Finite Group Databasebeta

← #13175902

Subgroups — #13175902

Order 1728.

1532 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723919C211yesmaximal, characteristic
2C3×#64086C211yesmaximal, characteristic
3C32×#878C211yesmaximal, characteristic
4#11723642C211yesmaximal, characteristic
5#11723640C211yesmaximal, characteristic
6C2×#88807C211yesmaximal, characteristic
7#11723595C211yesmaximal, characteristic
8#11723535C211yesmaximal, characteristic, complemented
9#11723455C211yesmaximal, characteristic, complemented
10#11723445C211yesmaximal, characteristic, complemented
11#11723445C211yesmaximal, characteristic, complemented
12#11723412C211yesmaximal, characteristic, complemented
13#11723409C211yesmaximal, characteristic, complemented
14#11723395C211yesmaximal, characteristic, complemented
15#11723393C211yesmaximal, characteristic, complemented
16#10596094—31nomaximal
17#10594035—34nomaximal
18C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C22×C3×C3⋊Dic3C2211yescharacteristic
21C22×C32×Dic3C2211yescharacteristic
22C2×C32×Dic6C2211yescharacteristic
23C2×#6293C2211yescharacteristic
24C2×#6293C2211yescharacteristic
25C33×C22⋊C4C2211yescharacteristic
26#88864C2211yescharacteristic
27C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
28#88857C2211yescharacteristic
29#88857C2211yescharacteristic
30C3×#3757C2211yescharacteristic
31C3×#3750C2211yescharacteristic
32C3×#3750C2211yescharacteristic
33C3×C4×C3⋊Dic3C2211yescharacteristic
34C3×C4×C3⋊Dic3C2211yescharacteristic
35C32×#217C2211yescharacteristic
36C32×C4⋊Dic3C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×Dic3⋊C4C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88807C2211yescharacteristic
42#88807C2211yescharacteristic
43#88807C2211yescharacteristic
44#88807C2211yescharacteristic
45#88807C2211yescharacteristic
46#88807C2211yescharacteristic
47#88806C2211yescharacteristic
48#88806C2211yescharacteristic
49Dic3⋊(C3⋊Dic3)C2211yescharacteristic
50Dic3⋊(C3⋊Dic3)C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×C22⋊Q8—31no—
54#64153—31no—
55#64153—31no—
56#64133—31no—
57C2×#3750—31no—
58#64098—31no—
59#64098—31no—
60#64098—31no—
61#64098—31no—
62#64097—31no—
63#64097—31no—
64#64087—31no—
65#64087—31no—
66#64087—34no—
67#64087—64no—
68#64086—31no—
69#64086D311yescharacteristic
70C3×#878D314yes—
71C3×#877—34no—
72#63982—34no—
73C2×Dic3⋊Dic3—34no—
74#63964—34no—
75#63955—34no—
76#63918—34no—
77#63867—34no—
78#63867—34no—
79#63866—34no—
80#63846—34no—
81#63845—34no—
82#63840—34no—
83#63839—34no—
84C23×C33C2311yescharacteristic
85C2×C33×C4C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C3⋊(C3⋊Dic3)—41no—
88C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92#6308—41no—
93C2×C3×C3⋊Dic3Q814yescomplemented
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C3×C3⋊Dic3C2311yescharacteristic
98C2×C32×Dic3—41no—
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102C2×C32×Dic3C2311yescharacteristic
103C32×Dic6—41no—
104#6293—41no—
105#6293—41no—
106#5623—94no—
107#5519—91no—
108C2×C32×Q8—31no—
109C22×C32×C4—31no—
110C22×C3⋊Dic3—31no—
111C22×C3⋊Dic3—34no—
112C22×C3⋊Dic3—64no—
113C22×C3⋊Dic3C2×D311yescharacteristic
114C2×#634—31no—
115C2×#634—34no—
116C2×#634—31no—
117C2×#634—31no—
118C2×#634—64no—
119C22×C3×Dic3—34no—
120C22×C3×Dic3C2×D314yes—
121C2×C3×Dic6C2×D314yes—
122C2×#627—34no—
123C2×#627—34no—
124C32×C4⋊C4—31no—
125C32×C4⋊C4—31no—
126C32×C4⋊C4—31no—
127C32×C22⋊C4—24no—
128C32×C22⋊C4—31no—
129C32×C22⋊C4C2×D314yes—
130C32×C22⋊C4C2×D311yescharacteristic
131#3757—31no—
132#3757—34no—
133#3757—64no—
134#3757C2×D311yescharacteristic
135C4⋊(C3⋊Dic3)—31no—
136C4⋊(C3⋊Dic3)—34no—
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)—31no—
139C4⋊(C3⋊Dic3)—64no—
140#3750—31no—
141#3750—31no—
142#3750—34no—
143#3750—34no—
144#3750—31no—
145#3750—31no—
146#3750—31no—
147#3750—31no—
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
157#3750C2×D311yescharacteristic
158C4×C3⋊Dic3—31no—
159C4×C3⋊Dic3—31no—
160C4×C3⋊Dic3C2×D311yescharacteristic
161C4×C3⋊Dic3C2×D311yescharacteristic
162C3×#217—34no—
163C3×#217C2×D314yes—
164C3×C4⋊Dic3C2×D314yes—
165C3×Dic3⋊C4—34no—
166C3×Dic3⋊C4—34no—
167C3×Dic3⋊C4C2×D314yes—
168C3×Dic3⋊C4C2×D314yes—
169C3×C4×Dic3—34no—
170C3×C4×Dic3—34no—
171#3724—34no—
172#3724—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178Dic3⋊Dic3—34no—
179Dic3⋊Dic3—34no—
180Dic3⋊Dic3—34no—
181Dic3⋊Dic3—34no—
182Dic3⋊Dic3—34no—
183Dic32—34no—
184Dic32—34no—
185C22×C33—21no—
186C22×C33—21no—
187C22×C33C2×Q812yes—
188C22×C33C2411yescharacteristic
189C33×C4—21no—
190C33×C4—21no—
191C3⋊(C3⋊Dic3)—21no—
192C3⋊(C3⋊Dic3)—21no—
193C3⋊(C3⋊Dic3)—21no—
194C3⋊(C3⋊Dic3)—21no—
195C3×C3⋊Dic3—21no—
196C3×C3⋊Dic3—21no—
197C3×C3⋊Dic3C2×Q812yes—
198C3×C3⋊Dic3C2×Q812yes—
199C32×Dic3—21no—
200C32×Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C3×C22⋊Q8—34no—
204C3×#117—91no—
205#944—94no—
206#944—94no—
207#926—91no—
208C2×C4⋊Dic3—91no—
209#924—94no—
210C2×Dic3⋊C4—94no—
211#890—91no—
212#890—91no—
213#889—94no—
214#889—94no—
215#889—94no—
216#889—94no—
217#888—94no—
218#888—94no—
219#888—91no—
220#888—91no—
221#878—94no—
222#878—91no—
223#878—94no—
224#878—91no—
225#878—91no—
226#878—184no—
227#878C3⋊D311yescharacteristic
228#877—34no—
229#877—94no—
230#870—91no—
231#870—91no—
232#869—91no—
233#869—91no—
234C23×C32—24no—
235C23×C32C22×D314yes—
236C23×C32C22×D311yescharacteristic
237C32×Q8—121no—
238C2×C32×C4—24no—
239C2×C32×C4—24no—
240C2×C32×C4—31no—
241C2×C32×C4—31no—
242C2×C32×C4—31no—
243C2×C32×C4—31no—
244C2×C32×C4—121no—
245C2×C32×C4C22×D314yes—
246C2×C32×C4C22×D314yes—
247C2×C32×C4C22×D311yescharacteristic
248C2×C32×C4C22×D311yescharacteristic
249C2×C3⋊Dic3—31no—
250C2×C3⋊Dic3—31no—
251C2×C3⋊Dic3—31no—
252C2×C3⋊Dic3—31no—
253C2×C3⋊Dic3—34no—
254C2×C3⋊Dic3—34no—
255C2×C3⋊Dic3—34no—
256C2×C3⋊Dic3—34no—
257C2×C3⋊Dic3—64no—
258C2×C3⋊Dic3—64no—
259C2×C3⋊Dic3—64no—
260C2×C3⋊Dic3—64no—
261C2×C3⋊Dic3—121no—
262C2×C3⋊Dic3—124no—
263C2×C3⋊Dic3—128no—
264C2×C3⋊Dic3Dic614yescomplemented
265C2×C3⋊Dic3C22×D311yescharacteristic
266C2×C3⋊Dic3C22×D311yescharacteristic
267C2×C3⋊Dic3C22×D311yescharacteristic
268C2×C3⋊Dic3C22×D311yescharacteristic
269#634—121no—
270#634—121no—
271#634—121no—
272#634—124no—
273#634—128no—
274C2×C3×Dic3—316no—
275C2×C3×Dic3—34no—
276C2×C3×Dic3—34no—
277C2×C3×Dic3—34no—
278C2×C3×Dic3—34no—
279C2×C3×Dic3—44no—
280C2×C3×Dic3C22×D314yes—
281C2×C3×Dic3C22×D314yes—
282C2×C3×Dic3C22×D314yes—
283C2×C3×Dic3C22×D314yes—
284C3×Dic6—44no—
285#627—124no—
286#627—124no—
287#544—271noSylow
288C2×C33—22no—
289C2×C33C22×Q811yescharacteristic
290C2×C33H(5,𝔽2)11yescharacteristic
291C2×C332−1+411yescharacteristic
292C2×C3×Q8—34no—
293C22×C3×C4—34no—
294C22×C3×C4—91no—
295C22×Dic3—34no—
296C22×Dic3—94no—
297C22×Dic3—91no—
298C22×Dic3—184no—
299C22×Dic3C2×C3⋊D311yescharacteristic
300C2×Dic6—94no—
301C2×Dic6—94no—
302C2×Dic6—94no—
303C2×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6—91no—
306C2×Dic6—184no—
307C2×Dic6C2×C3⋊D311yescharacteristic
308C3×C4⋊C4—34no—
309C3×C4⋊C4—34no—
310C3×C4⋊C4—34no—
311C3×C4⋊C4—91no—
312C3×C4⋊C4—91no—
313C3×C22⋊C4—24no—
314C3×C22⋊C4—34no—
315C3×C22⋊C4—91no—
316C3×C22⋊C4D3214yes—
317C3×C22⋊C4C2×C3⋊D311yescharacteristic
318C3×C42—91no—
319C3×C42—91no—
320#217—34no—
321#217—94no—
322#217—91no—
323#217—184no—
324#217C2×C3⋊D311yescharacteristic
325C4⋊Dic3—94no—
326C4⋊Dic3—94no—
327C4⋊Dic3—94no—
328C4⋊Dic3—91no—
329C4⋊Dic3—91no—
330C4⋊Dic3—91no—
331C4⋊Dic3—91no—
332C4⋊Dic3—91no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—184no—
338C4⋊Dic3C2×C3⋊D311yescharacteristic
339Dic3⋊C4—34no—
340Dic3⋊C4—34no—
341Dic3⋊C4—94no—
342Dic3⋊C4—94no—
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—94no—
352Dic3⋊C4—94no—
353Dic3⋊C4—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—184no—
360Dic3⋊C4—184no—
361Dic3⋊C4C2×C3⋊D311yescharacteristic
362Dic3⋊C4C2×C3⋊D311yescharacteristic
363C4×Dic3—34no—
364C4×Dic3—34no—
365C4×Dic3—94no—
366C4×Dic3—94no—
367C4×Dic3—91no—
368C4×Dic3—91no—
369C22×C32—21no—
370C22×C32—21no—
371C22×C32—24no—
372C22×C32—28no—
373C22×C32—24no—
374C22×C32—24no—
375C22×C32—44no—
376C22×C32—44no—
377C22×C32C2×Dic612yes—
378C22×C32D3×Q818yes—
379C22×C32C23×D314yes—
380C22×C32C23×D311yescharacteristic
381C32×C4—21no—
382C32×C4—21no—
383C32×C4—24no—
384C32×C4—24no—
385C32×C4—44no—
386C32×C4—44no—
387C32×C4—61no—
388C32×C4—61no—
389C32×C4—61no—
390C32×C4—61no—
391C3⋊Dic3—21no—
392C3⋊Dic3—21no—
393C3⋊Dic3—68no—
394C3⋊Dic3—64no—
395C3⋊Dic3—61no—
396C3⋊Dic3—61no—
397C3⋊Dic3—61no—
398C3⋊Dic3—61no—
399C3⋊Dic3—64no—
400C3⋊Dic3—64no—
401C3⋊Dic3—64no—
402C3⋊Dic3—124no—
403C3⋊Dic3—124no—
404C3⋊Dic3—124no—
405C3⋊Dic3C2×Dic612yes—
406C3⋊Dic3C2×Dic612yes—
407C3×Dic3—24no—
408C3×Dic3—24no—
409C3×Dic3—24no—
410C3×Dic3—24no—
411C3×Dic3—38no—
412C3×Dic3—38no—
413C3×Dic3—64no—
414C3×Dic3—64no—
415C4⋊Q8—271no—
416C4⋊Q8—271no—
417C4⋊Q8—271no—
418C4⋊Q8—271no—
419#125—271no—
420#125—271no—
421#125—271no—
422#125—271no—
423C22⋊Q8—31no—
424C22⋊Q8—271no—
425C22⋊Q8—271no—
426C22⋊Q8—271no—
427#117—91no—
428#117—271no—
429C2×C4⋊C4—271no—
430C33#54411yescharacteristic, complemented, Sylow
431C23×C3—24no—
432C23×C3C2×D3214yes—
433C23×C3C22×C3⋊D311yescharacteristic
434C3×Q8—124no—
435C2×C3×C4—24no—
436C2×C3×C4—24no—
437C2×C3×C4—34no—
438C2×C3×C4—34no—
439C2×C3×C4—34no—
440C2×C3×C4—34no—
441C2×C3×C4—91no—
442C2×C3×C4—91no—
443C2×C3×C4—94no—
444C2×C3×C4—91no—
445C2×C3×C4—91no—
446C2×C3×C4—124no—
447C2×C3×C4C2×D3214yes—
448C2×C3×C4C2×D3214yes—
449C2×C3×C4C22×C3⋊D311yescharacteristic
450C2×C3×C4C22×C3⋊D311yescharacteristic
451C2×Dic3—34no—
452C2×Dic3—34no—
453C2×Dic3—316no—
454C2×Dic3—34no—
455C2×Dic3—34no—
456C2×Dic3—41no—
457C2×Dic3—94no—
458C2×Dic3—94no—
459C2×Dic3—94no—
460C2×Dic3—94no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—184no—
466C2×Dic3—184no—
467C2×Dic3—184no—
468C2×Dic3—184no—
469C2×Dic3—364no—
470C2×Dic3—361no—
471C2×Dic3—368no—
472C2×Dic3C22×C3⋊D311yescharacteristic
473C2×Dic3C22×C3⋊D311yescharacteristic
474C2×Dic3C22×C3⋊D311yescharacteristic
475C2×Dic3C22×C3⋊D311yescharacteristic
476Dic6—41no—
477Dic6—361no—
478Dic6—361no—
479Dic6—364no—
480Dic6—364no—
481Dic6—364no—
482Dic6—361no—
483Dic6—368no—
484C2×C32—22no—
485C2×C32—24no—
486C2×C32—24no—
487C2×C32—24no—
488C2×C32—28no—
489C2×C32—48no—
490C2×C32C22×Dic611yescharacteristic
491C2×C32#100414yes—
492C2×C32C2×D3×Q814yes—
493C2×C32#100911yescharacteristic
494C2×C32#101014yes—
495C2×C32#101011yescharacteristic
496C2×Q8—31no—
497C2×Q8—271no—
498C2×Q8—271no—
499C2×Q8—271no—
500C22×C4—31no—
501C22×C4—91no—
502C22×C4—271no—
503C4⋊C4—31no—
504C4⋊C4—31no—
505C4⋊C4—31no—
506C4⋊C4—91no—
507C4⋊C4—91no—
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—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4—271no—
522C4⋊C4—271no—
523C22⋊C4—31no—
524C22⋊C4—91no—
525C22⋊C4—271no—
526C22⋊C4D3×C3⋊D311yescharacteristic
527C42—91no—
528C42—91no—
529C42—271no—
530C42—271no—
531C22×C3—24no—
532C22×C3—24no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—24no—
536C22×C3—28no—
537C22×C3—44no—
538C22×C3—44no—
539C22×C3D3×Dic618yes—
540C22×C3Q8×C3⋊D312yes—
541C22×C3C22×D3214yes—
542C22×C3C23×C3⋊D311yescharacteristic
543C3×C4—24no—
544C3×C4—24no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—44no—
548C3×C4—44no—
549C3×C4—64no—
550C3×C4—64no—
551C3×C4—64no—
552C3×C4—64no—
553C3×C4—92no—
554C3×C4—92no—
555C3×C4—181no—
556C3×C4—181no—
557Dic3—21no—
558Dic3—21no—
559Dic3—21no—
560Dic3—21no—
561Dic3—38no—
562Dic3—38no—
563Dic3—64no—
564Dic3—64no—
565Dic3—188no—
566Dic3—184no—
567Dic3—184no—
568Dic3—184no—
569Dic3—184no—
570Dic3—181no—
571Dic3—181no—
572Dic3—181no—
573Dic3—181no—
574Dic3—364no—
575Dic3—364no—
576Dic3—364no—
577C32—24no—
578C32#551911yescharacteristic, complemented
579C32#562314yescomplemented
580C23C2×D3×C3⋊D311yescharacteristic
581Q8—121no—
582Q8—1081no—
583Q8—1081no—
584Q8—1081no—
585C2×C4—31no—
586C2×C4—31no—
587C2×C4—31no—
588C2×C4—31no—
589C2×C4—91no—
590C2×C4—91no—
591C2×C4—94no—
592C2×C4—91no—
593C2×C4—91no—
594C2×C4—121no—
595C2×C4—271no—
596C2×C4—271no—
597C2×C4—271no—
598C2×C4—271no—
599C2×C4—1081no—
600C2×C4C2×D3×C3⋊D311yescharacteristic
601C2×C4C2×D3×C3⋊D311yescharacteristic
602C2×C3—28no—
603C2×C3—22no—
604C2×C3—24no—
605C2×C3—24no—
606C2×C3—24no—
607C2×C3—48no—
608C2×C3C2×D3×Dic614yes—
609C2×C3#6430914yes—
610C2×C3#6431414yes—
611C2×C3#6436111yescharacteristic
612C2×C3C2×Q8×C3⋊D311yescharacteristic
613C2×C3#6436711yescharacteristic
614C22—21no—
615C22—21no—
616C22C3⋊D3×Dic612yes—
617C22C22×D3×C3⋊D311yescharacteristic
618C4—21no—
619C4—21no—
620C4—61no—
621C4—61no—
622C4—61no—
623C4—61no—
624C4—92no—
625C4—92no—
626C4—181no—
627C4—181no—
628C4—541no—
629C4—541no—
630C4—541no—
631C4—541no—
632C3—24no—
633C3#1059403514yescomplemented
634C3#1059609411yescharacteristic, complemented
635C2—22no—
636C2C2×C3⋊D3×Dic611yescharacteristic
637C2#1172458211yescharacteristic
638C2#1172458811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC23×C33C23
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
8. #11723535C2semidirect
9. #11723455C2semidirect
10. #11723445C2semidirect
11. #11723445C2semidirect
12. #11723412C2semidirect
13. #11723409C2semidirect
14. #11723395C2semidirect
15. #11723393C2semidirect
93. C2×C3×C3⋊Dic3Q8semidirect
264. C2×C3⋊Dic3Dic6semidirect
430. C33#544semidirect
578. C32#5519semidirect
579. C32#5623semidirect
633. C3#10594035semidirect
634. C3#10596094semidirect

Maximal subgroups

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

subgroupindex
#117239192
C3×#640862
C32×#8782
#117236422
#117236402
C2×#888072
#117235952
#117235352
#117234552
#117234452
#117234122
#117234092
#117233952
#117233932
#105960943
#105940353