Finite Group Databasebeta

← #13151332

Subgroups — #13151332

Order 1728.

1851 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#88676C214yesmaximal, complemented
3C2×#88654C211yesmaximal, characteristic, complemented
4#11721850C211yesmaximal, characteristic, complemented
5#64149⋊C3C211yesmaximal, characteristic, complemented
6#11721317C214yesmaximal, complemented
7#11721158C211yesmaximal, characteristic, complemented
8#11721144C211yesmaximal, characteristic, complemented
9C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
10#10595300—31nomaximal
11D3×#940—31nomaximal
12C23×(C3⋊D3)⋊C3C2212yes—
13C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
14C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×#6235C2214yescomplemented
16C2×C4⋊#272C2211yescharacteristic, complemented
17C2×#638⋊C3C2214yescomplemented
18C2×#638⋊C3C2212yescomplemented
19D4×(C3⋊D3)⋊C3—24no—
20C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
21C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
22C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
23#88676—216no—
24C2×#6200C2214yescomplemented
25C2×#6200C2212yescomplemented
26C2×#6198C2214yescomplemented
27C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
28#88654—24no—
29C4×#1084C2211yescharacteristic, complemented
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
31#88447C2212yescomplemented
32C2×D4×C3⋊D3D311yescharacteristic, complemented
33C2×C3×D3×D4—31no—
34C2×#3811—34no—
35C2×D3×#67—34no—
36C2×#3802—31no—
37C2×D3×D12—31no—
38#64149D311yescharacteristic, complemented
39C3×#940—31no—
40C3×#940—31no—
41C3×#940—31no—
42C3×#940—61no—
43#63979—34no—
44#63979—34no—
45#63914—31no—
46#63911—31no—
47#63911—31no—
48#63911—31no—
49C4×#624—31no—
50C4×D3×Dic3—31no—
51C23×H(3,𝔽3)C2312yes—
52C22×#272C2312yes—
53C22×(C3⋊D3)⋊C3—28no—
54C22×(C3⋊D3)⋊C3—24no—
55C22×(C3⋊D3)⋊C3C2312yes—
56C22×(C3⋊D3)⋊C3C2312yes—
57C22×(C3⋊D3)⋊C3C2311yescharacteristic
58C2×((C3⋊D3)⋊C3)⋊C2—24no—
59C2×((C3⋊D3)⋊C3)⋊C2—28no—
60D4×H(3,𝔽3)—22no—
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62#6235—28no—
63C2×#1084C2312yes—
64C4⋊#272—22no—
65#638⋊C3—28no—
66#638⋊C3—24no—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C4⋊((C3⋊D3)⋊C3)—22no—
70C4⋊((C3⋊D3)⋊C3)—22no—
71C4×(C3⋊D3)⋊C3D414yescomplemented
72#6200—24no—
73#6200—28no—
74#6198—28no—
75((C3⋊D3)⋊C3)⋊C4D418yescomplemented
76C2×#940—91no—
77D3×#127—91no—
78D3×#127—91no—
79C23×C3⋊D3C2×D312yes—
80C23×C3×D3—32no—
81C22×D32—32no—
82C22×D32—32no—
83C2×C32×D4—21no—
84C2×C32×D4C2×D311yescharacteristic, complemented
85C2×C32×D4C2×D311yescharacteristic, complemented
86C2×#638C2×D314yescomplemented
87C2×#638C2×D312yescomplemented
88D4×C3⋊D3—24no—
89C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
90C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
91C2×C4×C3⋊D3C2×D311yescharacteristic
92C2×C3×#67—34no—
93C2×C3×#67—34no—
94C2×C3×#67—34no—
95C2×C3×#67—32no—
96C2×C3×#67—64no—
97C3×D3×D4—64no—
98C2×C3×D12—31no—
99C2×C3×D12—31no—
100C2×C3×D12—31no—
101C2×C3×D12—31no—
102C2×C3×D12—61no—
103C2×C3×C4×D3—31no—
104#3811—616no—
105D3×#67—616no—
106C2×#626—32no—
107C2×#626—34no—
108C2×#626—34no—
109C2×#626—34no—
110C2×#625—32no—
111C2×#625—34no—
112C2×#624—32no—
113C2×D3×Dic3—32no—
114#3802—64no—
115D3×D12—64no—
116C4×C3⋊Dic3C2×D311yescharacteristic, complemented
117C3×C4×Dic3—31no—
118C3×C4×Dic3—31no—
119C3×C4×Dic3—31no—
120C3×C4×Dic3—61no—
121Dic32—32no—
122Dic32—32no—
123C22×H(3,𝔽3)—24no—
124C22×H(3,𝔽3)—22no—
125C22×H(3,𝔽3)C2411yescharacteristic
126C2×#272—22no—
127C2×#272—24no—
128C2×(C3⋊D3)⋊C3—24no—
129C2×(C3⋊D3)⋊C3—24no—
130C2×(C3⋊D3)⋊C3—22no—
131C2×(C3⋊D3)⋊C3—22no—
132C2×(C3⋊D3)⋊C3—48no—
133C2×(C3⋊D3)⋊C3C2×D414yes—
134C2×(C3⋊D3)⋊C3C2×D412yes—
135((C3⋊D3)⋊C3)⋊C2—48no—
136C4×H(3,𝔽3)C2×D412yescomplemented
137#1084C2×D414yescomplemented
138(C3⋊Dic3)⋊C3C2×D414yescomplemented
139(C3⋊Dic3)⋊C3C2×D412yescomplemented
140C22×C3×D4—91no—
141C22×#67—94no—
142C2×D3×D4—31no—
143C2×D3×D4—91no—
144C2×D3×D4—91no—
145C2×D3×D4—91no—
146C2×D3×D4—94no—
147C2×D3×D4—94no—
148C22×D12—91no—
149C3×#127—91no—
150C3×#127—91no—
151#940—31no—
152#940—31no—
153#940—31no—
154#940—61no—
155#940—91no—
156#940—91no—
157#940—91no—
158#940—91no—
159#940—91no—
160#940—94no—
161#940—94no—
162#940—94no—
163C2×C4×Dic3—91no—
164#874—91no—
165#874—91no—
166C42×D3—91no—
167C42×D3—91no—
168C23×C32—22no—
169C23×C32C22×D312yes—
170C23×C32C22×D312yes—
171C22×C3⋊D3—28no—
172C22×C3⋊D3—24no—
173C22×C3⋊D3C22×D312yes—
174C22×C3⋊D3C22×D312yes—
175C22×C3⋊D3C22×D311yescharacteristic
176C22×C3×D3—31no—
177C22×C3×D3—32no—
178C22×C3×D3—32no—
179C22×C3×D3—32no—
180C22×C3×D3—32no—
181C22×C3×D3—62no—
182C22×C3×D3—68no—
183C22×C3×D3—64no—
184C2×D32—64no—
185C2×D32—68no—
186C2×D32—68no—
187C2×D32—64no—
188C32×D4—22no—
189C32×D4—22no—
190C32×D4—42no—
191C2×C32×C4—21no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194#638—28no—
195#638—24no—
196C2×C3⋊Dic3C22×D312yes—
197C2×C3⋊Dic3C22×D311yescharacteristic
198C4⋊(C3⋊D3)—22no—
199C4⋊(C3⋊D3)—22no—
200C4×C3⋊D3#6714yescomplemented
201C3×#67—68no—
202C3×#67—64no—
203C3×#67—68no—
204C3×#67—68no—
205C3×#67—128no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—62no—
211C3×D12—62no—
212C3×D12—62no—
213C3×D12—62no—
214C3×D12—62no—
215C3×D12—122no—
216C3×C4×D3—34no—
217#626—68no—
218#626—68no—
219#626—64no—
220#626—68no—
221#625—68no—
222#625—64no—
223#624—38no—
224D3×Dic3—38no—
225C2×#127—271noSylow
226C2×H(3,𝔽3)—42no—
227C2×H(3,𝔽3)C22×D412yes—
228C2×H(3,𝔽3)C22×D411yescharacteristic
229#272—42no—
230(C3⋊D3)⋊C3—42no—
231(C3⋊D3)⋊C3—42no—
232(C3⋊D3)⋊C3#12714yescomplemented
233C24×C3—92no—
234C23×D3—32no—
235C23×D3—92no—
236C23×D3—92no—
237C23×D3—92no—
238C23×D3—92no—
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4—61no—
242C2×C3×D4—91no—
243C2×C3×D4—91no—
244C2×C3×D4—92no—
245C2×C3×D4—94no—
246C2×C3×D4—94no—
247C2×C3×D4—91no—
248C2×C3×D4—184no—
249C2×C3×D4D3211yescharacteristic
250C22×C3×C4—91no—
251C2×#67—34no—
252C2×#67—34no—
253C2×#67—34no—
254C2×#67—32no—
255C2×#67—64no—
256C2×#67—92no—
257C2×#67—94no—
258C2×#67—94no—
259C2×#67—92no—
260C2×#67—94no—
261C2×#67—94no—
262C2×#67—92no—
263C2×#67—94no—
264C2×#67—1816no—
265C22×Dic3—92no—
266D3×D4—64no—
267D3×D4—184no—
268D3×D4—184no—
269D3×D4—184no—
270D3×D4—1816no—
271D3×D4—1816no—
272C2×D12—31no—
273C2×D12—31no—
274C2×D12—31no—
275C2×D12—31no—
276C2×D12—61no—
277C2×D12—91no—
278C2×D12—91no—
279C2×D12—94no—
280C2×D12—94no—
281C2×D12—92no—
282C2×D12—184no—
283C2×C4×D3—31no—
284C2×C4×D3—91no—
285C2×C4×D3—92no—
286C2×C4×D3—92no—
287C3×C42—91no—
288C3×C42—91no—
289C4×Dic3—31no—
290C4×Dic3—31no—
291C4×Dic3—31no—
292C4×Dic3—61no—
293C4×Dic3—91no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C22×C32—24no—
298C22×C32—24no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32—21no—
302C22×C32—44no—
303C22×C32—42no—
304C22×C32C23×D311yescharacteristic
305C22×C32C23×D311yescharacteristic
306C2×C3⋊D3—24no—
307C2×C3⋊D3—22no—
308C2×C3⋊D3—24no—
309C2×C3⋊D3—22no—
310C2×C3⋊D3—48no—
311C2×C3⋊D3D3×D414yes—
312C2×C3⋊D3C2×#6712yes—
313C2×C3×D3—34no—
314C2×C3×D3—32no—
315C2×C3×D3—64no—
316C2×C3×D3—64no—
317C2×C3×D3—62no—
318C2×C3×D3—64no—
319C2×C3×D3—62no—
320C2×C3×D3—64no—
321C2×C3×D3—62no—
322C2×C3×D3—62no—
323C2×C3×D3—122no—
324C2×C3×D3—124no—
325C2×C3×D3—128no—
326D32—128no—
327D32—128no—
328C32×C4—22no—
329C32×C4D3×D412yescomplemented
330C32×C4C2×#6712yescomplemented
331C3⋊Dic3D3×D414yescomplemented
332C3⋊Dic3C2×#6712yescomplemented
333C3×Dic3—34no—
334C3×Dic3—32no—
335C3×Dic3—34no—
336C3×Dic3—34no—
337C3×Dic3—64no—
338C22×D4—91no—
339C22×D4—271no—
340C22×D4—274no—
341#127—91no—
342#127—91no—
343#127—271no—
344#127—271no—
345#127—274no—
346C2×C42—271no—
347H(3,𝔽3)C2×#12711yescharacteristic, complemented, Sylow
348C23×C3—32no—
349C23×C3—32no—
350C23×C3—62no—
351C23×C3—91no—
352C23×C3—92no—
353C23×C3—92no—
354C23×C3—92no—
355C23×C3—188no—
356C23×C3—184no—
357C23×C3C2×D3212yes—
358C22×D3—31no—
359C22×D3—32no—
360C22×D3—32no—
361C22×D3—32no—
362C22×D3—32no—
363C22×D3—62no—
364C22×D3—64no—
365C22×D3—68no—
366C22×D3—91no—
367C22×D3—92no—
368C22×D3—92no—
369C22×D3—184no—
370C22×D3—188no—
371C22×D3—188no—
372C22×D3—184no—
373C22×D3—184no—
374C22×D3—188no—
375C22×D3—184no—
376C22×D3—188no—
377C3×D4—22no—
378C3×D4—62no—
379C3×D4—62no—
380C3×D4—122no—
381C3×D4—182no—
382C3×D4—182no—
383C3×D4—188no—
384C3×D4—188no—
385C3×D4—184no—
386C3×D4—182no—
387C2×C3×C4—31no—
388C2×C3×C4—31no—
389C2×C3×C4—61no—
390C2×C3×C4—94no—
391C2×C3×C4—91no—
392C2×C3×C4—92no—
393C2×C3×C4—92no—
394C2×C3×C4C2×D3211yescharacteristic
395#67—64no—
396#67—68no—
397#67—68no—
398#67—68no—
399#67—128no—
400#67—184no—
401#67—188no—
402#67—188no—
403#67—184no—
404#67—188no—
405#67—188no—
406#67—184no—
407#67—188no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—32no—
411C2×Dic3—31no—
412C2×Dic3—62no—
413C2×Dic3—91no—
414C2×Dic3—92no—
415C2×Dic3—98no—
416D12—62no—
417D12—62no—
418D12—62no—
419D12—62no—
420D12—122no—
421D12—182no—
422D12—182no—
423D12—188no—
424D12—184no—
425D12—188no—
426C4×D3—34no—
427C4×D3—94no—
428C4×D3—98no—
429C4×D3—98no—
430C2×C32—22no—
431C2×C32—21no—
432C2×C32—42no—
433C2×C32—42no—
434C2×C32—82no—
435C2×C32C2×D3×D412yes—
436C2×C32C2×D3×D412yes—
437C2×C32C2×D3×D411yescharacteristic
438C2×C32C22×#6711yescharacteristic
439C3⋊D3—42no—
440C3⋊D3—42no—
441C3⋊D3#94014yescomplemented
442C3×D3—34no—
443C3×D3—122no—
444C3×D3—122no—
445C3×D3—122no—
446C3×D3—122no—
447C3×D3—242no—
448C24—92no—
449C24—272no—
450C2×D4—94no—
451C2×D4—94no—
452C2×D4—92no—
453C2×D4—91no—
454C2×D4—91no—
455C2×D4—91no—
456C2×D4—184no—
457C2×D4—274no—
458C2×D4—274no—
459C2×D4—272no—
460C2×D4—544no—
461C2×D4—5416no—
462C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
463C22×C4—91no—
464C22×C4—272no—
465C42—91no—
466C42—91no—
467C42—272no—
468C22×C3—24no—
469C22×C3—22no—
470C22×C3—31no—
471C22×C3—31no—
472C22×C3—64no—
473C22×C3—64no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—61no—
477C22×C3—94no—
478C22×C3—92no—
479C22×C3—124no—
480C22×C3—122no—
481C22×C3—184no—
482C22×C3—184no—
483C22×C3—182no—
484C22×C3—182no—
485C22×C3—184no—
486C22×C3—182no—
487C22×C3—368no—
488C22×C3C22×D3211yescharacteristic
489C2×D3—34no—
490C2×D3—32no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—64no—
494C2×D3—64no—
495C2×D3—64no—
496C2×D3—64no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—94no—
500C2×D3—92no—
501C2×D3—122no—
502C2×D3—124no—
503C2×D3—128no—
504C2×D3—182no—
505C2×D3—184no—
506C2×D3—182no—
507C2×D3—184no—
508C2×D3—368no—
509C2×D3—368no—
510C2×D3—368no—
511C2×D3—368no—
512C3×C4—32no—
513C3×C4—32no—
514C3×C4—62no—
515C3×C4—94no—
516C3×C4—92no—
517C3×C4—94no—
518C3×C4D3×#6712yes—
519Dic3—34no—
520Dic3—34no—
521Dic3—34no—
522Dic3—32no—
523Dic3—64no—
524Dic3—92no—
525Dic3—94no—
526C32—21no—
527C32D3×#12711yescharacteristic, complemented
528C32C2×#94011yescharacteristic, complemented
529C23—91no—
530C23—92no—
531C23—92no—
532C23—92no—
533C23—184no—
534C23—188no—
535C23—548no—
536C23—544no—
537C23C2×((C3⋊D3)⋊C3)⋊C212yes—
538D4—22no—
539D4—184no—
540D4—188no—
541D4—188no—
542D4—182no—
543D4—182no—
544D4—182no—
545D4—548no—
546D4—548no—
547D4—544no—
548C2×C4—92no—
549C2×C4—92no—
550C2×C4—91no—
551C2×C4—94no—
552C2×C4—278no—
553C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
554C2×C3—32no—
555C2×C3—32no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—42no—
559C2×C3—62no—
560C2×C3—61no—
561C2×C3—94no—
562C2×C3—122no—
563C2×C3—122no—
564C2×C3—242no—
565C2×C3—362no—
566C2×C3—362no—
567C2×C3—362no—
568C2×C3D32×D412yes—
569C2×C3C2×D3×#6711yescharacteristic
570D3—34no—
571D3—94no—
572D3—122no—
573D3—122no—
574D3—122no—
575D3—122no—
576D3—242no—
577D3—362no—
578D3—362no—
579C22—24no—
580C22—22no—
581C22—94no—
582C22—92no—
583C22—184no—
584C22—184no—
585C22—184no—
586C22—182no—
587C22—182no—
588C22—182no—
589C22—368no—
590C22—1088no—
591C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
592C4—94no—
593C4—94no—
594C4—92no—
595C4#8867612yescomplemented
596C3—31no—
597C3—31no—
598C3—61no—
599C3D3×#94011yescharacteristic
600C2—42no—
601C2—94no—
602C2—362no—
603C2—362no—
604C2—362no—
605C2D4×((C3⋊D3)⋊C3)⋊C212yes—
606C2C2×#8867611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#127
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×(C3⋊D3)⋊C3C2semidirect
2. C2×#88676C2semidirect
3. C2×#88654C2semidirect
4. #11721850C2semidirect
5. #64149⋊C3C2semidirect
6. #11721317C2semidirect
7. #11721158C2semidirect
8. #11721144C2semidirect
9. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
14. C2×D4×H(3,𝔽3)C22semidirect
15. C2×#6235C22semidirect
16. C2×C4⋊#272C22semidirect
17. C2×#638⋊C3C22semidirect
18. C2×#638⋊C3C22semidirect
20. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
21. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
24. C2×#6200C22semidirect
25. C2×#6200C22semidirect
26. C2×#6198C22semidirect
29. C4×#1084C22semidirect
30. C4×(C3⋊Dic3)⋊C3C22semidirect
31. #88447C22semidirect
32. C2×D4×C3⋊D3D3semidirect
38. #64149D3semidirect
71. C4×(C3⋊D3)⋊C3D4semidirect
75. ((C3⋊D3)⋊C3)⋊C4D4semidirect
84. C2×C32×D4C2×D3semidirect
85. C2×C32×D4C2×D3semidirect
86. C2×#638C2×D3semidirect
87. C2×#638C2×D3semidirect
89. C2×C4⋊(C3⋊D3)C2×D3semidirect
90. C2×C4⋊(C3⋊D3)C2×D3semidirect
116. C4×C3⋊Dic3C2×D3semidirect
136. C4×H(3,𝔽3)C2×D4semidirect
137. #1084C2×D4semidirect
138. (C3⋊Dic3)⋊C3C2×D4semidirect
139. (C3⋊Dic3)⋊C3C2×D4semidirect
200. C4×C3⋊D3#67semidirect
232. (C3⋊D3)⋊C3#127semidirect
329. C32×C4D3×D4semidirect
330. C32×C4C2×#67semidirect
331. C3⋊Dic3D3×D4semidirect
332. C3⋊Dic3C2×#67semidirect
347. H(3,𝔽3)C2×#127semidirect
441. C3⋊D3#940semidirect
462. C2×D4((C3⋊D3)⋊C3)⋊C2semidirect
527. C32D3×#127semidirect
528. C32C2×#940semidirect
595. C4#88676semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
C2×#886762
C2×#886542
#117218502
#64149⋊C32
#117213172
#117211582
#117211442
C4×((C3⋊D3)⋊C3)⋊C42
#105953003
D3×#9403