Finite Group Databasebeta

← #13176498

Subgroups — #13176498

Order 1728.

1376 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723930C211yesmaximal, characteristic, complemented
2C3×#64098C211yesmaximal, characteristic, complemented
3C32×#892C211yesmaximal, characteristic
4#11723534C211yesmaximal, characteristic, complemented
5C4×#6293C211yesmaximal, characteristic, complemented
6#11723444C211yesmaximal, characteristic, complemented
7#11723432C211yesmaximal, characteristic
8#11723430C211yesmaximal, characteristic
9#11723428C211yesmaximal, characteristic
10#11723427C211yesmaximal, characteristic
11#11723415C211yesmaximal, characteristic
12#11723415C211yesmaximal, characteristic
13#11723412C211yesmaximal, characteristic, complemented
14#11723405C211yesmaximal, characteristic, complemented
15#11723389C211yesmaximal, characteristic, complemented
16#10595989—31nomaximal
17#10594598—34nomaximal
18C2×C32×C4×D3C2211yescharacteristic
19C2×#6293C2211yescharacteristic
20C2×#6293C2211yescharacteristic
21C33×C4⋊C4C2211yescharacteristic
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
23#88857C2211yescharacteristic
24#88857C2211yescharacteristic
25#88857C2211yescharacteristic
26#88857C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×#3750C2211yescharacteristic
30C3×#3750C2211yescharacteristic
31C3×#3750C2211yescharacteristic
32C3×#3750C2211yescharacteristic
33C3×C4×C3⋊Dic3C2211yescharacteristic
34C32×(C2×D3)⋊C4C2211yescharacteristic
35C32×(C2×D3)⋊C4C2211yescharacteristic
36C32×C4⋊Dic3C2211yescharacteristic
37C32×C4×Dic3C2211yescharacteristic
38C32×C4×Dic3C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88806C2211yescharacteristic
42#88806C2211yescharacteristic
43Dic3⋊(C3⋊Dic3)C2211yescharacteristic
44Dic3⋊(C3⋊Dic3)C2211yescharacteristic
45Dic3⋊(C3⋊Dic3)C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
48(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
49(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
50(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×#117—31no—
54#64133—31no—
55#64133—31no—
56#64098—31no—
57#64098—31no—
58#64098—31no—
59#64098—34no—
60#64098—64no—
61#64098D311yescharacteristic, complemented
62#64088—31no—
63#64088—31no—
64#64088—31no—
65#64088—31no—
66#64079—31no—
67#64078—31no—
68C4×#634—31no—
69C4×#634—31no—
70C3×#892D314yes—
71C3×#889—34no—
72#63918—34no—
73C4×#627—34no—
74#63867—34no—
75#63862—34no—
76#63861—34no—
77#63859—34no—
78#63858—34no—
79#63851—34no—
80#63851—34no—
81#63847—34no—
82#63846—34no—
83#63837—34no—
84C22×C32×D3C2311yescharacteristic
85C2×C33×C4C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C33×C4C2311yescharacteristic
88C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3×C3⋊Dic3C2311yescharacteristic
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C32×Dic3C2311yescharacteristic
97C2×C32×Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C32×C4×D3—22no—
100#6293—22no—
101#6293—22no—
102#5696—91no—
103#5518—94no—
104C22×C32×C4—31no—
105C2×#634—31no—
106C2×#634—31no—
107C2×C3×C4×D3C2×D314yes—
108C2×#627—34no—
109C2×#627—34no—
110C32×C4⋊C4—24no—
111C32×C4⋊C4—31no—
112C32×C4⋊C4C2×D314yes—
113C32×C4⋊C4C2×D311yescharacteristic
114C32×C22⋊C4—31no—
115C32×C22⋊C4—31no—
116C32×C42—31no—
117C32×C42—31no—
118#3757—31no—
119#3757—31no—
120#3757—31no—
121#3757—31no—
122C4⋊(C3⋊Dic3)—31no—
123C4⋊(C3⋊Dic3)—34no—
124C4⋊(C3⋊Dic3)—31no—
125C4⋊(C3⋊Dic3)—31no—
126C4⋊(C3⋊Dic3)—31no—
127C4⋊(C3⋊Dic3)—31no—
128C4⋊(C3⋊Dic3)—64no—
129C4⋊(C3⋊Dic3)C2×D311yescharacteristic
130#3750—31no—
131#3750—31no—
132#3750—31no—
133#3750—31no—
134#3750—34no—
135#3750—34no—
136#3750—34no—
137#3750—34no—
138#3750—31no—
139#3750—31no—
140#3750—31no—
141#3750—31no—
142#3750—64no—
143#3750—64no—
144#3750—64no—
145#3750—64no—
146#3750C2×D311yescharacteristic
147#3750C2×D311yescharacteristic
148#3750C2×D311yescharacteristic
149#3750C2×D311yescharacteristic
150C4×C3⋊Dic3—31no—
151C4×C3⋊Dic3—34no—
152C4×C3⋊Dic3—31no—
153C4×C3⋊Dic3—31no—
154C4×C3⋊Dic3—64no—
155C4×C3⋊Dic3C2×D311yescharacteristic
156C3×(C2×D3)⋊C4C2×D314yes—
157C3×(C2×D3)⋊C4C2×D314yes—
158C3×C4⋊Dic3—34no—
159C3×C4⋊Dic3C2×D314yes—
160C3×Dic3⋊C4—34no—
161C3×Dic3⋊C4—34no—
162C3×Dic3⋊C4—34no—
163C3×Dic3⋊C4—34no—
164C3×C4×Dic3—34no—
165C3×C4×Dic3C2×D314yes—
166C3×C4×Dic3C2×D314yes—
167#3724—34no—
168#3724—34no—
169Dic3⋊Dic3—34no—
170Dic3⋊Dic3—34no—
171Dic3⋊Dic3—34no—
172Dic3⋊Dic3—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175(C2×D3)⋊Dic3—34no—
176(C2×D3)⋊Dic3—34no—
177(C2×D3)⋊Dic3—34no—
178(C2×D3)⋊Dic3—34no—
179Dic32—34no—
180Dic32—34no—
181C22×C33C2411yescharacteristic
182C2×C32×D3—21no—
183C2×C32×D3—21no—
184C2×C32×D3—21no—
185C33×C4—21no—
186C33×C4—21no—
187C33×C4—21no—
188C3⋊(C3⋊Dic3)—21no—
189C3⋊(C3⋊Dic3)—21no—
190C3⋊(C3⋊Dic3)—21no—
191C3⋊(C3⋊Dic3)—21no—
192C3×C3⋊Dic3—21no—
193C3×C3⋊Dic3—21no—
194C3×C3⋊Dic3—21no—
195C3×C3⋊Dic3—21no—
196C32×Dic3—21no—
197C32×Dic3—21no—
198C32×Dic3#4112yes—
199C3×#125—91no—
200C3×#117—34no—
201#924—94no—
202#924—94no—
203#898—91no—
204#898—91no—
205#898—91no—
206#896—91no—
207#896—91no—
208#892C3⋊D311yescharacteristic
209#890—91no—
210#889—34no—
211#889—94no—
212#889—94no—
213#889—94no—
214#889—91no—
215#889—91no—
216#889—91no—
217#889—184no—
218#888—91no—
219#887—91no—
220#879—94no—
221#879—94no—
222#879—94no—
223#879—94no—
224#876—91no—
225#870—94no—
226#869—94no—
227C4×Dic6—94no—
228C4×Dic6—94no—
229C4×Dic6—91no—
230C23×C32—31no—
231C22×C3×D3C22×D314yes—
232C2×C32×C4—24no—
233C2×C32×C4—24no—
234C2×C32×C4—24no—
235C2×C32×C4—31no—
236C2×C32×C4—31no—
237C2×C32×C4—31no—
238C2×C32×C4—62no—
239C2×C32×C4C22×D314yes—
240C2×C32×C4C22×D314yes—
241C2×C32×C4C22×D314yes—
242C2×C32×C4C22×D311yescharacteristic
243C2×C32×C4C22×D311yescharacteristic
244C2×C32×C4C22×D311yescharacteristic
245C2×C3⋊Dic3—31no—
246C2×C3⋊Dic3—31no—
247C2×C3⋊Dic3—31no—
248C2×C3⋊Dic3—31no—
249C2×C3⋊Dic3—34no—
250C2×C3⋊Dic3—34no—
251C2×C3⋊Dic3—34no—
252C2×C3⋊Dic3—34no—
253C2×C3⋊Dic3—64no—
254C2×C3⋊Dic3—64no—
255C2×C3⋊Dic3—64no—
256C2×C3⋊Dic3—64no—
257C2×C3⋊Dic3C22×D311yescharacteristic
258C2×C3⋊Dic3C22×D311yescharacteristic
259C2×C3⋊Dic3C22×D311yescharacteristic
260C2×C3⋊Dic3C22×D311yescharacteristic
261#634—62no—
262#634—62no—
263C2×C3×Dic3—34no—
264C2×C3×Dic3—34no—
265C2×C3×Dic3—34no—
266C2×C3×Dic3—34no—
267C2×C3×Dic3C22×D314yes—
268C2×C3×Dic3C22×D314yes—
269C2×C3×Dic3C22×D314yes—
270C3×C4×D3—28no—
271#627—68no—
272#627—68no—
273#541—271noSylow
274C2×C33C2×#4111yescharacteristic
275C2×C332−1+411yescharacteristic
276C2×C332−1+411yescharacteristic
277C32×D3—41no—
278C22×C3×C4—34no—
279C2×C4×D3C2×C3⋊D311yescharacteristic
280C2×Dic6—94no—
281C2×Dic6—94no—
282C2×Dic6—91no—
283C2×Dic6—91no—
284C3×C4⋊C4—24no—
285C3×C4⋊C4—34no—
286C3×C4⋊C4—91no—
287C3×C4⋊C4—91no—
288C3×C4⋊C4—91no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—91no—
291C3×C4⋊C4D3214yes—
292C3×C4⋊C4C2×C3⋊D311yescharacteristic
293C3×C22⋊C4—34no—
294C3×C22⋊C4—34no—
295C3×C42—34no—
296C3×C42—34no—
297C3×C42—91no—
298#217—94no—
299#217—94no—
300#217—94no—
301#217—94no—
302(C2×D3)⋊C4—91no—
303(C2×D3)⋊C4—91no—
304(C2×D3)⋊C4—91no—
305(C2×D3)⋊C4—91no—
306(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
307(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
308C4⋊Dic3—34no—
309C4⋊Dic3—94no—
310C4⋊Dic3—94no—
311C4⋊Dic3—94no—
312C4⋊Dic3—94no—
313C4⋊Dic3—94no—
314C4⋊Dic3—91no—
315C4⋊Dic3—91no—
316C4⋊Dic3—91no—
317C4⋊Dic3—184no—
318C4⋊Dic3C2×C3⋊D311yescharacteristic
319Dic3⋊C4—34no—
320Dic3⋊C4—34no—
321Dic3⋊C4—34no—
322Dic3⋊C4—34no—
323Dic3⋊C4—94no—
324Dic3⋊C4—94no—
325Dic3⋊C4—94no—
326Dic3⋊C4—94no—
327Dic3⋊C4—94no—
328Dic3⋊C4—94no—
329Dic3⋊C4—94no—
330Dic3⋊C4—94no—
331Dic3⋊C4—91no—
332Dic3⋊C4—91no—
333Dic3⋊C4—91no—
334Dic3⋊C4—91no—
335Dic3⋊C4—91no—
336Dic3⋊C4—91no—
337Dic3⋊C4—91no—
338Dic3⋊C4—91no—
339Dic3⋊C4—91no—
340Dic3⋊C4—91no—
341Dic3⋊C4—184no—
342Dic3⋊C4—184no—
343Dic3⋊C4—184no—
344Dic3⋊C4—184no—
345C4×Dic3—34no—
346C4×Dic3—94no—
347C4×Dic3—94no—
348C4×Dic3—94no—
349C4×Dic3—91no—
350C4×Dic3—91no—
351C4×Dic3—91no—
352C4×Dic3—184no—
353C4×Dic3C2×C3⋊D311yescharacteristic
354C4×Dic3C2×C3⋊D311yescharacteristic
355C22×C32—24no—
356C22×C32—61no—
357C22×C32—61no—
358C22×C32—61no—
359C22×C32C23×D314yes—
360C22×C32C23×D311yescharacteristic
361C2×C3×D3—24no—
362C2×C3×D3—24no—
363C2×C3×D3—24no—
364C32×C4—21no—
365C32×C4—21no—
366C32×C4—21no—
367C32×C4—24no—
368C32×C4—24no—
369C32×C4—24no—
370C32×C4—32no—
371C32×C4—44no—
372C32×C4—44no—
373C32×C4—44no—
374C32×C4—61no—
375C32×C4—61no—
376C3⋊Dic3—21no—
377C3⋊Dic3—21no—
378C3⋊Dic3—21no—
379C3⋊Dic3—21no—
380C3⋊Dic3—68no—
381C3⋊Dic3—61no—
382C3⋊Dic3—61no—
383C3⋊Dic3—61no—
384C3⋊Dic3—64no—
385C3⋊Dic3—61no—
386C3⋊Dic3—64no—
387C3⋊Dic3—64no—
388C3⋊Dic3—64no—
389C3⋊Dic3—124no—
390C3⋊Dic3—124no—
391C3⋊Dic3—124no—
392C3×Dic3—24no—
393C3×Dic3—24no—
394C3×Dic3—64no—
395C3×Dic3—64no—
396C3×Dic3—64no—
397C3×Dic3—64no—
398C3×Dic3#23518yes—
399C4⋊Q8—271no—
400#126—271no—
401#126—271no—
402#126—271no—
403#126—271no—
404#125—91no—
405#125—271no—
406#125—271no—
407#125—271no—
408#125—271no—
409C22⋊Q8—271no—
410C22⋊Q8—271no—
411C4×Q8—271no—
412C4×Q8—271no—
413#117—31no—
414C33#54111yescharacteristic, complemented, Sylow
415C23×C3—34no—
416C22×D3C22×C3⋊D311yescharacteristic
417C2×C3×C4—24no—
418C2×C3×C4—24no—
419C2×C3×C4—24no—
420C2×C3×C4—34no—
421C2×C3×C4—34no—
422C2×C3×C4—34no—
423C2×C3×C4—68no—
424C2×C3×C4—91no—
425C2×C3×C4—91no—
426C2×C3×C4—91no—
427C2×C3×C4—91no—
428C2×C3×C4C2×D3214yes—
429C2×C3×C4C2×D3214yes—
430C2×C3×C4C2×D3214yes—
431C2×C3×C4C22×C3⋊D311yescharacteristic
432C2×C3×C4C22×C3⋊D311yescharacteristic
433C2×C3×C4C22×C3⋊D311yescharacteristic
434C2×Dic3—34no—
435C2×Dic3—34no—
436C2×Dic3—34no—
437C2×Dic3—34no—
438C2×Dic3—94no—
439C2×Dic3—94no—
440C2×Dic3—94no—
441C2×Dic3—94no—
442C2×Dic3—91no—
443C2×Dic3—91no—
444C2×Dic3—91no—
445C2×Dic3—91no—
446C2×Dic3—184no—
447C2×Dic3—184no—
448C2×Dic3—184no—
449C2×Dic3—184no—
450C2×Dic3C22×C3⋊D311yescharacteristic
451C2×Dic3C22×C3⋊D311yescharacteristic
452C2×Dic3C22×C3⋊D311yescharacteristic
453C4×D3—22no—
454Dic6—188no—
455Dic6—188no—
456Dic6—182no—
457Dic6—182no—
458C2×C32—24no—
459C2×C32—24no—
460C2×C32—24no—
461C2×C32—121no—
462C2×C32C2×#23514yes—
463C2×C32#100711yescharacteristic
464C2×C32D3×#4111yescharacteristic
465C2×C32#101014yes—
466C2×C32#101014yes—
467C2×C32#101011yescharacteristic
468C3×D3—44no—
469C2×Q8—271no—
470C2×Q8—271no—
471C22×C4—31no—
472C4⋊C4—31no—
473C4⋊C4—91no—
474C4⋊C4—91no—
475C4⋊C4—91no—
476C4⋊C4—91no—
477C4⋊C4—91no—
478C4⋊C4—271no—
479C4⋊C4—271no—
480C4⋊C4—271no—
481C4⋊C4—271no—
482C4⋊C4—271no—
483C4⋊C4—271no—
484C4⋊C4—271no—
485C4⋊C4—271no—
486C4⋊C4—271no—
487C4⋊C4—271no—
488C4⋊C4—271no—
489C4⋊C4—271no—
490C4⋊C4—271no—
491C4⋊C4D3×C3⋊D311yescharacteristic
492C22⋊C4—31no—
493C22⋊C4—31no—
494C22⋊C4—271no—
495C22⋊C4—271no—
496C22⋊C4—271no—
497C22⋊C4—271no—
498C42—31no—
499C42—31no—
500C42—91no—
501C42—271no—
502C42—271no—
503C42—271no—
504C22×C3—24no—
505C22×C3—64no—
506C22×C3—64no—
507C22×C3—64no—
508C22×C3C22×D3214yes—
509C22×C3C23×C3⋊D311yescharacteristic
510C2×D3—21no—
511C2×D3—21no—
512C2×D3—21no—
513C3×C4—24no—
514C3×C4—24no—
515C3×C4—24no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—38no—
520C3×C4—44no—
521C3×C4—44no—
522C3×C4—44no—
523C3×C4—64no—
524C3×C4—64no—
525C3×C4—181no—
526C3×C4—181no—
527C3×C4—181no—
528C3×C4—181no—
529Dic3—21no—
530Dic3—21no—
531Dic3—64no—
532Dic3—64no—
533Dic3—64no—
534Dic3—64no—
535Dic3—188no—
536Dic3—184no—
537Dic3—184no—
538Dic3—184no—
539Dic3—184no—
540Dic3—181no—
541Dic3—181no—
542Dic3—181no—
543Dic3—181no—
544Dic3—364no—
545Dic3—364no—
546Dic3—364no—
547Dic3#382812yes—
548C32—24no—
549C32#551814yescomplemented
550C32#569611yescharacteristic, complemented
551C23—31no—
552Q8—542no—
553Q8—542no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—62no—
558C2×C4—91no—
559C2×C4—91no—
560C2×C4—91no—
561C2×C4—91no—
562C2×C4—271no—
563C2×C4—271no—
564C2×C4—271no—
565C2×C4—271no—
566C2×C4C2×D3×C3⋊D311yescharacteristic
567C2×C4C2×D3×C3⋊D311yescharacteristic
568C2×C4C2×D3×C3⋊D311yescharacteristic
569C2×C3—24no—
570C2×C3—24no—
571C2×C3—24no—
572C2×C3—124no—
573C2×C3D3×#23514yes—
574C2×C3#6430914yes—
575C2×C3#6431514yes—
576C2×C3C2×#382811yescharacteristic
577C2×C3#6436711yescharacteristic
578C2×C3#6436711yescharacteristic
579D3—41no—
580C22—61no—
581C22—61no—
582C22—61no—
583C22C22×D3×C3⋊D311yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—32no—
588C4—61no—
589C4—61no—
590C4—181no—
591C4—181no—
592C4—181no—
593C4—181no—
594C4—541no—
595C4—541no—
596C4—541no—
597C4—541no—
598C3—24no—
599C3#1059459814yescomplemented
600C3#1059598911yescharacteristic, complemented
601C2—121no—
602C2D3×#382811yescharacteristic
603C2#1172458211yescharacteristic
604C2#1172459011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C32×D3C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#541
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723930C2semidirect
2. C3×#64098C2semidirect
4. #11723534C2semidirect
5. C4×#6293C2semidirect
6. #11723444C2semidirect
13. #11723412C2semidirect
14. #11723405C2semidirect
15. #11723389C2semidirect
61. #64098D3semidirect
414. C33#541semidirect
549. C32#5518semidirect
550. C32#5696semidirect
599. C3#10594598semidirect
600. C3#10595989semidirect

Maximal subgroups

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

subgroupindex
#117239302
C3×#640982
C32×#8922
#117235342
C4×#62932
#117234442
#117234322
#117234302
#117234282
#117234272
#117234152
#117234122
#117234052
#117233892
#105959893
#105945983