Finite Group Databasebeta

← #13178686

Subgroups — #13178686

Order 1728.

1612 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89046C212yesmaximal, complemented
2C3×#64149C212yesmaximal, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11723725C214yesmaximal, complemented
5#11723717C212yesmaximal, complemented
6C4×#6296C211yesmaximal, characteristic, complemented
7#11723697C211yesmaximal, characteristic, complemented
8#11723691C212yesmaximal, complemented
9#10595144—31nomaximal
10#10595143—32nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638C2214yes—
13C2×C3×#638C2214yes—
14C22×C3×C3⋊Dic3C2212yes—
15C2×C3×C4⋊(C3⋊D3)C2212yes—
16C2×#6296C2211yescharacteristic
17C2×#6295C2212yes—
18C2×#6295C2212yes—
19#89046—28no—
20C2×#6294C2214yes—
21C3×#3757C2212yes—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×C4×C3⋊Dic3C2211yescharacteristic
24C3×C4×C3⋊Dic3C2212yes—
25#88815C2211yescharacteristic
26#88815C2212yes—
27#88814C2214yes—
28C2×#3805—32no—
29C2×#3804—34no—
30#64149D312yescomplemented
31D4×C3⋊Dic3D311yescharacteristic, complemented
32C3×#940—32no—
33C3×#940—32no—
34C3×#940—62no—
35C3×D4×Dic3—32no—
36C3×D4×Dic3—61no—
37#63980—32no—
38#63978—34no—
39#63971—34no—
40#63970—34no—
41#63958—34no—
42C4×#627—32no—
43C4×#627—31no—
44#63914—31no—
45#63911—32no—
46#63898—32no—
47#63894—32no—
48Dic3×D12—32no—
49C23×C33C2312yes—
50C22×C3×C3⋊D3C2314yes—
51C33×D4—22no—
52C2×C33×C4C2311yescharacteristic
53C3×#638—28no—
54C3×#638—44no—
55C2×C3×C3⋊Dic3—24no—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2312yes—
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C3×C4⋊(C3⋊D3)—24no—
62#6296D414yescomplemented
63#6295—24no—
64#6295—42no—
65#6294—28no—
66#5718—92no—
67#5553—91no—
68C2×C32×D4—21no—
69C2×C32×D4—22no—
70C2×C32×D4—41no—
71C2×C32×D4C2×D312yes—
72C2×C32×D4C2×D311yescharacteristic
73C2×#638C2×D314yes—
74C2×#638C2×D314yes—
75C22×C3⋊Dic3C2×D312yes—
76C2×C4⋊(C3⋊D3)C2×D312yes—
77C2×C3×#67—34no—
78C2×C3×#67—34no—
79C2×C3×#67—34no—
80C2×C3×#67—34no—
81C2×C3×#67—64no—
82C2×C3×#67—64no—
83C22×C3×Dic3—34no—
84C22×C3×Dic3—62no—
85C2×C3×D12—32no—
86C2×C3×D12—32no—
87C2×C3×D12—62no—
88C2×#627—32no—
89C2×#627—31no—
90C2×#626—34no—
91C2×#626—34no—
92C2×#625—32no—
93C2×#625—32no—
94C2×#624—34no—
95#3805—68no—
96#3804—616no—
97C2×D3×Dic3—34no—
98C2×D3×Dic3—34no—
99#3757C2×D312yes—
100C4⋊(C3⋊Dic3)C2×D311yescharacteristic
101C4×C3⋊Dic3C2×D311yescharacteristic
102C4×C3⋊Dic3C2×D312yes—
103C3×#217—34no—
104C3×#217—62no—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—61no—
107C3×C4×Dic3—32no—
108C3×C4×Dic3—32no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—61no—
111C3×C4×Dic3—62no—
112#3724—32no—
113#3724—31no—
114Dic3⋊Dic3—32no—
115Dic3⋊Dic3—32no—
116Dic3⋊Dic3—32no—
117#3722—34no—
118(C2×D3)⋊Dic3—34no—
119(C2×D3)⋊Dic3—34no—
120C22×C33—22no—
121C22×C33—22no—
122C22×C33—22no—
123C22×C33C2411yescharacteristic
124C2×C3×C3⋊D3—24no—
125C2×C3×C3⋊D3—24no—
126C2×C3×C3⋊D3—24no—
127C33×C4#4112yescomplemented
128C3×C3⋊Dic3—22no—
129C3×C3⋊Dic3—22no—
130C3×C3⋊Dic3—21no—
131C3×C3⋊Dic3C2×D414yes—
132C3×C3⋊Dic3C2×D412yes—
133C2×#237—94no—
134C2×#235—92no—
135C3×#127—92no—
136C3×C4×D4—91no—
137#940—32no—
138#940—32no—
139#940—62no—
140#940—92no—
141#939—94no—
142#939—94no—
143#938—94no—
144D4×Dic3—32no—
145D4×Dic3—61no—
146D4×Dic3—92no—
147#930—92no—
148#893—92no—
149#884—94no—
150#874—91no—
151C4×D12—92no—
152C4×Dic6—92no—
153C4×Dic6—91no—
154C23×C32—22no—
155C23×C32—24no—
156C23×C32—42no—
157C23×C32C22×D314yes—
158C23×C32C22×D312yes—
159C22×C3⋊D3C22×D314yes—
160C22×C3×D3—34no—
161C22×C3×D3—34no—
162C22×C3×D3—64no—
163C32×D4—22no—
164C32×D4—24no—
165C32×D4—24no—
166C32×D4—44no—
167C32×D4—44no—
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4—41no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D312yes—
173#638—28no—
174#638—44no—
175C2×C3⋊Dic3—24no—
176C2×C3⋊Dic3C22×D312yes—
177C2×C3⋊Dic3C22×D312yes—
178C2×C3⋊Dic3C22×D311yescharacteristic
179C2×C3⋊Dic3C22×D312yes—
180C2×C3⋊Dic3C22×D311yescharacteristic
181C4⋊(C3⋊D3)—24no—
182C3×#67—68no—
183C3×#67—68no—
184C3×#67—128no—
185C3×#67—124no—
186C3×#67—124no—
187C3×#67—128no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—68no—
201C2×C3×Dic3—124no—
202C3×D12—64no—
203C3×D12—64no—
204C3×D12—124no—
205#627—38no—
206#627—34no—
207#626—68no—
208#626—124no—
209#625—64no—
210#625—122no—
211#624—68no—
212D3×Dic3—68no—
213D3×Dic3—68no—
214#550—271noSylow
215C2×C33—42no—
216C2×C33C22×D411yescharacteristic
217C2×C33C2×#4111yescharacteristic
218C2×C33H(5,𝔽2)11yescharacteristic
219C3×C3⋊D3—44no—
220C2×C3×D4—21no—
221C2×C3×D4—22no—
222C2×C3×D4—41no—
223C2×C3×D4—94no—
224C2×C3×D4—94no—
225C2×C3×D4—92no—
226C2×C3×D4D3212yes—
227C2×C3×D4D3211yescharacteristic
228C22×C3×C4—92no—
229C2×#67—34no—
230C2×#67—34no—
231C2×#67—34no—
232C2×#67—34no—
233C2×#67—64no—
234C2×#67—64no—
235C2×#67—94no—
236C2×#67—94no—
237C22×Dic3—34no—
238C22×Dic3—62no—
239C22×Dic3—94no—
240#237—1816no—
241#235—188no—
242C2×D12—32no—
243C2×D12—32no—
244C2×D12—62no—
245C2×D12—92no—
246C2×D12—92no—
247C2×C4×D3—94no—
248C2×C4×D3—94no—
249C2×Dic6—91no—
250C2×Dic6—92no—
251C3×C4⋊C4—91no—
252C3×C22⋊C4—92no—
253C3×C42—92no—
254C3×C42—91no—
255#217—34no—
256#217—62no—
257#217—94no—
258(C2×D3)⋊C4—94no—
259(C2×D3)⋊C4—94no—
260C4⋊Dic3—32no—
261C4⋊Dic3—61no—
262C4⋊Dic3—91no—
263C4⋊Dic3—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267C4×Dic3—32no—
268C4×Dic3—32no—
269C4×Dic3—32no—
270C4×Dic3—61no—
271C4×Dic3—62no—
272C22×C32—22no—
273C22×C32—24no—
274C22×C32—24no—
275C22×C32—22no—
276C22×C32—24no—
277C22×C32—24no—
278C22×C32—22no—
279C22×C32—21no—
280C22×C32—22no—
281C22×C32—41no—
282C22×C32—44no—
283C22×C32—44no—
284C22×C32—42no—
285C22×C32—42no—
286C22×C32—44no—
287C22×C32—44no—
288C22×C32—82no—
289C22×C32—82no—
290C22×C32C23×D311yescharacteristic
291C22×C32C23×D312yes—
292C2×C3⋊D3—24no—
293C2×C3⋊D3—24no—
294C2×C3⋊D3—24no—
295C2×C3×D3—64no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—64no—
300C2×C3×D3—64no—
301C2×C3×D3—124no—
302C2×C3×D3—124no—
303C2×C3×D3—124no—
304C32×C4—24no—
305C32×C4—22no—
306C32×C4—42no—
307C32×C4#23512yescomplemented
308C32×C4#23714yescomplemented
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3—21no—
312C3⋊Dic3D3×D414yes—
313C3⋊Dic3D3×D412yes—
314C3×Dic3—34no—
315C3×Dic3—34no—
316C3×Dic3—34no—
317C3×Dic3—62no—
318C3×Dic3—64no—
319C3×Dic3—64no—
320C3×Dic3—64no—
321C3×Dic3—62no—
322C3×Dic3—62no—
323C3×Dic3—62no—
324C3×Dic3—121no—
325C3×Dic3—122no—
326C2×#41—272no—
327#127—92no—
328#127—271no—
329#121—274no—
330#121—272no—
331C4×Q8—271no—
332C4×D4—91no—
333C4×D4—272no—
334C33#55011yescharacteristic, complemented, Sylow
335C23×C3—22no—
336C23×C3—24no—
337C23×C3—42no—
338C23×C3—94no—
339C23×C3C2×D3214yes—
340C23×C3C2×D3212yes—
341C22×D3—34no—
342C22×D3—34no—
343C22×D3—64no—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—22no—
347C3×D4—44no—
348C3×D4—44no—
349C3×D4—188no—
350C3×D4—184no—
351C3×D4—364no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—41no—
355C2×C3×C4—92no—
356C2×C3×C4—91no—
357C2×C3×C4—92no—
358C2×C3×C4—91no—
359C2×C3×C4—92no—
360C2×C3×C4—184no—
361C2×C3×C4C2×D3212yes—
362C2×C3×C4C2×D3211yescharacteristic
363#67—68no—
364#67—68no—
365#67—128no—
366#67—124no—
367#67—124no—
368#67—128no—
369#67—188no—
370#67—364no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—32no—
377C2×Dic3—34no—
378C2×Dic3—61no—
379C2×Dic3—62no—
380C2×Dic3—62no—
381C2×Dic3—61no—
382C2×Dic3—62no—
383C2×Dic3—68no—
384C2×Dic3—124no—
385C2×Dic3—188no—
386D12—64no—
387D12—64no—
388D12—124no—
389D12—184no—
390D12—362no—
391C4×D3—188no—
392C4×D3—188no—
393Dic6—94no—
394Dic6—98no—
395C2×C32—21no—
396C2×C32—22no—
397C2×C32—21no—
398C2×C32—22no—
399C2×C32—22no—
400C2×C32—21no—
401C2×C32—41no—
402C2×C32—41no—
403C2×C32—41no—
404C2×C32—42no—
405C2×C32—44no—
406C2×C32—44no—
407C2×C32—84no—
408C2×C32—84no—
409C2×C32C2×#23511yescharacteristic
410C2×C32C2×D3×D412yes—
411C2×C32C2×D3×D411yescharacteristic
412C2×C32C2×#23712yes—
413C2×C32#100412yes—
414C2×C32#100911yescharacteristic
415C3⋊D3—44no—
416C3×D3—124no—
417C3×D3—124no—
418C3×D3—244no—
419#41—548no—
420C2×Q8—271no—
421C2×D4—94no—
422C2×D4—94no—
423C2×D4—92no—
424C2×D4—272no—
425C2×D4—272no—
426C2×D4#111311yescharacteristic
427C22×C4—92no—
428C22×C4—274no—
429C4⋊C4—91no—
430C4⋊C4—271no—
431C4⋊C4—272no—
432C22⋊C4—92no—
433C22⋊C4—274no—
434C42—91no—
435C42—92no—
436C22×C3—24no—
437C22×C3—24no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—24no—
441C22×C3—24no—
442C22×C3—22no—
443C22×C3—21no—
444C22×C3—22no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—41no—
452C22×C3—82no—
453C22×C3—82no—
454C22×C3—184no—
455C22×C3—184no—
456C22×C3—184no—
457C22×C3C22×D3211yescharacteristic
458C22×C3C22×D3212yes—
459C2×D3—64no—
460C2×D3—64no—
461C2×D3—64no—
462C2×D3—64no—
463C2×D3—64no—
464C2×D3—64no—
465C2×D3—124no—
466C2×D3—124no—
467C2×D3—124no—
468C3×C4—24no—
469C3×C4—22no—
470C3×C4—42no—
471C3×C4—94no—
472C3×C4—92no—
473C3×C4—182no—
474C3×C4—181no—
475C3×C4—182no—
476C3×C4#380414yescomplemented
477C3×C4#380512yescomplemented
478Dic3—34no—
479Dic3—34no—
480Dic3—34no—
481Dic3—62no—
482Dic3—64no—
483Dic3—64no—
484Dic3—64no—
485Dic3—62no—
486Dic3—62no—
487Dic3—62no—
488Dic3—122no—
489Dic3—121no—
490C32—22no—
491C32—21no—
492C32—41no—
493C32#555311yescharacteristic, complemented
494C32#571812yescomplemented
495C23—94no—
496C23C2×#111312yes—
497Q8—274no—
498D4—22no—
499D4—188no—
500D4—184no—
501D4—364no—
502D4—544no—
503D4—1082no—
504C2×C4—91no—
505C2×C4—92no—
506C2×C4—91no—
507C2×C4—92no—
508C2×C4—92no—
509C2×C4—184no—
510C2×C4—548no—
511C2×C4C2×#111311yescharacteristic
512C2×C3—21no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—21no—
516C2×C3—22no—
517C2×C3—21no—
518C2×C3—44no—
519C2×C3—44no—
520C2×C3—42no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—84no—
525C2×C3—84no—
526C2×C3—364no—
527C2×C3D32×D412yes—
528C2×C3D32×D411yescharacteristic
529C2×C3#6431311yescharacteristic
530C2×C3#6431412yes—
531C2×C3C2×#380412yes—
532C2×C3C2×#380511yescharacteristic
533D3—124no—
534D3—124no—
535D3—244no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—184no—
540C22—184no—
541C22—184no—
542C22C22×#111311yescharacteristic
543C4—92no—
544C4—94no—
545C4—182no—
546C4—182no—
547C4—181no—
548C4#8904612yescomplemented
549C3—22no—
550C3—21no—
551C3—41no—
552C3#1059514312yescomplemented
553C3#1059514411yescharacteristic, complemented
554C2—42no—
555C2—364no—
556C2D4×#111311yescharacteristic
557C2#1172462311yescharacteristic
558C2C2×#8904611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#550
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89046C2semidirect
2. C3×#64149C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. #11723725C2semidirect
5. #11723717C2semidirect
6. C4×#6296C2semidirect
7. #11723697C2semidirect
8. #11723691C2semidirect
30. #64149D3semidirect
31. D4×C3⋊Dic3D3semidirect
62. #6296D4semidirect
127. C33×C4#41semidirect
307. C32×C4#235semidirect
308. C32×C4#237semidirect
334. C33#550semidirect
476. C3×C4#3804semidirect
477. C3×C4#3805semidirect
493. C32#5553semidirect
494. C32#5718semidirect
548. C4#89046semidirect
552. C3#10595143semidirect
553. C3#10595144semidirect

Maximal subgroups

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

subgroupindex
C2×#890462
C3×#641492
C3×D4×C3⋊Dic32
#117237252
#117237172
C4×#62962
#117236972
#117236912
#105951443
#105951433