Finite Group Databasebeta

← #13178312

Subgroups — #13178312

Order 1728.

1532 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C212yesmaximal, complemented
2C3×#64104C212yesmaximal, complemented
3C3×#64096C211yesmaximal, characteristic, complemented
4#11723697C212yesmaximal, complemented
5#11723697C211yesmaximal, characteristic, complemented
6C4×#6295C211yesmaximal, characteristic, complemented
7#11723687C214yesmaximal, complemented
8#11723685C212yesmaximal, complemented
9#10594527—31nomaximal
10#10594524—32nomaximal
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C2×C3×C4⋊(C3⋊D3)C2214yes—
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×#634C2211yescharacteristic
15C2×#6295C2214yes—
16C2×#6295C2212yes—
17C2×#6295C2211yescharacteristic
18C2×#6295C2212yes—
19C2×#6294C2214yes—
20#89041—28no—
21C33×C4⋊C4C2211yescharacteristic
22C3×#3752C2214yes—
23C3×#3750C2212yes—
24C3×C4×C3⋊Dic3C2212yes—
25C3×C4×C3⋊Dic3C2211yescharacteristic
26#88815C2211yescharacteristic
27#88814C2212yes—
28C2×#3799—32no—
29C2×#3796—32no—
30C2×#3795—32no—
31#64104D312yescomplemented
32#64096D311yescharacteristic, complemented
33C3×#895—32no—
34C3×#895—32no—
35C3×#895—62no—
36C3×#887—32no—
37C3×#887—61no—
38#63914—31no—
39#63914—32no—
40#63911—34no—
41#63911—32no—
42#63910—34no—
43C4×#626—32no—
44#63902—32no—
45C4×#625—31no—
46#63886—34no—
47#63885—34no—
48#63880—34no—
49C22×C3×C3⋊D3C2314yes—
50C22×C3×C3⋊D3C2312yes—
51C2×C33×C4C2312yes—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2312yes—
55C2×C3×C3⋊Dic3C2312yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4⋊(C3⋊D3)—24no—
58C3×C4⋊(C3⋊D3)—44no—
59C3×C4×C3⋊D3—24no—
60C3×#634D414yescomplemented
61#6295—28no—
62#6295—24no—
63#6295—22no—
64#6295—42no—
65#6294—28no—
66#5599—92no—
67#5577—91no—
68C2×C4⋊(C3⋊D3)C2×D312yes—
69C2×C4⋊(C3⋊D3)C2×D314yes—
70C2×C4×C3⋊D3C2×D312yes—
71C2×#634C2×D311yescharacteristic
72C2×C3×D12—32no—
73C2×C3×D12—32no—
74C2×C3×D12—34no—
75C2×C3×D12—34no—
76C2×C3×D12—62no—
77C2×C3×D12—64no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—62no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—61no—
83C2×#626—34no—
84C2×#626—34no—
85C2×#626—34no—
86C2×#626—32no—
87C2×#626—34no—
88C2×#625—34no—
89C2×#625—31no—
90C2×#625—32no—
91C2×#625—32no—
92C2×#624—34no—
93C2×D3×Dic3—34no—
94C2×D3×Dic3—34no—
95#3799—68no—
96#3796—68no—
97#3795—68no—
98C32×C4⋊C4—21no—
99C32×C4⋊C4—22no—
100C32×C4⋊C4—41no—
101C32×C4⋊C4C2×D312yes—
102C32×C4⋊C4C2×D311yescharacteristic
103#3752C2×D314yes—
104#3750C2×D312yes—
105C4×C3⋊Dic3C2×D311yescharacteristic
106C4×C3⋊Dic3C2×D312yes—
107C3×(C2×D3)⋊C4—34no—
108C3×(C2×D3)⋊C4—34no—
109C3×(C2×D3)⋊C4—64no—
110C3×Dic3⋊C4—34no—
111C3×Dic3⋊C4—62no—
112C3×C4×Dic3—34no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—62no—
115C3×C4×Dic3—61no—
116#3724—31no—
117Dic3⋊Dic3—32no—
118#3722—32no—
119(C2×D3)⋊Dic3—32no—
120(C2×D3)⋊Dic3—32no—
121C22×C33C2411yescharacteristic
122C2×C3×C3⋊D3—24no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3—24no—
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—24no—
128C33×C4—22no—
129C33×C4C2×D412yes—
130C3×C3⋊Dic3—22no—
131C3×C3⋊Dic3—21no—
132C3×C3⋊Dic3C2×D414yes—
133C3×C3⋊Dic3#4112yescomplemented
134C2×#239—92no—
135C2×#237—92no—
136C2×#235—92no—
137C3×#121—92no—
138C3×C4×Q8—91no—
139#940—94no—
140#940—92no—
141#939—94no—
142C4×#67—92no—
143#895—32no—
144#895—32no—
145#895—62no—
146#895—94no—
147#887—32no—
148#887—61no—
149#884—94no—
150#67⋊C4—94no—
151#874—91no—
152#874—92no—
153C4×D12—92no—
154C4×D12—91no—
155C22×C3⋊D3C22×D312yes—
156C22×C3⋊D3C22×D314yes—
157C22×C3×D3—32no—
158C22×C3×D3—32no—
159C22×C3×D3—34no—
160C22×C3×D3—34no—
161C22×C3×D3—62no—
162C22×C3×D3—64no—
163C2×C32×C4—21no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—24no—
167C2×C32×C4—41no—
168C2×C32×C4—42no—
169C2×C32×C4C22×D314yes—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D312yes—
173C2×C3⋊Dic3C22×D311yescharacteristic
174C2×C3⋊Dic3C22×D312yes—
175C2×C3⋊Dic3C22×D312yes—
176C2×C3⋊Dic3C22×D311yescharacteristic
177C4⋊(C3⋊D3)—24no—
178C4⋊(C3⋊D3)—44no—
179C4×C3⋊D3—24no—
180#634D1214yescomplemented
181C2×C3×Dic3—32no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—62no—
190C3×D12—64no—
191C3×D12—64no—
192C3×D12—124no—
193C3×D12—124no—
194C3×D12—124no—
195C3×D12—128no—
196C3×C4×D3—64no—
197C3×C4×D3—64no—
198C3×C4×D3—124no—
199C3×Dic6—38no—
200C3×Dic6—64no—
201#626—68no—
202#626—68no—
203#626—68no—
204#626—64no—
205#626—124no—
206#625—68no—
207#625—62no—
208#625—64no—
209#625—122no—
210#624—68no—
211D3×Dic3—68no—
212D3×Dic3—68no—
213#550—271noSylow
214C2×C33C22×D411yescharacteristic
215C2×C33C2×#4111yescharacteristic
216C2×C33H(5,𝔽2)11yescharacteristic
217C3×C3⋊D3—42no—
218C3×C3⋊D3—44no—
219C2×C3×Q8—91no—
220C2×C3×D4—92no—
221C2×C3×D4—94no—
222C22×C3×C4—92no—
223C2×#67—94no—
224C2×#67—94no—
225C2×#67—92no—
226C2×#67—94no—
227C2×#67—94no—
228C22×Dic3—94no—
229#239—188no—
230#237—188no—
231#235—188no—
232C2×D12—32no—
233C2×D12—32no—
234C2×D12—34no—
235C2×D12—34no—
236C2×D12—62no—
237C2×D12—64no—
238C2×D12—92no—
239C2×D12—91no—
240C2×D12—94no—
241C2×D12—92no—
242C2×C4×D3—32no—
243C2×C4×D3—32no—
244C2×C4×D3—62no—
245C2×C4×D3—94no—
246C2×C4×D3—94no—
247C2×Dic6—32no—
248C2×Dic6—61no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—41no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4D3212yes—
254C3×C4⋊C4D3211yescharacteristic
255C3×C22⋊C4—94no—
256C3×C42—92no—
257C3×C42—91no—
258#217—92no—
259(C2×D3)⋊C4—34no—
260(C2×D3)⋊C4—34no—
261(C2×D3)⋊C4—64no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264C4⋊Dic3—91no—
265Dic3⋊C4—34no—
266Dic3⋊C4—62no—
267Dic3⋊C4—92no—
268C4×Dic3—34no—
269C4×Dic3—32no—
270C4×Dic3—62no—
271C4×Dic3—61no—
272C22×C32—21no—
273C22×C32—22no—
274C22×C32—41no—
275C22×C32C23×D311yescharacteristic
276C22×C32C23×D312yes—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—24no—
279C2×C3⋊D3—22no—
280C2×C3⋊D3—24no—
281C2×C3⋊D3—22no—
282C2×C3⋊D3—24no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C2×C3×D3—62no—
286C2×C3×D3—64no—
287C2×C3×D3—62no—
288C2×C3×D3—64no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—64no—
292C2×C3×D3—62no—
293C2×C3×D3—64no—
294C2×C3×D3—64no—
295C2×C3×D3—62no—
296C2×C3×D3—122no—
297C2×C3×D3—122no—
298C2×C3×D3—124no—
299C2×C3×D3—124no—
300C2×C3×D3—124no—
301C32×C4—24no—
302C32×C4—22no—
303C32×C4—24no—
304C32×C4—22no—
305C32×C4—42no—
306C32×C4—44no—
307C32×C4—42no—
308C32×C4—82no—
309C32×C4C2×D1212yes—
310C32×C4D3×D414yes—
311C3⋊Dic3—22no—
312C3⋊Dic3—21no—
313C3⋊Dic3C2×D1214yes—
314C3⋊Dic3#23912yescomplemented
315C3×Dic3—38no—
316C3×Dic3—34no—
317C3×Dic3—64no—
318C3×Dic3—62no—
319C3×Dic3—62no—
320C3×Dic3—62no—
321C3×Dic3—62no—
322C3×Dic3—122no—
323C3×Dic3—121no—
324C2×#41—272no—
325#127—271no—
326#127—272no—
327#121—92no—
328#121—274no—
329C4×Q8—91no—
330C4×D4—272no—
331C4×D4—271no—
332C33#55011yescharacteristic, complemented, Sylow
333C23×C3—92no—
334C23×C3—94no—
335C22×D3—34no—
336C22×D3—34no—
337C22×D3—32no—
338C22×D3—32no—
339C22×D3—62no—
340C22×D3—64no—
341C3×Q8—94no—
342C3×D4—184no—
343C3×D4—364no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—91no—
351C2×C3×C4—92no—
352C2×C3×C4—92no—
353C2×C3×C4—91no—
354C2×C3×C4—184no—
355C2×C3×C4C2×D3214yes—
356C2×C3×C4C2×D3212yes—
357C2×C3×C4C2×D3212yes—
358C2×C3×C4C2×D3211yescharacteristic
359#67—188no—
360#67—188no—
361#67—188no—
362#67—184no—
363#67—364no—
364C2×Dic3—32no—
365C2×Dic3—34no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—61no—
370C2×Dic3—62no—
371C2×Dic3—61no—
372C2×Dic3—62no—
373C2×Dic3—188no—
374D12—64no—
375D12—64no—
376D12—124no—
377D12—124no—
378D12—124no—
379D12—128no—
380D12—184no—
381D12—182no—
382D12—188no—
383D12—362no—
384C4×D3—64no—
385C4×D3—64no—
386C4×D3—124no—
387C4×D3—188no—
388C4×D3—188no—
389Dic6—38no—
390Dic6—64no—
391C2×C32—22no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32C22×D1211yescharacteristic
401C2×C32C2×D3×D412yes—
402C2×C32#100412yes—
403C2×C32C2×#23911yescharacteristic
404C2×C32D3×#4112yes—
405C2×C32#100911yescharacteristic
406C3⋊D3—42no—
407C3⋊D3—44no—
408C3×D3—124no—
409C3×D3—122no—
410C3×D3—124no—
411C3×D3—122no—
412C3×D3—124no—
413C3×D3—244no—
414#41—548no—
415C2×Q8—91no—
416C2×D4—92no—
417C2×D4—94no—
418C2×D4—272no—
419C2×D4—271no—
420C2×D4—274no—
421C2×D4—272no—
422C22×C4—92no—
423C22×C4—274no—
424C4⋊C4—92no—
425C4⋊C4—271no—
426C4⋊C4#111311yescharacteristic
427C22⋊C4—94no—
428C22⋊C4—272no—
429C42—92no—
430C42—91no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—184no—
435C22×C3—182no—
436C22×C3—184no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3—184no—
440C22×C3C22×D3212yes—
441C22×C3C22×D3211yescharacteristic
442C2×D3—64no—
443C2×D3—62no—
444C2×D3—64no—
445C2×D3—62no—
446C2×D3—64no—
447C2×D3—62no—
448C2×D3—64no—
449C2×D3—62no—
450C2×D3—64no—
451C2×D3—64no—
452C2×D3—62no—
453C2×D3—64no—
454C2×D3—62no—
455C2×D3—122no—
456C2×D3—122no—
457C2×D3—124no—
458C2×D3—124no—
459C2×D3—124no—
460C3×C4—22no—
461C3×C4—24no—
462C3×C4—22no—
463C3×C4—24no—
464C3×C4—44no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4—82no—
468C3×C4—94no—
469C3×C4—92no—
470C3×C4—182no—
471C3×C4—181no—
472C3×C4D3×D1214yes—
473C3×C4#380212yes—
474Dic3—38no—
475Dic3—34no—
476Dic3—64no—
477Dic3—62no—
478Dic3—62no—
479Dic3—62no—
480Dic3—62no—
481Dic3—122no—
482Dic3—121no—
483C32—22no—
484C32—21no—
485C32—41no—
486C32#557711yescharacteristic, complemented
487C32#559912yescomplemented
488C23—94no—
489C23—92no—
490Q8—94no—
491D4—184no—
492D4—364no—
493D4—548no—
494D4—542no—
495D4—544no—
496D4—1082no—
497C2×C4—91no—
498C2×C4—92no—
499C2×C4—91no—
500C2×C4—92no—
501C2×C4—184no—
502C2×C4—548no—
503C2×C4C2×#111312yes—
504C2×C4C2×#111311yescharacteristic
505C2×C3—21no—
506C2×C3—22no—
507C2×C3—21no—
508C2×C3—22no—
509C2×C3—21no—
510C2×C3—22no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—362no—
515C2×C3—364no—
516C2×C3C2×D3×D1212yes—
517C2×C3C2×#380211yescharacteristic
518C2×C3#6431311yescharacteristic
519C2×C3#6431412yes—
520C2×C3D3×#23912yes—
521C2×C3#6431911yescharacteristic
522D3—124no—
523D3—122no—
524D3—124no—
525D3—122no—
526D3—124no—
527D3—244no—
528C22—182no—
529C22—184no—
530C22—182no—
531C22—184no—
532C22—182no—
533C22—184no—
534C22C22×#111311yescharacteristic
535C4—22no—
536C4—94no—
537C4—92no—
538C4—182no—
539C4—181no—
540C4C4⋊#111312yes—
541C3—22no—
542C3—21no—
543C3—41no—
544C3#1059452412yescomplemented
545C3#1059452711yescharacteristic, complemented
546C2—362no—
547C2—364no—
548C2C2×C4⋊#111311yescharacteristic
549C2#1172462311yescharacteristic
550C2#1172462611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4#1113
hypocenterC33#550
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89041C2semidirect
2. C3×#64104C2semidirect
3. C3×#64096C2semidirect
4. #11723697C2semidirect
5. #11723697C2semidirect
6. C4×#6295C2semidirect
7. #11723687C2semidirect
8. #11723685C2semidirect
31. #64104D3semidirect
32. #64096D3semidirect
60. C3×#634D4semidirect
133. C3×C3⋊Dic3#41semidirect
180. #634D12semidirect
314. C3⋊Dic3#239semidirect
332. C33#550semidirect
486. C32#5577semidirect
487. C32#5599semidirect
544. C3#10594524semidirect
545. C3#10594527semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C3×#641042
C3×#640962
#117236972
C4×#62952
#117236872
#117236852
#105945273
#105945243