Finite Group Databasebeta

← #13176445

Subgroups — #13176445

Order 1728.

1400 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723931C211yesmaximal, characteristic, complemented
2C3×#64105C211yesmaximal, characteristic, complemented
3C32×#896C211yesmaximal, characteristic, complemented
4#11723530C211yesmaximal, characteristic
5#11723518C211yesmaximal, characteristic
6#11723513C211yesmaximal, characteristic
7#11723448C211yesmaximal, characteristic, complemented
8#11723435C211yesmaximal, characteristic, complemented
9#11723430C211yesmaximal, characteristic, complemented
10#11723430C211yesmaximal, characteristic, complemented
11#11723425C211yesmaximal, characteristic, complemented
12#11723408C211yesmaximal, characteristic, complemented
13#11723408C211yesmaximal, characteristic, complemented
14#11723394C211yesmaximal, characteristic, complemented
15#11723393C211yesmaximal, characteristic, complemented
16#10596117—31nomaximal
17#10594589—34nomaximal
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C3×#634C2211yescharacteristic
20C2×C32×C4×D3C2211yescharacteristic
21C2×C32×Dic6C2211yescharacteristic
22C2×#6293C2211yescharacteristic
23C2×#6290C2211yescharacteristic
24C33×C4⋊C4C2211yescharacteristic
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
26C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
27C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
28#88857C2211yescharacteristic
29#88857C2211yescharacteristic
30C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
31C3×#3752C2211yescharacteristic
32C3×#3752C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×#3750C2211yescharacteristic
35C32×(C2×D3)⋊C4C2211yescharacteristic
36C32×(C2×D3)⋊C4C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×Dic3⋊C4C2211yescharacteristic
39#88807C2211yescharacteristic
40#88806C2211yescharacteristic
41#88806C2211yescharacteristic
42#88806C2211yescharacteristic
43#88806C2211yescharacteristic
44Dic3⋊(C3⋊Dic3)C2211yescharacteristic
45#88803C2211yescharacteristic
46#88803C2211yescharacteristic
47#88803C2211yescharacteristic
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×C22⋊Q8—31no—
54#64155—31no—
55#64153—31no—
56#64138—31no—
57#64133—31no—
58#64107—31no—
59#64107—31no—
60#64105D311yescharacteristic, complemented
61#64099—31no—
62#64099—34no—
63#64099—64no—
64#64098—31no—
65#64097—31no—
66#64094—31no—
67#64092—31no—
68#64088—31no—
69#64088—31no—
70C3×#896—34no—
71C3×#896D314yescomplemented
72#63915—34no—
73#63905—34no—
74#63905—34no—
75#63870—34no—
76#63861—34no—
77#63861—34no—
78#63861—34no—
79#63861—34no—
80#63856—34no—
81#63856—34no—
82#63839—34no—
83#63839—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
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C3×C4×C3⋊D3—41no—
97C3×#634—41no—
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101C32×C4×D3—41no—
102C32×Dic6—41no—
103#6293—41no—
104#6290—41no—
105#5646—94no—
106#5646—91no—
107C2×C32×Q8—31no—
108C22×C32×C4—31no—
109C2×#638—31no—
110C2×C4×C3⋊D3C2×D311yescharacteristic
111C2×#634—31no—
112C2×#634C2×D311yescharacteristic
113C2×C3×C4×D3—34no—
114C2×C3×C4×D3C2×D314yes—
115C2×C3×Dic6—34no—
116C2×C3×Dic6C2×D314yes—
117C2×#627—34no—
118C2×#625—34no—
119C32×C4⋊C4—24no—
120C32×C4⋊C4—31no—
121C32×C4⋊C4—31no—
122C32×C4⋊C4C2×D314yes—
123C32×C4⋊C4C2×D311yescharacteristic
124C32×C22⋊C4—31no—
125C32×C22⋊C4—31no—
126#3757—31no—
127#3757—31no—
128#3757—31no—
129#3752—31no—
130#3752—31no—
131#3752—31no—
132#3752C2×D311yescharacteristic
133#3752C2×D311yescharacteristic
134C4⋊(C3⋊Dic3)—31no—
135C4⋊(C3⋊Dic3)—31no—
136C4⋊(C3⋊Dic3)—31no—
137C4⋊(C3⋊Dic3)—34no—
138C4⋊(C3⋊Dic3)—34no—
139C4⋊(C3⋊Dic3)—34no—
140C4⋊(C3⋊Dic3)—31no—
141C4⋊(C3⋊Dic3)—64no—
142C4⋊(C3⋊Dic3)—64no—
143C4⋊(C3⋊Dic3)—64no—
144#3750—31no—
145#3750—31no—
146#3750—34no—
147#3750—34no—
148#3750—31no—
149#3750—31no—
150#3750—31no—
151#3750—31no—
152#3750—31no—
153#3750—64no—
154#3750—64no—
155#3750C2×D311yescharacteristic
156#3750C2×D311yescharacteristic
157C4×C3⋊Dic3—31no—
158C4×C3⋊Dic3—34no—
159C4×C3⋊Dic3—31no—
160C4×C3⋊Dic3—31no—
161C4×C3⋊Dic3—64no—
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⋊C4C2×D314yes—
169C3×Dic3⋊C4C2×D314yes—
170#3724—34no—
171#3724—34no—
172#3724—34no—
173#3724—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176(C2×D3)⋊Dic3—34no—
177(C2×D3)⋊Dic3—34no—
178(C2×D3)⋊Dic3—34no—
179(C2×D3)⋊Dic3—34no—
180(C2×D3)⋊Dic3—34no—
181(C2×D3)⋊Dic3—34no—
182Dic32—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—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—
200C3×C3⋊Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3—21no—
204C3×C22⋊Q8—34no—
205C3×C22⋊Q8—91no—
206#946—94no—
207#946—91no—
208#944—94no—
209#944—91no—
210#929—94no—
211#929—91no—
212#924—94no—
213#924—91no—
214#898—94no—
215#898—94no—
216#898—91no—
217#898—91no—
218#896—34no—
219#896C3⋊D311yescharacteristic, complemented
220#890—94no—
221#890—91no—
222#890—184no—
223#889—94no—
224#889—91no—
225#888—94no—
226#888—91no—
227#885—94no—
228#885—91no—
229#883—94no—
230#883—91no—
231#879—94no—
232#879—94no—
233#879—91no—
234#879—91no—
235C23×C32—31no—
236C22×C3⋊D3C22×D311yescharacteristic
237C22×C3×D3—34no—
238C22×C3×D3C22×D314yes—
239C32×Q8—121no—
240C2×C32×C4—24no—
241C2×C32×C4—24no—
242C2×C32×C4—24no—
243C2×C32×C4—31no—
244C2×C32×C4—31no—
245C2×C32×C4—31no—
246C2×C32×C4—121no—
247C2×C32×C4C22×D314yes—
248C2×C32×C4C22×D314yes—
249C2×C32×C4C22×D314yes—
250C2×C32×C4C22×D311yescharacteristic
251C2×C32×C4C22×D311yescharacteristic
252C2×C32×C4C22×D311yescharacteristic
253#638—121no—
254C2×C3⋊Dic3—31no—
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—34no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3—64no—
264C2×C3⋊Dic3—64no—
265C2×C3⋊Dic3—64no—
266C2×C3⋊Dic3C22×D311yescharacteristic
267C2×C3⋊Dic3C22×D311yescharacteristic
268C2×C3⋊Dic3C22×D311yescharacteristic
269C4×C3⋊D3—41no—
270#634—41no—
271#634—121no—
272C2×C3×Dic3—34no—
273C2×C3×Dic3—34no—
274C2×C3×Dic3—34no—
275C2×C3×Dic3C22×D314yes—
276C2×C3×Dic3C22×D314yes—
277C2×C3×Dic3C22×D314yes—
278C3×C4×D3—44no—
279C3×C4×D3—124no—
280C3×Dic6—44no—
281C3×Dic6—124no—
282#627—124no—
283#625—124no—
284#563—271noSylow
285C2×C33H(5,𝔽2)11yescharacteristic
286C2×C332−1+411yescharacteristic
287C2×C332−1+411yescharacteristic
288C3×C3⋊D3—41no—
289C32×D3—41no—
290C2×C3×Q8—34no—
291C2×C3×Q8—91no—
292C22×C3×C4—34no—
293C22×C3×C4—91no—
294C2×#67—94no—
295C2×#67—91no—
296C2×C4×D3—34no—
297C2×C4×D3C2×C3⋊D311yescharacteristic
298C2×Dic6—34no—
299C2×Dic6—94no—
300C2×Dic6—91no—
301C2×Dic6C2×C3⋊D311yescharacteristic
302C3×C4⋊C4—24no—
303C3×C4⋊C4—34no—
304C3×C4⋊C4—34no—
305C3×C4⋊C4—91no—
306C3×C4⋊C4—91no—
307C3×C4⋊C4D3214yes—
308C3×C4⋊C4C2×C3⋊D311yescharacteristic
309C3×C22⋊C4—34no—
310C3×C22⋊C4—34no—
311C3×C22⋊C4—91no—
312C3×C22⋊C4—91no—
313#217—94no—
314#217—94no—
315#217—94no—
316#217—91no—
317#217—91no—
318#217—91no—
319(C2×D3)⋊C4—34no—
320(C2×D3)⋊C4—34no—
321(C2×D3)⋊C4—94no—
322(C2×D3)⋊C4—94no—
323(C2×D3)⋊C4—94no—
324(C2×D3)⋊C4—91no—
325(C2×D3)⋊C4—91no—
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—94no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—184no—
338C4⋊Dic3—184no—
339C4⋊Dic3—184no—
340Dic3⋊C4—34no—
341Dic3⋊C4—34no—
342Dic3⋊C4—94no—
343Dic3⋊C4—94no—
344Dic3⋊C4—94no—
345Dic3⋊C4—94no—
346Dic3⋊C4—94no—
347Dic3⋊C4—94no—
348Dic3⋊C4—94no—
349Dic3⋊C4—91no—
350Dic3⋊C4—91no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—184no—
357Dic3⋊C4—184no—
358Dic3⋊C4C2×C3⋊D311yescharacteristic
359Dic3⋊C4C2×C3⋊D311yescharacteristic
360C4×Dic3—94no—
361C4×Dic3—94no—
362C4×Dic3—94no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—184no—
367C22×C32—24no—
368C22×C32—61no—
369C22×C32—61no—
370C22×C32—61no—
371C22×C32C23×D314yes—
372C22×C32C23×D311yescharacteristic
373C2×C3⋊D3—21no—
374C2×C3⋊D3—21no—
375C2×C3⋊D3—21no—
376C2×C3×D3—24no—
377C2×C3×D3—24no—
378C2×C3×D3—24no—
379C2×C3×D3—64no—
380C2×C3×D3—64no—
381C2×C3×D3—64no—
382C32×C4—28no—
383C32×C4—21no—
384C32×C4—21no—
385C32×C4—24no—
386C32×C4—21no—
387C32×C4—24no—
388C32×C4—24no—
389C32×C4—44no—
390C32×C4—44no—
391C32×C4—61no—
392C32×C4—61no—
393C32×C4—61no—
394C3⋊Dic3—21no—
395C3⋊Dic3—21no—
396C3⋊Dic3—21no—
397C3⋊Dic3—61no—
398C3⋊Dic3—61no—
399C3⋊Dic3—61no—
400C3⋊Dic3—61no—
401C3⋊Dic3—64no—
402C3⋊Dic3—64no—
403C3⋊Dic3—64no—
404C3⋊Dic3—64no—
405C3⋊Dic3—124no—
406C3⋊Dic3—124no—
407C3⋊Dic3—124no—
408C3⋊Dic3—124no—
409C3×Dic3—24no—
410C3×Dic3—24no—
411C3×Dic3—24no—
412C3×Dic3—64no—
413C3×Dic3—64no—
414C3×Dic3—64no—
415C4⋊Q8—271no—
416C4⋊Q8—271no—
417#126—271no—
418#126—271no—
419#126—271no—
420#126—271no—
421#125—271no—
422#125—271no—
423#124—271no—
424#123—271no—
425#123—271no—
426C22⋊Q8—31no—
427C22⋊Q8—91no—
428C22⋊Q8—271no—
429C22⋊Q8—271no—
430C33#56311yescharacteristic, complemented, Sylow
431C23×C3—34no—
432C23×C3—91no—
433C22×D3—34no—
434C22×D3C22×C3⋊D311yescharacteristic
435C3×Q8—124no—
436C3×Q8—361no—
437C2×C3×C4—24no—
438C2×C3×C4—24no—
439C2×C3×C4—24no—
440C2×C3×C4—34no—
441C2×C3×C4—34no—
442C2×C3×C4—34no—
443C2×C3×C4—91no—
444C2×C3×C4—91no—
445C2×C3×C4—91no—
446C2×C3×C4—124no—
447C2×C3×C4—361no—
448C2×C3×C4C2×D3214yes—
449C2×C3×C4C2×D3214yes—
450C2×C3×C4C2×D3214yes—
451C2×C3×C4C22×C3⋊D311yescharacteristic
452C2×C3×C4C22×C3⋊D311yescharacteristic
453C2×C3×C4C22×C3⋊D311yescharacteristic
454#67—361no—
455#67—364no—
456C2×Dic3—34no—
457C2×Dic3—34no—
458C2×Dic3—34no—
459C2×Dic3—94no—
460C2×Dic3—94no—
461C2×Dic3—94no—
462C2×Dic3—94no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—184no—
468C2×Dic3—184no—
469C2×Dic3—184no—
470C2×Dic3—184no—
471C2×Dic3C22×C3⋊D311yescharacteristic
472C2×Dic3C22×C3⋊D311yescharacteristic
473C2×Dic3C22×C3⋊D311yescharacteristic
474C4×D3—41no—
475C4×D3—124no—
476Dic6—41no—
477Dic6—124no—
478Dic6—361no—
479Dic6—364no—
480C2×C32—24no—
481C2×C32—24no—
482C2×C32—24no—
483C2×C32—121no—
484C2×C32#100414yes—
485C2×C32#100411yescharacteristic
486C2×C32#100714yes—
487C2×C32#100711yescharacteristic
488C2×C32#101014yes—
489C2×C32#101011yescharacteristic
490C3⋊D3—41no—
491C3×D3—44no—
492C3×D3—124no—
493C2×Q8—31no—
494C2×Q8—91no—
495C2×Q8—271no—
496C2×D4—271no—
497C22×C4—31no—
498C22×C4—91no—
499C4⋊C4—31no—
500C4⋊C4—31no—
501C4⋊C4—91no—
502C4⋊C4—91no—
503C4⋊C4—271no—
504C4⋊C4—271no—
505C4⋊C4—271no—
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⋊C4D3×C3⋊D311yescharacteristic
515C22⋊C4—31no—
516C22⋊C4—31no—
517C22⋊C4—91no—
518C22⋊C4—91no—
519C22⋊C4—271no—
520C22⋊C4—271no—
521C22⋊C4—271no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C42—271no—
526C42—271no—
527C42—271no—
528C22×C3—24no—
529C22×C3—64no—
530C22×C3—64no—
531C22×C3—64no—
532C22×C3—181no—
533C22×C3—181no—
534C22×C3—181no—
535C22×C3C22×D3214yes—
536C22×C3C23×C3⋊D311yescharacteristic
537C2×D3—21no—
538C2×D3—21no—
539C2×D3—21no—
540C2×D3—64no—
541C2×D3—64no—
542C2×D3—64no—
543C3×C4—28no—
544C3×C4—24no—
545C3×C4—24no—
546C3×C4—24no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—44no—
551C3×C4—44no—
552C3×C4—64no—
553C3×C4—64no—
554C3×C4—64no—
555C3×C4—181no—
556C3×C4—181no—
557C3×C4—181no—
558Dic3—21no—
559Dic3—21no—
560Dic3—21no—
561Dic3—64no—
562Dic3—64no—
563Dic3—64no—
564Dic3—184no—
565Dic3—184no—
566Dic3—184no—
567Dic3—184no—
568Dic3—181no—
569Dic3—181no—
570Dic3—181no—
571Dic3—181no—
572Dic3—364no—
573Dic3—364no—
574Dic3—364no—
575Dic3—364no—
576C32—24no—
577C32#564614yescomplemented
578C32#564611yescharacteristic, complemented
579C23—31no—
580C23—91no—
581Q8—121no—
582Q8—361no—
583Q8—1081no—
584D4—1081no—
585C2×C4—31no—
586C2×C4—31no—
587C2×C4—31no—
588C2×C4—91no—
589C2×C4—91no—
590C2×C4—91no—
591C2×C4—121no—
592C2×C4—271no—
593C2×C4—271no—
594C2×C4—271no—
595C2×C4—271no—
596C2×C4—361no—
597C2×C4C2×D3×C3⋊D311yescharacteristic
598C2×C4C2×D3×C3⋊D311yescharacteristic
599C2×C4C2×D3×C3⋊D311yescharacteristic
600C2×C3—24no—
601C2×C3—24no—
602C2×C3—24no—
603C2×C3—124no—
604C2×C3—361no—
605C2×C3#6429814yes—
606C2×C3#6431314yes—
607C2×C3#6431614yes—
608C2×C3#6436111yescharacteristic
609C2×C3#6436411yescharacteristic
610C2×C3#6436711yescharacteristic
611D3—41no—
612D3—124no—
613C22—61no—
614C22—61no—
615C22—61no—
616C22—181no—
617C22—181no—
618C22—181no—
619C22C22×D3×C3⋊D311yescharacteristic
620C4—21no—
621C4—21no—
622C4—21no—
623C4—61no—
624C4—61no—
625C4—61no—
626C4—181no—
627C4—181no—
628C4—181no—
629C4—541no—
630C4—541no—
631C4—541no—
632C4—541no—
633C3—24no—
634C3#1059458914yescomplemented
635C3#1059611711yescharacteristic, complemented
636C2—121no—
637C2—361no—
638C2#1172456311yescharacteristic
639C2#1172458711yescharacteristic
640C2#1172459111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×#6290C22
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. #11723931C2semidirect
2. C3×#64105C2semidirect
3. C32×#896C2semidirect
7. #11723448C2semidirect
8. #11723435C2semidirect
9. #11723430C2semidirect
10. #11723430C2semidirect
11. #11723425C2semidirect
12. #11723408C2semidirect
13. #11723408C2semidirect
14. #11723394C2semidirect
15. #11723393C2semidirect
60. #64105D3semidirect
71. C3×#896D3semidirect
219. #896C3⋊D3semidirect
430. C33#563semidirect
577. C32#5646semidirect
578. C32#5646semidirect
634. C3#10594589semidirect
635. C3#10596117semidirect

Maximal subgroups

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

subgroupindex
#117239312
C3×#641052
C32×#8962
#117235302
#117235182
#117235132
#117234482
#117234352
#117234302
#117234252
#117234082
#117233942
#117233932
#105961173
#105945893