Finite Group Databasebeta

← #13149383

Subgroups — #13149383

Order 1728.

658 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721757C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic
3#64098⋊C3C211yesmaximal, characteristic
4#11721098C211yesmaximal, characteristic
5#11721095C211yesmaximal, characteristic
6#11721093C211yesmaximal, characteristic
7#11721091C211yesmaximal, characteristic
8#11721091C211yesmaximal, characteristic
9#11721091C211yesmaximal, characteristic
10#11721087C211yesmaximal, characteristic
11#11721087C211yesmaximal, characteristic
12#11721087C211yesmaximal, characteristic
13#11721086C211yesmaximal, characteristic
14#11721086C211yesmaximal, characteristic
15#11721085C211yesmaximal, characteristic
16#10594354—31nomaximal
17#10594326—31nomaximal
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540C2211yescharacteristic
21#88540C2211yescharacteristic
22#88540C2211yescharacteristic
23#88540C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2211yescharacteristic
30#3750⋊C3C2211yescharacteristic
31#3750⋊C3C2211yescharacteristic
32#3750⋊C3C2211yescharacteristic
33#3750⋊C3C2211yescharacteristic
34#3750⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
37#88449C2211yescharacteristic
38#88449C2211yescharacteristic
39#88449C2211yescharacteristic
40#88449C2211yescharacteristic
41#88448C2211yescharacteristic
42#88448C2211yescharacteristic
43#88448C2211yescharacteristic
44#88448C2211yescharacteristic
45#88448C2211yescharacteristic
46#88448C2211yescharacteristic
47#88448C2211yescharacteristic
48#88448C2211yescharacteristic
49#88448C2211yescharacteristic
50#88448C2211yescharacteristic
51#88447C2211yescharacteristic
52#88447C2211yescharacteristic
53#64099D311yescharacteristic
54#64098D311yescharacteristic
55C3×#890—31no—
56C3×#889—31no—
57C3×#889—31no—
58C3×#889—31no—
59C3×#889—61no—
60#63872—31no—
61#63870—31no—
62#63870—31no—
63#63870—31no—
64#63869—31no—
65#63869—31no—
66#63869—31no—
67#63867—31no—
68#63867—31no—
69#63867—31no—
70#63867—31no—
71#63867—31no—
72#63847—31no—
73#63847—31no—
74#63847—31no—
75#63847—31no—
76#63846—31no—
77#63846—31no—
78#63846—31no—
79#63845—31no—
80#63845—31no—
81#63845—31no—
82#63845—31no—
83#63845—31no—
84C2×C4×H(3,𝔽3)C2311yescharacteristic
85C2×C4×H(3,𝔽3)C2311yescharacteristic
86C2×C4×H(3,𝔽3)C2311yescharacteristic
87C2×#1084C2311yescharacteristic
88C2×#1084C2311yescharacteristic
89C2×#1084C2311yescharacteristic
90C2×#1084C2311yescharacteristic
91C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
92C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
93C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
94C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
95C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
96C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
97C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
98C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
99C3⋊#564—91no—
100C3⋊#564—91no—
101C3⋊#564—91no—
102C32×C4⋊C4—21no—
103C32×C4⋊C4C2×D311yescharacteristic
104C32×C4⋊C4C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109#3750C2×D311yescharacteristic
110#3750C2×D311yescharacteristic
111#3750C2×D311yescharacteristic
112#3750C2×D311yescharacteristic
113#3750C2×D311yescharacteristic
114#3750C2×D311yescharacteristic
115C4×C3⋊Dic3C2×D311yescharacteristic
116C4×C3⋊Dic3C2×D311yescharacteristic
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—31no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—61no—
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4—31no—
127C3×Dic3⋊C4—31no—
128C3×Dic3⋊C4—31no—
129C3×Dic3⋊C4—31no—
130C3×Dic3⋊C4—31no—
131C3×Dic3⋊C4—31no—
132C3×Dic3⋊C4—31no—
133C3×Dic3⋊C4—31no—
134C3×Dic3⋊C4—31no—
135C3×Dic3⋊C4—31no—
136C3×Dic3⋊C4—31no—
137C3×Dic3⋊C4—31no—
138C3×Dic3⋊C4—61no—
139C3×Dic3⋊C4—61no—
140C3×Dic3⋊C4—61no—
141C3×Dic3⋊C4—61no—
142C3×C4×Dic3—31no—
143C3×C4×Dic3—31no—
144C3×C4×Dic3—31no—
145C3×C4×Dic3—31no—
146C3×C4×Dic3—61no—
147#3724—31no—
148#3724—31no—
149#3724—31no—
150#3724—31no—
151#3724—31no—
152#3724—31no—
153#3724—31no—
154#3724—31no—
155#3724—31no—
156#3724—31no—
157Dic3⋊Dic3—31no—
158Dic3⋊Dic3—31no—
159Dic3⋊Dic3—31no—
160Dic3⋊Dic3—31no—
161Dic3⋊Dic3—31no—
162Dic3⋊Dic3—31no—
163Dic3⋊Dic3—31no—
164Dic3⋊Dic3—31no—
165Dic3⋊Dic3—31no—
166Dic3⋊Dic3—31no—
167Dic3⋊Dic3—31no—
168Dic3⋊Dic3—31no—
169Dic3⋊Dic3—31no—
170Dic3⋊Dic3—31no—
171Dic3⋊Dic3—31no—
172Dic3⋊Dic3—31no—
173Dic3⋊Dic3—31no—
174Dic3⋊Dic3—31no—
175Dic32—31no—
176Dic32—31no—
177Dic32—31no—
178Dic32—31no—
179C22×H(3,𝔽3)C2411yescharacteristic
180C4×H(3,𝔽3)—21no—
181C4×H(3,𝔽3)—21no—
182C4×H(3,𝔽3)—21no—
183#1084—21no—
184#1084—21no—
185#1084—21no—
186#1084—21no—
187(C3⋊Dic3)⋊C3—21no—
188(C3⋊Dic3)⋊C3—21no—
189(C3⋊Dic3)⋊C3—21no—
190(C3⋊Dic3)⋊C3—21no—
191(C3⋊Dic3)⋊C3—21no—
192(C3⋊Dic3)⋊C3—21no—
193(C3⋊Dic3)⋊C3—21no—
194(C3⋊Dic3)⋊C3—21no—
195C3×#125—91no—
196C3×#125—91no—
197C3×#125—91no—
198#890—31no—
199#890—91no—
200#890—91no—
201#890—91no—
202#890—91no—
203#890—91no—
204#890—91no—
205#890—91no—
206#890—91no—
207#890—91no—
208#890—91no—
209#890—91no—
210#889—31no—
211#889—31no—
212#889—31no—
213#889—61no—
214#889—91no—
215#889—91no—
216#889—91no—
217#889—91no—
218#889—91no—
219#889—91no—
220#889—91no—
221#889—91no—
222#889—91no—
223#889—91no—
224#889—91no—
225#889—91no—
226#889—91no—
227#889—91no—
228#889—91no—
229#889—91no—
230#889—91no—
231#889—91no—
232#889—91no—
233#889—91no—
234#889—91no—
235#870—91no—
236#870—91no—
237#870—91no—
238#870—91no—
239#870—91no—
240#870—91no—
241C2×C32×C4—21no—
242C2×C32×C4—21no—
243C2×C32×C4—21no—
244C2×C32×C4C22×D311yescharacteristic
245C2×C32×C4C22×D311yescharacteristic
246C2×C32×C4C22×D311yescharacteristic
247C2×C32×C4C22×D311yescharacteristic
248C2×C32×C4C22×D311yescharacteristic
249C2×C32×C4C22×D311yescharacteristic
250C2×C3⋊Dic3C22×D311yescharacteristic
251C2×C3⋊Dic3C22×D311yescharacteristic
252C2×C3⋊Dic3C22×D311yescharacteristic
253C2×C3⋊Dic3C22×D311yescharacteristic
254C2×C3⋊Dic3C22×D311yescharacteristic
255C2×C3⋊Dic3C22×D311yescharacteristic
256C2×C3⋊Dic3C22×D311yescharacteristic
257C2×C3⋊Dic3C22×D311yescharacteristic
258C2×C3×Dic3—31no—
259C2×C3×Dic3—31no—
260C2×C3×Dic3—31no—
261C2×C3×Dic3—31no—
262C2×C3×Dic3—31no—
263C2×C3×Dic3—31no—
264C2×C3×Dic3—31no—
265C2×C3×Dic3—31no—
266C2×C3×Dic3—31no—
267C2×C3×Dic3—31no—
268C2×C3×Dic3—31no—
269C2×C3×Dic3—31no—
270C2×C3×Dic3—31no—
271C2×C3×Dic3—31no—
272C2×C3×Dic3—31no—
273C2×C3×Dic3—31no—
274C2×C3×Dic3—61no—
275C2×C3×Dic3—61no—
276C2×C3×Dic3—61no—
277C2×C3×Dic3—61no—
278#564—271noSylow
279C2×H(3,𝔽3)2−1+411yescharacteristic
280C2×H(3,𝔽3)2−1+411yescharacteristic
281C2×H(3,𝔽3)2−1+411yescharacteristic
282C3×C4⋊C4—31no—
283C3×C4⋊C4—31no—
284C3×C4⋊C4—61no—
285C3×C4⋊C4—91no—
286C3×C4⋊C4—91no—
287C3×C4⋊C4—91no—
288C3×C4⋊C4—91no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—91no—
291C3×C4⋊C4—91no—
292C3×C4⋊C4—91no—
293C3×C4⋊C4—91no—
294C3×C4⋊C4—91no—
295C3×C4⋊C4—91no—
296C3×C4⋊C4—91no—
297C3×C4⋊C4—91no—
298C3×C4⋊C4—91no—
299C3×C4⋊C4—91no—
300C3×C4⋊C4D3211yescharacteristic
301C3×C42—91no—
302C3×C42—91no—
303C3×C42—91no—
304C4⋊Dic3—31no—
305C4⋊Dic3—31no—
306C4⋊Dic3—31no—
307C4⋊Dic3—31no—
308C4⋊Dic3—31no—
309C4⋊Dic3—31no—
310C4⋊Dic3—61no—
311C4⋊Dic3—91no—
312C4⋊Dic3—91no—
313C4⋊Dic3—91no—
314C4⋊Dic3—91no—
315C4⋊Dic3—91no—
316C4⋊Dic3—91no—
317C4⋊Dic3—91no—
318C4⋊Dic3—91no—
319C4⋊Dic3—91no—
320C4⋊Dic3—91no—
321C4⋊Dic3—91no—
322C4⋊Dic3—91no—
323C4⋊Dic3—91no—
324C4⋊Dic3—91no—
325C4⋊Dic3—91no—
326C4⋊Dic3—91no—
327C4⋊Dic3—91no—
328C4⋊Dic3—91no—
329Dic3⋊C4—31no—
330Dic3⋊C4—31no—
331Dic3⋊C4—31no—
332Dic3⋊C4—31no—
333Dic3⋊C4—31no—
334Dic3⋊C4—31no—
335Dic3⋊C4—31no—
336Dic3⋊C4—31no—
337Dic3⋊C4—31no—
338Dic3⋊C4—31no—
339Dic3⋊C4—31no—
340Dic3⋊C4—31no—
341Dic3⋊C4—31no—
342Dic3⋊C4—31no—
343Dic3⋊C4—61no—
344Dic3⋊C4—61no—
345Dic3⋊C4—61no—
346Dic3⋊C4—61no—
347Dic3⋊C4—91no—
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—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—91no—
361Dic3⋊C4—91no—
362Dic3⋊C4—91no—
363Dic3⋊C4—91no—
364Dic3⋊C4—91no—
365Dic3⋊C4—91no—
366Dic3⋊C4—91no—
367Dic3⋊C4—91no—
368Dic3⋊C4—91no—
369Dic3⋊C4—91no—
370Dic3⋊C4—91no—
371Dic3⋊C4—91no—
372Dic3⋊C4—91no—
373Dic3⋊C4—91no—
374Dic3⋊C4—91no—
375Dic3⋊C4—91no—
376Dic3⋊C4—91no—
377Dic3⋊C4—91no—
378Dic3⋊C4—91no—
379Dic3⋊C4—91no—
380Dic3⋊C4—91no—
381C4×Dic3—31no—
382C4×Dic3—31no—
383C4×Dic3—31no—
384C4×Dic3—31no—
385C4×Dic3—61no—
386C4×Dic3—91no—
387C4×Dic3—91no—
388C4×Dic3—91no—
389C4×Dic3—91no—
390C4×Dic3—91no—
391C4×Dic3—91no—
392C4×Dic3—91no—
393C4×Dic3—91no—
394C22×C32—21no—
395C22×C32C23×D311yescharacteristic
396C22×C32C23×D311yescharacteristic
397C32×C4—21no—
398C32×C4—21no—
399C32×C4—21no—
400C32×C4—21no—
401C32×C4—21no—
402C32×C4—21no—
403C32×C4—41no—
404C32×C4—41no—
405C32×C4—41no—
406C3⋊Dic3—21no—
407C3⋊Dic3—21no—
408C3⋊Dic3—21no—
409C3⋊Dic3—21no—
410C3⋊Dic3—21no—
411C3⋊Dic3—21no—
412C3⋊Dic3—21no—
413C3⋊Dic3—21no—
414C3×Dic3—62no—
415C3×Dic3—61no—
416C3×Dic3—61no—
417C3×Dic3—61no—
418C3×Dic3—61no—
419C3×Dic3—61no—
420C3×Dic3—61no—
421C3×Dic3—61no—
422C3×Dic3—61no—
423C3×Dic3—61no—
424C3×Dic3—61no—
425C3×Dic3—61no—
426C3×Dic3—61no—
427C3×Dic3—61no—
428C3×Dic3—61no—
429C3×Dic3—61no—
430C3×Dic3—61no—
431C3×Dic3—121no—
432C3×Dic3—121no—
433C3×Dic3—121no—
434#125—91no—
435#125—91no—
436#125—91no—
437#125—271no—
438#125—271no—
439#125—271no—
440#125—271no—
441#125—271no—
442#125—271no—
443#125—271no—
444#125—271no—
445#125—271no—
446#125—271no—
447#125—271no—
448#125—271no—
449H(3,𝔽3)#56411yescharacteristic, complemented, Sylow
450C2×C3×C4—31no—
451C2×C3×C4—31no—
452C2×C3×C4—31no—
453C2×C3×C4—31no—
454C2×C3×C4—31no—
455C2×C3×C4—31no—
456C2×C3×C4—61no—
457C2×C3×C4—61no—
458C2×C3×C4—61no—
459C2×C3×C4—91no—
460C2×C3×C4—91no—
461C2×C3×C4—91no—
462C2×C3×C4—91no—
463C2×C3×C4—91no—
464C2×C3×C4—91no—
465C2×C3×C4—91no—
466C2×C3×C4—91no—
467C2×C3×C4—91no—
468C2×C3×C4—91no—
469C2×C3×C4—91no—
470C2×C3×C4—91no—
471C2×C3×C4C2×D3211yescharacteristic
472C2×C3×C4C2×D3211yescharacteristic
473C2×C3×C4C2×D3211yescharacteristic
474C2×Dic3—31no—
475C2×Dic3—31no—
476C2×Dic3—31no—
477C2×Dic3—31no—
478C2×Dic3—31no—
479C2×Dic3—31no—
480C2×Dic3—31no—
481C2×Dic3—31no—
482C2×Dic3—31no—
483C2×Dic3—31no—
484C2×Dic3—31no—
485C2×Dic3—31no—
486C2×Dic3—31no—
487C2×Dic3—31no—
488C2×Dic3—31no—
489C2×Dic3—31no—
490C2×Dic3—61no—
491C2×Dic3—61no—
492C2×Dic3—61no—
493C2×Dic3—61no—
494C2×Dic3—91no—
495C2×Dic3—91no—
496C2×Dic3—91no—
497C2×Dic3—91no—
498C2×Dic3—91no—
499C2×Dic3—91no—
500C2×Dic3—91no—
501C2×Dic3—91no—
502C2×C32—21no—
503C2×C32—21no—
504C2×C32—21no—
505C2×C32#100711yescharacteristic
506C2×C32#100711yescharacteristic
507C2×C32#101011yescharacteristic
508C2×C32#101011yescharacteristic
509C2×C32#101011yescharacteristic
510C2×C32#101011yescharacteristic
511C4⋊C4—91no—
512C4⋊C4—91no—
513C4⋊C4—91no—
514C4⋊C4—91no—
515C4⋊C4—91no—
516C4⋊C4—91no—
517C4⋊C4—91no—
518C4⋊C4—91no—
519C4⋊C4—91no—
520C4⋊C4—91no—
521C4⋊C4—91no—
522C4⋊C4—91no—
523C4⋊C4—91no—
524C4⋊C4—91no—
525C4⋊C4—91no—
526C4⋊C4—271no—
527C4⋊C4—271no—
528C4⋊C4—271no—
529C4⋊C4—271no—
530C4⋊C4—271no—
531C4⋊C4—271no—
532C4⋊C4—271no—
533C4⋊C4—271no—
534C4⋊C4—271no—
535C4⋊C4—271no—
536C4⋊C4—271no—
537C4⋊C4—271no—
538C4⋊C4—271no—
539C4⋊C4—271no—
540C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
541C42—91no—
542C42—91no—
543C42—91no—
544C42—271no—
545C42—271no—
546C22×C3—31no—
547C22×C3—31no—
548C22×C3—61no—
549C22×C3C22×D3211yescharacteristic
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—61no—
554C3×C4—61no—
555C3×C4—61no—
556C3×C4—61no—
557C3×C4—61no—
558C3×C4—61no—
559C3×C4—121no—
560C3×C4—121no—
561C3×C4—121no—
562C3×C4—181no—
563C3×C4—181no—
564C3×C4—181no—
565C3×C4—181no—
566C3×C4—181no—
567C3×C4—181no—
568C3×C4—181no—
569C3×C4—181no—
570C3×C4—181no—
571C3×C4—181no—
572C3×C4—181no—
573C3×C4—181no—
574Dic3—61no—
575Dic3—61no—
576Dic3—61no—
577Dic3—61no—
578Dic3—61no—
579Dic3—61no—
580Dic3—61no—
581Dic3—61no—
582Dic3—62no—
583Dic3—61no—
584Dic3—61no—
585Dic3—61no—
586Dic3—61no—
587Dic3—61no—
588Dic3—61no—
589Dic3—61no—
590Dic3—61no—
591Dic3—121no—
592Dic3—121no—
593Dic3—121no—
594Dic3—181no—
595Dic3—181no—
596Dic3—181no—
597Dic3—181no—
598Dic3—181no—
599Dic3—181no—
600Dic3—181no—
601Dic3—181no—
602C32—21no—
603C32C3⋊#56411yescharacteristic, complemented
604C32C3⋊#56411yescharacteristic, complemented
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—91no—
608C2×C4—91no—
609C2×C4—91no—
610C2×C4—91no—
611C2×C4—91no—
612C2×C4—91no—
613C2×C4—91no—
614C2×C4—91no—
615C2×C4—91no—
616C2×C4—91no—
617C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
618C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
619C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
620C2×C3—31no—
621C2×C3—31no—
622C2×C3—31no—
623C2×C3—31no—
624C2×C3—31no—
625C2×C3—31no—
626C2×C3—61no—
627C2×C3—61no—
628C2×C3—61no—
629C2×C3#6429711yescharacteristic
630C2×C3#6430911yescharacteristic
631C2×C3#6431511yescharacteristic
632C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—181no—
637C4—181no—
638C4—181no—
639C4—181no—
640C4—181no—
641C4—181no—
642C4—181no—
643C4—181no—
644C4—181no—
645C4—181no—
646C4—181no—
647C4—181no—
648C3—31no—
649C3—31no—
650C3—61no—
651C3#1059432611yescharacteristic
652C2#1172248211yescharacteristic
653C2#1172250211yescharacteristic
654C2#1172251611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#564
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
449. H(3,𝔽3)#564semidirect
603. C32C3⋊#564semidirect
604. C32C3⋊#564semidirect

Maximal subgroups

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

subgroupindex
#117217572
#64099⋊C32
#64098⋊C32
#117210982
#117210952
#117210932
#117210912
#117210872
#117210862
#117210852
#105943543
#105943263