Finite Group Databasebeta

← #13150387

Subgroups — #13150387

Order 1728.

1026 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×#6226C211yesmaximal, characteristic, complemented
2C2×#88638C211yesmaximal, characteristic, complemented
3C9×#930C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic, complemented
5#11721290C212yesmaximal, complemented
6#67⋊Dic9C212yesmaximal, complemented
7#11721042C212yesmaximal, complemented
8#11721024C212yesmaximal, complemented
9#11720964C211yesmaximal, characteristic, complemented
10#11720945C211yesmaximal, characteristic, complemented
11#11720913C211yesmaximal, characteristic, complemented
12#10594918—31nomaximal
13#10589700—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×#6226—24no—
17C2×#6226C2211yescharacteristic
18C2×#6226C2212yes—
19C2×#6226C2211yescharacteristic
20C2×C9×#67C2212yes—
21C2×C3×#611C2211yescharacteristic
22C2×C9×D12C2211yescharacteristic
23C2×C3×C4×D9C2211yescharacteristic
24C2×#6194C2212yes—
25C2×D3×Dic9C2212yes—
26#88638—24no—
27C3×#3676C2211yescharacteristic
28C9×(C2×D3)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2211yescharacteristic
30C9×C4⋊Dic3C2211yescharacteristic
31C3×Dic9⋊C4C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34(C2×D3)⋊Dic9C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3⋊Dic9C2212yes—
37#88442C2212yes—
38Dic3×Dic9C2212yes—
39C22×#634—31no—
40C2×#3795—31no—
41C3×#930D311yescharacteristic
42C3×C4×#67—31no—
43#63980—32no—
44#67⋊Dic3—32no—
45#63913—32no—
46#63896—32no—
47#63860—31no—
48#63857—31no—
49#63844—31no—
50C2×#3699—31no—
51C22×Dic18—31no—
52C22×Dic18—61no—
53C9×#121—31no—
54#63501—32no—
55#63498—31no—
56D4×Dic9—32no—
57C4×#611D311yescharacteristic, complemented
58#63459—31no—
59#63449—31no—
60#63447—32no—
61#63440—32no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9—24no—
69C2×C3⋊Dic9C2312yes—
70C2×C3⋊Dic9C2312yes—
71#6226—22no—
72#6226—24no—
73#6226D414yescomplemented
74C9×#67—24no—
75C3×#611—22no—
76C2×C9×Dic3C2312yes—
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C9×D12—22no—
81C3×C4×D9—22no—
82#6194—24no—
83D3×Dic9—24no—
84#5598—91no—
85#5552—91no—
86C22×C32×C4C2×D311yescharacteristic
87C22×C3⋊Dic3—32no—
88C2×#634—31no—
89C2×#634—31no—
90C2×#634—32no—
91C2×#634—64no—
92C2×C3×#67—31no—
93C2×C3×#67C2×D312yes—
94C2×C3×D12C2×D311yescharacteristic
95C2×C3×C4×D3—31no—
96C2×#625—32no—
97C2×D3×Dic3—32no—
98#3795—64no—
99C3×#217—31no—
100C3×(C2×D3)⋊C4—31no—
101C3×(C2×D3)⋊C4C2×D312yes—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×Dic3⋊C4—31no—
104C3×C4×Dic3—31no—
105#3724—32no—
106Dic3⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111C2×D4×C9—32no—
112C2×D4×C9—31no—
113C22×C4×C9—21no—
114C22×C4×C9C2×D311yescharacteristic
115C2×#611—32no—
116C2×#611C2×D311yescharacteristic
117C22×Dic9—32no—
118C22×Dic9—32no—
119C22×Dic9—62no—
120#3699—64no—
121C2×C4×D9C2×D311yescharacteristic
122C2×Dic18—31no—
123C2×Dic18—31no—
124C2×Dic18—32no—
125C2×Dic18—64no—
126C2×Dic18—61no—
127C2×Dic18—62no—
128C2×Dic18—61no—
129C2×Dic18—68no—
130C9×C4⋊C4—31no—
131C9×C22⋊C4—32no—
132#3676—32no—
133#3676—32no—
134#3676C2×D311yescharacteristic
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4C2×D311yescharacteristic
137C4⋊Dic9—32no—
138Dic9⋊C4—32no—
139Dic9⋊C4C2×D311yescharacteristic
140C4×Dic9—32no—
141C4×Dic9C2×D311yescharacteristic
142C22×C3×C9—21no—
143C22×C3×C9—21no—
144C22×C3×C9#4112yescomplemented
145C22×C3×C9C2411yescharacteristic
146C2×D3×C9—22no—
147C2×D3×C9—22no—
148C2×D3×C9—22no—
149C2×C3×D9—21no—
150C2×C3×D9—21no—
151C2×C3×D9—21no—
152C3×C4×C9—21no—
153C3×C4×C9C2×D412yes—
154C3⋊Dic9—22no—
155C3⋊Dic9C2×D414yes—
156C9×Dic3—22no—
157C3×Dic9—21no—
158C3×Dic9—21no—
159C3×Dic9—21no—
160C2×#237—91no—
161C2×#235—91no—
162C22×Dic6—91no—
163C22×Dic6—91no—
164C22×Dic6—181no—
165C3×#121—31no—
166C3×C4×D4—91no—
167#939—92no—
168#939—92no—
169#936—91no—
170D4×Dic3—92no—
171#930D911yescharacteristic, complemented
172C4×#67—31no—
173#924—91no—
174#897—91no—
175#896—92no—
176#887—91no—
177#885—92no—
178#885—92no—
179#67⋊C4—92no—
180#878—92no—
181#875—91no—
182C4×Dic6—91no—
183C23×C32C22×D311yescharacteristic
184C22×C3×D3—31no—
185C22×C3×D3C22×D312yes—
186C2×C32×C4—22no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C3⋊Dic3—32no—
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—64no—
192#634—34no—
193#634—64no—
194#634—62no—
195C3×#67—24no—
196C3×#67—62no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3C22×D312yes—
201C3×D12—22no—
202C3×C4×D3—62no—
203#625—64no—
204D3×Dic3—64no—
205C23×C9—21no—
206C23×C9—32no—
207C23×C9C22×D311yescharacteristic
208C22×D9C22×D311yescharacteristic
209D4×C9—64no—
210D4×C9—62no—
211C2×C4×C9—24no—
212C2×C4×C9—22no—
213C2×C4×C9—21no—
214C2×C4×C9—21no—
215C2×C4×C9—32no—
216C2×C4×C9C22×D311yescharacteristic
217C2×C4×C9C22×D311yescharacteristic
218#611—22no—
219#611—64no—
220C2×Dic9—32no—
221C2×Dic9—32no—
222C2×Dic9—62no—
223C2×Dic9—62no—
224C2×Dic9—64no—
225C2×Dic9—64no—
226C2×Dic9—68no—
227C2×Dic9C22×D311yescharacteristic
228C2×Dic9C22×D311yescharacteristic
229C2×Dic9C22×D311yescharacteristic
230C4×D9—22no—
231Dic18—34no—
232Dic18—64no—
233Dic18—64no—
234Dic18—68no—
235Dic18—64no—
236Dic18—62no—
237#548—271noSylow
238C2×C3×C9—22no—
239C2×C3×C9C22×D411yescharacteristic
240C2×C3×C9C2×#4111yescharacteristic
241C2×C3×C92−1+411yescharacteristic
242D3×C9—42no—
243C3×D9—41no—
244C2×C3×D4—32no—
245C2×C3×D4—31no—
246C2×C3×D4—91no—
247C22×C3×C4—21no—
248C22×C3×C4—91no—
249C22×C3×C4C2×D911yescharacteristic
250C22×C3×C4D3211yescharacteristic
251C2×#67—31no—
252C2×#67—92no—
253C2×#67—92no—
254C2×#67C2×D912yes—
255C22×Dic3—92no—
256C22×Dic3—92no—
257C22×Dic3—92no—
258C22×Dic3—182no—
259#237—184no—
260#235—184no—
261C2×D12C2×D911yescharacteristic
262C2×C4×D3—31no—
263C2×C4×D3—92no—
264C2×Dic6—91no—
265C2×Dic6—91no—
266C2×Dic6—91no—
267C2×Dic6—92no—
268C2×Dic6—92no—
269C2×Dic6—91no—
270C2×Dic6—181no—
271C2×Dic6—181no—
272C2×Dic6—182no—
273C2×Dic6—184no—
274C2×Dic6—184no—
275C2×Dic6—188no—
276C3×C4⋊C4—31no—
277C3×C4⋊C4—91no—
278C3×C22⋊C4—32no—
279C3×C22⋊C4—91no—
280C3×C22⋊C4—91no—
281C3×C42—91no—
282#217—31no—
283#217—92no—
284#217—92no—
285#217—92no—
286(C2×D3)⋊C4—31no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4C2×D912yes—
291C4⋊Dic3—92no—
292C4⋊Dic3C2×D911yescharacteristic
293Dic3⋊C4—31no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297C4×Dic3—31no—
298C4×Dic3—92no—
299C4×Dic3—92no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32#23712yes—
303C22×C32C23×D311yescharacteristic
304C2×C3×D3—22no—
305C2×C3×D3—22no—
306C2×C3×D3—22no—
307C2×C3×D3—61no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C32×C4—21no—
311C32×C4D3×D412yes—
312C3⋊Dic3—34no—
313C3⋊Dic3—62no—
314C3×Dic3—22no—
315C3×Dic3—61no—
316C3×Dic3—61no—
317C3×Dic3—61no—
318C22×C9—22no—
319C22×C9—22no—
320C22×C9—21no—
321C22×C9—21no—
322C22×C9—21no—
323C22×C9—22no—
324C22×C9—62no—
325C22×C9—62no—
326C22×C9—62no—
327C22×C9#23512yescomplemented
328C22×C9C23×D311yescharacteristic
329C2×D9—21no—
330C2×D9—21no—
331C2×D9—21no—
332C4×C9—22no—
333C4×C9—21no—
334C4×C9—22no—
335C4×C9—62no—
336C4×C9D3×D412yes—
337Dic9—21no—
338Dic9—21no—
339Dic9—21no—
340Dic9—34no—
341Dic9—64no—
342Dic9—64no—
343Dic9—62no—
344C2×#41—271no—
345C22×Q8—271no—
346#124—271no—
347#124—272no—
348C22⋊Q8—272no—
349C22⋊Q8—271no—
350#121—31no—
351#121—272no—
352C4×Q8—271no—
353C4×D4—91no—
354C4×D4—272no—
355C3×C9#54811yescharacteristic, complemented, Sylow
356C23×C3—21no—
357C23×C3—32no—
358C23×C3—91no—
359C23×C3C22×D911yescharacteristic
360C23×C3C2×D3211yescharacteristic
361C22×D3—31no—
362C22×D3C22×D912yes—
363C3×D4—64no—
364C3×D4—62no—
365C3×D4—182no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—32no—
372C2×C3×C4—91no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—182no—
376C2×C3×C4C22×D911yescharacteristic
377C2×C3×C4C22×D911yescharacteristic
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C2×D3211yescharacteristic
380#67—24no—
381#67—62no—
382#67—184no—
383#67—184no—
384C2×Dic3—31no—
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—92no—
388C2×Dic3—92no—
389C2×Dic3—92no—
390C2×Dic3—92no—
391C2×Dic3—182no—
392C2×Dic3—182no—
393C2×Dic3—184no—
394C2×Dic3—184no—
395C2×Dic3—184no—
396C2×Dic3—188no—
397C2×Dic3C22×D912yes—
398D12—22no—
399C4×D3—62no—
400C4×D3—184no—
401Dic6—94no—
402Dic6—94no—
403Dic6—184no—
404Dic6—184no—
405Dic6—184no—
406Dic6—188no—
407Dic6—184no—
408Dic6—182no—
409Dic6—182no—
410C2×C32—22no—
411C2×C32C2×D3×D411yescharacteristic
412C2×C32C2×#23711yescharacteristic
413C2×C32#101011yescharacteristic
414C3×D3—42no—
415C3×D3—121no—
416C2×C9—24no—
417C2×C9—22no—
418C2×C9—21no—
419C2×C9—21no—
420C2×C9—21no—
421C2×C9—122no—
422C2×C9C2×#23511yescharacteristic
423C2×C9C2×D3×D411yescharacteristic
424C2×C9#101011yescharacteristic
425D9—41no—
426#41—544no—
427C2×Q8—271no—
428C2×Q8—272no—
429C2×Q8—271no—
430C2×Q8—544no—
431C2×D4—32no—
432C2×D4—31no—
433C2×D4—91no—
434C2×D4—272no—
435C22×C4—91no—
436C22×C4—272no—
437C22×C4—272no—
438C22×C4D3×D911yescharacteristic
439C4⋊C4—31no—
440C4⋊C4—91no—
441C4⋊C4—272no—
442C4⋊C4—272no—
443C22⋊C4—32no—
444C22⋊C4—91no—
445C22⋊C4—91no—
446C22⋊C4—272no—
447C22⋊C4—272no—
448C22⋊C4—272no—
449C42—91no—
450C42—272no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—22no—
458C22×C3—21no—
459C22×C3—62no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3—181no—
463C22×C3—181no—
464C22×C3—181no—
465C22×C3#369912yescomplemented
466C22×C3C23×D911yescharacteristic
467C22×C3#379512yes—
468C22×C3C22×D3211yescharacteristic
469C2×D3—22no—
470C2×D3—22no—
471C2×D3—22no—
472C2×D3—61no—
473C2×D3—61no—
474C2×D3—61no—
475C3×C4—22no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—62no—
480C3×C4—181no—
481C3×C4—181no—
482C3×C4—181no—
483C3×C4D4×D912yes—
484C3×C4#381112yes—
485Dic3—22no—
486Dic3—61no—
487Dic3—61no—
488Dic3—61no—
489Dic3—94no—
490Dic3—94no—
491Dic3—184no—
492Dic3—184no—
493Dic3—182no—
494Dic3—182no—
495C32#559811yescharacteristic
496C9—21no—
497C9#555211yescharacteristic, complemented
498C23—32no—
499C23—91no—
500C23C2×D3×D911yescharacteristic
501Q8—274no—
502Q8—544no—
503Q8—542no—
504D4—64no—
505D4—62no—
506D4—182no—
507D4—544no—
508C2×C4—22no—
509C2×C4—32no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—91no—
513C2×C4—182no—
514C2×C4—272no—
515C2×C4—272no—
516C2×C4—544no—
517C2×C4—544no—
518C2×C4C2×D3×D911yescharacteristic
519C2×C4C2×D3×D911yescharacteristic
520C2×C3—24no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—122no—
527C2×C3—361no—
528C2×C3C2×D4×D911yescharacteristic
529C2×C3C2×#369911yescharacteristic
530C2×C3#6371611yescharacteristic
531C2×C3C2×#379511yescharacteristic
532C2×C3#6429811yescharacteristic
533C2×C3C2×#381111yescharacteristic
534D3—42no—
535D3—121no—
536C22—21no—
537C22—21no—
538C22—62no—
539C22—62no—
540C22—62no—
541C22—181no—
542C22—181no—
543C22—181no—
544C22#8863812yescomplemented
545C22C22×D3×D911yescharacteristic
546C4—21no—
547C4—62no—
548C4—181no—
549C4—181no—
550C4—181no—
551C4—274no—
552C4—542no—
553C4#8866812yes—
554C3—21no—
555C3#1058970011yescharacteristic, complemented
556C3#1059491811yescharacteristic
557C2—22no—
558C2—122no—
559C2—361no—
560C2C2×#8863811yescharacteristic
561C2#1172246511yescharacteristic
562C2C2×#8866811yescharacteristic
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×#6226C2semidirect
2. C2×#88638C2semidirect
3. C9×#930C2semidirect
4. C3×C4×#611C2semidirect
5. #11721290C2semidirect
6. #67⋊Dic9C2semidirect
7. #11721042C2semidirect
8. #11721024C2semidirect
9. #11720964C2semidirect
10. #11720945C2semidirect
11. #11720913C2semidirect
57. C4×#611D3semidirect
73. #6226D4semidirect
144. C22×C3×C9#41semidirect
171. #930D9semidirect
327. C22×C9#235semidirect
355. C3×C9#548semidirect
465. C22×C3#3699semidirect
497. C9#5552semidirect
544. C22#88638semidirect
555. C3#10589700semidirect

Maximal subgroups

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

subgroupindex
C22×#62262
C2×#886382
C9×#9302
C3×C4×#6112
#117212902
#67⋊Dic92
#117210422
#117210242
#117209642
#117209452
#117209132
#105949183
#105897003