristic
490C2×C32#101014yes—
491C2×C32#101011yescharacteristic
492C3⋊D3—41no—
493C3×D3—28no—
494C3×D3—124no—
495C2×Q8—271no—
496C2×D4—271no—
497C22×C4—31no—
498C22×C4—271no—
499C22×C4—271no—
500C22×C4—271no—
501C4⋊C4—31no—
502C4⋊C4—31no—
503C4⋊C4—91no—
504C4⋊C4—91no—
505C4⋊C4—91no—
506C4⋊C4—271no—
507C4⋊C4—271no—
508C4⋊C4—271no—
509C4⋊C4—271no—
510C4⋊C4—271no—
511C4⋊C4—271no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—271no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C22⋊C4—31no—
518C22⋊C4—31no—
519C22⋊C4—91no—
520C22⋊C4—91no—
521C22⋊C4—91no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C42—31no—
526C42—271no—
527C42—271no—
528C42—271no—
529C42D3×C3⋊D311yescharacteristic
530C22×C3—24no—
531C22×C3—38no—
532C22×C3—64no—
533C22×C3—64no—
534C22×C3—181no—
535C22×C3—181no—
536C22×C3—181no—
537C22×C3C22×D3214yes—
538C22×C3C23×C3⋊D311yescharacteristic
539C2×D3—21no—
540C2×D3—21no—
541C2×D3—64no—
542C2×D3—64no—
543C2×D3—64no—
544C2×D3#382812yes—
545C3×C4—28no—
546C3×C4—28no—
547C3×C4—24no—
548C3×C4—24no—
549C3×C4—24no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—44no—
554C3×C4—64no—
555C3×C4—64no—
556C3×C4—64no—
557C3×C4—181no—
558C3×C4—181no—
559C3×C4—181no—
560Dic3—21no—
561Dic3—21no—
562Dic3—21no—
563Dic3—64no—
564Dic3—64no—
565Dic3—64no—
566Dic3—98no—
567Dic3—92no—
568Dic3—188no—
569Dic3—188no—
570Dic3—184no—
571Dic3—184no—
572Dic3—184no—
573Dic3—181no—
574Dic3—181no—
575Dic3—181no—
576Dic3—364no—
577Dic3—364no—
578C32—24no—
579C32#553114yescomplemented
580C32#570811yescharacteristic, complemented
581C23—31no—
582C23—91no—
583Q8—542no—
584D4—542no—
585C2×C4—31no—
586C2×C4—31no—
587C2×C4—31no—
588C2×C4—62no—
589C2×C4—91no—
590C2×C4—91no—
591C2×C4—91no—
592C2×C4—271no—
593C2×C4—271no—
594C2×C4—271no—
595C2×C4—271no—
596C2×C4—542no—
597C2×C4—542no—
598C2×C4—542no—
599C2×C4C2×D3×C3⋊D311yescharacteristic
600C2×C4C2×D3×C3⋊D311yescharacteristic
601C2×C4C2×D3×C3⋊D311yescharacteristic
602C2×C3—24no—
603C2×C3—24no—
604C2×C3—24no—
605C2×C3—68no—
606C2×C3—361no—
607C2×C3#6429814yes—
608C2×C3D3×#23514yes—
609C2×C3#6430614yes—
610C2×C3C2×#382811yescharacteristic
611C2×C3#41×C3⋊D311yescharacteristic
612C2×C3#6436711yescharacteristic
613D3—22no—
614D3—124no—
615C22—32no—
616C22—61no—
617C22—61no—
618C22—181no—
619C22—181no—
620C22—181no—
621C22C22×D3×C3⋊D311yescharacteristic
622C4—21no—
623C4—21no—
624C4—21no—
625C4—61no—
626C4—61no—
627C4—61no—
628C4—181no—
629C4—181no—
630C4—181no—
631C4—272no—
632C4—541no—
633C4—541no—
634C4—541no—
635C3—24no—
636C3#1059391914yescomplemented
637C3#1059600211yescharacteristic, complemented
638C2—62no—
639C2—361no—
640C2#1172456311yescharacteristic
641C2D3×#382811yescharacteristic
642C2#1172457811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC2×#6290C22
hypercenterC42D3×C3⋊D3
hypocenterC33#552
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6308C2semidirect
2. C3×#64085C2semidirect
3. C32×#872C2semidirect
6. #11723497C2semidirect
7. #11723496C2semidirect
9. #11723464C2semidirect
10. #11723460C2semidirect
11. D3×#3750C2semidirect
12. #11723448C2semidirect
13. #11723444C2semidirect
14. #11723433C2semidirect
15. #11723409C2semidirect
64. #64085D3semidirect
71. C3×#872D3semidirect
230. #872C3⋊D3semidirect
431. C33#552semidirect
579. C32#5531semidirect
580. C32#5708semidirect
636. C3#10593919semidirect
637. C3#10596002semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C3×#640852
C32×#8722
#117235132
#117235082
#117234972
#117234962
#117234802
#117234642
#117234602
D3×#37502
#117234482
#117234442
#117234332
#117234092
#105960023
#105939193