Finite Group Databasebeta

← #13175897

Subgroups — #13175897

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#11723920C211yesmaximal, characteristic, complemented
2C3×#64092C211yesmaximal, characteristic
3C32×#878C211yesmaximal, characteristic, complemented
4#11723640C211yesmaximal, characteristic, complemented
5#11723605C211yesmaximal, characteristic
6#11723597C211yesmaximal, characteristic, complemented
7#11723591C211yesmaximal, characteristic
8#11723536C211yesmaximal, characteristic, complemented
9#11723435C211yesmaximal, characteristic, complemented
10#11723433C211yesmaximal, characteristic, complemented
11#11723418C211yesmaximal, characteristic, complemented
12#11723407C211yesmaximal, characteristic, complemented
13#11723402C211yesmaximal, characteristic, complemented
14#11723400C211yesmaximal, characteristic, complemented
15#11723393C211yesmaximal, characteristic, complemented
16#10596117—31nomaximal
17#10594042—34nomaximal
18C2×C3×#638C2211yescharacteristic
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C22×C32×Dic3C2211yescharacteristic
21C2×C32×Dic6C2211yescharacteristic
22C2×#6293C2211yescharacteristic
23C2×#6293C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25#88864C2211yescharacteristic
26#88864C2211yescharacteristic
27C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
28#88857C2211yescharacteristic
29#88857C2211yescharacteristic
30C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
31C3×#3757C2211yescharacteristic
32C3×#3752C2211yescharacteristic
33C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
34C3×#3750C2211yescharacteristic
35C32×#217C2211yescharacteristic
36C32×C4⋊Dic3C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×Dic3⋊C4C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88806C2211yescharacteristic
42#88806C2211yescharacteristic
43#88806C2211yescharacteristic
44#88806C2211yescharacteristic
45Dic3⋊(C3⋊Dic3)C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47#88803C2211yescharacteristic
48#88803C2211yescharacteristic
49#88803C2211yescharacteristic
50#88803C2211yescharacteristic
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#64105—31no—
61#64099—31no—
62#64098—31no—
63#64097—31no—
64#64094—31no—
65#64092D311yescharacteristic
66#64088—31no—
67#64088—31no—
68#64088—34no—
69#64088—64no—
70C3×#883—34no—
71C3×#878D314yescomplemented
72#63964—34no—
73#63961—34no—
74#63959—34no—
75#63956—34no—
76#63919—34no—
77#63862—34no—
78#63858—34no—
79#63856—34no—
80#63851—34no—
81#63847—34no—
82#63845—34no—
83#63839—34no—
84C23×C33C2311yescharacteristic
85C22×C3×C3⋊D3C2311yescharacteristic
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
92C3×#638—41no—
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C3×C4×C3⋊D3—41no—
97C2×C32×Dic3—41no—
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102C32×Dic6—41no—
103#6293—41no—
104#6293—41no—
105#5651—91no—
106#5646—94no—
107C2×C32×Q8—31no—
108C22×C32×C4—31no—
109C2×#638C2×D311yescharacteristic
110C2×C4×C3⋊D3C2×D311yescharacteristic
111C2×#634—31no—
112C2×#634—31no—
113C2×C3×#67—34no—
114C22×C3×Dic3C2×D314yes—
115C2×C3×C4×D3—34no—
116C2×C3×Dic6C2×D314yes—
117C2×#627—34no—
118C2×#627—34no—
119C32×C4⋊C4—31no—
120C32×C4⋊C4—31no—
121C32×C4⋊C4—31no—
122C32×C22⋊C4—24no—
123C32×C22⋊C4—31no—
124C32×C22⋊C4C2×D314yes—
125C32×C22⋊C4C2×D311yescharacteristic
126#3757—31no—
127#3757—31no—
128#3757—34no—
129#3757—34no—
130#3757—64no—
131#3757—64no—
132#3757C2×D311yescharacteristic
133#3752—31no—
134#3752—31no—
135#3752—31no—
136#3752—31no—
137#3752C2×D311yescharacteristic
138C4⋊(C3⋊Dic3)—31no—
139C4⋊(C3⋊Dic3)—34no—
140C4⋊(C3⋊Dic3)—31no—
141C4⋊(C3⋊Dic3)—31no—
142C4⋊(C3⋊Dic3)—64no—
143C4⋊(C3⋊Dic3)C2×D311yescharacteristic
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—31no—
154#3750—64no—
155#3750—64no—
156#3750C2×D311yescharacteristic
157C4×C3⋊Dic3—31no—
158C4×C3⋊Dic3—34no—
159C4×C3⋊Dic3—31no—
160C4×C3⋊Dic3—31no—
161C4×C3⋊Dic3—64no—
162C3×#217—34no—
163C3×#217C2×D314yes—
164C3×(C2×D3)⋊C4—34no—
165C3×C4⋊Dic3—34no—
166C3×C4⋊Dic3C2×D314yes—
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—
176Dic3⋊Dic3—34no—
177Dic3⋊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×C33—21no—
185C22×C33—21no—
186C22×C33—21no—
187C22×C33C2411yescharacteristic
188C2×C3×C3⋊D3—21no—
189C2×C3×C3⋊D3—21no—
190C2×C3×C3⋊D3—21no—
191C33×C4—21no—
192C33×C4—21no—
193C3⋊(C3⋊Dic3)—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×#123—91no—
205C3×C22⋊Q8—34no—
206#946—94no—
207#944—94no—
208#936—91no—
209#935—91no—
210#929—94no—
211#924—94no—
212#924—91no—
213#924—91no—
214#898—94no—
215#898—94no—
216#896—94no—
217#890—94no—
218#889—94no—
219#889—91no—
220#889—91no—
221#888—94no—
222#888—91no—
223#888—91no—
224#885—94no—
225#883—34no—
226#879—94no—
227#879—94no—
228#879—91no—
229#879—91no—
230#879—91no—
231#879—91no—
232#879—184no—
233#878—91no—
234#878C3⋊D311yescharacteristic, complemented
235C23×C32—24no—
236C23×C32C22×D314yes—
237C23×C32C22×D311yescharacteristic
238C22×C3⋊D3C22×D311yescharacteristic
239C22×C3×D3—34no—
240C32×Q8—121no—
241C2×C32×C4—24no—
242C2×C32×C4—24no—
243C2×C32×C4—31no—
244C2×C32×C4—31no—
245C2×C32×C4—31no—
246C2×C32×C4—31no—
247C2×C32×C4—121no—
248C2×C32×C4C22×D314yes—
249C2×C32×C4C22×D314yes—
250C2×C32×C4C22×D311yescharacteristic
251C2×C32×C4C22×D311yescharacteristic
252#638—41no—
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⋊Dic3C22×D311yescharacteristic
266C2×C3⋊Dic3C22×D311yescharacteristic
267C2×C3⋊Dic3C22×D311yescharacteristic
268C4×C3⋊D3—41no—
269#634—121no—
270#634—121no—
271C3×#67—124no—
272C2×C3×Dic3—34no—
273C2×C3×Dic3—34no—
274C2×C3×Dic3—34no—
275C2×C3×Dic3—44no—
276C2×C3×Dic3C22×D314yes—
277C2×C3×Dic3C22×D314yes—
278C2×C3×Dic3C22×D314yes—
279C2×C3×Dic3C22×D314yes—
280C3×C4×D3—124no—
281C3×Dic6—44no—
282#627—124no—
283#627—124no—
284#563—271noSylow
285C2×C33—41no—
286C2×C33H(5,𝔽2)11yescharacteristic
287C2×C332−1+411yescharacteristic
288C2×C332−1+411yescharacteristic
289C3×C3⋊D3—41no—
290C2×C3×Q8—34no—
291C2×C3×D4—91no—
292C22×C3×C4—34no—
293C22×C3×C4—91no—
294C2×#67—34no—
295C22×Dic3C2×C3⋊D311yescharacteristic
296C2×C4×D3—34no—
297C2×Dic6—94no—
298C2×Dic6—94no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C2×Dic6C2×C3⋊D311yescharacteristic
302C3×C4⋊C4—34no—
303C3×C4⋊C4—34no—
304C3×C4⋊C4—34no—
305C3×C4⋊C4—91no—
306C3×C4⋊C4—91no—
307C3×C22⋊C4—24no—
308C3×C22⋊C4—34no—
309C3×C22⋊C4—91no—
310C3×C22⋊C4—91no—
311C3×C22⋊C4D3214yes—
312C3×C22⋊C4C2×C3⋊D311yescharacteristic
313#217—34no—
314#217—94no—
315#217—94no—
316#217—91no—
317#217—91no—
318#217—91no—
319#217—91no—
320#217—91no—
321#217—91no—
322#217—184no—
323#217—184no—
324#217C2×C3⋊D311yescharacteristic
325(C2×D3)⋊C4—34no—
326(C2×D3)⋊C4—94no—
327(C2×D3)⋊C4—94no—
328(C2×D3)⋊C4—94no—
329(C2×D3)⋊C4—94no—
330C4⋊Dic3—34no—
331C4⋊Dic3—94no—
332C4⋊Dic3—94no—
333C4⋊Dic3—94no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—184no—
338C4⋊Dic3C2×C3⋊D311yescharacteristic
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—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—21no—
368C22×C32—21no—
369C22×C32—21no—
370C22×C32—24no—
371C22×C32—24no—
372C22×C32—24no—
373C22×C32—24no—
374C22×C32—44no—
375C22×C32—44no—
376C22×C32—44no—
377C22×C32C23×D314yes—
378C22×C32C23×D311yescharacteristic
379C2×C3⋊D3—21no—
380C2×C3⋊D3—21no—
381C2×C3⋊D3—21no—
382C2×C3×D3—64no—
383C2×C3×D3—64no—
384C2×C3×D3—64no—
385C32×C4—21no—
386C32×C4—21no—
387C32×C4—24no—
388C32×C4—24no—
389C32×C4—44no—
390C32×C4—44no—
391C32×C4—61no—
392C32×C4—61no—
393C32×C4—61no—
394C32×C4—61no—
395C3⋊Dic3—21no—
396C3⋊Dic3—21no—
397C3⋊Dic3—21no—
398C3⋊Dic3—61no—
399C3⋊Dic3—61no—
400C3⋊Dic3—61no—
401C3⋊Dic3—61no—
402C3⋊Dic3—64no—
403C3⋊Dic3—64no—
404C3⋊Dic3—64no—
405C3⋊Dic3—64no—
406C3⋊Dic3—124no—
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—271no—
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—91no—
427#123—271no—
428C22⋊Q8—31no—
429C22⋊Q8—271no—
430C22⋊Q8—271no—
431C22⋊Q8—271no—
432C33#56311yescharacteristic, complemented, Sylow
433C23×C3—24no—
434C23×C3—91no—
435C23×C3C2×D3214yes—
436C23×C3C22×C3⋊D311yescharacteristic
437C22×D3—34no—
438C3×Q8—124no—
439C3×D4—361no—
440C2×C3×C4—24no—
441C2×C3×C4—24no—
442C2×C3×C4—34no—
443C2×C3×C4—34no—
444C2×C3×C4—34no—
445C2×C3×C4—34no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—124no—
450C2×C3×C4—361no—
451C2×C3×C4C2×D3214yes—
452C2×C3×C4C2×D3214yes—
453C2×C3×C4C22×C3⋊D311yescharacteristic
454C2×C3×C4C22×C3⋊D311yescharacteristic
455#67—124no—
456C2×Dic3—34no—
457C2×Dic3—34no—
458C2×Dic3—34no—
459C2×Dic3—41no—
460C2×Dic3—94no—
461C2×Dic3—94no—
462C2×Dic3—94no—
463C2×Dic3—94no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—91no—
468C2×Dic3—184no—
469C2×Dic3—184no—
470C2×Dic3—184no—
471C2×Dic3—184no—
472C2×Dic3C22×C3⋊D311yescharacteristic
473C2×Dic3C22×C3⋊D311yescharacteristic
474C2×Dic3C22×C3⋊D311yescharacteristic
475C2×Dic3C22×C3⋊D311yescharacteristic
476C4×D3—124no—
477Dic6—41no—
478Dic6—361no—
479Dic6—361no—
480Dic6—364no—
481Dic6—364no—
482C2×C32—24no—
483C2×C32—24no—
484C2×C32—24no—
485C2×C32—41no—
486C2×C32—44no—
487C2×C32—84no—
488C2×C32#100414yes—
489C2×C32#100411yescharacteristic
490C2×C32#100714yes—
491C2×C32#101014yes—
492C2×C32#101011yescharacteristic
493C2×C32#101011yescharacteristic
494C3⋊D3—41no—
495C3×D3—124no—
496C2×Q8—31no—
497C2×Q8—271no—
498C2×Q8—271no—
499C2×D4—91no—
500C22×C4—31no—
501C22×C4—91no—
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⋊C4—271no—
518C22⋊C4—31no—
519C22⋊C4—91no—
520C22⋊C4—91no—
521C22⋊C4—271no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C22⋊C4—271no—
526C22⋊C4—271no—
527C22⋊C4D3×C3⋊D311yescharacteristic
528C42—271no—
529C42—271no—
530C42—271no—
531C22×C3—24no—
532C22×C3—24no—
533C22×C3—24no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—24no—
538C22×C3—44no—
539C22×C3—44no—
540C22×C3—44no—
541C22×C3—181no—
542C22×C3—181no—
543C22×C3—181no—
544C22×C3C22×D3214yes—
545C22×C3C23×C3⋊D311yescharacteristic
546C2×D3—64no—
547C2×D3—64no—
548C2×D3—64no—
549C3×C4—24no—
550C3×C4—24no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—44no—
554C3×C4—44no—
555C3×C4—64no—
556C3×C4—64no—
557C3×C4—64no—
558C3×C4—64no—
559C3×C4—181no—
560C3×C4—181no—
561C3×C4—181no—
562Dic3—21no—
563Dic3—21no—
564Dic3—21no—
565Dic3—21no—
566Dic3—64no—
567Dic3—64no—
568Dic3—64no—
569Dic3—184no—
570Dic3—184no—
571Dic3—184no—
572Dic3—184no—
573Dic3—181no—
574Dic3—181no—
575Dic3—181no—
576Dic3—181no—
577Dic3—364no—
578Dic3—364no—
579Dic3—364no—
580Dic3—364no—
581C32—24no—
582C32#564614yescomplemented
583C32#565111yescharacteristic, complemented
584C23—91no—
585C23C2×D3×C3⋊D311yescharacteristic
586Q8—121no—
587Q8—1081no—
588Q8—1081no—
589D4—361no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—31no—
593C2×C4—31no—
594C2×C4—91no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—121no—
598C2×C4—271no—
599C2×C4—271no—
600C2×C4—271no—
601C2×C4—271no—
602C2×C4—361no—
603C2×C4C2×D3×C3⋊D311yescharacteristic
604C2×C4C2×D3×C3⋊D311yescharacteristic
605C2×C3—24no—
606C2×C3—24no—
607C2×C3—24no—
608C2×C3—41no—
609C2×C3—44no—
610C2×C3—84no—
611C2×C3—361no—
612C2×C3#6429714yes—
613C2×C3#6430914yes—
614C2×C3#6431314yes—
615C2×C3#6436111yescharacteristic
616C2×C3#6436411yescharacteristic
617C2×C3#6436711yescharacteristic
618D3—124no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—181no—
623C22—181no—
624C22—181no—
625C22C22×D3×C3⋊D311yescharacteristic
626C4—21no—
627C4—21no—
628C4—61no—
629C4—61no—
630C4—61no—
631C4—61no—
632C4—181no—
633C4—181no—
634C4—181no—
635C4—541no—
636C4—541no—
637C4—541no—
638C4—541no—
639C3—24no—
640C3#1059404214yescomplemented
641C3#1059611711yescharacteristic, complemented
642C2—41no—
643C2—361no—
644C2#1172456611yescharacteristic
645C2#1172458211yescharacteristic
646C2#1172458711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×C3×#638C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723920C2semidirect
3. C32×#878C2semidirect
4. #11723640C2semidirect
6. #11723597C2semidirect
8. #11723536C2semidirect
9. #11723435C2semidirect
10. #11723433C2semidirect
11. #11723418C2semidirect
12. #11723407C2semidirect
13. #11723402C2semidirect
14. #11723400C2semidirect
15. #11723393C2semidirect
71. C3×#878D3semidirect
234. #878C3⋊D3semidirect
432. C33#563semidirect
582. C32#5646semidirect
583. C32#5651semidirect
640. C3#10594042semidirect
641. C3#10596117semidirect

Maximal subgroups

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

subgroupindex
#117239202
C3×#640922
C32×#8782
#117236402
#117236052
#117235972
#117235912
#117235362
#117234352
#117234332
#117234182
#117234072
#117234022
#117234002
#117233932
#105961173
#105940423