Finite Group Databasebeta

← #13170484

Subgroups — #13170484

Order 1728.

1286 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64306C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic, complemented
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
4#11723350C212yesmaximal, complemented
5C3×#63814C212yesmaximal, complemented
6#11722823C212yesmaximal, complemented
7#11722821C212yesmaximal, complemented
8#11722816C212yesmaximal, complemented
9D32×C3⋊C8C212yesmaximal, complemented
10D3×#948—32nomaximal
11#10594669—31nomaximal
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C32×#235C2212yes—
14C3×#3811C412yes—
15C3×#3805C412yes—
16C3×#3802C411yescharacteristic
17C3×C4×D32C2212yes—
18C3×#3798C2212yes—
19C3×#3797C411yescharacteristic
20C3×#3796C412yes—
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
22C3×#3748C2212yescomplemented
23C2×C32×C3⋊C8C2211yescharacteristic, complemented
24#88809C2212yescomplemented
25#88808C2212yes—
26#88786C2214yescomplemented
27C3⋊D3×C3⋊C8C2211yescharacteristic
28C3⋊D3×C3⋊C8C2211yescharacteristic
29C3⋊D3×C3⋊C8C2211yescharacteristic, complemented
30C3⋊D3×C3⋊C8C2211yescharacteristic, complemented
31D3×C3⋊(C3⋊C8)C2214yescomplemented
32C3×#3711C2214yescomplemented
33C3×D3×C3⋊C8C2214yescomplemented
34C3×D3×#41—32no—
35#64306D311yescharacteristic
36#64157—32no—
37C2×C8×C3⋊D3—31no—
38C3×#948—32no—
39C3×#901—32no—
40#63818—31no—
41#63815—32no—
42#63815—32no—
43#63815—34no—
44#63815—62no—
45#63814—34no—
46#63814D312yescomplemented
47D3×#208—32no—
48D3×#208—34no—
49C2×D3×C3⋊C8—32no—
50C2×D3×C3⋊C8—34no—
51C2×D3×C3⋊C8—61no—
52#63805—32no—
53#63803—32no—
54#63791—32no—
55D32×C8—32no—
56C22×C3×C3⋊D3C2×C411yescharacteristic
57C2×C3×D32C2×C412yes—
58C2×C33×C4C2311yescharacteristic
59C2×C3×C3⋊Dic3C2×C411yescharacteristic
60C3×C4×C3⋊D3C2×C411yescharacteristic
61C3×C4×C3⋊D3C2×C411yescharacteristic
62C3×C4×C3⋊D3C2311yescharacteristic
63C3×C4×C3⋊D3C2311yescharacteristic
64C32×#67C2×C414yes—
65C32×D12C2×C412yes—
66C32×C4×D3C2314yes—
67C32×Dic6C2×C412yes—
68C3×#627C2×C412yes—
69C3×#626C2×C414yes—
70C3×#625C2×C412yes—
71C3×#624C2×C412yes—
72C3×D3×Dic3C2×C414yes—
73C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
74C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
75C3×C3⋊(C3⋊C8)C2314yes—
76C32×C3⋊C8C2311yescharacteristic
77C32×C3⋊C8C2311yescharacteristic
78C2×#948—91no—
79D3×#131—92no—
80C32×#41—32no—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C3×#239—32no—
83C3×D3×Q8—32no—
84C3×#237—34no—
85C3×#237—32no—
86C3×D3×D4—34no—
87C3×D3×D4—32no—
88C3×#235—22no—
89C3×#235—34no—
90C3×#235C2×D312yes—
91C3×#235C2×D312yes—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—34no—
94C2×C3×C4×D3—61no—
95#3811Dic312yes—
96#3805Dic312yes—
97#3802Dic311yescharacteristic
98C4×D32C2×D312yes—
99#3798C2×D312yes—
100#3797Dic311yescharacteristic
101#3796Dic312yes—
102C2×C32×C8—31no—
103#3748—34no—
104#3748C2×D312yescomplemented
105C2×C3⋊(C3⋊C8)—32no—
106C2×C3⋊(C3⋊C8)—31no—
107C2×C3⋊(C3⋊C8)—34no—
108C2×C3⋊(C3⋊C8)—61no—
109C2×C3⋊(C3⋊C8)—62no—
110C2×C3⋊(C3⋊C8)—121no—
111C8×C3⋊D3—31no—
112C8×C3⋊D3—31no—
113C8×C3⋊D3—31no—
114C8×C3⋊D3—31no—
115C3×#208—32no—
116C3×#208—32no—
117C3×#208—34no—
118C3×#208—62no—
119C2×C3×C3⋊C8—21no—
120C2×C3×C3⋊C8—34no—
121C2×C3×C3⋊C8C2×D312yescomplemented
122C3×ΓL(1,𝔽25)—34no—
123C3×D3×C8—34no—
124#3712—32no—
125#3711—34no—
126#3711—34no—
127#3711—34no—
128#3711—64no—
129#3711C2×D314yescomplemented
130#3710—32no—
131D3×C3⋊C8—34no—
132D3×C3⋊C8—34no—
133D3×C3⋊C8—34no—
134D3×C3⋊C8—32no—
135D3×C3⋊C8—32no—
136D3×C3⋊C8—32no—
137D3×C3⋊C8—32no—
138D3×C3⋊C8—61no—
139D3×C3⋊C8—61no—
140D3×C3⋊C8—61no—
141D3×C3⋊C8—61no—
142D3×C3⋊C8—64no—
143D3×C3⋊C8C2×D314yescomplemented
144C22×C33C22×C411yescharacteristic
145C2×C3×C3⋊D3—22no—
146C2×C3×C3⋊D3C22×C411yescharacteristic
147C2×C3×C3⋊D3C22×C411yescharacteristic
148C2×C32×D3C22×C414yes—
149C3×D32—24no—
150C33×C4C22×C411yescharacteristic
151C33×C4C2411yescharacteristic
152C3×C3⋊Dic3C22×C411yescharacteristic
153C3×C3⋊Dic3C22×C411yescharacteristic
154C32×Dic3C22×C414yes—
155C2×C3×#41—91no—
156D3×#41—32no—
157C3×#131—92no—
158#948—32no—
159#948—92no—
160#948—92no—
161#948—92no—
162#948—92no—
163#948—182no—
164C2×#208—91no—
165C2×#208—92no—
166C22×C3⋊C8—92no—
167C22×C3⋊C8—91no—
168#907—94no—
169#907—92no—
170D3×M16—94no—
171D3×M16—92no—
172#901—32no—
173#901—94no—
174C2×D3×C8—94no—
175C2×D3×C8—92no—
176C2×D3×C8—181no—
177C22×C3⋊D3C2×Dic311yescharacteristic
178C22×C3×D3—32no—
179C22×C3×D3—34no—
180C22×C3×D3—61no—
181C2×D32C2×Dic312yes—
182C32×Q8—32no—
183C32×D4—34no—
184C32×D4—32no—
185C2×C32×C4—21no—
186C2×C32×C4—22no—
187C2×C32×C4—34no—
188C2×C32×C4—41no—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C3⋊Dic3C2×Dic311yescharacteristic
192C4×C3⋊D3C2×Dic311yescharacteristic
193C4×C3⋊D3C2×Dic311yescharacteristic
194C4×C3⋊D3C22×D311yescharacteristic
195C4×C3⋊D3C22×D311yescharacteristic
196C3×#67—24no—
197C3×#67—34no—
198C3×#67—34no—
199C3×#67C4×D314yes—
200C3×#67C2×Dic314yes—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—61no—
204C3×D12—22no—
205C3×D12—34no—
206C3×D12C4×D312yes—
207C3×D12C2×Dic312yes—
208C3×C4×D3—24no—
209C3×C4×D3—34no—
210C3×C4×D3—32no—
211C3×C4×D3—32no—
212C3×C4×D3—32no—
213C3×C4×D3—32no—
214C3×C4×D3—34no—
215C3×C4×D3—61no—
216C3×C4×D3—61no—
217C3×C4×D3—61no—
218C3×C4×D3—61no—
219C3×C4×D3C22×D314yes—
220C3×C4×D3C22×D314yes—
221C3×Dic6—22no—
222C3×Dic6—34no—
223C3×Dic6C4×D312yes—
224C3×Dic6C2×Dic312yes—
225#627C2×Dic312yes—
226#626C2×Dic314yes—
227#625C2×Dic312yes—
228#624C2×Dic312yes—
229D3×Dic3C2×Dic314yes—
230C32×C8—31no—
231C32×C8—31no—
232C3⋊(C3⋊C8)—31no—
233C3⋊(C3⋊C8)—32no—
234C3⋊(C3⋊C8)—32no—
235C3⋊(C3⋊C8)—31no—
236C3⋊(C3⋊C8)—62no—
237C3⋊(C3⋊C8)—61no—
238C3⋊(C3⋊C8)—61no—
239C3⋊(C3⋊C8)—62no—
240C3⋊(C3⋊C8)—121no—
241C3⋊(C3⋊C8)—121no—
242C3⋊(C3⋊C8)C22×D314yes—
243C3×C3⋊C8—21no—
244C3×C3⋊C8—21no—
245C3×C3⋊C8—34no—
246C3×C3⋊C8—34no—
247C3×C3⋊C8—64no—
248C3×C3⋊C8C22×D312yes—
249C3×C3⋊C8C22×D312yes—
250C2×#131—271noSylow
251C2×C33—21no—
252C2×C33C23×C411yescharacteristic
253C3×C3⋊D3—21no—
254C3×C3⋊D3#13112yescomplemented
255C32×D3—24no—
256C3×#41—32no—
257C3×#41—32no—
258C3×#41—62no—
259C3×#41—92no—
260C3×#41—92no—
261C3×#41—92no—
262C2×C3×Q8—91no—
263C2×C3×D4—92no—
264C2×C3×D4—91no—
265C22×C3×C4—92no—
266C22×C3×C4—91no—
267#239—32no—
268D3×Q8—32no—
269#237—34no—
270#237—32no—
271D3×D4—34no—
272D3×D4—32no—
273#235—34no—
274#235D3212yes—
275C2×C4×D3—32no—
276C2×C4×D3—34no—
277C2×C4×D3—61no—
278C3×M16—94no—
279C3×M16—92no—
280C2×C3×C8—32no—
281C2×C3×C8—61no—
282C2×C3×C8—94no—
283#208—32no—
284#208—32no—
285#208—34no—
286#208—62no—
287#208—92no—
288#208—94no—
289#208—94no—
290#208—184no—
291C2×C3⋊C8—34no—
292C2×C3⋊C8—92no—
293C2×C3⋊C8—94no—
294C2×C3⋊C8—94no—
295C2×C3⋊C8—92no—
296C2×C3⋊C8—91no—
297C2×C3⋊C8—91no—
298C2×C3⋊C8—91no—
299C2×C3⋊C8—91no—
300C2×C3⋊C8—91no—
301C2×C3⋊C8—184no—
302C2×C3⋊C8—181no—
303C2×C3⋊C8—182no—
304C2×C3⋊C8—361no—
305C2×C3⋊C8D3211yescharacteristic, complemented
306ΓL(1,𝔽25)—34no—
307ΓL(1,𝔽25)—94no—
308ΓL(1,𝔽25)—94no—
309D3×C8—34no—
310D3×C8—94no—
311D3×C8—92no—
312D3×C8—92no—
313D3×C8—92no—
314D3×C8—92no—
315D3×C8—94no—
316D3×C8—181no—
317D3×C8—181no—
318D3×C8—181no—
319D3×C8—181no—
320C22×C32—21no—
321C22×C32—22no—
322C22×C32—34no—
323C22×C32—41no—
324C22×C32C2×C4×D312yes—
325C22×C32C22×Dic311yescharacteristic
326C2×C3⋊D3—22no—
327C2×C3⋊D3C22×Dic311yescharacteristic
328C2×C3⋊D3C22×Dic311yescharacteristic
329C2×C3×D3—24no—
330C2×C3×D3—32no—
331C2×C3×D3—32no—
332C2×C3×D3—61no—
333C2×C3×D3—61no—
334C2×C3×D3—68no—
335C2×C3×D3—64no—
336C2×C3×D3—64no—
337C2×C3×D3C2×C4×D314yes—
338C2×C3×D3C22×Dic314yes—
339D32—24no—
340C32×C4—21no—
341C32×C4—22no—
342C32×C4—21no—
343C32×C4—22no—
344C32×C4—34no—
345C32×C4—41no—
346C32×C4—41no—
347C32×C4C2×C4×D312yes—
348C32×C4C22×Dic311yescharacteristic
349C32×C4C23×D312yes—
350C32×C4C23×D311yescharacteristic
351C3⋊Dic3C22×Dic311yescharacteristic
352C3⋊Dic3C22×Dic311yescharacteristic
353C3×Dic3—24no—
354C3×Dic3—32no—
355C3×Dic3—32no—
356C3×Dic3—61no—
357C3×Dic3—61no—
358C3×Dic3C2×C4×D314yes—
359C3×Dic3C22×Dic314yes—
360C2×#41—91no—
361#131—92no—
362#131—272no—
363#131—272no—
364#131—272no—
365C2×M16—272no—
366C2×M16—271no—
367C22×C8—271no—
368C22×C8—272no—
369C33C2×#13111yescharacteristic, complemented, Sylow
370C23×C3—92no—
371C23×C3—91no—
372C22×D3—32no—
373C22×D3—34no—
374C22×D3—61no—
375C3×Q8—32no—
376C3×Q8—32no—
377C3×Q8—62no—
378C3×Q8—92no—
379C3×D4—34no—
380C3×D4—34no—
381C3×D4—32no—
382C3×D4—32no—
383C3×D4—62no—
384C3×D4—64no—
385C3×D4—94no—
386C3×D4—92no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—34no—
390C2×C3×C4—34no—
391C2×C3×C4—41no—
392C2×C3×C4—64no—
393C2×C3×C4—92no—
394C2×C3×C4—94no—
395C2×C3×C4—91no—
396C2×C3×C4—91no—
397C2×C3×C4—91no—
398C2×C3×C4—91no—
399C2×C3×C4—91no—
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3212yes—
402#67—34no—
403#67—34no—
404#67D3×Dic314yes—
405C2×Dic3—32no—
406C2×Dic3—34no—
407C2×Dic3—61no—
408D12—34no—
409D12D3×Dic312yes—
410C4×D3—34no—
411C4×D3—32no—
412C4×D3—32no—
413C4×D3—32no—
414C4×D3—32no—
415C4×D3—34no—
416C4×D3—61no—
417C4×D3—61no—
418C4×D3—61no—
419C4×D3—61no—
420C4×D3C2×D3214yes—
421Dic6—34no—
422Dic6D3×Dic312yes—
423C3×C8—32no—
424C3×C8—32no—
425C3×C8—61no—
426C3×C8—61no—
427C3×C8—94no—
428C3⋊C8—34no—
429C3⋊C8—34no—
430C3⋊C8—64no—
431C3⋊C8—92no—
432C3⋊C8—92no—
433C3⋊C8—91no—
434C3⋊C8—91no—
435C3⋊C8—181no—
436C3⋊C8—181no—
437C3⋊C8—182no—
438C3⋊C8—182no—
439C3⋊C8—361no—
440C3⋊C8—361no—
441C3⋊C8C2×D3211yescharacteristic
442C3⋊C8C2×D3211yescharacteristic
443C2×C32—21no—
444C2×C32—22no—
445C2×C32—22no—
446C2×C32—21no—
447C2×C32—22no—
448C2×C32—41no—
449C2×C32—42no—
450C2×C32—42no—
451C2×C32—64no—
452C2×C32C22×C4×D312yes—
453C2×C32C23×Dic311yescharacteristic
454C3⋊D3—21no—
455C3⋊D3#94812yescomplemented
456C3×D3—24no—
457C3×D3—24no—
458C3×D3—34no—
459C3×D3—44no—
460C3×D3—62no—
461C3×D3—62no—
462C3×D3—62no—
463#41—32no—
464#41—92no—
465#41—92no—
466#41—92no—
467C2×Q8—91no—
468C2×D4—92no—
469C2×D4—91no—
470C22×C4—92no—
471C22×C4—91no—
472M16—94no—
473M16—92no—
474M16—272no—
475M16—274no—
476C2×C8—31no—
477C2×C8—94no—
478C2×C8—271no—
479C2×C8—271no—
480C2×C8—271no—
481C2×C8—271no—
482C2×C8—272no—
483C2×C8—274no—
484C2×C8—271no—
485C22×C3—21no—
486C22×C3—22no—
487C22×C3—34no—
488C22×C3—34no—
489C22×C3—41no—
490C22×C3—64no—
491C22×C3—91no—
492C22×C3—91no—
493C22×C3—182no—
494C22×C3—184no—
495C22×C3C4×D3211yescharacteristic
496C22×C3C2×D3×Dic312yes—
497C2×D3—32no—
498C2×D3—32no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3—68no—
502C2×D3—64no—
503C2×D3—64no—
504C2×D3C2×D3×Dic314yes—
505C3×C4—22no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—22no—
509C3×C4—34no—
510C3×C4—34no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—64no—
514C3×C4—91no—
515C3×C4—91no—
516C3×C4C4×D3211yescharacteristic
517C3×C4C2×D3×Dic312yes—
518C3×C4C22×D3211yescharacteristic
519C3×C4C22×D3212yes—
520Dic3—32no—
521Dic3—32no—
522Dic3—61no—
523Dic3—61no—
524Dic3C2×D3×Dic314yes—
525C32—21no—
526C32—22no—
527C32—41no—
528C32D3×#13112yescomplemented
529C32C2×#94811yescharacteristic, complemented
530C23—92no—
531C23—91no—
532Q8—32no—
533Q8—92no—
534D4—34no—
535D4—32no—
536D4—94no—
537D4—92no—
538C2×C4—34no—
539C2×C4—92no—
540C2×C4—94no—
541C2×C4—91no—
542C2×C4—91no—
543C2×C4—91no—
544C2×C4—91no—
545C2×C4—91no—
546C2×C4D3311yescharacteristic
547C8—31no—
548C8—31no—
549C8—94no—
550C8—271no—
551C8—271no—
552C2×C3—22no—
553C2×C3—21no—
554C2×C3—22no—
555C2×C3—21no—
556C2×C3—22no—
557C2×C3—42no—
558C2×C3—42no—
559C2×C3—41no—
560C2×C3—64no—
561C2×C3—64no—
562C2×C3—92no—
563C2×C3—124no—
564C2×C3—181no—
565C2×C3C2×C4×D3211yescharacteristic
566C2×C3C22×D3×Dic312yes—
567D3—24no—
568D3—34no—
569D3—62no—
570D3—62no—
571D3—62no—
572C22—34no—
573C22—91no—
574C22—91no—
575C22—182no—
576C22—184no—
577C22D32×Dic311yescharacteristic
578C4—34no—
579C4—91no—
580C4—91no—
581C4D32×Dic311yescharacteristic
582C4C2×D3311yescharacteristic
583C3—21no—
584C3—22no—
585C3—41no—
586C3#1059466911yescharacteristic, complemented
587C3D3×#94812yescomplemented
588C2—21no—
589C2—64no—
590C2—92no—
591C2—181no—
592C2C2×D32×Dic311yescharacteristic
subgroupG/H
center, FrattiniC4C2×D33
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C23
prime radicalC3×#64306C2
cosocleC33×C4C24
hypercenterC2×C4D33
hypocenterC33C2×#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723656C2semidirect
3. C2×C3⋊D3×C3⋊C8C2semidirect
4. #11723350C2semidirect
5. C3×#63814C2semidirect
6. #11722823C2semidirect
7. #11722821C2semidirect
8. #11722816C2semidirect
9. D32×C3⋊C8C2semidirect
21. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
22. C3×#3748C22semidirect
23. C2×C32×C3⋊C8C22semidirect
24. #88809C22semidirect
26. #88786C22semidirect
29. C3⋊D3×C3⋊C8C22semidirect
30. C3⋊D3×C3⋊C8C22semidirect
31. D3×C3⋊(C3⋊C8)C22semidirect
32. C3×#3711C22semidirect
33. C3×D3×C3⋊C8C22semidirect
46. #63814D3semidirect
104. #3748C2×D3semidirect
121. C2×C3×C3⋊C8C2×D3semidirect
129. #3711C2×D3semidirect
143. D3×C3⋊C8C2×D3semidirect
254. C3×C3⋊D3#131semidirect
305. C2×C3⋊C8D32semidirect
369. C33C2×#131semidirect
455. C3⋊D3#948semidirect
528. C32D3×#131semidirect
529. C32C2×#948semidirect
586. C3#10594669semidirect
587. C3D3×#948semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
#117236562
C2×C3⋊D3×C3⋊C82
#117233502
C3×#638142
#117228232
#117228212
#117228162
D32×C3⋊C82
D3×#9483
#105946693