Finite Group Databasebeta

← #13168645

Subgroups — #13168645

Order 1728.

1582 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89018C211yesmaximal, characteristic
2#11723689C211yesmaximal, characteristic, complemented
3#11723484C212yesmaximal, complemented
4C3×#63874C211yesmaximal, characteristic, complemented
5C3×Dic3×Dic6C212yesmaximal, complemented
6#11722937C214yesmaximal, complemented
7#11722918C212yesmaximal, complemented
8#11722912C212yesmaximal, complemented
9#10595353—31nomaximal
10#10593890—32nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6294C2212yes—
15C2×#6292C2212yes—
16C2×#6289C2214yes—
17C2×#6289C2212yes—
18#89018C418yescomplemented
19C3×C4⋊(C3⋊Dic3)C2212yes—
20C32×C4×Dic3C2212yes—
21#88815C2212yes—
22#88804C2214yes—
23C3×#3724C2211yescharacteristic
24C3×#3724C2211yescharacteristic
25C3×Dic3⋊Dic3C2214yes—
26C3×#3722C2212yes—
27C3×Dic32C2214yes—
28C2×#3832—31no—
29C2×#3799—32no—
30C2×#3799—61no—
31C4×C4⋊(C3⋊D3)—32no—
32C3×Q8×Dic3—32no—
33C3×#872—32no—
34C3×C4×Dic6—32no—
35#63900—31no—
36#63900—32no—
37#63894—32no—
38#63894—32no—
39#63894—34no—
40#63894—62no—
41D3×C4⋊Dic3—32no—
42#63883—34no—
43#63881—34no—
44#63880—34no—
45#63874—32no—
46#63874D311yescharacteristic, complemented
47Dic3×Dic6D312yescomplemented
48#63836—34no—
49C22×C3⋊(C3⋊D3)C2312yes—
50C22×C3×C3⋊D3C2311yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C4⋊(C3⋊(C3⋊D3))C2×C414yescomplemented
53C2×C3×C3⋊Dic3C2314yes—
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C3×C4×C3⋊D3C2×C414yescomplemented
56C2×C32×Dic3C2312yes—
57C2×C32×Dic3C2312yes—
58C2×C32×Dic3C2312yes—
59C32×Dic6C2×C414yescomplemented
60#6294—42no—
61#6292C2×C418yescomplemented
62#6289—44no—
63#6289C2×C418yescomplemented
64#5574—92no—
65#5523—91no—
66C2×C32×Q8—31no—
67#3832—38no—
68C2×C4⋊(C3⋊D3)—32no—
69C2×C4⋊(C3⋊D3)—32no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—62no—
73C2×C4⋊(C3⋊D3)—121no—
74C2×C4×C3⋊D3—32no—
75C2×C4×C3⋊D3—34no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—61no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6C2×D312yes—
81C2×#626—34no—
82C2×#626—62no—
83C2×#624—34no—
84C2×#624—32no—
85C2×#624—34no—
86C2×#624—34no—
87C2×#624—62no—
88C2×#624—64no—
89C2×D3×Dic3—34no—
90#3799—316no—
91#3799—68no—
92C32×C42—32no—
93#3752—34no—
94C4⋊(C3⋊Dic3)C2×D312yes—
95C3×(C2×D3)⋊C4—32no—
96C3×(C2×D3)⋊C4—32no—
97C3×C4⋊Dic3—32no—
98C3×C4⋊Dic3—32no—
99C3×C4⋊Dic3—34no—
100C3×C4⋊Dic3—62no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—34no—
103C3×Dic3⋊C4—32no—
104C3×C4×Dic3—22no—
105C3×C4×Dic3—34no—
106C3×C4×Dic3—34no—
107C3×C4×Dic3C2×D312yes—
108C3×C4×Dic3C2×D312yes—
109#3724—32no—
110#3724C2×D311yescharacteristic
111#3724C2×D311yescharacteristic
112Dic3⋊Dic3—34no—
113Dic3⋊Dic3C2×D314yes—
114#3722—34no—
115#3722—34no—
116#3722—64no—
117#3722C2×D312yes—
118Dic32C2×D314yes—
119C22×C33C2411yescharacteristic
120C2×C3⋊(C3⋊D3)—22no—
121C2×C3⋊(C3⋊D3)—22no—
122C2×C3⋊(C3⋊D3)C22×C414yes—
123C2×C3×C3⋊D3—21no—
124C2×C3×C3⋊D3—21no—
125C2×C3×C3⋊D3C22×C412yes—
126C33×C4C22×C412yes—
127C3×C3⋊Dic3—24no—
128C3×C3⋊Dic3C22×C412yes—
129C32×Dic3—22no—
130C32×Dic3—22no—
131C32×Dic3C22×C414yes—
132C2×#239—92no—
133C2×#239—181no—
134C2×#235—91no—
135C3×C4×Q8—92no—
136C3×#117—91no—
137Q8×Dic3—32no—
138C2×C4⋊Dic3—91no—
139#893—94no—
140#893—92no—
141#892—92no—
142D3×C4⋊C4—92no—
143D3×C4⋊C4—94no—
144D3×C4⋊C4—92no—
145#67⋊C4—94no—
146C4×D12—94no—
147C4×D12—92no—
148C4×D12—92no—
149C4×D12—182no—
150#872—32no—
151#872—94no—
152C4×Dic6—32no—
153C22×C3⋊D3—32no—
154C22×C3⋊D3—34no—
155C22×C3⋊D3—64no—
156C22×C3⋊D3—62no—
157C22×C3⋊D3—122no—
158C22×C3⋊D3C22×D311yescharacteristic
159C22×C3×D3—32no—
160C22×C3×D3—61no—
161C32×Q8—34no—
162C2×C32×C4—22no—
163C2×C32×C4—21no—
164C2×C32×C4—32no—
165C2×C32×C4—32no—
166C2×C32×C4—32no—
167C2×C32×C4—41no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D312yes—
170C2×C3⋊Dic3C22×D314yes—
171C2×C3⋊Dic3C22×D311yescharacteristic
172C4⋊(C3⋊D3)—38no—
173C4⋊(C3⋊D3)—38no—
174C4⋊(C3⋊D3)—34no—
175C4⋊(C3⋊D3)—64no—
176C4⋊(C3⋊D3)—68no—
177C4⋊(C3⋊D3)—124no—
178C4×C3⋊D3—38no—
179C4×C3⋊D3—124no—
180C4×C3⋊D3C4×D314yescomplemented
181C2×C3×Dic3—22no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—34no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3C22×D312yes—
190C2×C3×Dic3C22×D312yes—
191C2×C3×Dic3C22×D312yes—
192C2×C3×Dic3C22×D312yes—
193C2×C3×Dic3C22×D314yes—
194C3×C4×D3—38no—
195C3×C4×D3—64no—
196C3×Dic6—24no—
197C3×Dic6C4×D318yescomplemented
198#626—316no—
199#626—68no—
200#624—316no—
201#624—68no—
202#624—124no—
203#624—124no—
204#624—128no—
205#624—122no—
206D3×Dic3—124no—
207#520—271noSylow
208C2×C33C23×C411yescharacteristic
209C2×C33H(5,𝔽2)11yescharacteristic
210C2×C332−1+411yescharacteristic
211C3⋊(C3⋊D3)—24no—
212C3×C3⋊D3—22no—
213C2×C3×Q8—32no—
214C2×C3×Q8—61no—
215C22×C3×C4—91no—
216C2×#67—92no—
217C22×Dic3—92no—
218#239—916no—
219#239—188no—
220#235—98no—
221C2×D12—94no—
222C2×D12—91no—
223C2×D12—92no—
224C2×D12—182no—
225C2×D12—181no—
226C2×D12—182no—
227C2×D12—361no—
228C2×C4×D3—32no—
229C2×C4×D3—61no—
230C2×C4×D3—94no—
231C2×C4×D3—92no—
232C2×C4×D3—94no—
233C2×C4×D3—94no—
234C2×C4×D3—94no—
235C2×C4×D3—94no—
236C2×C4×D3—182no—
237C2×C4×D3—184no—
238C2×Dic6D3211yescharacteristic
239C3×C4⋊C4—91no—
240C3×C4⋊C4—94no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C42—32no—
245C3×C42—32no—
246C3×C42—62no—
247C3×C42—94no—
248(C2×D3)⋊C4—32no—
249(C2×D3)⋊C4—32no—
250(C2×D3)⋊C4—94no—
251(C2×D3)⋊C4—94no—
252(C2×D3)⋊C4—94no—
253(C2×D3)⋊C4—184no—
254C4⋊Dic3—32no—
255C4⋊Dic3—32no—
256C4⋊Dic3—34no—
257C4⋊Dic3—62no—
258C4⋊Dic3—92no—
259Dic3⋊C4—34no—
260Dic3⋊C4—32no—
261Dic3⋊C4—32no—
262Dic3⋊C4—94no—
263C4×Dic3—34no—
264C4×Dic3—34no—
265C4×Dic3D3212yes—
266C22×C32—22no—
267C22×C32—21no—
268C22×C32—41no—
269C22×C32C23×D312yes—
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—21no—
272C2×C3⋊D3—21no—
273C2×C3⋊D3—38no—
274C2×C3⋊D3—34no—
275C2×C3⋊D3—64no—
276C2×C3⋊D3—68no—
277C2×C3⋊D3—64no—
278C2×C3⋊D3—62no—
279C2×C3⋊D3—64no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—124no—
282C2×C3⋊D3—124no—
283C2×C3⋊D3—124no—
284C2×C3⋊D3—122no—
285C2×C3⋊D3—122no—
286C2×C3⋊D3—242no—
287C2×C3⋊D3—242no—
288C2×C3⋊D3C2×C4×D312yes—
289C2×C3×D3—34no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—121no—
294C2×C3×D3—121no—
295C32×C4—24no—
296C32×C4—22no—
297C32×C4—34no—
298C32×C4—42no—
299C32×C4—62no—
300C32×C4—62no—
301C32×C4C2×C4×D312yes—
302C32×C4C2×C4×D314yes—
303C3⋊Dic3—24no—
304C3⋊Dic3C2×C4×D312yes—
305C3×Dic3—24no—
306C3×Dic3—22no—
307C3×Dic3—22no—
308C3×Dic3—22no—
309C3×Dic3—22no—
310C3×Dic3—24no—
311C3×Dic3—34no—
312C3×Dic3—42no—
313C3×Dic3—62no—
314C3×Dic3—64no—
315C3×Dic3—64no—
316C3×Dic3—124no—
317C3×Dic3C2×C4×D318yes—
318C2×#41—271no—
319C4×Q8—92no—
320C4×D4—272no—
321C4×D4—274no—
322#117—91no—
323#117—272no—
324C2×C4⋊C4—272no—
325C2×C4⋊C4—271no—
326C33#52011yescharacteristic, complemented, Sylow
327C23×C3—91no—
328C22×D3—32no—
329C22×D3—61no—
330C22×D3—94no—
331C22×D3—92no—
332C22×D3—184no—
333C22×D3—182no—
334C22×D3—362no—
335C3×Q8—38no—
336C3×Q8—64no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—34no—
340C2×C3×C4—32no—
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—41no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—91no—
349C2×C3×C4—94no—
350C2×C3×C4—94no—
351C2×C3×C4C2×D3212yes—
352C2×C3×C4C2×D3211yescharacteristic
353#67—98no—
354C2×Dic3—32no—
355C2×Dic3—34no—
356C2×Dic3—34no—
357C2×Dic3—61no—
358C2×Dic3—64no—
359C2×Dic3—362no—
360C2×Dic3C2×D3212yes—
361C2×Dic3C2×D3212yes—
362C2×Dic3C2×D3212yes—
363D12—916no—
364D12—98no—
365D12—94no—
366D12—188no—
367D12—184no—
368D12—188no—
369D12—364no—
370C4×D3—38no—
371C4×D3—64no—
372C4×D3—916no—
373C4×D3—98no—
374C4×D3—188no—
375C4×D3—364no—
376C4×D3—364no—
377C4×D3—364no—
378C4×D3—364no—
379C4×D3—368no—
380Dic6#62414yescomplemented
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—22no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—41no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32C22×C4×D311yescharacteristic
391C2×C32C22×C4×D312yes—
392C2×C32#100712yes—
393C2×C32#100911yescharacteristic
394C2×C32#100912yes—
395C2×C32#101011yescharacteristic
396C3⋊D3—22no—
397C3⋊D3—64no—
398C3⋊D3—68no—
399C3⋊D3—124no—
400C3⋊D3—128no—
401C3⋊D3—244no—
402C3×D3—64no—
403C3×D3—122no—
404#41—278no—
405C2×Q8—31no—
406C2×D4—272no—
407C2×D4—271no—
408C22×C4—91no—
409C22×C4—272no—
410C22×C4—274no—
411C22×C4—272no—
412C4⋊C4—91no—
413C4⋊C4—94no—
414C4⋊C4—91no—
415C4⋊C4—92no—
416C4⋊C4—272no—
417C22⋊C4—92no—
418C22⋊C4—274no—
419C42—32no—
420C42—94no—
421C22×C3—22no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—92no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3C22×D3212yes—
428C22×C3C22×D3211yescharacteristic
429C2×D3—34no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—98no—
434C2×D3—94no—
435C2×D3—121no—
436C2×D3—121no—
437C2×D3—184no—
438C2×D3—188no—
439C2×D3—184no—
440C2×D3—184no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—364no—
444C2×D3—364no—
445C2×D3—362no—
446C2×D3—364no—
447C2×D3—362no—
448C2×D3—722no—
449C2×D3—722no—
450C3×C4—22no—
451C3×C4—24no—
452C3×C4—38no—
453C3×C4—42no—
454C3×C4—64no—
455C3×C4—64no—
456C3×C4—62no—
457C3×C4—62no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—92no—
461C3×C4—122no—
462C3×C4—184no—
463C3×C4C4×D3214yes—
464C3×C4C2×#62412yes—
465Dic3—22no—
466Dic3—22no—
467Dic3—34no—
468Dic3—62no—
469Dic3—64no—
470Dic3—64no—
471Dic3—124no—
472Dic3C2×#62414yes—
473C32—21no—
474C32—22no—
475C32—41no—
476C32#552311yescharacteristic, complemented
477C32#557412yescomplemented
478C23—91no—
479C23—272no—
480Q8—34no—
481D4—278no—
482D4—274no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—32no—
486C2×C4—91no—
487C2×C4—94no—
488C2×C4—94no—
489C2×C4—278no—
490C2×C4—1084no—
491C2×C4—1082no—
492C2×C4D3311yescharacteristic
493C2×C3—22no—
494C2×C3—21no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—22no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—182no—
503C2×C3#6429712yes—
504C2×C3C2×C4×D3212yes—
505C2×C3#6430812yes—
506C2×C3#6431611yescharacteristic
507C2×C3#6432011yescharacteristic
508C2×C3C22×#62411yescharacteristic
509D3—64no—
510D3—122no—
511D3—184no—
512D3—188no—
513D3—364no—
514D3—368no—
515D3—724no—
516C22—92no—
517C22—181no—
518C22—181no—
519C22—274no—
520C22—542no—
521C22—542no—
522C22C2×D3311yescharacteristic
523C4—34no—
524C4—62no—
525C4—62no—
526C4—92no—
527C4—184no—
528C4D3×#62412yes—
529C3—21no—
530C3—22no—
531C3—41no—
532C3#1059389012yescomplemented
533C3#1059535311yescharacteristic, complemented
534C2—182no—
535C2—544no—
536C2#1172439011yescharacteristic
537C2#1172439511yescharacteristic
538C2C2×D3×#62411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×#89018C2
hypercenterC2×C4D33
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723689C2semidirect
3. #11723484C2semidirect
4. C3×#63874C2semidirect
5. C3×Dic3×Dic6C2semidirect
6. #11722937C2semidirect
7. #11722918C2semidirect
8. #11722912C2semidirect
18. #89018C4semidirect
46. #63874D3semidirect
47. Dic3×Dic6D3semidirect
52. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
55. C3×C4×C3⋊D3C2×C4semidirect
59. C32×Dic6C2×C4semidirect
61. #6292C2×C4semidirect
63. #6289C2×C4semidirect
180. C4×C3⋊D3C4×D3semidirect
197. C3×Dic6C4×D3semidirect
326. C33#520semidirect
380. Dic6#624semidirect
476. C32#5523semidirect
477. C32#5574semidirect
532. C3#10593890semidirect
533. C3#10595353semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
#117236892
#117234842
C3×#638742
C3×Dic3×Dic62
#117229372
#117229182
#117229122
#105953533
#105938903