Finite Group Databasebeta

← #13176451

Subgroups — #13176451

Order 1728.

1416 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723939C211yesmaximal, characteristic, complemented
2C3×#64105C211yesmaximal, characteristic, complemented
3C32×#888C211yesmaximal, characteristic, complemented
4#11723504C211yesmaximal, characteristic
5#11723489C211yesmaximal, characteristic, complemented
6#11723477C211yesmaximal, characteristic, complemented
7#11723453C211yesmaximal, characteristic, complemented
8#11723451C211yesmaximal, characteristic, complemented
9#11723438C211yesmaximal, characteristic, complemented
10#11723435C211yesmaximal, characteristic, complemented
11#11723433C211yesmaximal, characteristic, complemented
12#11723417C211yesmaximal, characteristic
13#11723409C211yesmaximal, characteristic, complemented
14#11723405C211yesmaximal, characteristic, complemented
15#11723392C211yesmaximal, characteristic
16#10596206—31nomaximal
17#10594340—34nomaximal
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C3×#634C2211yescharacteristic
20C2×C32×Dic6C2211yescharacteristic
21C2×C32×Dic6C2211yescharacteristic
22C2×#6292C2211yescharacteristic
23C2×#6289C2211yescharacteristic
24C33×C4⋊C4C2211yescharacteristic
25#88859C2211yescharacteristic
26#88859C2211yescharacteristic
27#88859C2211yescharacteristic
28C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
29#88857C2211yescharacteristic
30C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
31C3×#3752C2211yescharacteristic
32C3×#3752C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×#3750C2211yescharacteristic
35C32×C4⋊Dic3C2211yescharacteristic
36C32×Dic3⋊C4C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×C4×Dic3C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88806C2211yescharacteristic
42#88806C2211yescharacteristic
43#88806C2211yescharacteristic
44#88806C2211yescharacteristic
45#88806C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47#88804C2211yescharacteristic
48#88804C2211yescharacteristic
49#88803C2211yescharacteristic
50#88803C2211yescharacteristic
51#88803C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×C4⋊Q8—31no—
54#64156—31no—
55#64155—31no—
56#64155—31no—
57#64153—31no—
58#64107—31no—
59#64107—31no—
60#64107—34no—
61#64107—64no—
62#64105—31no—
63#64105D311yescharacteristic, complemented
64#64103—31no—
65#64103—31no—
66#64098—31no—
67#64098—31no—
68#64085—31no—
69#64085—31no—
70C3×#896—34no—
71C3×#888D314yescomplemented
72#63907—34no—
73#63888—34no—
74#63888—34no—
75#63873—34no—
76#63868—34no—
77#63862—34no—
78#63859—34no—
79#63856—34no—
80#63854—34no—
81#63850—34no—
82#63847—34no—
83#63845—34no—
84C22×C3⋊(C3⋊D3)C2311yescharacteristic
85C22×C3×C3⋊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⋊Dic3C2311yescharacteristic
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C3×C4×C3⋊D3—41no—
96C3×#634—41no—
97C2×C32×Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101C32×Dic6—41no—
102C32×Dic6—41no—
103#6292—41no—
104#6289—41no—
105#5735—94no—
106#5646—91no—
107C2×C32×Q8—31no—
108C2×C32×Q8—31no—
109C2×C4⋊(C3⋊D3)—31no—
110C2×C4×C3⋊D3—31no—
111C2×C4×C3⋊D3C2×D311yescharacteristic
112C2×#634C2×D311yescharacteristic
113C2×C3×C4×D3—34no—
114C2×C3×Dic6—34no—
115C2×C3×Dic6C2×D314yes—
116C2×C3×Dic6C2×D314yes—
117C2×#626—34no—
118C2×#624—34no—
119C32×C4⋊C4—24no—
120C32×C4⋊C4—31no—
121C32×C4⋊C4—31no—
122C32×C4⋊C4—31no—
123C32×C4⋊C4C2×D314yes—
124C32×C4⋊C4C2×D311yescharacteristic
125C32×C42—31no—
126#3752—31no—
127#3752—31no—
128#3752—31no—
129#3752—31no—
130#3752—31no—
131#3752—31no—
132#3752—34no—
133#3752—34no—
134#3752—34no—
135#3752—31no—
136#3752—31no—
137#3752—64no—
138#3752—64no—
139#3752—64no—
140#3752C2×D311yescharacteristic
141#3752C2×D311yescharacteristic
142C4⋊(C3⋊Dic3)—31no—
143C4⋊(C3⋊Dic3)—34no—
144C4⋊(C3⋊Dic3)—31no—
145C4⋊(C3⋊Dic3)—64no—
146#3750—31no—
147#3750—34no—
148#3750—31no—
149#3750—31no—
150#3750—31no—
151#3750—31no—
152#3750—31no—
153#3750—31no—
154#3750—31no—
155#3750—64no—
156#3750C2×D311yescharacteristic
157#3750C2×D311yescharacteristic
158C4×C3⋊Dic3—31no—
159C4×C3⋊Dic3—34no—
160C4×C3⋊Dic3—31no—
161C4×C3⋊Dic3—64no—
162C3×(C2×D3)⋊C4—34no—
163C3×(C2×D3)⋊C4—34no—
164C3×C4⋊Dic3C2×D314yes—
165C3×Dic3⋊C4—34no—
166C3×Dic3⋊C4—34no—
167C3×Dic3⋊C4C2×D314yes—
168C3×Dic3⋊C4C2×D314yes—
169C3×C4×Dic3C2×D314yes—
170#3724—34no—
171#3724—34no—
172#3724—34no—
173#3724—34no—
174#3724—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178#3722—34no—
179#3722—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×C3×C3⋊D3—21no—
189C2×C3×C3⋊D3—21no—
190C2×C3×C3⋊D3—21no—
191C33×C4—21no—
192C33×C4—21no—
193C33×C4—21no—
194C3⋊(C3⋊Dic3)—21no—
195C3⋊(C3⋊Dic3)—21no—
196C3⋊(C3⋊Dic3)—21no—
197C3×C3⋊Dic3—21no—
198C3×C3⋊Dic3—21no—
199C3×C3⋊Dic3—21no—
200C32×Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3—21no—
204C3×C4⋊Q8—34no—
205C3×C22⋊Q8—91no—
206#947—94no—
207#946—94no—
208#946—94no—
209#946—91no—
210#944—94no—
211#944—91no—
212#929—91no—
213#924—91no—
214#898—94no—
215#898—94no—
216#898—91no—
217#898—91no—
218#898—184no—
219#896—34no—
220#896—94no—
221#896—91no—
222#894—94no—
223#894—94no—
224#890—91no—
225#889—94no—
226#889—94no—
227#889—91no—
228#888C3⋊D311yescharacteristic, complemented
229#885—91no—
230#883—91no—
231#879—91no—
232#879—91no—
233#876—94no—
234#876—94no—
235C22×C3⋊D3—31no—
236C22×C3⋊D3—34no—
237C22×C3⋊D3—64no—
238C22×C3⋊D3C22×D311yescharacteristic
239C22×C3×D3—34no—
240C32×Q8—121no—
241C32×Q8—121no—
242C2×C32×C4—24no—
243C2×C32×C4—24no—
244C2×C32×C4—24no—
245C2×C32×C4—31no—
246C2×C32×C4—31no—
247C2×C32×C4—31no—
248C2×C32×C4—31no—
249C2×C32×C4C22×D314yes—
250C2×C32×C4C22×D314yes—
251C2×C32×C4C22×D314yes—
252C2×C32×C4C22×D311yescharacteristic
253C2×C32×C4C22×D311yescharacteristic
254C2×C32×C4C22×D311yescharacteristic
255C2×C3⋊Dic3—31no—
256C2×C3⋊Dic3—31no—
257C2×C3⋊Dic3—31no—
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⋊Dic3C22×D311yescharacteristic
265C2×C3⋊Dic3C22×D311yescharacteristic
266C2×C3⋊Dic3C22×D311yescharacteristic
267C4⋊(C3⋊D3)—121no—
268C4×C3⋊D3—41no—
269C4×C3⋊D3—121no—
270#634—41no—
271C2×C3×Dic3—34no—
272C2×C3×Dic3—34no—
273C2×C3×Dic3—34no—
274C2×C3×Dic3C22×D314yes—
275C2×C3×Dic3C22×D314yes—
276C2×C3×Dic3C22×D314yes—
277C2×C3×Dic3C22×D314yes—
278C3×C4×D3—124no—
279C3×Dic6—44no—
280C3×Dic6—44no—
281C3×Dic6—124no—
282#626—124no—
283#624—124no—
284#563—271noSylow
285C2×C33H(5,𝔽2)11yescharacteristic
286C2×C332−1+411yescharacteristic
287C2×C332−1+411yescharacteristic
288C3⋊(C3⋊D3)—41no—
289C3×C3⋊D3—41no—
290C2×C3×Q8—34no—
291C2×C3×Q8—34no—
292C2×C3×Q8—91no—
293C22×C3×C4—91no—
294C2×#67—91no—
295C2×D12—94no—
296C2×C4×D3—34no—
297C2×C4×D3—94no—
298C2×C4×D3—91no—
299C2×Dic6—34no—
300C2×Dic6C2×C3⋊D311yescharacteristic
301C2×Dic6C2×C3⋊D311yescharacteristic
302C3×C4⋊C4—24no—
303C3×C4⋊C4—34no—
304C3×C4⋊C4—34no—
305C3×C4⋊C4—34no—
306C3×C4⋊C4—91no—
307C3×C4⋊C4—91no—
308C3×C4⋊C4D3214yes—
309C3×C4⋊C4C2×C3⋊D311yescharacteristic
310C3×C22⋊C4—91no—
311C3×C22⋊C4—91no—
312C3×C42—34no—
313#217—91no—
314#217—91no—
315#217—91no—
316(C2×D3)⋊C4—34no—
317(C2×D3)⋊C4—34no—
318(C2×D3)⋊C4—94no—
319(C2×D3)⋊C4—94no—
320(C2×D3)⋊C4—94no—
321(C2×D3)⋊C4—94no—
322(C2×D3)⋊C4—94no—
323(C2×D3)⋊C4—94no—
324(C2×D3)⋊C4—94no—
325(C2×D3)⋊C4—94no—
326(C2×D3)⋊C4—91no—
327(C2×D3)⋊C4—91no—
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4—91no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—184no—
332(C2×D3)⋊C4—184no—
333(C2×D3)⋊C4—184no—
334C4⋊Dic3—94no—
335C4⋊Dic3—94no—
336C4⋊Dic3—91no—
337C4⋊Dic3—91no—
338C4⋊Dic3—91no—
339C4⋊Dic3—184no—
340C4⋊Dic3C2×C3⋊D311yescharacteristic
341Dic3⋊C4—34no—
342Dic3⋊C4—34no—
343Dic3⋊C4—94no—
344Dic3⋊C4—94no—
345Dic3⋊C4—94no—
346Dic3⋊C4—94no—
347Dic3⋊C4—94no—
348Dic3⋊C4—94no—
349Dic3⋊C4—94no—
350Dic3⋊C4—94no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—184no—
359Dic3⋊C4C2×C3⋊D311yescharacteristic
360Dic3⋊C4C2×C3⋊D311yescharacteristic
361C4×Dic3—94no—
362C4×Dic3—94no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—184no—
366C4×Dic3C2×C3⋊D311yescharacteristic
367C22×C32—24no—
368C22×C32C23×D314yes—
369C22×C32C23×D311yescharacteristic
370C2×C3⋊D3—21no—
371C2×C3⋊D3—21no—
372C2×C3⋊D3—21no—
373C2×C3⋊D3—61no—
374C2×C3⋊D3—61no—
375C2×C3⋊D3—61no—
376C2×C3⋊D3—64no—
377C2×C3⋊D3—64no—
378C2×C3⋊D3—64no—
379C2×C3⋊D3—124no—
380C2×C3⋊D3—124no—
381C2×C3⋊D3—124no—
382C2×C3×D3—64no—
383C2×C3×D3—64no—
384C2×C3×D3—64no—
385C32×C4—21no—
386C32×C4—21no—
387C32×C4—21no—
388C32×C4—24no—
389C32×C4—24no—
390C32×C4—24no—
391C32×C4—44no—
392C32×C4—44no—
393C32×C4—44no—
394C32×C4—61no—
395C32×C4—61no—
396C32×C4—61no—
397C32×C4—61no—
398C3⋊Dic3—21no—
399C3⋊Dic3—21no—
400C3⋊Dic3—21no—
401C3⋊Dic3—61no—
402C3⋊Dic3—61no—
403C3⋊Dic3—61no—
404C3⋊Dic3—64no—
405C3⋊Dic3—64no—
406C3⋊Dic3—64no—
407C3⋊Dic3—124no—
408C3⋊Dic3—124no—
409C3⋊Dic3—124no—
410C3×Dic3—24no—
411C3×Dic3—24no—
412C3×Dic3—24no—
413C3×Dic3—24no—
414C3×Dic3—64no—
415C3×Dic3—64no—
416C3×Dic3—64no—
417C4⋊Q8—31no—
418C4⋊Q8—271no—
419#126—271no—
420#126—271no—
421#126—271no—
422#126—271no—
423#125—271no—
424#125—271no—
425#124—271no—
426#123—271no—
427#123—271no—
428C22⋊Q8—91no—
429C22⋊Q8—271no—
430C22⋊Q8—271no—
431C22⋊Q8—271no—
432C33#56311yescharacteristic, complemented, Sylow
433C23×C3—91no—
434C22×D3—34no—
435C22×D3—94no—
436C22×D3—91no—
437C22×D3—184no—
438C3×Q8—124no—
439C3×Q8—124no—
440C3×Q8—361no—
441C2×C3×C4—24no—
442C2×C3×C4—24no—
443C2×C3×C4—24no—
444C2×C3×C4—34no—
445C2×C3×C4—34no—
446C2×C3×C4—34no—
447C2×C3×C4—34no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4—91no—
451C2×C3×C4—361no—
452C2×C3×C4C2×D3214yes—
453C2×C3×C4C2×D3214yes—
454C2×C3×C4C2×D3214yes—
455C2×C3×C4C22×C3⋊D311yescharacteristic
456C2×C3×C4C22×C3⋊D311yescharacteristic
457C2×C3×C4C22×C3⋊D311yescharacteristic
458#67—361no—
459C2×Dic3—34no—
460C2×Dic3—34no—
461C2×Dic3—34no—
462C2×Dic3—94no—
463C2×Dic3—94no—
464C2×Dic3—94no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—91no—
468C2×Dic3—184no—
469C2×Dic3—184no—
470C2×Dic3—184no—
471C2×Dic3C22×C3⋊D311yescharacteristic
472C2×Dic3C22×C3⋊D311yescharacteristic
473C2×Dic3C22×C3⋊D311yescharacteristic
474C2×Dic3C22×C3⋊D311yescharacteristic
475D12—364no—
476C4×D3—124no—
477C4×D3—361no—
478C4×D3—364no—
479Dic6—41no—
480Dic6—41no—
481Dic6—124no—
482C2×C32—24no—
483C2×C32—24no—
484C2×C32—24no—
485C2×C32#100414yes—
486C2×C32#100411yescharacteristic
487C2×C32#100714yes—
488C2×C32#100714yes—
489C2×C32#100711yescharacteristic
490C2×C32#101011yescharacteristic
491C3⋊D3—41no—
492C3⋊D3—121no—
493C3⋊D3—124no—
494C3⋊D3—244no—
495C3×D3—124no—
496C2×Q8—31no—
497C2×Q8—31no—
498C2×Q8—91no—
499C2×D4—271no—
500C22×C4—91no—
501C22×C4—271no—
502C4⋊C4—31no—
503C4⋊C4—31no—
504C4⋊C4—31no—
505C4⋊C4—91no—
506C4⋊C4—91no—
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—
517C4⋊C4D3×C3⋊D311yescharacteristic
518C22⋊C4—91no—
519C22⋊C4—91no—
520C22⋊C4—271no—
521C22⋊C4—271no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C22⋊C4—271no—
526C22⋊C4—271no—
527C22⋊C4—271no—
528C42—31no—
529C42—271no—
530C42—271no—
531C22×C3—24no—
532C22×C3—181no—
533C22×C3—181no—
534C22×C3—181no—
535C22×C3C22×D3214yes—
536C22×C3C23×C3⋊D311yescharacteristic
537C2×D3—64no—
538C2×D3—64no—
539C2×D3—64no—
540C2×D3—184no—
541C2×D3—184no—
542C2×D3—184no—
543C2×D3—181no—
544C2×D3—181no—
545C2×D3—181no—
546C2×D3—364no—
547C2×D3—364no—
548C2×D3—364no—
549C3×C4—24no—
550C3×C4—24no—
551C3×C4—24no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—44no—
556C3×C4—44no—
557C3×C4—44no—
558C3×C4—64no—
559C3×C4—64no—
560C3×C4—64no—
561C3×C4—64no—
562C3×C4—181no—
563C3×C4—181no—
564C3×C4—181no—
565Dic3—21no—
566Dic3—21no—
567Dic3—21no—
568Dic3—21no—
569Dic3—64no—
570Dic3—64no—
571Dic3—64no—
572Dic3—184no—
573Dic3—184no—
574Dic3—184no—
575Dic3—181no—
576Dic3—181no—
577Dic3—181no—
578Dic3—364no—
579Dic3—364no—
580Dic3—364no—
581C32—24no—
582C32#564611yescharacteristic, complemented
583C32#573514yescomplemented
584C23—91no—
585C23—271no—
586Q8—121no—
587Q8—121no—
588Q8—361no—
589D4—1081no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—31no—
593C2×C4—31no—
594C2×C4—91no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—271no—
598C2×C4—271no—
599C2×C4—271no—
600C2×C4—361no—
601C2×C4—1081no—
602C2×C4C2×D3×C3⋊D311yescharacteristic
603C2×C4C2×D3×C3⋊D311yescharacteristic
604C2×C4C2×D3×C3⋊D311yescharacteristic
605C2×C3—24no—
606C2×C3—24no—
607C2×C3—24no—
608C2×C3—361no—
609C2×C3#6429714yes—
610C2×C3#6431314yes—
611C2×C3#6431614yes—
612C2×C3#6436111yescharacteristic
613C2×C3#6436411yescharacteristic
614C2×C3#6436411yescharacteristic
615D3—124no—
616D3—364no—
617D3—361no—
618D3—724no—
619C22—181no—
620C22—181no—
621C22—181no—
622C22—541no—
623C22—541no—
624C22—541no—
625C22C22×D3×C3⋊D311yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—61no—
630C4—61no—
631C4—61no—
632C4—61no—
633C4—181no—
634C4—181no—
635C4—181no—
636C4—541no—
637C4—541no—
638C4—541no—
639C3—24no—
640C3#1059434014yescomplemented
641C3#1059620611yescharacteristic, complemented
642C2—361no—
643C2—1081no—
644C2#1172456611yescharacteristic
645C2#1172458711yescharacteristic
646C2#1172459111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×#6292C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723939C2semidirect
2. C3×#64105C2semidirect
3. C32×#888C2semidirect
5. #11723489C2semidirect
6. #11723477C2semidirect
7. #11723453C2semidirect
8. #11723451C2semidirect
9. #11723438C2semidirect
10. #11723435C2semidirect
11. #11723433C2semidirect
13. #11723409C2semidirect
14. #11723405C2semidirect
63. #64105D3semidirect
71. C3×#888D3semidirect
228. #888C3⋊D3semidirect
432. C33#563semidirect
582. C32#5646semidirect
583. C32#5735semidirect
640. C3#10594340semidirect
641. C3#10596206semidirect

Maximal subgroups

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

subgroupindex
#117239392
C3×#641052
C32×#8882
#117235042
#117234892
#117234772
#117234532
#117234512
#117234382
#117234352
#117234332
#117234172
#117234092
#117234052
#117233922
#105962063
#105943403