Finite Group Databasebeta

← #13148985

Subgroups — #13148985

Order 1728.

968 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88634C212yesmaximal, complemented
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×#63457C211yesmaximal, characteristic, complemented
4C9×#888C211yesmaximal, characteristic, complemented
5#11721007C211yesmaximal, characteristic, complemented
6C4×#6192C211yesmaximal, characteristic, complemented
7Dic3×D36C211yesmaximal, characteristic, complemented
8D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#11720984C212yesmaximal, complemented
10#11720981C212yesmaximal, complemented
11#11720941C212yesmaximal, complemented
12#10594373—31nomaximal
13#10589828—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C3×D36C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C9×Dic6C2212yes—
20C2×#6192C2212yes—
21C2×#6192C2211yescharacteristic
22C2×Dic3×D9C2212yes—
23C2×Dic3×D9C2212yes—
24#88634—28no—
25C3×C9×C4⋊C4C2211yescharacteristic
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27#88534C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C9×C4⋊Dic3C2211yescharacteristic
30C9×Dic3⋊C4C2212yes—
31C4×C9×Dic3C2211yescharacteristic
32#88445C2211yescharacteristic
33(C2×D9)⋊Dic3C2211yescharacteristic
34(C2×D9)⋊Dic3C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3⋊Dic9C2211yescharacteristic
37Dic9⋊Dic3C2212yes—
38C2×#3796—32no—
39C4⋊C4×C3⋊D3—31no—
40C3×#895—31no—
41C3×#888D311yescharacteristic
42#63916—31no—
43C4×#626—31no—
44Dic3×D12—31no—
45D3×C4⋊Dic3—31no—
46#63887—32no—
47#63884—32no—
48#63853—32no—
49C2×#3701—32no—
50C9×C4⋊Q8—31no—
51#63508—32no—
52#63457—31no—
53#63457D311yescharacteristic, complemented
54#63456—32no—
55#63456—32no—
56C4⋊C4×D9—31no—
57C4⋊C4×D9—31no—
58C4⋊C4×D9—61no—
59C4×D36—31no—
60C4×D36—31no—
61C22×C3⋊D9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2311yescharacteristic
68C4×C3⋊D9—24no—
69C4×C3⋊D9—22no—
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×D36—24no—
75C3×D36—22no—
76C3×C4×D9—22no—
77C9×Dic6—24no—
78#6192—24no—
79#6192D414yescomplemented
80Dic3×D9—24no—
81Dic3×D9—24no—
82#5726—91no—
83#5611—91no—
84C2×C4×C3⋊D3—32no—
85C2×C4×C3⋊D3—31no—
86C2×C3×D12—32no—
87C2×C3×D12—31no—
88C2×C3×C4×D3—31no—
89C2×C3×Dic6C2×D312yes—
90C2×#626—31no—
91C2×#626—32no—
92C2×D3×Dic3—32no—
93C2×D3×Dic3—32no—
94#3796—68no—
95C32×C4⋊C4C2×D311yescharacteristic
96C4⋊(C3⋊Dic3)—31no—
97#3750—32no—
98C3×(C2×D3)⋊C4—32no—
99C3×C4⋊Dic3C2×D311yescharacteristic
100C3×Dic3⋊C4C2×D312yes—
101C3×C4×Dic3C2×D311yescharacteristic
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—31no—
104#3722—31no—
105(C2×D3)⋊Dic3—32no—
106(C2×D3)⋊Dic3—31no—
107(C2×D3)⋊Dic3—32no—
108C2×Q8×C9—32no—
109#3701—68no—
110C2×D36—31no—
111C2×D36—32no—
112C2×D36C2×D311yescharacteristic
113C2×D36C2×D312yes—
114C2×C4×D9—32no—
115C2×C4×D9—32no—
116C2×C4×D9—31no—
117C2×C4×D9—32no—
118C2×C4×D9—61no—
119C2×C4×D9—62no—
120C2×C4×D9C2×D311yescharacteristic
121C9×C4⋊C4—21no—
122C9×C4⋊C4—32no—
123C9×C4⋊C4—31no—
124C9×C4⋊C4C2×D311yescharacteristic
125C42×C9—31no—
126(C2×D9)⋊C4—32no—
127(C2×D9)⋊C4—31no—
128(C2×D9)⋊C4—32no—
129(C2×D9)⋊C4—31no—
130(C2×D9)⋊C4C2×D312yes—
131C4⋊Dic9—31no—
132C4⋊Dic9—31no—
133C4⋊Dic9—61no—
134Dic9⋊C4—32no—
135Dic9⋊C4—32no—
136Dic9⋊C4—62no—
137C22×C3×C9C2411yescharacteristic
138C2×C3⋊D9—21no—
139C2×C3⋊D9—21no—
140C2×C3⋊D9C2×D412yes—
141C2×C3×D9—21no—
142C2×C3×D9—21no—
143C2×C3×D9—22no—
144C2×C3×D9—22no—
145C2×C3×D9—22no—
146C2×C3×D9C2×D412yes—
147C3×C4×C9—22no—
148C3×C4×C9#4112yescomplemented
149C3⋊Dic9—22no—
150C3⋊Dic9—21no—
151C9×Dic3—22no—
152C9×Dic3—21no—
153C9×Dic3C2×D412yes—
154C3×Dic9—21no—
155C2×#239—92no—
156C2×#237—92no—
157C3×C4⋊Q8—31no—
158C3×#121—91no—
159#946—92no—
160#938—91no—
161#935—92no—
162D4×Dic3—91no—
163C4×#67—91no—
164C2×C4⋊Dic3—91no—
165#895—31no—
166#895—91no—
167#894—92no—
168#894—92no—
169D3×C4⋊C4—91no—
170D3×C4⋊C4—91no—
171D3×C4⋊C4—91no—
172D3×C4⋊C4—181no—
173#888D911yescharacteristic, complemented
174#883—92no—
175#878—92no—
176C4×D12—91no—
177C4×D12—91no—
178C22×C3⋊D3—31no—
179C22×C3×D3—31no—
180C22×C3×D3—32no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—32no—
185C4×C3⋊D3—62no—
186C4×C3⋊D3—64no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3C22×D311yescharacteristic
189C2×C3×Dic3C22×D312yes—
190C2×C3×Dic3C22×D311yescharacteristic
191C3×D12—64no—
192C3×D12—62no—
193C3×C4×D3—62no—
194C3×Dic6—24no—
195#626—34no—
196#626—64no—
197D3×Dic3—64no—
198D3×Dic3—64no—
199C22×D9—31no—
200C22×D9—61no—
201C22×D9C22×D312yes—
202C22×D9C22×D311yescharacteristic
203Q8×C9—64no—
204C2×C4×C9—21no—
205C2×C4×C9—22no—
206C2×C4×C9—31no—
207C2×C4×C9—32no—
208C2×C4×C9—31no—
209C2×C4×C9C22×D312yes—
210C2×C4×C9C22×D311yescharacteristic
211C2×Dic9—31no—
212C2×Dic9—32no—
213C2×Dic9—61no—
214C2×Dic9—62no—
215C2×Dic9C22×D311yescharacteristic
216D36—24no—
217D36—22no—
218D36—34no—
219D36—64no—
220C4×D9—22no—
221C4×D9—64no—
222C4×D9—64no—
223C4×D9—64no—
224C4×D9—64no—
225C4×D9—62no—
226C4×D9—124no—
227#547—271noSylow
228C2×C3×C9C22×D411yescharacteristic
229C2×C3×C9C2×#4111yescharacteristic
230C2×C3×C92−1+411yescharacteristic
231C3⋊D9—22no—
232C3×D9—22no—
233C3×D9—42no—
234C2×C3×Q8—32no—
235C2×C3×D4—91no—
236C2×C3×D4—92no—
237C22×C3×C4—91no—
238C2×#67—91no—
239C2×#67—92no—
240C22×Dic3—92no—
241C22×Dic3—92no—
242#239—188no—
243#237—188no—
244C2×D12—31no—
245C2×D12—32no—
246C2×D12—91no—
247C2×D12—92no—
248C2×C4×D3—31no—
249C2×C4×D3—92no—
250C2×C4×D3—92no—
251C2×C4×D3—92no—
252C2×C4×D3—91no—
253C2×C4×D3—91no—
254C2×C4×D3—92no—
255C2×C4×D3—181no—
256C2×C4×D3—182no—
257C2×Dic6C2×D912yes—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—31no—
260C3×C4⋊C4—32no—
261C3×C4⋊C4C2×D911yescharacteristic
262C3×C4⋊C4D3211yescharacteristic
263C3×C22⋊C4—92no—
264C3×C42—31no—
265#217—92no—
266#217—92no—
267#217—91no—
268(C2×D3)⋊C4—32no—
269(C2×D3)⋊C4—91no—
270(C2×D3)⋊C4—91no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—91no—
274C4⋊Dic3—91no—
275C4⋊Dic3—91no—
276C4⋊Dic3—91no—
277C4⋊Dic3—92no—
278C4⋊Dic3—181no—
279C4⋊Dic3C2×D911yescharacteristic
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—91no—
284Dic3⋊C4—182no—
285Dic3⋊C4C2×D912yes—
286C4×Dic3C2×D911yescharacteristic
287C22×C32C23×D311yescharacteristic
288C2×C3⋊D3—32no—
289C2×C3⋊D3—61no—
290C2×C3⋊D3—61no—
291C2×C3×D3—32no—
292C2×C3×D3—61no—
293C2×C3×D3—61no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C32×C4—22no—
298C32×C4#23912yes—
299C3⋊Dic3—61no—
300C3⋊Dic3—62no—
301C3×Dic3—22no—
302C3×Dic3—21no—
303C3×Dic3—61no—
304C3×Dic3D3×D412yes—
305C22×C9—21no—
306C22×C9C23×D311yescharacteristic
307C2×D9—21no—
308C2×D9—21no—
309C2×D9—22no—
310C2×D9—22no—
311C2×D9—22no—
312C2×D9—32no—
313C2×D9—62no—
314C2×D9—62no—
315C2×D9—62no—
316C2×D9—61no—
317C2×D9—61no—
318C2×D9D3×D412yes—
319C4×C9—22no—
320C4×C9—22no—
321C4×C9—32no—
322C4×C9—42no—
323C4×C9—62no—
324C4×C9—61no—
325C4×C9#23712yescomplemented
326Dic9—21no—
327Dic9—62no—
328Dic9—62no—
329Dic9—61no—
330Dic9—122no—
331C2×#41—272no—
332C4⋊Q8—31no—
333#123—272no—
334#123—272no—
335C22⋊Q8—272no—
336#121—91no—
337#121—271no—
338C4×D4—271no—
339C4×D4—271no—
340C2×C4⋊C4—271no—
341C2×C4⋊C4—271no—
342C3×C9#54711yescharacteristic, complemented, Sylow
343C23×C3—91no—
344C23×C3—92no—
345C22×D3—31no—
346C22×D3—32no—
347C22×D3—91no—
348C22×D3—91no—
349C22×D3—181no—
350C3×Q8—64no—
351C3×D4—184no—
352C3×D4—182no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—31no—
356C2×C3×C4—32no—
357C2×C3×C4—31no—
358C2×C3×C4—91no—
359C2×C3×C4—182no—
360C2×C3×C4C22×D912yes—
361C2×C3×C4C22×D911yescharacteristic
362C2×C3×C4C2×D3212yes—
363C2×C3×C4C2×D3211yescharacteristic
364#67—94no—
365#67—184no—
366C2×Dic3—31no—
367C2×Dic3—91no—
368C2×Dic3—92no—
369C2×Dic3—91no—
370C2×Dic3—92no—
371C2×Dic3—181no—
372C2×Dic3—182no—
373C2×Dic3—184no—
374C2×Dic3—184no—
375C2×Dic3C22×D911yescharacteristic
376C2×Dic3C22×D912yes—
377C2×Dic3C22×D911yescharacteristic
378D12—64no—
379D12—62no—
380D12—94no—
381D12—184no—
382C4×D3—62no—
383C4×D3—184no—
384C4×D3—184no—
385C4×D3—184no—
386C4×D3—184no—
387C4×D3—184no—
388C4×D3—182no—
389C4×D3—182no—
390C4×D3—364no—
391Dic6—24no—
392C2×C32C2×D3×D411yescharacteristic
393C2×C32C2×#23911yescharacteristic
394C2×C32#100711yescharacteristic
395C3⋊D3—62no—
396C3×D3—62no—
397C3×D3—122no—
398C2×C9—21no—
399C2×C9—21no—
400C2×C9—21no—
401C2×C9C2×D3×D411yescharacteristic
402C2×C9C2×#23711yescharacteristic
403C2×C9#101011yescharacteristic
404D9—22no—
405D9—42no—
406D9—64no—
407D9—62no—
408#41—548no—
409C2×Q8—32no—
410C2×D4—91no—
411C2×D4—92no—
412C2×D4—271no—
413C2×D4—272no—
414C22×C4—91no—
415C22×C4—272no—
416C22×C4—272no—
417C22×C4—271no—
418C22×C4—272no—
419C4⋊C4—31no—
420C4⋊C4—32no—
421C4⋊C4—272no—
422C4⋊C4—271no—
423C4⋊C4—271no—
424C4⋊C4—272no—
425C4⋊C4D3×D911yescharacteristic
426C22⋊C4—92no—
427C22⋊C4—271no—
428C22⋊C4—271no—
429C22⋊C4—272no—
430C22⋊C4—272no—
431C42—31no—
432C22×C3—21no—
433C22×C3—92no—
434C22×C3—181no—
435C22×C3—181no—
436C22×C3—182no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3C23×D911yescharacteristic
440C22×C3C22×D3211yescharacteristic
441C2×D3—32no—
442C2×D3—61no—
443C2×D3—61no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—92no—
448C2×D3—92no—
449C2×D3—182no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—181no—
453C2×D3—181no—
454C2×D3—181no—
455C2×D3—181no—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—32no—
460C3×C4—42no—
461C3×C4—62no—
462C3×C4—61no—
463C3×C4—181no—
464C3×C4#370112yescomplemented
465C3×C4#379612yes—
466Dic3—22no—
467Dic3—21no—
468Dic3—61no—
469Dic3—182no—
470Dic3—181no—
471Dic3—182no—
472Dic3—182no—
473Dic3—181no—
474Dic3—362no—
475Dic3D4×D912yes—
476C32#572611yescharacteristic
477C9—21no—
478C9#561111yescharacteristic, complemented
479C23—91no—
480C23—92no—
481C23—271no—
482Q8—64no—
483D4—184no—
484D4—182no—
485D4—274no—
486D4—544no—
487C2×C4—31no—
488C2×C4—31no—
489C2×C4—32no—
490C2×C4—91no—
491C2×C4—182no—
492C2×C4—271no—
493C2×C4—272no—
494C2×C4—544no—
495C2×C4—544no—
496C2×C4—544no—
497C2×C4—542no—
498C2×C4C2×D3×D912yes—
499C2×C4C2×D3×D911yescharacteristic
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—182no—
504C2×C3—362no—
505C2×C3C2×D4×D911yescharacteristic
506C2×C3C2×#370111yescharacteristic
507C2×C3#6371311yescharacteristic
508C2×C3C2×#379611yescharacteristic
509C2×C3D32×D411yescharacteristic
510C2×C3#6431511yescharacteristic
511D3—62no—
512D3—122no—
513D3—184no—
514D3—182no—
515D3—182no—
516C22—92no—
517C22—181no—
518C22—181no—
519C22—182no—
520C22—182no—
521C22—182no—
522C22—272no—
523C22—541no—
524C22—541no—
525C22C22×D3×D911yescharacteristic
526C4—22no—
527C4—32no—
528C4—61no—
529C4—62no—
530C4—181no—
531C4—541no—
532C4—542no—
533C4#8863412yescomplemented
534C3—21no—
535C3#1058982811yescharacteristic, complemented
536C3#1059437311yescharacteristic
537C2—182no—
538C2—362no—
539C2—542no—
540C2C2×#8863411yescharacteristic
541C2D3×D4×D911yescharacteristic
542C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#547
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88634C2semidirect
2. C4⋊C4×C3⋊D9C2semidirect
3. C3×#63457C2semidirect
4. C9×#888C2semidirect
5. #11721007C2semidirect
6. C4×#6192C2semidirect
7. Dic3×D36C2semidirect
8. D9×C4⋊Dic3C2semidirect
9. #11720984C2semidirect
10. #11720981C2semidirect
11. #11720941C2semidirect
53. #63457D3semidirect
79. #6192D4semidirect
148. C3×C4×C9#41semidirect
173. #888D9semidirect
325. C4×C9#237semidirect
342. C3×C9#547semidirect
464. C3×C4#3701semidirect
478. C9#5611semidirect
533. C4#88634semidirect
535. C3#10589828semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
C4⋊C4×C3⋊D92
C3×#634572
C9×#8882
#117210072
C4×#61922
Dic3×D362
D9×C4⋊Dic32
#117209842
#117209812
#117209412
#105943733
#105898283