Finite Group Databasebeta

← #13170485

Subgroups — #13170485

Order 1728.

1276 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64306C211yesmaximal, characteristic
2C2×#88808C211yesmaximal, characteristic, complemented
3C3⋊D3×#208C211yesmaximal, characteristic, complemented
4#11723349C212yesmaximal, complemented
5C3×#63815C212yesmaximal, complemented
6#11722822C214yesmaximal, complemented
7#11722816C212yesmaximal, complemented
8D32×C3⋊C8C212yesmaximal, complemented
9D3×#948—32nomaximal
10#10594707—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C32×#235C2212yes—
13C3×#3811C412yes—
14C3×#3805C412yes—
15C3×#3802C411yescharacteristic
16C3×C4×D32C2212yes—
17C3×#3798C2212yes—
18C3×#3797C411yescharacteristic
19C3×#3796C412yes—
20#88855C2211yescharacteristic, complemented
21C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
22C32×#208C2211yescharacteristic, complemented
23#88808C2211yescharacteristic
24#88808C2211yescharacteristic
25#88808C2212yescomplemented
26#88787C2212yescomplemented
27#88786C2214yescomplemented
28C3⋊D3×C3⋊C8C2212yes—
29D3×C3⋊(C3⋊C8)C2214yescomplemented
30C3×#3711C2214yescomplemented
31C3×D3×C3⋊C8C2214yescomplemented
32C3×D3×#41—32no—
33#64306D311yescharacteristic
34#64157—32no—
35M16×C3⋊D3—31no—
36C3×#948—32no—
37C3×#907—32no—
38C2×#3710—31no—
39#63815—34no—
40#63815D312yescomplemented
41#63814—32no—
42#63814—32no—
43#63814—34no—
44#63814—62no—
45D3×#208—32no—
46D3×#208—34no—
47D3×#208—61no—
48C2×D3×C3⋊C8—32no—
49C2×D3×C3⋊C8—34no—
50#63804—34no—
51#63791—32no—
52D32×C8—32no—
53C22×C3×C3⋊D3C2×C411yescharacteristic
54C2×C3×D32C2×C412yes—
55C2×C33×C4C2311yescharacteristic
56C2×C3×C3⋊Dic3C2×C411yescharacteristic
57C3×C4×C3⋊D3C2×C411yescharacteristic
58C3×C4×C3⋊D3C2×C411yescharacteristic
59C3×C4×C3⋊D3C2311yescharacteristic
60C3×C4×C3⋊D3C2311yescharacteristic
61C32×#67C2×C414yes—
62C32×D12C2×C412yes—
63C32×C4×D3C2314yes—
64C32×Dic6C2×C412yes—
65C3×#627C2×C412yes—
66C3×#626C2×C414yes—
67C3×#625C2×C412yes—
68C3×#624C2×C412yes—
69C3×D3×Dic3C2×C414yes—
70C3⋊(C3⋊(C3⋊C8))C2312yes—
71C3×C3⋊(C3⋊C8)C2312yes—
72C3×C3⋊(C3⋊C8)C2312yes—
73C32×C3⋊C8C2312yes—
74C2×#948—91no—
75D3×#131—92no—
76C32×#41—32no—
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C3×#239—32no—
79C3×D3×Q8—32no—
80C3×#237—34no—
81C3×#237—32no—
82C3×D3×D4—34no—
83C3×D3×D4—32no—
84C3×#235—22no—
85C3×#235—34no—
86C3×#235C2×D312yes—
87C3×#235C2×D312yes—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—34no—
90C2×C3×C4×D3—61no—
91#3811Dic312yes—
92#3805Dic312yes—
93#3802Dic311yescharacteristic
94C4×D32C2×D312yes—
95#3798C2×D312yes—
96#3797Dic311yescharacteristic
97#3796Dic312yes—
98C32×M16—31no—
99#3748—32no—
100#3748—31no—
101#3748—34no—
102#3748—61no—
103#3748—62no—
104#3748—121no—
105C2×C3⋊(C3⋊C8)—34no—
106C2×C3⋊(C3⋊C8)C2×D312yescomplemented
107#3743—32no—
108C8×C3⋊D3—32no—
109C3×#208—21no—
110C3×#208—34no—
111C3×#208C2×D312yescomplemented
112C2×C3×C3⋊C8—32no—
113C2×C3×C3⋊C8—32no—
114C2×C3×C3⋊C8—34no—
115C2×C3×C3⋊C8—62no—
116C3×ΓL(1,𝔽25)—34no—
117C3×D3×C8—34no—
118#3711—34no—
119#3711—34no—
120#3711—34no—
121#3711—62no—
122#3711—64no—
123#3711C2×D314yescomplemented
124#3710—31no—
125#3710—31no—
126#3710—32no—
127D3×C3⋊C8—34no—
128D3×C3⋊C8—34no—
129D3×C3⋊C8—32no—
130D3×C3⋊C8—32no—
131D3×C3⋊C8—32no—
132D3×C3⋊C8—32no—
133D3×C3⋊C8—34no—
134D3×C3⋊C8—62no—
135D3×C3⋊C8—64no—
136D3×C3⋊C8C2×D314yescomplemented
137C22×C33C22×C411yescharacteristic
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3C22×C411yescharacteristic
140C2×C3×C3⋊D3C22×C411yescharacteristic
141C2×C32×D3C22×C414yes—
142C3×D32—24no—
143C33×C4C22×C411yescharacteristic
144C33×C4C2411yescharacteristic
145C3×C3⋊Dic3C22×C411yescharacteristic
146C3×C3⋊Dic3C22×C411yescharacteristic
147C32×Dic3C22×C414yes—
148C2×C3×#41—91no—
149D3×#41—32no—
150C3×#131—92no—
151#948—32no—
152#948—94no—
153#948—92no—
154#948—92no—
155#948—182no—
156C2×#208—92no—
157C2×#208—91no—
158C22×C3⋊C8—91no—
159C22×C3⋊C8—92no—
160#907—32no—
161#907—94no—
162D3×M16—94no—
163D3×M16—92no—
164D3×M16—181no—
165#901—94no—
166#901—92no—
167C2×D3×C8—94no—
168C2×D3×C8—92no—
169C22×C3⋊D3C2×Dic311yescharacteristic
170C22×C3×D3—32no—
171C22×C3×D3—34no—
172C22×C3×D3—61no—
173C2×D32C2×Dic312yes—
174C32×Q8—32no—
175C32×D4—34no—
176C32×D4—32no—
177C2×C32×C4—22no—
178C2×C32×C4—21no—
179C2×C32×C4—34no—
180C2×C32×C4—41no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3C2×Dic311yescharacteristic
184C4×C3⋊D3C2×Dic311yescharacteristic
185C4×C3⋊D3C2×Dic311yescharacteristic
186C4×C3⋊D3C22×D311yescharacteristic
187C4×C3⋊D3C22×D311yescharacteristic
188C3×#67—24no—
189C3×#67—34no—
190C3×#67—34no—
191C3×#67C4×D314yes—
192C3×#67C2×Dic314yes—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—61no—
196C3×D12—22no—
197C3×D12—34no—
198C3×D12C4×D312yes—
199C3×D12C2×Dic312yes—
200C3×C4×D3—24no—
201C3×C4×D3—34no—
202C3×C4×D3—32no—
203C3×C4×D3—32no—
204C3×C4×D3—32no—
205C3×C4×D3—32no—
206C3×C4×D3—34no—
207C3×C4×D3—61no—
208C3×C4×D3—61no—
209C3×C4×D3—61no—
210C3×C4×D3—61no—
211C3×C4×D3C22×D314yes—
212C3×C4×D3C22×D314yes—
213C3×Dic6—22no—
214C3×Dic6—34no—
215C3×Dic6C4×D312yes—
216C3×Dic6C2×Dic312yes—
217#627C2×Dic312yes—
218#626C2×Dic314yes—
219#625C2×Dic312yes—
220#624C2×Dic312yes—
221D3×Dic3C2×Dic314yes—
222C32×C8—32no—
223C3⋊(C3⋊C8)—32no—
224C3⋊(C3⋊C8)—34no—
225C3⋊(C3⋊C8)—64no—
226C3⋊(C3⋊C8)—62no—
227C3⋊(C3⋊C8)—122no—
228C3⋊(C3⋊C8)C22×D312yes—
229C3⋊(C3⋊C8)C22×D312yes—
230C3×C3⋊C8—22no—
231C3×C3⋊C8—32no—
232C3×C3⋊C8—32no—
233C3×C3⋊C8—32no—
234C3×C3⋊C8—32no—
235C3×C3⋊C8—62no—
236C3×C3⋊C8—62no—
237C3×C3⋊C8C22×D314yes—
238C2×#131—271noSylow
239C2×C33—21no—
240C2×C33C23×C411yescharacteristic
241C3×C3⋊D3—21no—
242C3×C3⋊D3#13112yescomplemented
243C32×D3—24no—
244C3×#41—32no—
245C3×#41—32no—
246C3×#41—62no—
247C3×#41—92no—
248C3×#41—92no—
249C3×#41—92no—
250C2×C3×Q8—91no—
251C2×C3×D4—92no—
252C2×C3×D4—91no—
253C22×C3×C4—92no—
254C22×C3×C4—91no—
255#239—32no—
256D3×Q8—32no—
257#237—34no—
258#237—32no—
259D3×D4—34no—
260D3×D4—32no—
261#235—34no—
262#235D3212yes—
263C2×C4×D3—32no—
264C2×C4×D3—34no—
265C2×C4×D3—61no—
266C3×M16—32no—
267C3×M16—61no—
268C3×M16—94no—
269C2×C3×C8—94no—
270C2×C3×C8—92no—
271#208—34no—
272#208—94no—
273#208—94no—
274#208—92no—
275#208—91no—
276#208—92no—
277#208—184no—
278#208—181no—
279#208—182no—
280#208—361no—
281#208D3211yescharacteristic, complemented
282C2×C3⋊C8—32no—
283C2×C3⋊C8—32no—
284C2×C3⋊C8—34no—
285C2×C3⋊C8—62no—
286C2×C3⋊C8—91no—
287C2×C3⋊C8—91no—
288C2×C3⋊C8—92no—
289C2×C3⋊C8—94no—
290C2×C3⋊C8—94no—
291C2×C3⋊C8—92no—
292C2×C3⋊C8—184no—
293ΓL(1,𝔽25)—34no—
294ΓL(1,𝔽25)—94no—
295ΓL(1,𝔽25)—94no—
296ΓL(1,𝔽25)—182no—
297D3×C8—34no—
298D3×C8—92no—
299D3×C8—92no—
300D3×C8—92no—
301D3×C8—92no—
302D3×C8—94no—
303D3×C8—94no—
304D3×C8—182no—
305C22×C32—21no—
306C22×C32—22no—
307C22×C32—34no—
308C22×C32—41no—
309C22×C32C2×C4×D312yes—
310C22×C32C22×Dic311yescharacteristic
311C2×C3⋊D3—22no—
312C2×C3⋊D3C22×Dic311yescharacteristic
313C2×C3⋊D3C22×Dic311yescharacteristic
314C2×C3×D3—24no—
315C2×C3×D3—32no—
316C2×C3×D3—32no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—68no—
320C2×C3×D3—64no—
321C2×C3×D3—122no—
322C2×C3×D3C2×C4×D314yes—
323C2×C3×D3C22×Dic314yes—
324D32—24no—
325C32×C4—21no—
326C32×C4—22no—
327C32×C4—21no—
328C32×C4—22no—
329C32×C4—34no—
330C32×C4—41no—
331C32×C4—41no—
332C32×C4C2×C4×D312yes—
333C32×C4C22×Dic311yescharacteristic
334C32×C4C23×D312yes—
335C32×C4C23×D311yescharacteristic
336C3⋊Dic3C22×Dic311yescharacteristic
337C3⋊Dic3C22×Dic311yescharacteristic
338C3×Dic3—24no—
339C3×Dic3—32no—
340C3×Dic3—32no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3C2×C4×D314yes—
344C3×Dic3C22×Dic314yes—
345C2×#41—91no—
346#131—92no—
347#131—274no—
348#131—272no—
349C2×M16—271no—
350C2×M16—272no—
351C22×C8—272no—
352C22×C8—271no—
353C33C2×#13111yescharacteristic, complemented, Sylow
354C23×C3—92no—
355C23×C3—91no—
356C22×D3—32no—
357C22×D3—34no—
358C22×D3—61no—
359C3×Q8—32no—
360C3×Q8—32no—
361C3×Q8—62no—
362C3×Q8—92no—
363C3×D4—34no—
364C3×D4—34no—
365C3×D4—32no—
366C3×D4—32no—
367C3×D4—62no—
368C3×D4—64no—
369C3×D4—94no—
370C3×D4—92no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—34no—
374C2×C3×C4—34no—
375C2×C3×C4—41no—
376C2×C3×C4—64no—
377C2×C3×C4—92no—
378C2×C3×C4—94no—
379C2×C3×C4—91no—
380C2×C3×C4—91no—
381C2×C3×C4—91no—
382C2×C3×C4—91no—
383C2×C3×C4—91no—
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C2×D3212yes—
386#67—34no—
387#67—34no—
388#67D3×Dic314yes—
389C2×Dic3—32no—
390C2×Dic3—34no—
391C2×Dic3—61no—
392D12—34no—
393D12D3×Dic312yes—
394C4×D3—34no—
395C4×D3—32no—
396C4×D3—32no—
397C4×D3—32no—
398C4×D3—32no—
399C4×D3—34no—
400C4×D3—61no—
401C4×D3—61no—
402C4×D3—61no—
403C4×D3—61no—
404C4×D3C2×D3214yes—
405Dic6—34no—
406Dic6D3×Dic312yes—
407C3×C8—34no—
408C3×C8—62no—
409C3×C8—92no—
410C3×C8—92no—
411C3⋊C8—32no—
412C3⋊C8—32no—
413C3⋊C8—32no—
414C3⋊C8—32no—
415C3⋊C8—62no—
416C3⋊C8—62no—
417C3⋊C8—94no—
418C3⋊C8—92no—
419C3⋊C8—182no—
420C3⋊C8—184no—
421C3⋊C8—362no—
422C3⋊C8C2×D3212yes—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—22no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32—64no—
431C2×C32—81no—
432C2×C32C22×C4×D312yes—
433C2×C32C23×Dic311yescharacteristic
434C3⋊D3—21no—
435C3⋊D3#94812yescomplemented
436C3×D3—24no—
437C3×D3—24no—
438C3×D3—34no—
439C3×D3—44no—
440C3×D3—62no—
441C3×D3—62no—
442C3×D3—121no—
443#41—32no—
444#41—92no—
445#41—92no—
446#41—92no—
447C2×Q8—91no—
448C2×D4—92no—
449C2×D4—91no—
450C22×C4—92no—
451C22×C4—91no—
452M16—31no—
453M16—94no—
454M16—272no—
455M16—274no—
456M16—271no—
457C2×C8—94no—
458C2×C8—92no—
459C2×C8—272no—
460C2×C8—271no—
461C2×C8—271no—
462C2×C8—272no—
463C2×C8—274no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—34no—
467C22×C3—34no—
468C22×C3—41no—
469C22×C3—64no—
470C22×C3—91no—
471C22×C3—91no—
472C22×C3—182no—
473C22×C3—184no—
474C22×C3C4×D3211yescharacteristic
475C22×C3C2×D3×Dic312yes—
476C2×D3—32no—
477C2×D3—32no—
478C2×D3—61no—
479C2×D3—61no—
480C2×D3—68no—
481C2×D3—64no—
482C2×D3—122no—
483C2×D3C2×D3×Dic314yes—
484C3×C4—22no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—22no—
488C3×C4—34no—
489C3×C4—34no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—64no—
493C3×C4—91no—
494C3×C4—91no—
495C3×C4C4×D3211yescharacteristic
496C3×C4C2×D3×Dic312yes—
497C3×C4C22×D3211yescharacteristic
498C3×C4C22×D3212yes—
499Dic3—32no—
500Dic3—32no—
501Dic3—61no—
502Dic3—61no—
503Dic3C2×D3×Dic314yes—
504C32—22no—
505C32—21no—
506C32—41no—
507C32D3×#13112yescomplemented
508C32C2×#94811yescharacteristic, complemented
509C23—92no—
510C23—91no—
511Q8—32no—
512Q8—92no—
513D4—34no—
514D4—32no—
515D4—94no—
516D4—92no—
517C2×C4—34no—
518C2×C4—92no—
519C2×C4—94no—
520C2×C4—91no—
521C2×C4—91no—
522C2×C4—91no—
523C2×C4—91no—
524C2×C4—91no—
525C2×C4D3311yescharacteristic
526C8—32no—
527C8—92no—
528C8—92no—
529C8—272no—
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—22no—
534C2×C3—41no—
535C2×C3—42no—
536C2×C3—41no—
537C2×C3—64no—
538C2×C3—64no—
539C2×C3—81no—
540C2×C3—92no—
541C2×C3—124no—
542C2×C3—181no—
543C2×C3C2×C4×D3211yescharacteristic
544C2×C3C22×D3×Dic312yes—
545D3—24no—
546D3—34no—
547D3—62no—
548D3—62no—
549D3—121no—
550C22—34no—
551C22—91no—
552C22—91no—
553C22—182no—
554C22—184no—
555C22D32×Dic311yescharacteristic
556C4—34no—
557C4—91no—
558C4—91no—
559C4D32×Dic311yescharacteristic
560C4C2×D3311yescharacteristic
561C3—21no—
562C3—22no—
563C3—41no—
564C3#1059470711yescharacteristic, complemented
565C3D3×#94812yescomplemented
566C2—21no—
567C2—64no—
568C2—92no—
569C2—181no—
570C2C2×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. C2×#88808C2semidirect
3. C3⋊D3×#208C2semidirect
4. #11723349C2semidirect
5. C3×#63815C2semidirect
6. #11722822C2semidirect
7. #11722816C2semidirect
8. D32×C3⋊C8C2semidirect
20. #88855C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. C32×#208C22semidirect
25. #88808C22semidirect
26. #88787C22semidirect
27. #88786C22semidirect
29. D3×C3⋊(C3⋊C8)C22semidirect
30. C3×#3711C22semidirect
31. C3×D3×C3⋊C8C22semidirect
40. #63815D3semidirect
106. C2×C3⋊(C3⋊C8)C2×D3semidirect
111. C3×#208C2×D3semidirect
123. #3711C2×D3semidirect
136. D3×C3⋊C8C2×D3semidirect
242. C3×C3⋊D3#131semidirect
281. #208D32semidirect
353. C33C2×#131semidirect
435. C3⋊D3#948semidirect
507. C32D3×#131semidirect
508. C32C2×#948semidirect
564. C3#10594707semidirect
565. C3D3×#948semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
C2×#888082
C3⋊D3×#2082
#117233492
C3×#638152
#117228222
#117228162
D32×C3⋊C82
D3×#9483
#105947073