Finite Group Databasebeta

← #13147823

Subgroups — #13147823

Order 1728.

1099 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88648C211yesmaximal, characteristic
2C4×#6229C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4#64081⋊C3C211yesmaximal, characteristic, complemented
5#11721138C212yesmaximal, complemented
6#11721132C211yesmaximal, characteristic, complemented
7#11721129C211yesmaximal, characteristic, complemented
8#11721122C212yesmaximal, complemented
9#11721121C212yesmaximal, complemented
10#11721119C212yesmaximal, complemented
11#11721083C211yesmaximal, characteristic, complemented
12#10593890—31nomaximal
13#10593889—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#6198C2212yes—
19C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22#88648C418yescomplemented
23C42×H(3,𝔽3)C2211yescharacteristic
24C4⋊#1084C2211yescharacteristic
25#88540C2212yes—
26C4×#1084C2212yes—
27#3752⋊C3C2212yes—
28#3752⋊C3C2211yescharacteristic
29#3752⋊C3C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31#3750⋊C3C2211yescharacteristic
32#3750⋊C3C2211yescharacteristic
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34#88450C2212yes—
35#88449C2212yes—
36#88448C2212yes—
37#88447C2212yes—
38C2×#3799—31no—
39C2×#3796—31no—
40C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
41#64081D311yescharacteristic, complemented
42C3×C4×D12—31no—
43C3×#872—31no—
44C3×C4×Dic6—31no—
45C3×C4×Dic6—31no—
46C3×C4×Dic6—61no—
47#63904—32no—
48#63900—31no—
49#63898—31no—
50#63894—31no—
51D3×C4⋊Dic3—31no—
52#63883—32no—
53#63881—32no—
54#63880—32no—
55#63880—32no—
56#63877—32no—
57D3×Dic3⋊C4—32no—
58#63843—31no—
59Dic3×Dic6—31no—
60#63836—32no—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×#1084C2312yes—
67C2×#1084C2312yes—
68#6229C2×C414yescomplemented
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
74C4×(C3⋊D3)⋊C3—42no—
75C4×(C3⋊D3)⋊C3C2×C414yescomplemented
76#6198C2×C418yescomplemented
77((C3⋊D3)⋊C3)⋊C4—42no—
78((C3⋊D3)⋊C3)⋊C4—42no—
79((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
80#5574—91no—
81#5546—91no—
82#5523—91no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D312yes—
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×C3×D12—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—32no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—61no—
92C2×#626—32no—
93C2×#626—32no—
94C2×#624—32no—
95C2×#624—32no—
96C2×#624—32no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100#3799—38no—
101#3796—38no—
102C32×C42—21no—
103C32×C42C2×D311yescharacteristic
104C32×C42C2×D311yescharacteristic
105#3752C2×D311yescharacteristic
106#3752C2×D311yescharacteristic
107#3752C2×D312yes—
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109#3750C2×D311yescharacteristic
110#3750C2×D311yescharacteristic
111C4×C3⋊Dic3C2×D311yescharacteristic
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—31no—
114C3×(C2×D3)⋊C4—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—61no—
119C3×Dic3⋊C4—31no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—31no—
123C3×Dic3⋊C4—62no—
124C3×C4×Dic3—32no—
125C3×C4×Dic3—31no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—62no—
128#3724—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132#3722—32no—
133(C2×D3)⋊Dic3—32no—
134Dic32—32no—
135Dic32—32no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—21no—
140C2×(C3⋊D3)⋊C3—22no—
141C2×(C3⋊D3)⋊C3C22×C414yes—
142C2×(C3⋊D3)⋊C3C22×C412yes—
143C4×H(3,𝔽3)—21no—
144C4×H(3,𝔽3)—21no—
145C4×H(3,𝔽3)C22×C412yes—
146#1084—22no—
147#1084C22×C414yes—
148(C3⋊Dic3)⋊C3—21no—
149(C3⋊Dic3)⋊C3—22no—
150(C3⋊Dic3)⋊C3—21no—
151(C3⋊Dic3)⋊C3C22×C412yes—
152C2×#239—91no—
153C2×#237—91no—
154C2×#235—91no—
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C3×#117—91no—
158Q8×Dic3—91no—
159D4×Dic3—91no—
160C4×#67—92no—
161C2×C4⋊Dic3—91no—
162C2×Dic3⋊C4—92no—
163#893—91no—
164#893—92no—
165#892—92no—
166D3×C4⋊C4—92no—
167D3×C4⋊C4—92no—
168D3×C4⋊C4—91no—
169D3×C4⋊C4—91no—
170#887—91no—
171#67⋊C4—92no—
172#67⋊C4—92no—
173#67⋊C4—92no—
174#877—92no—
175C4×D12—31no—
176C4×D12—91no—
177C4×D12—91no—
178C4×D12—92no—
179#872—31no—
180#872—91no—
181#872—92no—
182C4×Dic6—31no—
183C4×Dic6—31no—
184C4×Dic6—61no—
185C4×Dic6—91no—
186C22×C3⋊D3C22×D312yes—
187C22×C3⋊D3C22×D311yescharacteristic
188C22×C3×D3—32no—
189C22×C3×D3—31no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D311yescharacteristic
202C2×C3⋊Dic3C22×D311yescharacteristic
203C4⋊(C3⋊D3)C4×D314yescomplemented
204C4×C3⋊D3—42no—
205C4×C3⋊D3C4×D314yescomplemented
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—62no—
216C3×D12—34no—
217C3×C4×D3—34no—
218C3×C4×D3—122no—
219C3×Dic6—34no—
220C3×Dic6—34no—
221C3×Dic6—64no—
222#626—38no—
223#626—38no—
224#624—38no—
225#624—122no—
226#624—122no—
227D3×Dic3—38no—
228D3×Dic3—122no—
229D3×Dic3—122no—
230#520—271noSylow
231C2×H(3,𝔽3)C23×C411yescharacteristic
232C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
233C2×H(3,𝔽3)2−1+411yescharacteristic
234(C3⋊D3)⋊C3—24no—
235(C3⋊D3)⋊C3—22no—
236C2×C3×Q8—91no—
237C2×C3×D4—91no—
238C22×C3×C4—91no—
239C22×C3×C4—92no—
240C2×#67—92no—
241C2×#67—92no—
242C22×Dic3—92no—
243C22×Dic3—92no—
244C22×Dic3—92no—
245#239—98no—
246#237—98no—
247#235—98no—
248C2×D12—31no—
249C2×D12—91no—
250C2×D12—92no—
251C2×C4×D3—31no—
252C2×C4×D3—32no—
253C2×C4×D3—91no—
254C2×C4×D3—92no—
255C2×C4×D3—92no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—92no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×Dic6—31no—
262C2×Dic6—31no—
263C2×Dic6—61no—
264C3×C4⋊C4—91no—
265C3×C4⋊C4—91no—
266C3×C4⋊C4—91no—
267C3×C4⋊C4—92no—
268C3×C4⋊C4—91no—
269C3×C22⋊C4—92no—
270C3×C22⋊C4—91no—
271C3×C22⋊C4—91no—
272C3×C42—31no—
273C3×C42—31no—
274C3×C42—61no—
275C3×C42—91no—
276C3×C42—92no—
277C3×C42D3211yescharacteristic
278#217—92no—
279(C2×D3)⋊C4—31no—
280(C2×D3)⋊C4—31no—
281(C2×D3)⋊C4—32no—
282(C2×D3)⋊C4—91no—
283(C2×D3)⋊C4—91no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287C4⋊Dic3—31no—
288C4⋊Dic3—31no—
289C4⋊Dic3—31no—
290C4⋊Dic3—61no—
291C4⋊Dic3—91no—
292C4⋊Dic3—92no—
293C4⋊Dic3—92no—
294Dic3⋊C4—32no—
295Dic3⋊C4—32no—
296Dic3⋊C4—31no—
297Dic3⋊C4—31no—
298Dic3⋊C4—62no—
299Dic3⋊C4—91no—
300Dic3⋊C4—91no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305C4×Dic3—32no—
306C4×Dic3—32no—
307C4×Dic3—31no—
308C4×Dic3—62no—
309C4×Dic3—91no—
310C4×Dic3—92no—
311C4×Dic3—92no—
312C4×Dic3—92no—
313C22×C32—21no—
314C22×C32C23×D311yescharacteristic
315C22×C32C23×D311yescharacteristic
316C2×C3⋊D3—22no—
317C2×C3⋊D3—21no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3C2×C4×D314yes—
321C2×C3⋊D3C2×C4×D312yes—
322C2×C3×D3—34no—
323C2×C3×D3—32no—
324C2×C3×D3—61no—
325C2×C3×D3—62no—
326C2×C3×D3—62no—
327C2×C3×D3—61no—
328C32×C4—22no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—22no—
334C32×C4—41no—
335C32×C4C2×C4×D312yes—
336C32×C4C2×C4×D312yes—
337C3⋊Dic3—22no—
338C3⋊Dic3—21no—
339C3⋊Dic3—21no—
340C3⋊Dic3C2×C4×D312yes—
341C3×Dic3—32no—
342C3×Dic3—34no—
343C3×Dic3—34no—
344C3×Dic3—64no—
345C3×Dic3—61no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—61no—
350C3×Dic3—122no—
351C2×#41—271no—
352C4×Q8—91no—
353C4×Q8—271no—
354C4×D4—91no—
355C4×D4—272no—
356C4×D4—271no—
357C4×D4—272no—
358#117—91no—
359#117—272no—
360C2×C4⋊C4—272no—
361C2×C4⋊C4—271no—
362H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
363C23×C3—92no—
364C23×C3—91no—
365C22×D3—31no—
366C22×D3—32no—
367C22×D3—92no—
368C22×D3—91no—
369C3×Q8—94no—
370C3×D4—94no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—94no—
381C2×C3×C4—91no—
382C2×C3×C4—92no—
383C2×C3×C4—91no—
384C2×C3×C4—92no—
385C2×C3×C4—91no—
386C2×C3×C4—92no—
387C2×C3×C4—362no—
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391#67—98no—
392#67—98no—
393C2×Dic3—32no—
394C2×Dic3—32no—
395C2×Dic3—31no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—32no—
401C2×Dic3—62no—
402C2×Dic3—62no—
403C2×Dic3—91no—
404C2×Dic3—91no—
405C2×Dic3—91no—
406C2×Dic3—92no—
407C2×Dic3—98no—
408C2×Dic3—362no—
409C2×Dic3—362no—
410D12—34no—
411D12—94no—
412D12—98no—
413C4×D3—34no—
414C4×D3—94no—
415C4×D3—98no—
416C4×D3—98no—
417C4×D3—122no—
418C4×D3—362no—
419C4×D3—362no—
420C4×D3—362no—
421C4×D3—362no—
422C4×D3—362no—
423Dic6—34no—
424Dic6—34no—
425Dic6—64no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32C22×C4×D311yescharacteristic
430C2×C32C22×C4×D311yescharacteristic
431C2×C32#100411yescharacteristic
432C2×C32#100911yescharacteristic
433C2×C32#101011yescharacteristic
434C2×C32#101011yescharacteristic
435C3⋊D3—22no—
436C3⋊D3—24no—
437C3×D3—64no—
438C3×D3—62no—
439#41—278no—
440C2×Q8—91no—
441C2×D4—91no—
442C2×D4—272no—
443C22×C4—91no—
444C22×C4—92no—
445C22×C4—272no—
446C22×C4—272no—
447C22×C4—272no—
448C4⋊C4—91no—
449C4⋊C4—91no—
450C4⋊C4—92no—
451C4⋊C4—91no—
452C4⋊C4—91no—
453C4⋊C4—272no—
454C4⋊C4—272no—
455C22⋊C4—91no—
456C22⋊C4—91no—
457C22⋊C4—92no—
458C22⋊C4—272no—
459C42—91no—
460C42—92no—
461C42—272no—
462C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
463C22×C3—31no—
464C22×C3—31no—
465C22×C3—61no—
466C22×C3—94no—
467C22×C3—92no—
468C22×C3—181no—
469C22×C3—182no—
470C22×C3—182no—
471C22×C3—181no—
472C22×C3C22×D3211yescharacteristic
473C2×D3—32no—
474C2×D3—34no—
475C2×D3—61no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—61no—
479C2×D3—94no—
480C2×D3—92no—
481C2×D3—182no—
482C2×D3—181no—
483C2×D3—181no—
484C2×D3—182no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—32no—
488C3×C4—32no—
489C3×C4—61no—
490C3×C4—61no—
491C3×C4—61no—
492C3×C4—61no—
493C3×C4—62no—
494C3×C4—62no—
495C3×C4—92no—
496C3×C4—94no—
497C3×C4—121no—
498C3×C4—181no—
499C3×C4—182no—
500C3×C4—181no—
501C3×C4—182no—
502C3×C4C4×D3212yes—
503Dic3—34no—
504Dic3—34no—
505Dic3—32no—
506Dic3—64no—
507Dic3—62no—
508Dic3—62no—
509Dic3—61no—
510Dic3—62no—
511Dic3—61no—
512Dic3—92no—
513Dic3—122no—
514Dic3—182no—
515Dic3—181no—
516Dic3—181no—
517C32—21no—
518C32#552311yescharacteristic, complemented
519C32#554611yescharacteristic, complemented
520C23—91no—
521C23—92no—
522Q8—94no—
523D4—94no—
524D4—278no—
525C2×C4—92no—
526C2×C4—91no—
527C2×C4—94no—
528C2×C4—91no—
529C2×C4—91no—
530C2×C4—92no—
531C2×C4—92no—
532C2×C4—278no—
533C2×C4—362no—
534C2×C4—1082no—
535C2×C4—1082no—
536C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
537C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
538C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—61no—
546C2×C3—61no—
547C2×C3—61no—
548C2×C3—182no—
549C2×C3—184no—
550C2×C3#6429811yescharacteristic
551C2×C3C2×C4×D3211yescharacteristic
552C2×C3#6430711yescharacteristic
553D3—62no—
554D3—64no—
555D3—184no—
556D3—182no—
557C22—92no—
558C22—94no—
559C22—181no—
560C22—182no—
561C22—182no—
562C22—181no—
563C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
564C4—21no—
565C4—21no—
566C4—92no—
567C4—94no—
568C4—181no—
569C4—182no—
570C4—182no—
571C4—181no—
572C4C4×((C3⋊D3)⋊C3)⋊C212yes—
573C3—31no—
574C3—31no—
575C3—61no—
576C3#1059388711yescharacteristic
577C2—182no—
578C2—184no—
579C2#1172248311yescharacteristic
580C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
581C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88648C2
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#6229C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #64081⋊C3C2semidirect
5. #11721138C2semidirect
6. #11721132C2semidirect
7. #11721129C2semidirect
8. #11721122C2semidirect
9. #11721121C2semidirect
10. #11721119C2semidirect
11. #11721083C2semidirect
22. #88648C4semidirect
40. C4×C4⋊(C3⋊D3)D3semidirect
41. #64081D3semidirect
68. #6229C2×C4semidirect
73. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
75. C4×(C3⋊D3)⋊C3C2×C4semidirect
76. #6198C2×C4semidirect
79. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
203. C4⋊(C3⋊D3)C4×D3semidirect
205. C4×C3⋊D3C4×D3semidirect
362. H(3,𝔽3)#520semidirect
518. C32#5523semidirect
519. C32#5546semidirect

Maximal subgroups

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

subgroupindex
C2×#886482
C4×#62292
C4×C4⋊((C3⋊D3)⋊C3)2
#64081⋊C32
#117211382
#117211322
#117211292
#117211222
#117211212
#117211192
#117210832
#105938903
#105938893