Finite Group Databasebeta

← #13177671

Subgroups — #13177671

Order 1728.

1596 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723976C211yesmaximal, characteristic
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic
3C32×#935C211yesmaximal, characteristic
4#11723641C211yesmaximal, characteristic, complemented
5#11723641C211yesmaximal, characteristic, complemented
6C2×#88807C211yesmaximal, characteristic, complemented
7#11723598C211yesmaximal, characteristic, complemented
8#11723597C211yesmaximal, characteristic, complemented
9#11723596C211yesmaximal, characteristic, complemented
10#11723595C211yesmaximal, characteristic, complemented
11#11723594C211yesmaximal, characteristic, complemented
12#11723445C211yesmaximal, characteristic, complemented
13#11723402C211yesmaximal, characteristic, complemented
14#11723401C211yesmaximal, characteristic, complemented
15#11723390C211yesmaximal, characteristic, complemented
16#10596121—31nomaximal
17#10595117—34nomaximal
18C2×C33×D4C2211yescharacteristic
19C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
20C22×C3×C3⋊Dic3C2211yescharacteristic
21C22×C3×C3⋊Dic3C2211yescharacteristic
22C22×C32×Dic3C2211yescharacteristic
23C2×#6293C2211yescharacteristic
24#88864C2211yescharacteristic
25#88864C2211yescharacteristic
26#88864C2211yescharacteristic
27#88857C2211yescharacteristic
28#88857C2211yescharacteristic
29C3×#3757C2211yescharacteristic
30C3×#3757C2211yescharacteristic
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×C4×C3⋊Dic3C2211yescharacteristic
33C32×#217C2211yescharacteristic
34C32×#217C2211yescharacteristic
35C32×#217C2211yescharacteristic
36C32×Dic3⋊C4C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38#88807C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88807C2211yescharacteristic
42#88807C2211yescharacteristic
43#88806C2211yescharacteristic
44#88806C2211yescharacteristic
45#88806C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47Dic3⋊(C3⋊Dic3)C2211yescharacteristic
48Dic3⋊(C3⋊Dic3)C2211yescharacteristic
49Dic3×C3⋊Dic3C2211yescharacteristic
50Dic3×C3⋊Dic3C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×#123—31no—
54#64144—31no—
55#64144—34no—
56#64144—64no—
57D4×C3⋊Dic3D311yescharacteristic
58#64135—31no—
59C2×#3750—31no—
60#64099—31no—
61#64098—31no—
62#64098—31no—
63#64096—31no—
64#64088—31no—
65#64088—31no—
66#64087—31no—
67#64087—31no—
68#64086—31no—
69#64086—31no—
70C3×#935D314yes—
71C3×D4×Dic3—34no—
72C2×Dic3⋊Dic3—34no—
73#63964—34no—
74#63964—34no—
75#63957—34no—
76#63956—34no—
77#63956—34no—
78#63955—34no—
79#63955—34no—
80#63867—34no—
81#63847—34no—
82#63846—34no—
83#63837—34no—
84C23×C33C2311yescharacteristic
85C23×C33C2311yescharacteristic
86C33×D4—22no—
87C2×C33×C4C2311yescharacteristic
88C2×C3⋊(C3⋊Dic3)—22no—
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
93C2×C3×C3⋊Dic3—22no—
94C2×C3×C3⋊Dic3—22no—
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C3×C3⋊Dic3C2311yescharacteristic
98C2×C3×C3⋊Dic3C2311yescharacteristic
99C2×C32×Dic3—22no—
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102C2×C32×Dic3C2311yescharacteristic
103C2×C32×Dic3C2311yescharacteristic
104#6293—22no—
105#5650—94no—
106#5541—91no—
107C2×C32×D4—24no—
108C2×C32×D4C2×D314yes—
109C2×C32×D4C2×D311yescharacteristic
110C22×C32×C4—31no—
111C22×C3⋊Dic3—31no—
112C22×C3⋊Dic3—34no—
113C22×C3⋊Dic3—64no—
114C22×C3⋊Dic3C2×D311yescharacteristic
115C22×C3⋊Dic3C2×D311yescharacteristic
116C2×#634—31no—
117C22×C3×Dic3—34no—
118C22×C3×Dic3—34no—
119C22×C3×Dic3C2×D314yes—
120C2×#627—34no—
121C32×C4⋊C4—31no—
122C32×C4⋊C4—31no—
123C32×C22⋊C4—31no—
124C32×C22⋊C4—31no—
125C32×C22⋊C4—31no—
126#3757—31no—
127#3757—31no—
128#3757—31no—
129#3757—34no—
130#3757—34no—
131#3757—34no—
132#3757—64no—
133#3757—64no—
134#3757—64no—
135#3757C2×D311yescharacteristic
136#3757C2×D311yescharacteristic
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)—31no—
139C4⋊(C3⋊Dic3)—31no—
140C4⋊(C3⋊Dic3)C2×D311yescharacteristic
141#3750—31no—
142#3750—31no—
143#3750—34no—
144#3750—34no—
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—64no—
154#3750—64no—
155C4×C3⋊Dic3—31no—
156C4×C3⋊Dic3—31no—
157C4×C3⋊Dic3—31no—
158C4×C3⋊Dic3—31no—
159C4×C3⋊Dic3C2×D311yescharacteristic
160C3×#217—34no—
161C3×#217—34no—
162C3×#217C2×D314yes—
163C3×#217C2×D314yes—
164C3×#217C2×D314yes—
165C3×C4⋊Dic3—34no—
166C3×Dic3⋊C4C2×D314yes—
167C3×Dic3⋊C4C2×D314yes—
168C3×C4×Dic3—34no—
169#3724—34no—
170#3724—34no—
171#3724—34no—
172Dic3⋊Dic3—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178Dic3⋊Dic3—34no—
179Dic3⋊Dic3—34no—
180Dic32—34no—
181Dic32—34no—
182Dic32—34no—
183Dic32—34no—
184C22×C33—21no—
185C22×C33—21no—
186C22×C33—21no—
187C22×C33—21no—
188C22×C33—21no—
189C22×C33#4112yes—
190C22×C33C2411yescharacteristic
191C33×C4—21no—
192C3⋊(C3⋊Dic3)—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#4112yes—
200C32×Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3—21no—
204C3×#123—34no—
205C3×C4×D4—91no—
206#935—94no—
207#935—91no—
208#935—184no—
209#935C3⋊D311yescharacteristic
210D4×Dic3—34no—
211#926—94no—
212#926—91no—
213C2×C4⋊Dic3—91no—
214#924—91no—
215#924—91no—
216C2×Dic3⋊C4—94no—
217#890—94no—
218#889—94no—
219#889—94no—
220#889—91no—
221#889—91no—
222#887—94no—
223#879—94no—
224#879—94no—
225#879—91no—
226#879—91no—
227#878—94no—
228#878—94no—
229#877—94no—
230#877—94no—
231#877—91no—
232#877—91no—
233#870—91no—
234C4×Dic6—91no—
235C23×C32—24no—
236C23×C32—24no—
237C23×C32C22×D314yes—
238C23×C32C22×D314yes—
239C23×C32C22×D311yescharacteristic
240C23×C32C22×D311yescharacteristic
241C32×D4—22no—
242C32×D4—28no—
243C32×D4—48no—
244C2×C32×C4—24no—
245C2×C32×C4—31no—
246C2×C32×C4—31no—
247C2×C32×C4—31no—
248C2×C32×C4—31no—
249C2×C32×C4—62no—
250C2×C32×C4C22×D314yes—
251C2×C32×C4C22×D311yescharacteristic
252C2×C3⋊Dic3—22no—
253C2×C3⋊Dic3—22no—
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⋊Dic3—62no—
267C2×C3⋊Dic3—68no—
268C2×C3⋊Dic3—128no—
269C2×C3⋊Dic3C22×D311yescharacteristic
270C2×C3⋊Dic3C22×D311yescharacteristic
271C2×C3⋊Dic3C22×D311yescharacteristic
272C2×C3⋊Dic3C22×D311yescharacteristic
273#634—62no—
274C2×C3×Dic3—28no—
275C2×C3×Dic3—34no—
276C2×C3×Dic3—34no—
277C2×C3×Dic3—34no—
278C2×C3×Dic3—34no—
279C2×C3×Dic3—68no—
280C2×C3×Dic3—68no—
281C2×C3×Dic3C22×D314yes—
282C2×C3×Dic3C22×D314yes—
283C2×C3×Dic3C22×D314yes—
284C2×C3×Dic3C22×D314yes—
285#627—68no—
286#552—271noSylow
287C2×C33—22no—
288C2×C33—41no—
289C2×C33C2×#4111yescharacteristic
290C2×C33C2×#4111yescharacteristic
291C2×C332−1+411yescharacteristic
292C2×C3×D4—24no—
293C2×C3×D4D3214yes—
294C2×C3×D4C2×C3⋊D311yescharacteristic
295C22×C3×C4—34no—
296C22×C3×C4—91no—
297C22×C3×C4—91no—
298C22×Dic3—34no—
299C22×Dic3—34no—
300C22×Dic3—94no—
301C22×Dic3—91no—
302C22×Dic3—184no—
303C22×Dic3C2×C3⋊D311yescharacteristic
304C2×Dic6—94no—
305C2×Dic6—91no—
306C3×C4⋊C4—34no—
307C3×C4⋊C4—34no—
308C3×C4⋊C4—91no—
309C3×C22⋊C4—34no—
310C3×C22⋊C4—34no—
311C3×C22⋊C4—34no—
312C3×C22⋊C4—91no—
313C3×C22⋊C4—91no—
314C3×C42—91no—
315#217—34no—
316#217—34no—
317#217—94no—
318#217—94no—
319#217—94no—
320#217—91no—
321#217—91no—
322#217—91no—
323#217—184no—
324#217—184no—
325#217—184no—
326#217C2×C3⋊D311yescharacteristic
327#217C2×C3⋊D311yescharacteristic
328#217C2×C3⋊D311yescharacteristic
329C4⋊Dic3—34no—
330C4⋊Dic3—94no—
331C4⋊Dic3—94no—
332C4⋊Dic3—94no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—91no—
338Dic3⋊C4—94no—
339Dic3⋊C4—94no—
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—34no—
361C4×Dic3—94no—
362C4×Dic3—94no—
363C4×Dic3—94no—
364C4×Dic3—94no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C4×Dic3—91no—
369C22×C32—21no—
370C22×C32—21no—
371C22×C32—21no—
372C22×C32—21no—
373C22×C32—21no—
374C22×C32—24no—
375C22×C32—28no—
376C22×C32—24no—
377C22×C32—24no—
378C22×C32—24no—
379C22×C32—24no—
380C22×C32—24no—
381C22×C32—44no—
382C22×C32—44no—
383C22×C32—44no—
384C22×C32—44no—
385C22×C32—44no—
386C22×C32#23512yes—
387C22×C32#23718yes—
388C22×C32C23×D314yes—
389C22×C32C23×D311yescharacteristic
390C32×C4—21no—
391C32×C4—24no—
392C32×C4—44no—
393C32×C4—61no—
394C32×C4—61no—
395C32×C4—61no—
396C32×C4—61no—
397C3⋊Dic3—21no—
398C3⋊Dic3—21no—
399C3⋊Dic3—21no—
400C3⋊Dic3—61no—
401C3⋊Dic3—61no—
402C3⋊Dic3—61no—
403C3⋊Dic3—61no—
404C3⋊Dic3—64no—
405C3⋊Dic3—64no—
406C3⋊Dic3—64no—
407C3⋊Dic3—64no—
408C3⋊Dic3—124no—
409C3⋊Dic3—124no—
410C3⋊Dic3—124no—
411C3⋊Dic3—124no—
412C3⋊Dic3#23712yes—
413C3×Dic3—24no—
414C3×Dic3—24no—
415C3×Dic3—24no—
416C3×Dic3—24no—
417C3×Dic3—38no—
418C3×Dic3—64no—
419C3×Dic3—64no—
420C3×Dic3—64no—
421#126—271no—
422#126—271no—
423#125—271no—
424#125—271no—
425#125—271no—
426#123—31no—
427#123—271no—
428C22⋊Q8—271no—
429C22⋊Q8—271no—
430C4×Q8—271no—
431C4×D4—91no—
432#117—271no—
433#117—271no—
434#117—271no—
435C2×C4⋊C4—271no—
436C33#55211yescharacteristic, complemented, Sylow
437C23×C3—24no—
438C23×C3—24no—
439C23×C3C2×D3214yes—
440C23×C3C2×D3214yes—
441C23×C3C22×C3⋊D311yescharacteristic
442C23×C3C22×C3⋊D311yescharacteristic
443C3×D4—28no—
444C3×D4—22no—
445C3×D4—48no—
446C2×C3×C4—24no—
447C2×C3×C4—34no—
448C2×C3×C4—34no—
449C2×C3×C4—34no—
450C2×C3×C4—34no—
451C2×C3×C4—68no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4—91no—
455C2×C3×C4—91no—
456C2×C3×C4—182no—
457C2×C3×C4—182no—
458C2×C3×C4C2×D3214yes—
459C2×C3×C4C22×C3⋊D311yescharacteristic
460C2×Dic3—22no—
461C2×Dic3—34no—
462C2×Dic3—34no—
463C2×Dic3—34no—
464C2×Dic3—34no—
465C2×Dic3—68no—
466C2×Dic3—68no—
467C2×Dic3—94no—
468C2×Dic3—94no—
469C2×Dic3—94no—
470C2×Dic3—94no—
471C2×Dic3—91no—
472C2×Dic3—91no—
473C2×Dic3—91no—
474C2×Dic3—91no—
475C2×Dic3—184no—
476C2×Dic3—184no—
477C2×Dic3—184no—
478C2×Dic3—184no—
479C2×Dic3—188no—
480C2×Dic3—182no—
481C2×Dic3—368no—
482C2×Dic3C22×C3⋊D311yescharacteristic
483C2×Dic3C22×C3⋊D311yescharacteristic
484C2×Dic3C22×C3⋊D311yescharacteristic
485C2×Dic3C22×C3⋊D311yescharacteristic
486Dic6—182no—
487Dic6—188no—
488C2×C32—22no—
489C2×C32—24no—
490C2×C32—24no—
491C2×C32—24no—
492C2×C32—28no—
493C2×C32—48no—
494C2×C32—41no—
495C2×C32—44no—
496C2×C32—84no—
497C2×C32C2×#23511yescharacteristic
498C2×C32C2×#23714yes—
499C2×C32C2×#23711yescharacteristic
500C2×C32D3×#4114yes—
501C2×C32#101014yes—
502C2×C32#101011yescharacteristic
503C2×Q8—271no—
504C2×D4D3×C3⋊D311yescharacteristic
505C22×C4—31no—
506C22×C4—91no—
507C22×C4—91no—
508C22×C4—271no—
509C4⋊C4—31no—
510C4⋊C4—31no—
511C4⋊C4—91no—
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—
523C4⋊C4—271no—
524C4⋊C4—271no—
525C22⋊C4—31no—
526C22⋊C4—31no—
527C22⋊C4—31no—
528C22⋊C4—91no—
529C22⋊C4—91no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C22⋊C4—271no—
533C42—91no—
534C42—271no—
535C42—271no—
536C42—271no—
537C42—271no—
538C22×C3—24no—
539C22×C3—24no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—24no—
543C22×C3—24no—
544C22×C3—24no—
545C22×C3—21no—
546C22×C3—21no—
547C22×C3—21no—
548C22×C3—24no—
549C22×C3—28no—
550C22×C3—44no—
551C22×C3—44no—
552C22×C3—44no—
553C22×C3—44no—
554C22×C3—44no—
555C22×C3#380418yes—
556C22×C3#383012yes—
557C22×C3C22×D3214yes—
558C22×C3C23×C3⋊D311yescharacteristic
559C3×C4—24no—
560C3×C4—21no—
561C3×C4—44no—
562C3×C4—64no—
563C3×C4—64no—
564C3×C4—64no—
565C3×C4—64no—
566C3×C4—92no—
567C3×C4—181no—
568C3×C4—181no—
569C3×C4—181no—
570Dic3—21no—
571Dic3—21no—
572Dic3—21no—
573Dic3—21no—
574Dic3—38no—
575Dic3—64no—
576Dic3—64no—
577Dic3—64no—
578Dic3—184no—
579Dic3—184no—
580Dic3—184no—
581Dic3—184no—
582Dic3—181no—
583Dic3—181no—
584Dic3—181no—
585Dic3—181no—
586Dic3—364no—
587Dic3—364no—
588Dic3—364no—
589Dic3—364no—
590C32—24no—
591C32#554111yescharacteristic, complemented
592C32#565014yescomplemented
593C23C2×D3×C3⋊D311yescharacteristic
594C23C2×D3×C3⋊D311yescharacteristic
595Q8—542no—
596D4—22no—
597C2×C4—31no—
598C2×C4—31no—
599C2×C4—31no—
600C2×C4—31no—
601C2×C4—62no—
602C2×C4—91no—
603C2×C4—91no—
604C2×C4—91no—
605C2×C4—91no—
606C2×C4—182no—
607C2×C4—182no—
608C2×C4—271no—
609C2×C4—271no—
610C2×C4—271no—
611C2×C4—271no—
612C2×C4—542no—
613C2×C4C2×D3×C3⋊D311yescharacteristic
614C2×C3—22no—
615C2×C3—28no—
616C2×C3—24no—
617C2×C3—24no—
618C2×C3—24no—
619C2×C3—48no—
620C2×C3—41no—
621C2×C3—44no—
622C2×C3—84no—
623C2×C3#6430914yes—
624C2×C3D3×#23714yes—
625C2×C3C2×#380414yes—
626C2×C3C2×#383011yescharacteristic
627C2×C3#41×C3⋊D311yescharacteristic
628C2×C3#6436711yescharacteristic
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22#8902812yes—
635C22C22×D3×C3⋊D311yescharacteristic
636C4—21no—
637C4—61no—
638C4—61no—
639C4—61no—
640C4—61no—
641C4—92no—
642C4—181no—
643C4—181no—
644C4—181no—
645C4—541no—
646C4—541no—
647C4—541no—
648C4—541no—
649C3—24no—
650C3#1059511714yescomplemented
651C3#1059612111yescharacteristic, complemented
652C2—22no—
653C2—41no—
654C2#1172458211yescharacteristic
655C2C3⋊D3×#23711yescharacteristic
656C2C2×#8902811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#552
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11723641C2semidirect
5. #11723641C2semidirect
6. C2×#88807C2semidirect
7. #11723598C2semidirect
8. #11723597C2semidirect
9. #11723596C2semidirect
10. #11723595C2semidirect
11. #11723594C2semidirect
12. #11723445C2semidirect
13. #11723402C2semidirect
14. #11723401C2semidirect
15. #11723390C2semidirect
436. C33#552semidirect
591. C32#5541semidirect
592. C32#5650semidirect
650. C3#10595117semidirect
651. C3#10596121semidirect

Maximal subgroups

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

subgroupindex
#117239762
C3×D4×C3⋊Dic32
C32×#9352
#117236412
C2×#888072
#117235982
#117235972
#117235962
#117235952
#117235942
#117234452
#117234022
#117234012
#117233902
#105961213
#105951173