Finite Group Databasebeta

← #13150971

Subgroups — #13150971

Order 1728.

1024 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88662C212yesmaximal, complemented
2#11721838C211yesmaximal, characteristic, complemented
3C9×#940C211yesmaximal, characteristic, complemented
4C3×D4×Dic9C211yesmaximal, characteristic, complemented
5#67×Dic9C212yesmaximal, complemented
6#11721263C212yesmaximal, complemented
7#11721262C212yesmaximal, complemented
8#11721045C211yesmaximal, characteristic, complemented
9D12×Dic9C211yesmaximal, characteristic, complemented
10C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
11#11720970C211yesmaximal, characteristic, complemented
12#10595139—31nomaximal
13#10589818—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×#67C2212yes—
18C2×C9×#67C2212yes—
19C22×C3×Dic9C2212yes—
20C2×C9×D12C2211yescharacteristic
21C2×#6195C2212yes—
22#88662—28no—
23C2×D3×Dic9C2212yes—
24C2×(C3⋊D9)⋊C4C2211yescharacteristic
25C2×(C3⋊D9)⋊C4C2211yescharacteristic
26C2×#6189C2212yes—
27#88551C2212yes—
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C3×#3676C2212yes—
30C4×C9×Dic3C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34(C2×D3)⋊Dic9C2212yes—
35Dic3⋊Dic9C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2211yescharacteristic
38C2×#3804—32no—
39#64147—31no—
40C3×#940D311yescharacteristic
41C3×D4×Dic3—31no—
42Dic3×#67—32no—
43#63959—32no—
44#63958—32no—
45#63918—31no—
46#63916—31no—
47Dic3×D12—31no—
48C4×#624—31no—
49C2×#3699—32no—
50C9×#127—31no—
51#63500—31no—
52#63500—32no—
53#63500—31no—
54#63500—61no—
55#63498—32no—
56D4×Dic9—32no—
57D4×Dic9—31no—
58D4×Dic9D311yescharacteristic, complemented
59C42×D9—31no—
60#63431—31no—
61C23×C3×C9C2312yes—
62C22×C3⋊D9C2311yescharacteristic
63C22×D3×C9C2312yes—
64C3×D4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66#6233—24no—
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69C4×C3⋊D9—22no—
70C9×#67—24no—
71C9×#67—24no—
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2311yescharacteristic
74C2×C3×Dic9—24no—
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C9×D12—22no—
79#6195—24no—
80D3×Dic9—24no—
81(C3⋊D9)⋊C4—22no—
82(C3⋊D9)⋊C4D414yescomplemented
83#6189—24no—
84#5716—91no—
85#5548—91no—
86C2×C32×D4C2×D311yescharacteristic
87C2×#638—32no—
88C2×C4×C3⋊D3—31no—
89C2×C3×#67C2×D312yes—
90C2×C3×#67C2×D312yes—
91C22×C3×Dic3—32no—
92C2×C3×D12C2×D311yescharacteristic
93C2×#627—32no—
94C2×#626—32no—
95C2×#624—31no—
96C2×#624—31no—
97#3804—68no—
98C2×D3×Dic3—32no—
99#3757—32no—
100C4⋊(C3⋊Dic3)—31no—
101C3×#217—32no—
102C3×C4⋊Dic3—31no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3C2×D311yescharacteristic
105Dic3⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108Dic32—31no—
109Dic32—31no—
110C2×D4×C9—21no—
111C2×D4×C9—32no—
112C2×D4×C9—32no—
113C2×D4×C9—31no—
114C2×D4×C9C2×D311yescharacteristic
115C2×#611—32no—
116C2×#611—32no—
117C2×#611—62no—
118C22×Dic9—32no—
119C22×Dic9C2×D312yes—
120#3699—68no—
121C2×C4×D9—31no—
122C2×C4×D9—31no—
123C2×C4×D9—31no—
124C2×C4×D9—61no—
125C2×Dic18—32no—
126C42×C9—31no—
127#3676—32no—
128#3676—32no—
129#3676—32no—
130#3676—62no—
131#3676C2×D312yes—
132C4⋊Dic9—31no—
133C4⋊Dic9—32no—
134C4⋊Dic9—61no—
135C4⋊Dic9C2×D311yescharacteristic
136C4×Dic9—31no—
137C4×Dic9—31no—
138C4×Dic9C2×D311yescharacteristic
139C22×C3×C9—22no—
140C22×C3×C9—22no—
141C22×C3×C9—22no—
142C22×C3×C9C2411yescharacteristic
143C2×C3⋊D9—21no—
144C2×C3⋊D9—21no—
145C2×C3⋊D9C2×D412yes—
146C2×D3×C9—22no—
147C2×D3×C9—22no—
148C2×D3×C9—22no—
149C3×C4×C9#4112yescomplemented
150C3⋊Dic9—22no—
151C3⋊Dic9—21no—
152C9×Dic3C2×D412yes—
153C9×Dic3#4112yescomplemented
154C3×Dic9—22no—
155C3×Dic9—21no—
156C3×Dic9C2×D412yes—
157C2×#237—92no—
158C2×#235—92no—
159C3×#127—31no—
160C3×C4×D4—91no—
161#940D911yescharacteristic, complemented
162#938—91no—
163#938—92no—
164#938—91no—
165#938—91no—
166#938—181no—
167#936—92no—
168D4×Dic3—31no—
169D4×Dic3—91no—
170D4×Dic3—92no—
171C4×#67—92no—
172#895—91no—
173#888—91no—
174#885—92no—
175#884—92no—
176C4×D12—91no—
177C42×D3—91no—
178C42×D3—91no—
179#869—91no—
180C23×C32C22×D312yes—
181C22×C3⋊D3—31no—
182C22×C3×D3C22×D312yes—
183C32×D4—22no—
184C2×C32×C4C22×D311yescharacteristic
185#638—64no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—32no—
188C4×C3⋊D3—62no—
189C3×#67—24no—
190C3×#67—24no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3C22×D311yescharacteristic
196C2×C3×Dic3C22×D311yescharacteristic
197C3×D12—22no—
198#627—64no—
199#626—64no—
200#624—34no—
201#624—62no—
202D3×Dic3—64no—
203C23×C9—22no—
204C23×C9—32no—
205C23×C9C22×D312yes—
206C22×D9—31no—
207C22×D9—61no—
208D4×C9—22no—
209D4×C9—42no—
210D4×C9—64no—
211D4×C9—64no—
212D4×C9—62no—
213C2×C4×C9—21no—
214C2×C4×C9—31no—
215C2×C4×C9—31no—
216C2×C4×C9C22×D311yescharacteristic
217#611—64no—
218#611—64no—
219#611—124no—
220C2×Dic9—24no—
221C2×Dic9—31no—
222C2×Dic9—32no—
223C2×Dic9—62no—
224C2×Dic9—61no—
225C2×Dic9—64no—
226C2×Dic9C22×D311yescharacteristic
227C2×Dic9C22×D312yes—
228C2×Dic9C22×D311yescharacteristic
229C4×D9—34no—
230C4×D9—62no—
231C4×D9—62no—
232C4×D9—122no—
233Dic18—64no—
234#532—271noSylow
235C2×C3×C9—42no—
236C2×C3×C9C22×D411yescharacteristic
237C2×C3×C9C2×#4111yescharacteristic
238C2×C3×C9C2×#4111yescharacteristic
239C3⋊D9—22no—
240D3×C9—42no—
241C2×C3×D4—21no—
242C2×C3×D4—32no—
243C2×C3×D4—32no—
244C2×C3×D4—31no—
245C2×C3×D4C2×D911yescharacteristic
246C2×C3×D4D3211yescharacteristic
247C22×C3×C4—92no—
248C2×#67—92no—
249C2×#67—92no—
250C2×#67—92no—
251C2×#67—182no—
252C2×#67C2×D912yes—
253C2×#67C2×D912yes—
254C22×Dic3—32no—
255C22×Dic3—92no—
256#237—188no—
257#235—188no—
258C2×D12—92no—
259C2×D12C2×D911yescharacteristic
260C2×C4×D3—91no—
261C2×C4×D3—91no—
262C2×C4×D3—91no—
263C2×C4×D3—92no—
264C2×C4×D3—91no—
265C2×C4×D3—91no—
266C2×C4×D3—91no—
267C2×C4×D3—181no—
268C2×Dic6—92no—
269C2×Dic6—92no—
270C3×C4⋊C4—91no—
271C3×C22⋊C4—92no—
272C3×C42—31no—
273C3×C42—91no—
274#217—32no—
275#217—92no—
276#217—92no—
277#217—92no—
278#217—92no—
279#217—182no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282C4⋊Dic3—31no—
283C4⋊Dic3—92no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286C4⋊Dic3—181no—
287Dic3⋊C4—92no—
288C4×Dic3—31no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3C2×D911yescharacteristic
294C22×C32—22no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32C23×D311yescharacteristic
298C2×C3⋊D3—32no—
299C2×C3⋊D3—61no—
300C2×C3⋊D3—61no—
301C2×C3×D3—22no—
302C2×C3×D3—22no—
303C2×C3×D3—22no—
304C32×C4#23712yes—
305C3⋊Dic3—62no—
306C3⋊Dic3—61no—
307C3×Dic3—32no—
308C3×Dic3—62no—
309C3×Dic3—61no—
310C3×Dic3D3×D412yes—
311C3×Dic3#23712yes—
312C22×C9—22no—
313C22×C9—22no—
314C22×C9—22no—
315C22×C9—21no—
316C22×C9—42no—
317C22×C9—42no—
318C22×C9—42no—
319C22×C9—62no—
320C22×C9—62no—
321C22×C9—62no—
322C22×C9C23×D311yescharacteristic
323C2×D9—32no—
324C2×D9—62no—
325C2×D9—61no—
326C2×D9—61no—
327C2×D9—121no—
328C2×D9—121no—
329C4×C9—22no—
330C4×C9—32no—
331C4×C9—32no—
332C4×C9#23512yescomplemented
333Dic9—22no—
334Dic9—21no—
335Dic9—61no—
336Dic9—62no—
337Dic9—121no—
338Dic9—122no—
339Dic9D3×D412yes—
340C2×#41—272no—
341C4⋊Q8—271no—
342#127—31no—
343#124—272no—
344#121—272no—
345#121—271no—
346#121—271no—
347C4×D4—91no—
348C4×D4—271no—
349C4×D4—272no—
350C2×C42—271no—
351C3×C9#53211yescharacteristic, complemented, Sylow
352C23×C3—22no—
353C23×C3—32no—
354C23×C3C22×D912yes—
355C23×C3C2×D3212yes—
356C22×D3—91no—
357C22×D3—91no—
358C22×D3—181no—
359C22×D3C22×D912yes—
360C3×D4—22no—
361C3×D4—22no—
362C3×D4—42no—
363C3×D4—64no—
364C3×D4—64no—
365C3×D4—62no—
366C2×C3×C4—21no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—92no—
372C2×C3×C4—184no—
373C2×C3×C4C22×D911yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375#67—24no—
376#67—24no—
377#67—184no—
378#67—184no—
379#67—184no—
380#67—364no—
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—32no—
384C2×Dic3—64no—
385C2×Dic3—91no—
386C2×Dic3—92no—
387C2×Dic3—91no—
388C2×Dic3—92no—
389C2×Dic3—182no—
390C2×Dic3—181no—
391C2×Dic3—184no—
392C2×Dic3C22×D911yescharacteristic
393C2×Dic3C22×D911yescharacteristic
394D12—22no—
395D12—184no—
396C4×D3—94no—
397C4×D3—94no—
398C4×D3—182no—
399C4×D3—184no—
400C4×D3—182no—
401C4×D3—182no—
402C4×D3—182no—
403C4×D3—362no—
404Dic6—184no—
405Dic6—184no—
406C2×C32—42no—
407C2×C32C2×D3×D411yescharacteristic
408C2×C32C2×#23711yescharacteristic
409C2×C32C2×#23711yescharacteristic
410C3⋊D3—62no—
411C3×D3—42no—
412C2×C9—21no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—42no—
416C2×C9—82no—
417C2×C9—122no—
418C2×C9C2×#23511yescharacteristic
419C2×C9C2×D3×D411yescharacteristic
420C2×C9D3×#4111yescharacteristic
421D9—62no—
422D9—122no—
423#41—548no—
424C2×Q8—272no—
425C2×D4—32no—
426C2×D4—32no—
427C2×D4—31no—
428C2×D4—272no—
429C2×D4—272no—
430C2×D4D3×D911yescharacteristic
431C22×C4—92no—
432C22×C4—271no—
433C22×C4—271no—
434C22×C4—272no—
435C22×C4—271no—
436C4⋊C4—91no—
437C4⋊C4—272no—
438C4⋊C4—271no—
439C22⋊C4—92no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C22⋊C4—272no—
443C42—31no—
444C42—91no—
445C42—271no—
446C42—271no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—21no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—62no—
458C22×C3—62no—
459C22×C3—62no—
460C22×C3C23×D911yescharacteristic
461C22×C3C22×D3211yescharacteristic
462C2×D3—22no—
463C2×D3—22no—
464C2×D3—22no—
465C2×D3—92no—
466C2×D3—92no—
467C2×D3—182no—
468C2×D3—181no—
469C2×D3—181no—
470C2×D3—181no—
471C2×D3—181no—
472C2×D3—361no—
473C2×D3—361no—
474C3×C4—22no—
475C3×C4—32no—
476C3×C4—32no—
477C3×C4—92no—
478C3×C4—182no—
479C3×C4—181no—
480C3×C4#369912yescomplemented
481C3×C4#380412yes—
482Dic3—32no—
483Dic3—62no—
484Dic3—61no—
485Dic3—182no—
486Dic3—181no—
487Dic3—182no—
488Dic3—181no—
489Dic3—361no—
490Dic3—362no—
491Dic3D4×D912yes—
492Dic3#369912yescomplemented
493C32#571611yescharacteristic
494C9—21no—
495C9#554811yescharacteristic, complemented
496C23—32no—
497C23—271no—
498C23C2×D3×D912yes—
499Q8—544no—
500D4—22no—
501D4—64no—
502D4—64no—
503D4—62no—
504D4—544no—
505D4—544no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—91no—
509C2×C4—91no—
510C2×C4—92no—
511C2×C4—184no—
512C2×C4—274no—
513C2×C4—271no—
514C2×C4—272no—
515C2×C4—542no—
516C2×C4—544no—
517C2×C4—542no—
518C2×C4C2×D3×D911yescharacteristic
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—42no—
523C2×C3—42no—
524C2×C3—82no—
525C2×C3—122no—
526C2×C3C2×D4×D911yescharacteristic
527C2×C3C2×#369911yescharacteristic
528C2×C3C2×#369911yescharacteristic
529C2×C3D32×D411yescharacteristic
530C2×C3D3×#23711yescharacteristic
531C2×C3C2×#380411yescharacteristic
532D3—42no—
533D3—182no—
534D3—182no—
535D3—362no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—62no—
540C22—62no—
541C22—62no—
542C22—272no—
543C22—541no—
544C22—541no—
545C22C22×D3×D911yescharacteristic
546C4—32no—
547C4—32no—
548C4—92no—
549C4—181no—
550C4—182no—
551C4—541no—
552C4—542no—
553C4#8866212yescomplemented
554C3—21no—
555C3#1058981811yescharacteristic, complemented
556C3#1059513911yescharacteristic
557C2—42no—
558C2—122no—
559C2—542no—
560C2D3×D4×D911yescharacteristic
561C2D3×#369911yescharacteristic
562C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88662C2semidirect
2. #11721838C2semidirect
3. C9×#940C2semidirect
4. C3×D4×Dic9C2semidirect
5. #67×Dic9C2semidirect
6. #11721263C2semidirect
7. #11721262C2semidirect
8. #11721045C2semidirect
9. D12×Dic9C2semidirect
10. C4×(C3⋊D9)⋊C4C2semidirect
11. #11720970C2semidirect
58. D4×Dic9D3semidirect
82. (C3⋊D9)⋊C4D4semidirect
149. C3×C4×C9#41semidirect
153. C9×Dic3#41semidirect
161. #940D9semidirect
332. C4×C9#235semidirect
351. C3×C9#532semidirect
480. C3×C4#3699semidirect
492. Dic3#3699semidirect
495. C9#5548semidirect
553. C4#88662semidirect
555. C3#10589818semidirect

Maximal subgroups

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

subgroupindex
C2×#886622
#117218382
C9×#9402
C3×D4×Dic92
#67×Dic92
#117212632
#117212622
#117210452
D12×Dic92
C4×(C3⋊D9)⋊C42
#117209702
#105951393
#105898183