Finite Group Databasebeta

← #13148976

Subgroups — #13148976

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×#88640C212yesmaximal, complemented
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C9×#893C211yesmaximal, characteristic, complemented
4C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
5#11721045C211yesmaximal, characteristic, complemented
6#11721036C211yesmaximal, characteristic, complemented
7#11721031C212yesmaximal, complemented
8C4×#6192C211yesmaximal, characteristic, complemented
9#11720970C211yesmaximal, characteristic, complemented
10#11720957C212yesmaximal, complemented
11#11720955C212yesmaximal, complemented
12#10594515—31nomaximal
13#10589653—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C9×D12C2211yescharacteristic
17C2×C4×D3×C9C2212yes—
18C2×C3×C4×D9C2211yescharacteristic
19C2×C3×C4×D9C2212yes—
20C2×#6195C2212yes—
21C2×#6194C2212yes—
22C2×#6192C2211yescharacteristic
23C2×#6189C2212yes—
24#88640—28no—
25C3×C9×C4⋊C4C2211yescharacteristic
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27#88534C2212yes—
28C9×(C2×D3)⋊C4C2212yes—
29C4×C9×Dic3C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×Dic9⋊C4C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34#88445C2211yescharacteristic
35(C2×D9)⋊Dic3C2211yescharacteristic
36Dic3⋊Dic9C2212yes—
37Dic3⋊Dic9C2211yescharacteristic
38C2×#3798—32no—
39C4⋊C4×C3⋊D3—31no—
40C3×#893D311yescharacteristic
41C3×D3×C4⋊C4—31no—
42#63918—31no—
43#63916—31no—
44#63908—31no—
45#63905—32no—
46C4×#626—31no—
47#63853—32no—
48#63851—32no—
49C2×#3697—32no—
50C4×D4×C9—31no—
51#63500—31no—
52#63500—31no—
53#63486—32no—
54C4⋊C4×D9—31no—
55C4⋊C4×D9—61no—
56C4⋊C4×D9D311yescharacteristic, complemented
57#63445—32no—
58#63445—32no—
59C4×D36—31no—
60#63431—31no—
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2312yes—
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×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C9×D12—22no—
75C4×D3×C9—24no—
76C3×C4×D9—24no—
77C3×C4×D9—22no—
78#6195—24no—
79#6194—24no—
80#6192D414yescomplemented
81#6189—24no—
82#5551—91no—
83#5507—91no—
84C2×C4×C3⋊D3—32no—
85C2×C4×C3⋊D3—31no—
86C2×C3×D12C2×D311yescharacteristic
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3C2×D312yes—
90C2×#627—32no—
91C2×#626—31no—
92C2×#626—32no—
93C2×#625—32no—
94#3798—68no—
95C32×C4⋊C4C2×D311yescharacteristic
96C4⋊(C3⋊Dic3)—31no—
97#3750—32no—
98C3×(C2×D3)⋊C4C2×D312yes—
99C3×C4⋊Dic3—31no—
100C3×Dic3⋊C4—32no—
101C3×C4×Dic3C2×D311yescharacteristic
102Dic3⋊Dic3—31no—
103Dic3⋊Dic3—32no—
104#3722—31no—
105(C2×D3)⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—31no—
108C2×D4×C9—31no—
109C22×C4×C9—32no—
110C2×#611—32no—
111C2×#611—32no—
112#3697—68no—
113C2×D36—31no—
114C2×C4×D9—31no—
115C2×C4×D9—32no—
116C2×C4×D9—61no—
117C2×C4×D9—62no—
118C2×C4×D9C2×D311yescharacteristic
119C2×C4×D9C2×D312yes—
120C2×Dic18—32no—
121C9×C4⋊C4—21no—
122C9×C4⋊C4C2×D311yescharacteristic
123C9×C22⋊C4—32no—
124C42×C9—31no—
125#3676—32no—
126#3676—32no—
127(C2×D9)⋊C4—31no—
128(C2×D9)⋊C4—31no—
129C4⋊Dic9—31no—
130C4⋊Dic9—32no—
131C4⋊Dic9—31no—
132C4⋊Dic9—61no—
133C4⋊Dic9C2×D311yescharacteristic
134Dic9⋊C4—32no—
135Dic9⋊C4—62no—
136Dic9⋊C4C2×D312yes—
137C22×C3×C9C2411yescharacteristic
138C2×C3⋊D9—21no—
139C2×C3⋊D9—21no—
140C2×C3⋊D9C2×D412yes—
141C2×D3×C9—22no—
142C2×D3×C9—22no—
143C2×D3×C9—22no—
144C2×C3×D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9C2×D412yes—
147C3×C4×C9—22no—
148C3×C4×C9#4112yescomplemented
149C3⋊Dic9—22no—
150C3⋊Dic9—21no—
151C9×Dic3—21no—
152C9×Dic3C2×D412yes—
153C3×Dic9—22no—
154C3×Dic9—21no—
155C2×#235—92no—
156C2×#235—92no—
157C3×C4×D4—31no—
158C2×C3×C4⋊C4—91no—
159#938—91no—
160#938—91no—
161#930—91no—
162#929—92no—
163C4×#67—91no—
164#924—92no—
165#896—92no—
166#895—91no—
167#894—92no—
168#893D911yescharacteristic, complemented
169D3×C4⋊C4—31no—
170D3×C4⋊C4—91no—
171D3×C4⋊C4—91no—
172D3×C4⋊C4—181no—
173#888—91no—
174#883—92no—
175#883—92no—
176C4×D12—91no—
177#869—91no—
178C22×C3⋊D3—31no—
179C22×C3×D3—31no—
180C22×C3×D3C22×D312yes—
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×Dic3—32no—
189C2×C3×Dic3C22×D311yescharacteristic
190C2×C3×Dic3C22×D311yescharacteristic
191C3×D12—22no—
192C3×C4×D3—24no—
193C3×C4×D3—64no—
194C3×C4×D3—62no—
195#627—64no—
196#626—34no—
197#626—64no—
198#625—64no—
199C23×C9—32no—
200C22×D9—31no—
201C22×D9—61no—
202C22×D9C22×D311yescharacteristic
203D4×C9—62no—
204C2×C4×C9—21no—
205C2×C4×C9—22no—
206C2×C4×C9—31no—
207C2×C4×C9—31no—
208C2×C4×C9—64no—
209C2×C4×C9C22×D312yes—
210C2×C4×C9C22×D311yescharacteristic
211#611—64no—
212#611—64no—
213C2×Dic9—31no—
214C2×Dic9—32no—
215C2×Dic9—61no—
216C2×Dic9—62no—
217C2×Dic9C22×D311yescharacteristic
218C2×Dic9C22×D312yes—
219D36—34no—
220C4×D9—24no—
221C4×D9—22no—
222C4×D9—64no—
223C4×D9—64no—
224C4×D9—62no—
225C4×D9—124no—
226Dic18—64no—
227#547—271noSylow
228C2×C3×C9C22×D411yescharacteristic
229C2×C3×C9C2×#4111yescharacteristic
230C2×C3×C92−1+411yescharacteristic
231C3⋊D9—22no—
232D3×C9—42no—
233C3×D9—22no—
234C2×C3×D4—31no—
235C22×C3×C4—32no—
236C22×C3×C4—91no—
237C22×C3×C4—92no—
238C2×#67—92no—
239C2×#67—92no—
240C2×#67—92no—
241C2×#67—91no—
242#235—188no—
243#235—188no—
244C2×D12—91no—
245C2×D12—92no—
246C2×D12C2×D911yescharacteristic
247C2×C4×D3—31no—
248C2×C4×D3—32no—
249C2×C4×D3—92no—
250C2×C4×D3—91no—
251C2×C4×D3—91no—
252C2×C4×D3—92no—
253C2×C4×D3—181no—
254C2×C4×D3—182no—
255C2×C4×D3C2×D912yes—
256C2×Dic6—92no—
257C2×Dic6—92no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—91no—
260C3×C4⋊C4—92no—
261C3×C4⋊C4C2×D911yescharacteristic
262C3×C4⋊C4D3211yescharacteristic
263C3×C22⋊C4—32no—
264C3×C42—31no—
265#217—92no—
266#217—92no—
267#217—91no—
268(C2×D3)⋊C4—91no—
269(C2×D3)⋊C4—91no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—91no—
273(C2×D3)⋊C4C2×D912yes—
274C4⋊Dic3—31no—
275C4⋊Dic3—92no—
276C4⋊Dic3—91no—
277C4⋊Dic3—91no—
278C4⋊Dic3—91no—
279C4⋊Dic3—181no—
280Dic3⋊C4—32no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—91no—
285Dic3⋊C4—182no—
286C4×Dic3C2×D911yescharacteristic
287C22×C32C23×D311yescharacteristic
288C2×C3⋊D3—32no—
289C2×C3⋊D3—61no—
290C2×C3⋊D3—61no—
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C2×C3×D3—22no—
294C2×C3×D3—32no—
295C2×C3×D3—61no—
296C2×C3×D3—61no—
297C32×C4—22no—
298C32×C4#23512yes—
299C3⋊Dic3—61no—
300C3⋊Dic3—62no—
301C3×Dic3—21no—
302C3×Dic3—61no—
303C3×Dic3—62no—
304C3×Dic3D3×D412yes—
305C22×C9—21no—
306C22×C9—62no—
307C22×C9—62no—
308C22×C9—62no—
309C22×C9C23×D311yescharacteristic
310C2×D9—21no—
311C2×D9—21no—
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—61no—
324C4×C9#23512yescomplemented
325Dic9—21no—
326Dic9—22no—
327Dic9—62no—
328Dic9—61no—
329Dic9—62no—
330Dic9—122no—
331C2×#41—272no—
332C4⋊Q8—271no—
333#123—272no—
334#123—272no—
335C22⋊Q8—272no—
336#121—271no—
337#121—271no—
338C4×D4—31no—
339C4×D4—271no—
340C2×C4⋊C4—91no—
341C2×C4⋊C4—271no—
342C3×C9#54711yescharacteristic, complemented, Sylow
343C23×C3—32no—
344C23×C3—91no—
345C22×D3—31no—
346C22×D3—91no—
347C22×D3—91no—
348C22×D3—181no—
349C22×D3C22×D912yes—
350C3×D4—62no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—64no—
356C2×C3×C4—91no—
357C2×C3×C4—92no—
358C2×C3×C4—184no—
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—
366#67—184no—
367#67—184no—
368C2×Dic3—31no—
369C2×Dic3—32no—
370C2×Dic3—91no—
371C2×Dic3—92no—
372C2×Dic3—91no—
373C2×Dic3—92no—
374C2×Dic3—181no—
375C2×Dic3—182no—
376C2×Dic3C22×D911yescharacteristic
377C2×Dic3C22×D911yescharacteristic
378D12—22no—
379D12—94no—
380D12—184no—
381C4×D3—24no—
382C4×D3—64no—
383C4×D3—62no—
384C4×D3—184no—
385C4×D3—184no—
386C4×D3—184no—
387C4×D3—182no—
388C4×D3—182no—
389C4×D3—364no—
390Dic6—184no—
391Dic6—184no—
392C2×C32C2×#23511yescharacteristic
393C2×C32C2×D3×D411yescharacteristic
394C2×C32#101011yescharacteristic
395C3⋊D3—62no—
396C3×D3—42no—
397C3×D3—62no—
398C2×C9—21no—
399C2×C9—21no—
400C2×C9—21no—
401C2×C9—122no—
402C2×C9C2×#23511yescharacteristic
403C2×C9C2×D3×D411yescharacteristic
404C2×C9#100711yescharacteristic
405D9—22no—
406D9—64no—
407D9—62no—
408#41—548no—
409C2×Q8—272no—
410C2×D4—31no—
411C2×D4—271no—
412C2×D4—272no—
413C2×D4—272no—
414C22×C4—32no—
415C22×C4—91no—
416C22×C4—92no—
417C22×C4—271no—
418C22×C4—272no—
419C4⋊C4—91no—
420C4⋊C4—92no—
421C4⋊C4—271no—
422C4⋊C4—272no—
423C4⋊C4—271no—
424C4⋊C4—272no—
425C4⋊C4D3×D911yescharacteristic
426C22⋊C4—32no—
427C22⋊C4—271no—
428C22⋊C4—271no—
429C22⋊C4—272no—
430C22⋊C4—272no—
431C42—31no—
432C22×C3—21no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—62no—
436C22×C3—92no—
437C22×C3—181no—
438C22×C3—181no—
439C22×C3C23×D911yescharacteristic
440C22×C3C22×D3211yescharacteristic
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—32no—
445C2×D3—61no—
446C2×D3—61no—
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—61no—
462C3×C4—181no—
463C3×C4—182no—
464C3×C4#369712yescomplemented
465C3×C4#379812yes—
466Dic3—21no—
467Dic3—61no—
468Dic3—62no—
469Dic3—182no—
470Dic3—181no—
471Dic3—182no—
472Dic3—181no—
473Dic3—182no—
474Dic3—362no—
475Dic3D4×D912yes—
476C32#555111yescharacteristic
477C9—21no—
478C9#550711yescharacteristic, complemented
479C23—32no—
480C23—91no—
481C23—271no—
482Q8—544no—
483D4—62no—
484D4—274no—
485D4—544no—
486D4—544no—
487C2×C4—31no—
488C2×C4—31no—
489C2×C4—64no—
490C2×C4—91no—
491C2×C4—92no—
492C2×C4—184no—
493C2×C4—182no—
494C2×C4—271no—
495C2×C4—272no—
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—122no—
504C2×C3—182no—
505C2×C3C2×#369711yescharacteristic
506C2×C3C2×D4×D911yescharacteristic
507C2×C3#6371611yescharacteristic
508C2×C3C2×#379811yescharacteristic
509C2×C3D32×D411yescharacteristic
510C2×C3#6431511yescharacteristic
511D3—42no—
512D3—62no—
513D3—184no—
514D3—182no—
515D3—182no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—92no—
520C22—181no—
521C22—181no—
522C22—272no—
523C22—541no—
524C22—541no—
525C22C22×D3×D911yescharacteristic
526C4—22no—
527C4—32no—
528C4—61no—
529C4—181no—
530C4—182no—
531C4—541no—
532C4—542no—
533C4#8864012yescomplemented
534C3—21no—
535C3#1058965311yescharacteristic, complemented
536C3#1059451511yescharacteristic
537C2—122no—
538C2—182no—
539C2—542no—
540C2C2×#8864011yescharacteristic
541C2D3×D4×D911yescharacteristic
542C2#1172250811yescharacteristic
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×#88640C2semidirect
2. C4⋊C4×C3⋊D9C2semidirect
3. C9×#893C2semidirect
4. C3×C4⋊C4×D9C2semidirect
5. #11721045C2semidirect
6. #11721036C2semidirect
7. #11721031C2semidirect
8. C4×#6192C2semidirect
9. #11720970C2semidirect
10. #11720957C2semidirect
11. #11720955C2semidirect
56. C4⋊C4×D9D3semidirect
80. #6192D4semidirect
148. C3×C4×C9#41semidirect
168. #893D9semidirect
324. C4×C9#235semidirect
342. C3×C9#547semidirect
464. C3×C4#3697semidirect
478. C9#5507semidirect
533. C4#88640semidirect
535. C3#10589653semidirect

Maximal subgroups

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

subgroupindex
C2×#886402
C4⋊C4×C3⋊D92
C9×#8932
C3×C4⋊C4×D92
#117210452
#117210362
#117210312
C4×#61922
#117209702
#117209572
#117209552
#105945153
#105896533