Finite Group Databasebeta

← #13150193

Subgroups — #13150193

Order 1728.

964 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
1C22×C9×Dic6C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic, complemented
5#11721277C212yesmaximal, complemented
6#11721240C212yesmaximal, complemented
7#11721003C212yesmaximal, complemented
8#11720986C212yesmaximal, complemented
9#11720953C211yesmaximal, characteristic, complemented
10#11720944C211yesmaximal, characteristic, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10594902—31nomaximal
13#10589919—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C22×C9×Dic3C2212yes—
19C2×C3×C4×D9C2211yescharacteristic
20C2×C9×Dic6—24no—
21C2×C9×Dic6C2212yes—
22C2×C9×Dic6C2211yescharacteristic
23C2×C9×Dic6C2211yescharacteristic
24C2×#6192C2212yes—
25C2×(C3⋊D9)⋊C4C2212yes—
26#88635—24no—
27#88536C2212yes—
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C3×#3676C2211yescharacteristic
30C3×(C2×D9)⋊C4C2211yescharacteristic
31C3×Dic9⋊C4C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33#88445C2212yes—
34#88445C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3⋊Dic9C2212yes—
37#88442C2212yes—
38Dic3×Dic9C2212yes—
39C22×C3×Dic6D311yescharacteristic
40C2×#3799—31no—
41#64139—31no—
42C3×C4×#67—31no—
43#63978—32no—
44#63952—32no—
45#63907—32no—
46#63888—32no—
47#63864—31no—
48#63856—31no—
49Dic3×Dic6—31no—
50C22×Q8×C9—31no—
51C2×#3701—31no—
52#63509—32no—
53#63509—31no—
54#63508—31no—
55#63508—32no—
56Q8×Dic9—31no—
57#63492—31no—
58#63492—32no—
59#63492—61no—
60C4×#611—32no—
61C4×#611D311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68#6233—24no—
69C2×C3⋊Dic9C2312yes—
70C4⋊(C3⋊D9)—22no—
71C3×#611—22no—
72C2×C9×Dic3—24no—
73C2×C9×Dic3C2312yes—
74C2×C9×Dic3C2312yes—
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C3×C4×D9—22no—
79C9×Dic6—22no—
80C9×Dic6—24no—
81C9×Dic6D414yescomplemented
82#6192—24no—
83(C3⋊D9)⋊C4—24no—
84#5817—91no—
85#5552—91no—
86C22×C32×C4C2×D311yescharacteristic
87C2×#638—32no—
88C2×C4⋊(C3⋊D3)—31no—
89C2×C3×#67—31no—
90C22×C3×Dic3C2×D312yes—
91C2×C3×C4×D3—31no—
92C2×C3×Dic6—24no—
93C2×C3×Dic6C2×D311yescharacteristic
94C2×C3×Dic6C2×D312yes—
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#626—32no—
97C2×#624—32no—
98#3799—64no—
99#3752—32no—
100C4⋊(C3⋊Dic3)—31no—
101C3×#217—31no—
102C3×(C2×D3)⋊C4—31no—
103C3×Dic3⋊C4—31no—
104C3×C4×Dic3—31no—
105#3724—32no—
106Dic3⋊Dic3—32no—
107#3722—32no—
108#3722—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111C2×Q8×C9—31no—
112C2×Q8×C9—32no—
113C2×Q8×C9—31no—
114C2×Q8×C9—64no—
115C22×C4×C9—21no—
116C22×C4×C9—32no—
117C22×C4×C9C2×D311yescharacteristic
118C2×#611—32no—
119C2×#611—62no—
120C2×#611C2×D311yescharacteristic
121#3701—64no—
122C2×D36—32no—
123C2×D36—31no—
124C2×D36—61no—
125C2×C4×D9—32no—
126C2×C4×D9C2×D311yescharacteristic
127#3676C2×D311yescharacteristic
128(C2×D9)⋊C4—32no—
129(C2×D9)⋊C4—32no—
130(C2×D9)⋊C4—32no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—62no—
133(C2×D9)⋊C4C2×D311yescharacteristic
134C4⋊Dic9—31no—
135C4⋊Dic9—32no—
136C4⋊Dic9—61no—
137Dic9⋊C4—32no—
138Dic9⋊C4C2×D311yescharacteristic
139C4×Dic9—32no—
140C4×Dic9C2×D311yescharacteristic
141C22×C3×C9—21no—
142C22×C3×C9—21no—
143C22×C3×C9#4112yescomplemented
144C22×C3×C9C2411yescharacteristic
145C2×C3⋊D9—22no—
146C2×C3⋊D9—22no—
147C2×C3⋊D9—22no—
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9—21no—
151C3×C4×C9—21no—
152C3×C4×C9C2×D412yes—
153C3⋊Dic9—22no—
154C9×Dic3—22no—
155C9×Dic3C2×D414yes—
156C3×Dic9—21no—
157C3×Dic9—21no—
158C3×Dic9—21no—
159C22×C3×Q8—31no—
160C2×#239—91no—
161C2×#235—91no—
162C22×Dic6D911yescharacteristic, complemented
163C3×C4×D4—91no—
164#947—91no—
165#947—92no—
166#946—92no—
167#946—91no—
168Q8×Dic3—91no—
169#939—92no—
170#930—92no—
171#930—91no—
172#930—91no—
173#930—181no—
174C4×#67—31no—
175C4×#67—92no—
176#924—91no—
177#896—92no—
178#885—92no—
179#67⋊C4—92no—
180#875—91no—
181C4×Dic6—91no—
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3—32no—
184C22×C3×D3—31no—
185C2×C32×C4—22no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188#638—64no—
189C2×C3⋊Dic3—32no—
190C4⋊(C3⋊D3)—62no—
191C3×#67—62no—
192C2×C3×Dic3—24no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3C22×D312yes—
197C2×C3×Dic3C22×D312yes—
198C3×C4×D3—62no—
199C3×Dic6—22no—
200C3×Dic6—24no—
201C3×Dic6#6714yes—
202#626—64no—
203#624—64no—
204C23×C9—21no—
205C23×C9C22×D311yescharacteristic
206C22×D9—32no—
207C22×D9—62no—
208C22×D9C22×D311yescharacteristic
209Q8×C9—34no—
210Q8×C9—64no—
211Q8×C9—62no—
212C2×C4×C9—22no—
213C2×C4×C9—21no—
214C2×C4×C9—21no—
215C2×C4×C9—32no—
216C2×C4×C9—32no—
217C2×C4×C9—42no—
218C2×C4×C9—64no—
219C2×C4×C9C22×D311yescharacteristic
220C2×C4×C9C22×D311yescharacteristic
221#611—22no—
222#611—64no—
223#611—124no—
224C2×Dic9—32no—
225C2×Dic9—62no—
226C2×Dic9C22×D311yescharacteristic
227C2×Dic9C22×D311yescharacteristic
228C2×Dic9C22×D311yescharacteristic
229D36—64no—
230D36—62no—
231D36—122no—
232C4×D9—22no—
233C4×D9—64no—
234#548—271noSylow
235C2×C3×C9—22no—
236C2×C3×C9C22×D411yescharacteristic
237C2×C3×C9C2×#4111yescharacteristic
238C2×C3×C92−1+411yescharacteristic
239C3⋊D9—42no—
240C3×D9—41no—
241C2×C3×Q8—31no—
242C2×C3×Q8—32no—
243C2×C3×Q8—31no—
244C2×C3×Q8—64no—
245C2×C3×D4—91no—
246C22×C3×C4—21no—
247C22×C3×C4—32no—
248C22×C3×C4—91no—
249C22×C3×C4C2×D911yescharacteristic
250C22×C3×C4D3211yescharacteristic
251C2×#67—31no—
252C2×#67—92no—
253C2×#67—92no—
254C2×#67—92no—
255C2×#67—182no—
256C22×Dic3C2×D912yes—
257#239—184no—
258#235—184no—
259C2×D12—92no—
260C2×D12—91no—
261C2×D12—91no—
262C2×D12—181no—
263C2×C4×D3—31no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×Dic6—24no—
267C2×Dic6C2×D911yescharacteristic
268C2×Dic6C2×D912yes—
269C2×Dic6C2×D911yescharacteristic
270C3×C4⋊C4—91no—
271C3×C22⋊C4—91no—
272C3×C22⋊C4—91no—
273C3×C42—91no—
274#217—31no—
275#217—92no—
276(C2×D3)⋊C4—31no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—182no—
285C4⋊Dic3—92no—
286C4⋊Dic3—91no—
287C4⋊Dic3—91no—
288C4⋊Dic3—181no—
289Dic3⋊C4—31no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293C4×Dic3—31no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32#23912yes—
299C22×C32C23×D311yescharacteristic
300C2×C3⋊D3—62no—
301C2×C3⋊D3—62no—
302C2×C3⋊D3—62no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C32×C4—21no—
307C32×C4C2×#6712yes—
308C3⋊Dic3—62no—
309C3×Dic3—22no—
310C3×Dic3—61no—
311C3×Dic3—61no—
312C3×Dic3—61no—
313C3×Dic3C2×#6714yes—
314C22×C9—21no—
315C22×C9—21no—
316C22×C9—21no—
317C22×C9—22no—
318C22×C9—41no—
319C22×C9—41no—
320C22×C9#23512yescomplemented
321C22×C9C23×D311yescharacteristic
322C2×D9—21no—
323C2×D9—21no—
324C2×D9—21no—
325C2×D9—62no—
326C2×D9—62no—
327C2×D9—62no—
328C2×D9—122no—
329C2×D9—122no—
330C2×D9—122no—
331C4×C9—21no—
332C4×C9—22no—
333C4×C9—34no—
334C4×C9—41no—
335C4×C9—62no—
336C4×C9D3×D412yes—
337Dic9—21no—
338Dic9—21no—
339Dic9—21no—
340Dic9—62no—
341Dic9—122no—
342C2×#41—271no—
343C22×Q8—31no—
344#124—271no—
345#124—272no—
346C22⋊Q8—272no—
347C22⋊Q8—271no—
348#121—272no—
349#121—271no—
350C4×Q8—271no—
351C4×D4—91no—
352C4×D4—272no—
353C3×C9#54811yescharacteristic, complemented, Sylow
354C23×C3—21no—
355C23×C3—91no—
356C23×C3C22×D911yescharacteristic
357C23×C3C2×D3211yescharacteristic
358C22×D3—31no—
359C22×D3—92no—
360C22×D3—92no—
361C22×D3—182no—
362C3×Q8—34no—
363C3×Q8—64no—
364C3×Q8—62no—
365C3×D4—182no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—32no—
371C2×C3×C4—32no—
372C2×C3×C4—42no—
373C2×C3×C4—64no—
374C2×C3×C4—91no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—182no—
378C2×C3×C4C22×D911yescharacteristic
379C2×C3×C4C22×D911yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382#67—62no—
383#67—184no—
384#67—184no—
385#67—184no—
386#67—364no—
387C2×Dic3—24no—
388C2×Dic3—31no—
389C2×Dic3—31no—
390C2×Dic3—31no—
391C2×Dic3—92no—
392C2×Dic3—92no—
393C2×Dic3—182no—
394C2×Dic3C22×D912yes—
395C2×Dic3C22×D912yes—
396D12—184no—
397D12—182no—
398D12—182no—
399D12—362no—
400C4×D3—62no—
401C4×D3—184no—
402C4×D3—184no—
403Dic6—22no—
404Dic6—24no—
405Dic6#61114yescomplemented
406C2×C32—22no—
407C2×C32C2×#23911yescharacteristic
408C2×C32#100711yescharacteristic
409C2×C32C22×#6711yescharacteristic
410C3⋊D3—122no—
411C3×D3—121no—
412C2×C9—22no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—42no—
417C2×C9C2×#23511yescharacteristic
418C2×C9C2×D3×D411yescharacteristic
419C2×C9#101011yescharacteristic
420D9—41no—
421D9—122no—
422D9—242no—
423#41—544no—
424C2×Q8—31no—
425C2×Q8—32no—
426C2×Q8—31no—
427C2×Q8—64no—
428C2×D4—91no—
429C2×D4—272no—
430C2×D4—272no—
431C2×D4—271no—
432C22×C4—32no—
433C22×C4—91no—
434C22×C4—272no—
435C22×C4D3×D911yescharacteristic
436C4⋊C4—91no—
437C4⋊C4—272no—
438C4⋊C4—272no—
439C4⋊C4—271no—
440C22⋊C4—91no—
441C22⋊C4—91no—
442C22⋊C4—272no—
443C22⋊C4—272no—
444C22⋊C4—272no—
445C22⋊C4—272no—
446C42—91no—
447C42—272no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—22no—
453C22×C3—21no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—181no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3#370112yescomplemented
460C22×C3C23×D911yescharacteristic
461C22×C3#379912yes—
462C22×C3C22×D3211yescharacteristic
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—182no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—182no—
472C2×D3—362no—
473C2×D3—362no—
474C2×D3—362no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—22no—
478C3×C4—34no—
479C3×C4—41no—
480C3×C4—62no—
481C3×C4—181no—
482C3×C4—181no—
483C3×C4—181no—
484C3×C4C2×#61112yes—
485C3×C4D3×#6712yes—
486Dic3—22no—
487Dic3—61no—
488Dic3—61no—
489Dic3—61no—
490Dic3—182no—
491Dic3—182no—
492Dic3—362no—
493Dic3C2×#61114yes—
494C32#581711yescharacteristic
495C9—21no—
496C9#555211yescharacteristic, complemented
497C23—91no—
498C23—272no—
499C23C2×D3×D911yescharacteristic
500Q8—34no—
501Q8—64no—
502Q8—62no—
503D4—182no—
504D4—544no—
505D4—544no—
506D4—542no—
507C2×C4—22no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4—64no—
511C2×C4—91no—
512C2×C4—91no—
513C2×C4—91no—
514C2×C4—182no—
515C2×C4—272no—
516C2×C4—544no—
517C2×C4C2×D3×D911yescharacteristic
518C2×C4C2×D3×D911yescharacteristic
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—361no—
526C2×C3C2×#370111yescharacteristic
527C2×C3#6371311yescharacteristic
528C2×C3C22×#61111yescharacteristic
529C2×C3#6429711yescharacteristic
530C2×C3C2×#379911yescharacteristic
531C2×C3C2×D3×#6711yescharacteristic
532D3—121no—
533D3—362no—
534D3—362no—
535D3—722no—
536C22—21no—
537C22—21no—
538C22—181no—
539C22—181no—
540C22—181no—
541C22—542no—
542C22—542no—
543C22—542no—
544C22#8863512yescomplemented
545C22C22×D3×D911yescharacteristic
546C4—21no—
547C4—34no—
548C4—62no—
549C4—181no—
550C4—181no—
551C4—181no—
552C4—542no—
553C4D3×#61112yes—
554C3—21no—
555C3#1058991911yescharacteristic, complemented
556C3#1059490211yescharacteristic
557C2—22no—
558C2—361no—
559C2—1082no—
560C2#1172245811yescharacteristic
561C2C2×#8863511yescharacteristic
562C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#548
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C9×Dic6C2semidirect
2. C2×#88635C2semidirect
3. #11721819C2semidirect
4. C3×C4×#611C2semidirect
5. #11721277C2semidirect
6. #11721240C2semidirect
7. #11721003C2semidirect
8. #11720986C2semidirect
9. #11720953C2semidirect
10. #11720944C2semidirect
11. Dic6×Dic9C2semidirect
61. C4×#611D3semidirect
81. C9×Dic6D4semidirect
143. C22×C3×C9#41semidirect
162. C22×Dic6D9semidirect
320. C22×C9#235semidirect
353. C3×C9#548semidirect
405. Dic6#611semidirect
459. C22×C3#3701semidirect
496. C9#5552semidirect
544. C22#88635semidirect
555. C3#10589919semidirect

Maximal subgroups

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

subgroupindex
C22×C9×Dic62
C2×#886352
#117218192
C3×C4×#6112
#117212772
#117212402
#117210032
#117209862
#117209532
#117209442
Dic6×Dic92
#105949023
#105899193