Finite Group Databasebeta

← #13170494

Subgroups — #13170494

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
1#11724618C211yesmaximal, characteristic
2#11723354C212yesmaximal, complemented
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#63818C211yesmaximal, characteristic, complemented
5C3×#63817C212yesmaximal, complemented
6#11722827C212yesmaximal, complemented
7#11722826C212yesmaximal, complemented
8#11722820C212yesmaximal, complemented
9#11722814C212yesmaximal, complemented
10D3×#907—32nomaximal
11#10594669—31nomaximal
12C3×#3828C2212yes—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14#89049C412yes—
15#89046C412yes—
16C4⋊#1113C411yescharacteristic
17C4×#1113C2212yes—
18#89042C2212yes—
19#89041C412yes—
20#89040C411yescharacteristic
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
22C32×#208C2212yescomplemented
23C2×C32×C3⋊C8C2211yescharacteristic, complemented
24#88787C2214yescomplemented
25C3⋊D3×C3⋊C8C2211yescharacteristic
26C3⋊D3×C3⋊C8C2211yescharacteristic
27C3⋊D3×C3⋊C8C2214yescomplemented
28C3⋊D3×C3⋊C8C2211yescharacteristic, complemented
29C3⋊D3×C3⋊C8C2211yescharacteristic, complemented
30C3×#3712C2214yescomplemented
31C3×#3712C2212yescomplemented
32C3×#3710C2212yes—
33C3×#3710C2214yescomplemented
34#64306—31no—
35D3×#235—32no—
36#64116—32no—
37C2×C8×C3⋊D3—31no—
38C3×#907—32no—
39C3×D3×M16—32no—
40C3×#901—32no—
41#63818D311yescharacteristic, complemented
42#63817D312yescomplemented
43#63815—32no—
44#63815—32no—
45#63815—62no—
46C2×D3×C3⋊C8—32no—
47C2×D3×C3⋊C8—61no—
48#63805—32no—
49#63805—34no—
50#63804—34no—
51#63803—32no—
52#63791—32no—
53D3×ΓL(1,𝔽25)—34no—
54D32×C8—32no—
55D32×C8—34no—
56C22×C3×C3⋊D3C2×C411yescharacteristic
57C2×#1113C2×C412yes—
58C2×C33×C4C2311yescharacteristic
59C3×#638C2×C414yes—
60C2×C3×C3⋊Dic3C2×C411yescharacteristic
61C3×C4⋊(C3⋊D3)C2×C412yes—
62C3×C4×C3⋊D3C2×C411yescharacteristic
63C3×C4×C3⋊D3C2×C411yescharacteristic
64C3×C4×C3⋊D3C2311yescharacteristic
65C3×C4×C3⋊D3C2314yes—
66C3×C4×C3⋊D3C2311yescharacteristic
67C3×#634C2×C412yes—
68#6296C2×C412yes—
69#6295C2×C414yes—
70#6295C2×C412yes—
71#6294C2×C412yes—
72#6294C2×C414yes—
73C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
74C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
75C32×C3⋊C8C2314yes—
76C32×C3⋊C8C2311yescharacteristic
77C32×C3⋊C8C2311yescharacteristic
78D3×#131—92no—
79C2×#907—91no—
80#3828C2×D312yes—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C3×#235—32no—
83C3×#235—32no—
84C3×#235—62no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—61no—
87#3811—32no—
88D3×#67—34no—
89#3805—32no—
90#3804—34no—
91#3802—31no—
92D3×D12—32no—
93C4×D32—34no—
94C4×D32—32no—
95#3799—32no—
96#3798—34no—
97#3798—32no—
98#3797—31no—
99#3796—32no—
100#3795—32no—
101D3×Dic6—32no—
102C32×M16—32no—
103C2×C32×C8—31no—
104C2×C3⋊(C3⋊C8)—32no—
105C2×C3⋊(C3⋊C8)—31no—
106C2×C3⋊(C3⋊C8)—61no—
107C2×C3⋊(C3⋊C8)—62no—
108C2×C3⋊(C3⋊C8)—121no—
109#3743—34no—
110C8×C3⋊D3—31no—
111C8×C3⋊D3—34no—
112C8×C3⋊D3—31no—
113C8×C3⋊D3—31no—
114C8×C3⋊D3—31no—
115C3×#208—22no—
116C3×#208C2×D312yescomplemented
117C3×#208C2×D312yescomplemented
118C2×C3×C3⋊C8—21no—
119C2×C3×C3⋊C8C2×D312yescomplemented
120C3×ΓL(1,𝔽25)—34no—
121C3×ΓL(1,𝔽25)—34no—
122C3×ΓL(1,𝔽25)—34no—
123C3×D3×C8—34no—
124C3×D3×C8—34no—
125C3×D3×C8—34no—
126#3712C2×D314yescomplemented
127#3712C2×D312yescomplemented
128#3711—34no—
129#3711—34no—
130#3711—64no—
131#3710C2×D312yes—
132#3710C2×D314yescomplemented
133D3×C3⋊C8—32no—
134D3×C3⋊C8—34no—
135D3×C3⋊C8—34no—
136D3×C3⋊C8—32no—
137D3×C3⋊C8—32no—
138D3×C3⋊C8—32no—
139D3×C3⋊C8—61no—
140D3×C3⋊C8—61no—
141D3×C3⋊C8—61no—
142D3×C3⋊C8—61no—
143D3×C3⋊C8—64no—
144C22×C33C22×C411yescharacteristic
145C2×C3×C3⋊D3—22no—
146C2×C3×C3⋊D3C22×C414yes—
147C2×C3×C3⋊D3C22×C411yescharacteristic
148C2×C3×C3⋊D3C22×C411yescharacteristic
149#1113—24no—
150C33×C4C22×C411yescharacteristic
151C33×C4C2411yescharacteristic
152C3×C3⋊Dic3C22×C411yescharacteristic
153C3×C3⋊Dic3C22×C414yes—
154C3×C3⋊Dic3C22×C411yescharacteristic
155D3×#41—92no—
156C2×#235—91no—
157C3×#131—92no—
158C2×C3×M16—91no—
159#948—92no—
160C22×C3⋊C8—91no—
161#907—32no—
162#907—92no—
163#907—92no—
164#907—94no—
165#907—92no—
166#907—92no—
167#907—182no—
168D3×M16—32no—
169D3×M16—94no—
170#901—32no—
171#901—94no—
172C2×ΓL(1,𝔽25)—92no—
173C2×D3×C8—92no—
174C2×D3×C8—94no—
175C2×D3×C8—92no—
176C2×D3×C8—181no—
177C22×C3⋊D3C4×D311yescharacteristic
178C22×C3×D3—32no—
179C22×C3×D3—61no—
180C2×D32—34no—
181C2×D32—32no—
182C2×C32×C4—21no—
183C2×C32×C4—22no—
184C2×C32×C4—41no—
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187#638C4×D314yes—
188C2×C3⋊Dic3C4×D311yescharacteristic
189C4⋊(C3⋊D3)C4×D312yes—
190C4×C3⋊D3C4×D311yescharacteristic
191C4×C3⋊D3C4×D311yescharacteristic
192C4×C3⋊D3C22×D311yescharacteristic
193C4×C3⋊D3C22×D314yes—
194C4×C3⋊D3C22×D311yescharacteristic
195#634C4×D312yes—
196C3×#67—34no—
197C3×#67—34no—
198C3×#67—64no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—61no—
201C3×D12—32no—
202C3×D12—32no—
203C3×D12—62no—
204C3×C4×D3—32no—
205C3×C4×D3—32no—
206C3×C4×D3—32no—
207C3×C4×D3—32no—
208C3×C4×D3—34no—
209C3×C4×D3—34no—
210C3×C4×D3—61no—
211C3×C4×D3—61no—
212C3×C4×D3—64no—
213C3×C4×D3—61no—
214C3×C4×D3—61no—
215C3×Dic6—32no—
216C3×Dic6—32no—
217C3×Dic6—62no—
218#627—34no—
219#627—32no—
220#626—34no—
221#626—34no—
222#626—34no—
223#625—34no—
224#625—32no—
225#624—32no—
226#624—34no—
227D3×Dic3—34no—
228D3×Dic3—34no—
229D3×Dic3—34no—
230C32×C8—31no—
231C32×C8—34no—
232C32×C8—31no—
233C3⋊(C3⋊C8)—32no—
234C3⋊(C3⋊C8)—31no—
235C3⋊(C3⋊C8)—32no—
236C3⋊(C3⋊C8)—31no—
237C3⋊(C3⋊C8)—61no—
238C3⋊(C3⋊C8)—62no—
239C3⋊(C3⋊C8)—61no—
240C3⋊(C3⋊C8)—62no—
241C3⋊(C3⋊C8)—121no—
242C3⋊(C3⋊C8)—121no—
243C3×C3⋊C8—21no—
244C3×C3⋊C8—21no—
245C3×C3⋊C8—24no—
246C3×C3⋊C8C22×D312yes—
247C3×C3⋊C8C22×D312yes—
248C3×C3⋊C8C22×D314yes—
249C3×C3⋊C8C22×D314yes—
250C2×#131—271noSylow
251C2×C33—21no—
252C2×C33C23×C411yescharacteristic
253C3×C3⋊D3—21no—
254C3×C3⋊D3—24no—
255C3×C3⋊D3#13112yescomplemented
256C3×#41—92no—
257C22×C3×C4—91no—
258C2×#67—92no—
259#239—92no—
260D3×Q8—92no—
261#237—92no—
262#237—94no—
263D3×D4—92no—
264D3×D4—94no—
265#235—32no—
266#235—32no—
267#235—62no—
268#235—92no—
269#235—94no—
270#235—92no—
271#235—92no—
272C2×D12—91no—
273C2×C4×D3—32no—
274C2×C4×D3—61no—
275C2×C4×D3—92no—
276C2×C4×D3—94no—
277C2×Dic6—91no—
278C3×M16—32no—
279C3×M16—32no—
280C3×M16—62no—
281C3×M16—94no—
282C3×M16—92no—
283C2×C3×C8—32no—
284C2×C3×C8—61no—
285C2×C3×C8—92no—
286C2×C3×C8—94no—
287#208—94no—
288#208D3212yescomplemented
289C2×C3⋊C8—92no—
290C2×C3⋊C8—91no—
291C2×C3⋊C8—91no—
292C2×C3⋊C8—94no—
293C2×C3⋊C8—91no—
294C2×C3⋊C8—91no—
295C2×C3⋊C8—91no—
296C2×C3⋊C8—181no—
297C2×C3⋊C8—182no—
298C2×C3⋊C8—361no—
299C2×C3⋊C8D3211yescharacteristic, complemented
300ΓL(1,𝔽25)—34no—
301ΓL(1,𝔽25)—34no—
302ΓL(1,𝔽25)—34no—
303ΓL(1,𝔽25)—94no—
304ΓL(1,𝔽25)—94no—
305ΓL(1,𝔽25)—184no—
306D3×C8—34no—
307D3×C8—34no—
308D3×C8—34no—
309D3×C8—92no—
310D3×C8—92no—
311D3×C8—92no—
312D3×C8—94no—
313D3×C8—94no—
314D3×C8—92no—
315D3×C8—181no—
316D3×C8—181no—
317D3×C8—181no—
318D3×C8—181no—
319D3×C8—184no—
320C22×C32—21no—
321C22×C32—22no—
322C22×C32—41no—
323C22×C32C2×C4×D312yes—
324C22×C32C2×C4×D311yescharacteristic
325C2×C3⋊D3—22no—
326C2×C3⋊D3C2×C4×D314yes—
327C2×C3⋊D3C2×C4×D311yescharacteristic
328C2×C3⋊D3C2×C4×D311yescharacteristic
329C2×C3×D3—32no—
330C2×C3×D3—34no—
331C2×C3×D3—34no—
332C2×C3×D3—32no—
333C2×C3×D3—61no—
334C2×C3×D3—61no—
335C2×C3×D3—64no—
336C2×C3×D3—64no—
337C2×C3×D3—64no—
338D32—68no—
339D32—64no—
340C32×C4—21no—
341C32×C4—22no—
342C32×C4—22no—
343C32×C4—21no—
344C32×C4—41no—
345C32×C4—41no—
346C32×C4C2×C4×D311yescharacteristic
347C32×C4C2×C4×D312yes—
348C32×C4C23×D311yescharacteristic
349C32×C4C23×D312yes—
350C3⋊Dic3C2×C4×D311yescharacteristic
351C3⋊Dic3C2×C4×D314yes—
352C3⋊Dic3C2×C4×D311yescharacteristic
353C3×Dic3—32no—
354C3×Dic3—34no—
355C3×Dic3—34no—
356C3×Dic3—32no—
357C3×Dic3—64no—
358C3×Dic3—61no—
359C3×Dic3—61no—
360C2×#41—271no—
361#131—92no—
362#131—272no—
363#131—272no—
364#131—272no—
365C2×M16—91no—
366C2×M16—272no—
367C22×C8—271no—
368C22×C8—272no—
369C33C2×#13111yescharacteristic, complemented, Sylow
370C23×C3—91no—
371C22×D3—32no—
372C22×D3—61no—
373C22×D3—92no—
374C22×D3—94no—
375C3×Q8—92no—
376C3×D4—94no—
377C3×D4—92no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—41no—
381C2×C3×C4—91no—
382C2×C3×C4—91no—
383C2×C3×C4—91no—
384C2×C3×C4—94no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3212yes—
389#67—34no—
390#67—34no—
391#67—64no—
392#67—94no—
393#67—94no—
394#67—94no—
395C2×Dic3—32no—
396C2×Dic3—61no—
397C2×Dic3—92no—
398C2×Dic3—94no—
399D12—32no—
400D12—32no—
401D12—62no—
402D12—92no—
403D12—94no—
404C4×D3—32no—
405C4×D3—34no—
406C4×D3—34no—
407C4×D3—32no—
408C4×D3—32no—
409C4×D3—32no—
410C4×D3—61no—
411C4×D3—61no—
412C4×D3—61no—
413C4×D3—61no—
414C4×D3—64no—
415C4×D3—94no—
416C4×D3—94no—
417C4×D3—94no—
418Dic6—32no—
419Dic6—32no—
420Dic6—62no—
421Dic6—92no—
422Dic6—94no—
423C3×C8—32no—
424C3×C8—34no—
425C3×C8—34no—
426C3×C8—32no—
427C3×C8—61no—
428C3×C8—61no—
429C3×C8—64no—
430C3⋊C8—92no—
431C3⋊C8—91no—
432C3⋊C8—92no—
433C3⋊C8—91no—
434C3⋊C8—181no—
435C3⋊C8—182no—
436C3⋊C8—181no—
437C3⋊C8—182no—
438C3⋊C8—361no—
439C3⋊C8—361no—
440C3⋊C8C2×D3211yescharacteristic
441C3⋊C8C2×D3214yes—
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×C32C22×C4×D311yescharacteristic
452C2×C32C22×C4×D312yes—
453C3⋊D3—21no—
454C3⋊D3—24no—
455C3⋊D3#90712yescomplemented
456C3×D3—34no—
457C3×D3—62no—
458C3×D3—62no—
459C3×D3—64no—
460C3×D3—64no—
461C3×D3—62no—
462C3×D3—124no—
463#41—92no—
464#41—272no—
465#41—272no—
466#41—272no—
467C2×Q8—271no—
468C2×D4—271no—
469C2×D4—272no—
470C22×C4—91no—
471C22×C4—272no—
472M16—32no—
473M16—92no—
474M16—94no—
475M16—274no—
476C2×C8—31no—
477C2×C8—92no—
478C2×C8—94no—
479C2×C8—271no—
480C2×C8—271no—
481C2×C8—271no—
482C2×C8—271no—
483C2×C8—274no—
484C2×C8—271no—
485C22×C3—21no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—91no—
489C22×C3—94no—
490C22×C3—91no—
491C22×C3—182no—
492C22×C3C4×D3212yes—
493C22×C3C4×D3211yescharacteristic
494C2×D3—32no—
495C2×D3—32no—
496C2×D3—34no—
497C2×D3—34no—
498C2×D3—61no—
499C2×D3—61no—
500C2×D3—64no—
501C2×D3—64no—
502C2×D3—64no—
503C2×D3—188no—
504C2×D3—184no—
505C3×C4—21no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—21no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—91no—
512C3×C4—94no—
513C3×C4—91no—
514C3×C4C4×D3211yescharacteristic
515C3×C4C4×D3212yes—
516C3×C4C22×D3211yescharacteristic
517C3×C4C22×D3212yes—
518Dic3—34no—
519Dic3—34no—
520Dic3—32no—
521Dic3—32no—
522Dic3—64no—
523Dic3—61no—
524Dic3—61no—
525C32—21no—
526C32—22no—
527C32—41no—
528C32C2×#90711yescharacteristic, complemented
529C32D3×#13112yescomplemented
530C23—91no—
531C23—272no—
532Q8—92no—
533Q8—272no—
534D4—94no—
535D4—92no—
536D4—272no—
537D4—274no—
538C2×C4—91no—
539C2×C4—91no—
540C2×C4—94no—
541C2×C4—91no—
542C2×C4—91no—
543C2×C4—91no—
544C2×C4—272no—
545C2×C4—274no—
546C2×C4D3311yescharacteristic
547C8—31no—
548C8—31no—
549C8—34no—
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—92no—
561C2×C3—181no—
562C2×C3—184no—
563C2×C3C2×C4×D3211yescharacteristic
564C2×C3C2×C4×D3212yes—
565D3—34no—
566D3—62no—
567D3—62no—
568D3—64no—
569D3—62no—
570D3—64no—
571D3—124no—
572C22—91no—
573C22—91no—
574C22—94no—
575C22—182no—
576C22—544no—
577C22#8894911yescharacteristic
578C4—91no—
579C4—91no—
580C4—94no—
581C4#8894911yescharacteristic
582C4C2×D3311yescharacteristic
583C3—21no—
584C3—22no—
585C3—41no—
586C3#1059466911yescharacteristic, complemented
587C3D3×#90712yescomplemented
588C2—21no—
589C2—92no—
590C2—181no—
591C2—184no—
592C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC4C2×D33
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C23
prime radical#11724618C2
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. #11723354C2semidirect
3. C2×C3⋊D3×C3⋊C8C2semidirect
4. C3×#63818C2semidirect
5. C3×#63817C2semidirect
6. #11722827C2semidirect
7. #11722826C2semidirect
8. #11722820C2semidirect
9. #11722814C2semidirect
21. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
22. C32×#208C22semidirect
23. C2×C32×C3⋊C8C22semidirect
24. #88787C22semidirect
27. C3⋊D3×C3⋊C8C22semidirect
28. C3⋊D3×C3⋊C8C22semidirect
29. C3⋊D3×C3⋊C8C22semidirect
30. C3×#3712C22semidirect
31. C3×#3712C22semidirect
33. C3×#3710C22semidirect
41. #63818D3semidirect
42. #63817D3semidirect
116. C3×#208C2×D3semidirect
117. C3×#208C2×D3semidirect
119. C2×C3×C3⋊C8C2×D3semidirect
126. #3712C2×D3semidirect
127. #3712C2×D3semidirect
132. #3710C2×D3semidirect
255. C3×C3⋊D3#131semidirect
288. #208D32semidirect
299. C2×C3⋊C8D32semidirect
369. C33C2×#131semidirect
455. C3⋊D3#907semidirect
528. C32C2×#907semidirect
529. C32D3×#131semidirect
586. C3#10594669semidirect
587. C3D3×#907semidirect

Maximal subgroups

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

subgroupindex
#117246182
#117233542
C2×C3⋊D3×C3⋊C82
C3×#638182
C3×#638172
#117228272
#117228262
#117228202
#117228142
D3×#9073
#105946693