Finite Group Databasebeta

← #13148980

Subgroups — #13148980

Order 1728.

970 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×#88635C212yesmaximal, complemented
2#11721751C211yesmaximal, characteristic, complemented
3C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
4C9×#888C211yesmaximal, characteristic, complemented
5#11721004C211yesmaximal, characteristic, complemented
6C4×#6192C211yesmaximal, characteristic, complemented
7#11721000C211yesmaximal, characteristic, complemented
8#11720997C212yesmaximal, complemented
9#11720993C211yesmaximal, characteristic, complemented
10#11720986C212yesmaximal, complemented
11#11720938C212yesmaximal, complemented
12#10594517—31nomaximal
13#10589828—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4⋊(C3⋊D9)C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×C9×Dic6C2212yes—
20C2×#6192C2212yes—
21C2×#6192C2211yescharacteristic
22C2×(C3⋊D9)⋊C4C2212yes—
23C2×(C3⋊D9)⋊C4C2212yes—
24#88635—28no—
25C3×C9×C4⋊C4C2211yescharacteristic
26#88536C2212yes—
27C9×C4⋊Dic3C2211yescharacteristic
28C9×Dic3⋊C4C2212yes—
29C4×C9×Dic3C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×Dic9⋊C4C2212yes—
32#88445C2212yes—
33#88445C2212yes—
34#88445C2211yescharacteristic
35(C2×D9)⋊Dic3C2211yescharacteristic
36Dic3⋊Dic9C2211yescharacteristic
37#88442C2212yes—
38C2×#3799—32no—
39#64104—31no—
40C3×D3×C4⋊C4—31no—
41C3×#888D311yescharacteristic
42#63909—31no—
43C4×#626—31no—
44#63900—31no—
45#63897—32no—
46#63894—31no—
47#63888—32no—
48#63850—32no—
49C2×#3701—32no—
50C9×C4⋊Q8—31no—
51#63508—32no—
52#63457—31no—
53#63457—31no—
54#63457—61no—
55#63456—32no—
56#63456—32no—
57C4⋊C4×D9—31no—
58C4⋊C4×D9D311yescharacteristic, complemented
59C4×D36—31no—
60C4×D36—31no—
61C22×C3⋊D9C2312yes—
62C22×C3⋊D9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C4⋊(C3⋊D9)—24no—
68C4⋊(C3⋊D9)—22no—
69C4×C3⋊D9—22no—
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C3×C4×D9—24no—
76C3×C4×D9—22no—
77C9×Dic6—24no—
78#6192—24no—
79#6192D414yescomplemented
80(C3⋊D9)⋊C4—24no—
81(C3⋊D9)⋊C4—24no—
82#5726—91no—
83#5507—91no—
84C2×C4⋊(C3⋊D3)—32no—
85C2×C4⋊(C3⋊D3)—31no—
86C2×C4×C3⋊D3—31no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—31no—
89C2×C3×Dic6C2×D312yes—
90C2×#626—31no—
91C2×#626—32no—
92C2×#624—32no—
93C2×#624—32no—
94#3799—68no—
95C32×C4⋊C4C2×D311yescharacteristic
96#3752—32no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4C2×D312yes—
101C3×C4×Dic3C2×D311yescharacteristic
102#3724—32no—
103Dic3⋊Dic3—31no—
104#3722—32no—
105#3722—32no—
106#3722—31no—
107(C2×D3)⋊Dic3—31no—
108C2×Q8×C9—32no—
109#3701—68no—
110C2×D36—31no—
111C2×D36—32no—
112C2×D36—31no—
113C2×D36—32no—
114C2×D36—61no—
115C2×D36—62no—
116C2×C4×D9—31no—
117C2×C4×D9—32no—
118C2×C4×D9—32no—
119C2×C4×D9—61no—
120C2×C4×D9C2×D311yescharacteristic
121C2×C4×D9C2×D312yes—
122C9×C4⋊C4—21no—
123C9×C4⋊C4—32no—
124C9×C4⋊C4—31no—
125C9×C4⋊C4C2×D311yescharacteristic
126C42×C9—31no—
127(C2×D9)⋊C4—31no—
128(C2×D9)⋊C4—32no—
129(C2×D9)⋊C4—32no—
130(C2×D9)⋊C4—32no—
131(C2×D9)⋊C4—31no—
132(C2×D9)⋊C4—62no—
133C4⋊Dic9—31no—
134C4⋊Dic9C2×D311yescharacteristic
135Dic9⋊C4—32no—
136Dic9⋊C4C2×D312yes—
137C22×C3×C9C2411yescharacteristic
138C2×C3⋊D9—21no—
139C2×C3⋊D9—21no—
140C2×C3⋊D9—22no—
141C2×C3⋊D9—22no—
142C2×C3⋊D9—22no—
143C2×C3⋊D9C2×D412yes—
144C2×C3×D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9C2×D412yes—
147C3×C4×C9—22no—
148C3×C4×C9#4112yescomplemented
149C3⋊Dic9—21no—
150C9×Dic3—22no—
151C9×Dic3—21no—
152C9×Dic3C2×D412yes—
153C3×Dic9—22no—
154C3×Dic9—21no—
155C2×#239—92no—
156C2×#235—92no—
157C3×C4⋊Q8—31no—
158C2×C3×C4⋊C4—91no—
159#946—92no—
160#930—91no—
161#929—92no—
162C4×#67—91no—
163#896—92no—
164#895—91no—
165#895—91no—
166#895—91no—
167#895—181no—
168#894—92no—
169#894—92no—
170#894—92no—
171#893—91no—
172D3×C4⋊C4—31no—
173D3×C4⋊C4—91no—
174D3×C4⋊C4—91no—
175#888D911yescharacteristic, complemented
176C4×D12—91no—
177C4×D12—91no—
178C22×C3⋊D3—31no—
179C22×C3⋊D3—32no—
180C22×C3×D3—31no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3—31no—
184C4⋊(C3⋊D3)—64no—
185C4⋊(C3⋊D3)—62no—
186C4×C3⋊D3—62no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3C22×D311yescharacteristic
190C2×C3×Dic3C22×D312yes—
191C2×C3×Dic3C22×D311yescharacteristic
192C3×C4×D3—64no—
193C3×C4×D3—62no—
194C3×Dic6—24no—
195#626—34no—
196#626—64no—
197#624—64no—
198#624—64no—
199C22×D9—31no—
200C22×D9—32no—
201C22×D9—61no—
202C22×D9—62no—
203C22×D9C22×D311yescharacteristic
204Q8×C9—64no—
205C2×C4×C9—21no—
206C2×C4×C9—22no—
207C2×C4×C9—31no—
208C2×C4×C9—32no—
209C2×C4×C9—31no—
210C2×C4×C9C22×D312yes—
211C2×C4×C9C22×D311yescharacteristic
212C2×Dic9—31no—
213C2×Dic9—61no—
214C2×Dic9C22×D312yes—
215C2×Dic9C22×D311yescharacteristic
216D36—34no—
217D36—64no—
218D36—64no—
219D36—62no—
220D36—122no—
221D36—124no—
222C4×D9—24no—
223C4×D9—22no—
224C4×D9—64no—
225C4×D9—64no—
226C4×D9—62no—
227C4×D9—122no—
228#547—271noSylow
229C2×C3×C9C22×D411yescharacteristic
230C2×C3×C9C2×#4111yescharacteristic
231C2×C3×C92−1+411yescharacteristic
232C3⋊D9—22no—
233C3⋊D9—42no—
234C3×D9—22no—
235C2×C3×Q8—32no—
236C22×C3×C4—91no—
237C22×C3×C4—92no—
238C2×#67—91no—
239C2×#67—92no—
240#239—188no—
241#235—188no—
242C2×D12—91no—
243C2×D12—92no—
244C2×D12—91no—
245C2×D12—92no—
246C2×D12—91no—
247C2×D12—92no—
248C2×D12—181no—
249C2×D12—182no—
250C2×C4×D3—31no—
251C2×C4×D3—32no—
252C2×C4×D3—92no—
253C2×C4×D3—92no—
254C2×C4×D3—91no—
255C2×C4×D3—91no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—181no—
259C2×Dic6C2×D912yes—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—32no—
263C3×C4⋊C4—91no—
264C3×C4⋊C4—92no—
265C3×C4⋊C4C2×D911yescharacteristic
266C3×C4⋊C4D3211yescharacteristic
267C3×C42—31no—
268#217—91no—
269(C2×D3)⋊C4—91no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—91no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—91no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—182no—
279C4⋊Dic3—31no—
280C4⋊Dic3—91no—
281C4⋊Dic3C2×D911yescharacteristic
282Dic3⋊C4—32no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—91no—
286Dic3⋊C4C2×D912yes—
287C4×Dic3C2×D911yescharacteristic
288C22×C32C23×D311yescharacteristic
289C2×C3⋊D3—32no—
290C2×C3⋊D3—61no—
291C2×C3⋊D3—61no—
292C2×C3⋊D3—62no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—62no—
295C2×C3×D3—32no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C32×C4—22no—
299C32×C4#23912yes—
300C3⋊Dic3—61no—
301C3×Dic3—22no—
302C3×Dic3—21no—
303C3×Dic3—62no—
304C3×Dic3—61no—
305C3×Dic3D3×D412yes—
306C22×C9—21no—
307C22×C9C23×D311yescharacteristic
308C2×D9—21no—
309C2×D9—21no—
310C2×D9—32no—
311C2×D9—62no—
312C2×D9—61no—
313C2×D9—61no—
314C2×D9—62no—
315C2×D9—62no—
316C2×D9—62no—
317C2×D9—121no—
318C2×D9—121no—
319C2×D9—122no—
320C2×D9—122no—
321C2×D9—122no—
322C2×D9D3×D412yes—
323C4×C9—22no—
324C4×C9—22no—
325C4×C9—32no—
326C4×C9—42no—
327C4×C9—62no—
328C4×C9—61no—
329C4×C9#23512yescomplemented
330Dic9—21no—
331Dic9—22no—
332Dic9—61no—
333Dic9—121no—
334C2×#41—272no—
335C4⋊Q8—31no—
336#123—272no—
337#123—272no—
338C22⋊Q8—272no—
339#121—271no—
340#121—271no—
341C4×D4—271no—
342C4×D4—271no—
343C2×C4⋊C4—91no—
344C2×C4⋊C4—271no—
345C3×C9#54711yescharacteristic, complemented, Sylow
346C23×C3—91no—
347C22×D3—31no—
348C22×D3—91no—
349C22×D3—91no—
350C22×D3—92no—
351C22×D3—92no—
352C22×D3—181no—
353C22×D3—182no—
354C3×Q8—64no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—31no—
358C2×C3×C4—32no—
359C2×C3×C4—31no—
360C2×C3×C4—91no—
361C2×C3×C4—92no—
362C2×C3×C4—184no—
363C2×C3×C4—182no—
364C2×C3×C4C22×D912yes—
365C2×C3×C4C22×D911yescharacteristic
366C2×C3×C4C2×D3212yes—
367C2×C3×C4C2×D3211yescharacteristic
368#67—94no—
369#67—184no—
370C2×Dic3—31no—
371C2×Dic3—32no—
372C2×Dic3—91no—
373C2×Dic3—91no—
374C2×Dic3—181no—
375C2×Dic3C22×D911yescharacteristic
376C2×Dic3C22×D912yes—
377C2×Dic3C22×D911yescharacteristic
378D12—94no—
379D12—184no—
380D12—184no—
381D12—184no—
382D12—182no—
383D12—182no—
384D12—362no—
385D12—364no—
386C4×D3—64no—
387C4×D3—62no—
388C4×D3—184no—
389C4×D3—184no—
390C4×D3—184no—
391C4×D3—182no—
392C4×D3—182no—
393C4×D3—184no—
394C4×D3—362no—
395Dic6—24no—
396C2×C32C2×D3×D411yescharacteristic
397C2×C32C2×#23911yescharacteristic
398C2×C32#100711yescharacteristic
399C3⋊D3—62no—
400C3⋊D3—122no—
401C3×D3—62no—
402C2×C9—21no—
403C2×C9—21no—
404C2×C9—21no—
405C2×C9C2×#23511yescharacteristic
406C2×C9C2×D3×D411yescharacteristic
407C2×C9#100711yescharacteristic
408D9—22no—
409D9—62no—
410D9—122no—
411D9—122no—
412D9—242no—
413#41—548no—
414C2×Q8—32no—
415C2×D4—271no—
416C2×D4—272no—
417C2×D4—271no—
418C2×D4—272no—
419C22×C4—91no—
420C22×C4—92no—
421C22×C4—272no—
422C22×C4—272no—
423C22×C4—271no—
424C4⋊C4—31no—
425C4⋊C4—32no—
426C4⋊C4—91no—
427C4⋊C4—92no—
428C4⋊C4—272no—
429C4⋊C4—271no—
430C4⋊C4D3×D911yescharacteristic
431C22⋊C4—271no—
432C22⋊C4—272no—
433C22⋊C4—272no—
434C22⋊C4—271no—
435C22⋊C4—272no—
436C42—31no—
437C22×C3—21no—
438C22×C3—92no—
439C22×C3—181no—
440C22×C3—181no—
441C22×C3C23×D911yescharacteristic
442C22×C3C22×D3211yescharacteristic
443C2×D3—32no—
444C2×D3—61no—
445C2×D3—61no—
446C2×D3—92no—
447C2×D3—92no—
448C2×D3—182no—
449C2×D3—181no—
450C2×D3—181no—
451C2×D3—181no—
452C2×D3—181no—
453C2×D3—182no—
454C2×D3—182no—
455C2×D3—182no—
456C2×D3—182no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—361no—
460C2×D3—361no—
461C2×D3—362no—
462C2×D3—362no—
463C2×D3—362no—
464C3×C4—22no—
465C3×C4—22no—
466C3×C4—22no—
467C3×C4—32no—
468C3×C4—42no—
469C3×C4—62no—
470C3×C4—61no—
471C3×C4—182no—
472C3×C4—181no—
473C3×C4#370112yescomplemented
474C3×C4#379912yes—
475Dic3—22no—
476Dic3—21no—
477Dic3—61no—
478Dic3—62no—
479Dic3—181no—
480Dic3—181no—
481Dic3—361no—
482Dic3D4×D912yes—
483C32#572611yescharacteristic
484C9—21no—
485C9#550711yescharacteristic, complemented
486C23—91no—
487C23—271no—
488C23—272no—
489Q8—64no—
490D4—274no—
491D4—544no—
492D4—544no—
493D4—542no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—32no—
497C2×C4—91no—
498C2×C4—92no—
499C2×C4—184no—
500C2×C4—182no—
501C2×C4—271no—
502C2×C4—544no—
503C2×C4—544no—
504C2×C4—542no—
505C2×C4C2×D3×D912yes—
506C2×C4C2×D3×D911yescharacteristic
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—182no—
511C2×C3C2×D4×D911yescharacteristic
512C2×C3C2×#370111yescharacteristic
513C2×C3#6371311yescharacteristic
514C2×C3C2×#379911yescharacteristic
515C2×C3D32×D411yescharacteristic
516C2×C3#6431611yescharacteristic
517D3—62no—
518D3—182no—
519D3—182no—
520D3—362no—
521D3—362no—
522D3—362no—
523D3—722no—
524C22—92no—
525C22—181no—
526C22—181no—
527C22—272no—
528C22—541no—
529C22—541no—
530C22—542no—
531C22—542no—
532C22—542no—
533C22C22×D3×D911yescharacteristic
534C4—22no—
535C4—32no—
536C4—61no—
537C4—62no—
538C4—181no—
539C4—182no—
540C4—541no—
541C4#8863512yescomplemented
542C3—21no—
543C3#1058982811yescharacteristic, complemented
544C3#1059451711yescharacteristic
545C2—182no—
546C2—542no—
547C2—1082no—
548C2C2×#8863511yescharacteristic
549C2D3×D4×D911yescharacteristic
550C2#1172250911yescharacteristic
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×#88635C2semidirect
2. #11721751C2semidirect
3. C3×C4⋊C4×D9C2semidirect
4. C9×#888C2semidirect
5. #11721004C2semidirect
6. C4×#6192C2semidirect
7. #11721000C2semidirect
8. #11720997C2semidirect
9. #11720993C2semidirect
10. #11720986C2semidirect
11. #11720938C2semidirect
58. C4⋊C4×D9D3semidirect
79. #6192D4semidirect
148. C3×C4×C9#41semidirect
175. #888D9semidirect
329. C4×C9#235semidirect
345. C3×C9#547semidirect
473. C3×C4#3701semidirect
485. C9#5507semidirect
541. C4#88635semidirect
543. C3#10589828semidirect

Maximal subgroups

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

subgroupindex
C2×#886352
#117217512
C3×C4⋊C4×D92
C9×#8882
#117210042
C4×#61922
#117210002
#117209972
#117209932
#117209862
#117209382
#105945173
#105898283