Finite Group Databasebeta

← #13150187

Subgroups — #13150187

Order 1728.

944 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic, complemented
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic, complemented
5#11721270C212yesmaximal, complemented
6#11721263C212yesmaximal, complemented
7#11721036C211yesmaximal, characteristic, complemented
8#11720976C212yesmaximal, complemented
9#11720964C211yesmaximal, characteristic, complemented
10#11720955C212yesmaximal, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10594899—31nomaximal
13#10589933—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×#67C2212yes—
17C2×C3×#611C2211yescharacteristic
18C9×#235—24no—
19C2×C9×D12C2211yescharacteristic
20C2×C4×D3×C9C2212yes—
21C2×C3×C4×D9C2211yescharacteristic
22C2×C9×Dic6C2211yescharacteristic
23C2×#6194C2212yes—
24C2×Dic3×D9C2212yes—
25C2×#6189C2212yes—
26D9×Dic6—24no—
27#88551C2212yes—
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29#88534C2212yes—
30C3×#3676C2211yescharacteristic
31C3×(C2×D9)⋊C4C2211yescharacteristic
32C3×Dic9⋊C4C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34(C2×D3)⋊Dic9C2212yes—
35(C2×D3)⋊Dic9C2212yes—
36(C2×D3)⋊Dic9C2212yes—
37Dic3⋊Dic9C2212yes—
38Dic3×Dic9C2212yes—
39C2×C3×#235D311yescharacteristic
40C2×D3×Dic6—31no—
41#64133—31no—
42C3×C4×#67—31no—
43#63970—32no—
44#63959—32no—
45#63908—31no—
46#63879—32no—
47#63860—31no—
48#63851—32no—
49Dic3×Dic6—31no—
50C2×C9×#41—31no—
51C2×Q8×D9—31no—
52Q8×Dic9—31no—
53#63500—32no—
54#63500—31no—
55#63498—32no—
56#63498—31no—
57C4×#611—32no—
58C4×#611D311yescharacteristic, complemented
59#63486—31no—
60#63486—32no—
61#63486—61no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2312yes—
70#6226—22no—
71C9×#67—24no—
72C3×#611—22no—
73C2×C9×Dic3C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C9×D12—22no—
78C4×D3×C9—24no—
79C3×C4×D9—22no—
80C9×Dic6D414yescomplemented
81#6194—24no—
82Dic3×D9—24no—
83#6189—24no—
84#5831—91no—
85#5552—91no—
86C22×C32×C4C2×D311yescharacteristic
87C2×#634—31no—
88C2×C3×#67—31no—
89C2×C3×#67C2×D312yes—
90C3×#235—24no—
91C2×C3×D12C2×D311yescharacteristic
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3C2×D312yes—
94C2×C3×Dic6C2×D311yescharacteristic
95C2×#627—32no—
96C2×#625—32no—
97C2×D3×Dic3—32no—
98D3×Dic6—64no—
99#3757—32no—
100C4⋊(C3⋊Dic3)—31no—
101#3750—32no—
102C3×#217—31no—
103C3×(C2×D3)⋊C4—31no—
104C3×Dic3⋊C4—31no—
105C3×C4×Dic3—31no—
106Dic3⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111C9×#41—64no—
112C2×Q8×C9—31no—
113C2×D4×C9—32no—
114C2×D4×C9—31no—
115C22×C4×C9—21no—
116C22×C4×C9—32no—
117C22×C4×C9C2×D311yescharacteristic
118C2×#611—32no—
119C2×#611C2×D311yescharacteristic
120Q8×D9—64no—
121C2×C4×D9—32no—
122C2×C4×D9C2×D311yescharacteristic
123C2×Dic18—31no—
124C2×Dic18—32no—
125C2×Dic18—61no—
126#3676—32no—
127#3676—32no—
128#3676—32no—
129#3676—32no—
130#3676—62no—
131#3676C2×D311yescharacteristic
132(C2×D9)⋊C4C2×D311yescharacteristic
133C4⋊Dic9—31no—
134C4⋊Dic9—32no—
135C4⋊Dic9—61no—
136Dic9⋊C4—32no—
137Dic9⋊C4—62no—
138Dic9⋊C4C2×D311yescharacteristic
139C4×Dic9—32no—
140C4×Dic9C2×D311yescharacteristic
141C22×C3×C9—21no—
142C22×C3×C9—21no—
143C22×C3×C9—21no—
144C22×C3×C9C2411yescharacteristic
145C2×D3×C9—22no—
146C2×D3×C9—22no—
147C2×D3×C9—22no—
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9#4112yescomplemented
151C3×C4×C9—21no—
152C3×C4×C9C2×D412yes—
153C3⋊Dic9—22no—
154C3⋊Dic9—22no—
155C9×Dic3C2×D414yes—
156C3×Dic9—21no—
157C3×Dic9—21no—
158C3×Dic9—21no—
159C2×C3×#41—31no—
160C2×D3×Q8—91no—
161C2×#235D911yescharacteristic, complemented
162C22×Dic6—91no—
163C3×C4×D4—91no—
164Q8×Dic3—91no—
165#939—92no—
166#938—91no—
167#938—92no—
168#936—91no—
169#936—92no—
170#930—91no—
171C4×#67—31no—
172C4×#67—92no—
173#924—92no—
174#924—91no—
175#924—91no—
176#924—181no—
177#896—92no—
178#885—92no—
179#67⋊C4—92no—
180#875—91no—
181C4×Dic6—91no—
182C23×C32C22×D311yescharacteristic
183C22×C3×D3—31no—
184C22×C3×D3C22×D312yes—
185C2×C32×C4—22no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—32no—
190#634—62no—
191C3×#67—24no—
192C3×#67—62no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3C22×D312yes—
197C3×D12—22no—
198C3×C4×D3—24no—
199C3×C4×D3—62no—
200C3×Dic6#6714yes—
201#627—64no—
202#625—64no—
203D3×Dic3—64no—
204C23×C9—21no—
205C23×C9—32no—
206C23×C9C22×D311yescharacteristic
207C22×D9C22×D311yescharacteristic
208Q8×C9—34no—
209D4×C9—64no—
210D4×C9—62no—
211C2×C4×C9—22no—
212C2×C4×C9—21no—
213C2×C4×C9—21no—
214C2×C4×C9—32no—
215C2×C4×C9—42no—
216C2×C4×C9—64no—
217C2×C4×C9C22×D311yescharacteristic
218C2×C4×C9C22×D311yescharacteristic
219#611—22no—
220#611—64no—
221C2×Dic9—32no—
222C2×Dic9—32no—
223C2×Dic9—62no—
224C2×Dic9—62no—
225C2×Dic9C22×D311yescharacteristic
226C2×Dic9C22×D311yescharacteristic
227C2×Dic9C22×D311yescharacteristic
228C4×D9—22no—
229C4×D9—64no—
230Dic18—64no—
231Dic18—62no—
232Dic18—122no—
233#548—271noSylow
234C2×C3×C9—41no—
235C2×C3×C9C22×D411yescharacteristic
236C2×C3×C9C2×#4111yescharacteristic
237C2×C3×C92−1+411yescharacteristic
238D3×C9—42no—
239C3×D9—22no—
240C3×#41—64no—
241C2×C3×Q8—31no—
242C2×C3×D4—32no—
243C2×C3×D4—31no—
244C2×C3×D4—91no—
245C22×C3×C4—21no—
246C22×C3×C4—32no—
247C22×C3×C4—91no—
248C22×C3×C4C2×D911yescharacteristic
249C22×C3×C4D3211yescharacteristic
250C2×#67—31no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67C2×D912yes—
254C22×Dic3—92no—
255D3×Q8—184no—
256#235—24no—
257C2×D12C2×D911yescharacteristic
258C2×C4×D3—31no—
259C2×C4×D3—92no—
260C2×C4×D3C2×D912yes—
261C2×Dic6—92no—
262C2×Dic6—91no—
263C2×Dic6—91no—
264C2×Dic6—92no—
265C2×Dic6—181no—
266C2×Dic6—184no—
267C2×Dic6C2×D911yescharacteristic
268C3×C4⋊C4—91no—
269C3×C22⋊C4—91no—
270C3×C22⋊C4—91no—
271C3×C42—91no—
272#217—31no—
273#217—92no—
274#217—92no—
275#217—92no—
276#217—92no—
277#217—92no—
278#217—182no—
279(C2×D3)⋊C4—31no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283C4⋊Dic3—92no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286C4⋊Dic3—181no—
287Dic3⋊C4—31no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—182no—
292C4×Dic3—31no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32C23×D311yescharacteristic
299C2×C3×D3—22no—
300C2×C3×D3—22no—
301C2×C3×D3—22no—
302C2×C3×D3—32no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C32×C4—21no—
306C32×C4C2×#6712yes—
307C3⋊Dic3—62no—
308C3⋊Dic3—62no—
309C3×Dic3—61no—
310C3×Dic3—61no—
311C3×Dic3—61no—
312C3×Dic3C2×#6714yes—
313C22×C9—22no—
314C22×C9—21no—
315C22×C9—21no—
316C22×C9—21no—
317C22×C9—21no—
318C22×C9—41no—
319C22×C9—41no—
320C22×C9—62no—
321C22×C9—62no—
322C22×C9—62no—
323C22×C9C23×D311yescharacteristic
324C2×D9—21no—
325C2×D9—21no—
326C2×D9#23512yescomplemented
327C4×C9—21no—
328C4×C9—22no—
329C4×C9—34no—
330C4×C9—41no—
331C4×C9D3×D412yes—
332Dic9—21no—
333Dic9—21no—
334Dic9—21no—
335Dic9—62no—
336Dic9—62no—
337Dic9—122no—
338Dic9—122no—
339C2×#41—31no—
340C22×Q8—271no—
341#124—271no—
342#124—272no—
343C22⋊Q8—272no—
344C22⋊Q8—271no—
345#121—271no—
346#121—272no—
347C4×Q8—271no—
348C4×D4—91no—
349C4×D4—272no—
350C3×C9#54811yescharacteristic, complemented, Sylow
351C23×C3—21no—
352C23×C3—32no—
353C23×C3—91no—
354C23×C3C22×D911yescharacteristic
355C23×C3C2×D3211yescharacteristic
356C22×D3—31no—
357C22×D3C22×D912yes—
358C3×Q8—34no—
359C3×D4—64no—
360C3×D4—62no—
361C3×D4—182no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—32no—
367C2×C3×C4—42no—
368C2×C3×C4—64no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—182no—
373C2×C3×C4C22×D911yescharacteristic
374C2×C3×C4C22×D911yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377#67—24no—
378#67—62no—
379#67—184no—
380#67—184no—
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—92no—
385C2×Dic3—92no—
386C2×Dic3—92no—
387C2×Dic3—92no—
388C2×Dic3—182no—
389C2×Dic3—182no—
390C2×Dic3—184no—
391C2×Dic3C22×D912yes—
392D12—22no—
393C4×D3—24no—
394C4×D3—62no—
395C4×D3—184no—
396Dic6—184no—
397Dic6—182no—
398Dic6—182no—
399Dic6—184no—
400Dic6—362no—
401Dic6#61114yescomplemented
402C2×C32—41no—
403C2×C32D3×#4111yescharacteristic
404C2×C32#101011yescharacteristic
405C2×C32C22×#6711yescharacteristic
406C3×D3—42no—
407C3×D3—62no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—42no—
412C2×C9—41no—
413C2×C9—122no—
414C2×C9C2×#23511yescharacteristic
415C2×C9C2×D3×D411yescharacteristic
416C2×C9#101011yescharacteristic
417D9—22no—
418#41—64no—
419C2×Q8—31no—
420C2×Q8—272no—
421C2×Q8—271no—
422C2×Q8—544no—
423C2×D4—32no—
424C2×D4—31no—
425C2×D4—91no—
426C2×D4—272no—
427C22×C4—32no—
428C22×C4—91no—
429C22×C4—272no—
430C22×C4D3×D911yescharacteristic
431C4⋊C4—91no—
432C4⋊C4—272no—
433C4⋊C4—272no—
434C4⋊C4—271no—
435C22⋊C4—91no—
436C22⋊C4—91no—
437C22⋊C4—272no—
438C22⋊C4—272no—
439C22⋊C4—272no—
440C22⋊C4—272no—
441C42—91no—
442C42—272no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—62no—
454C22×C3—62no—
455C22×C3—62no—
456C22×C3—92no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3C23×D911yescharacteristic
460C22×C3C22×D3211yescharacteristic
461C2×D3—22no—
462C2×D3—22no—
463C2×D3—22no—
464C2×D3—32no—
465C2×D3—61no—
466C2×D3—61no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—22no—
470C3×C4—34no—
471C3×C4—41no—
472C3×C4—181no—
473C3×C4—181no—
474C3×C4—181no—
475C3×C4C2×#61112yes—
476C3×C4D3×#6712yes—
477Dic3—61no—
478Dic3—61no—
479Dic3—61no—
480Dic3—182no—
481Dic3—182no—
482Dic3—182no—
483Dic3—182no—
484Dic3—362no—
485Dic3—362no—
486Dic3C2×#61114yes—
487C32#583111yescharacteristic
488C9—21no—
489C9#555211yescharacteristic, complemented
490C23—32no—
491C23—91no—
492C23C2×D3×D911yescharacteristic
493Q8—34no—
494Q8—544no—
495Q8—542no—
496D4—64no—
497D4—62no—
498D4—182no—
499D4—544no—
500C2×C4—22no—
501C2×C4—32no—
502C2×C4—64no—
503C2×C4—91no—
504C2×C4—91no—
505C2×C4—91no—
506C2×C4—182no—
507C2×C4—272no—
508C2×C4—272no—
509C2×C4—544no—
510C2×C4C2×D3×D911yescharacteristic
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—42no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—122no—
519C2×C3—182no—
520C2×C3#41×D911yescharacteristic
521C2×C3#6371611yescharacteristic
522C2×C3C22×#61111yescharacteristic
523C2×C3#6429811yescharacteristic
524C2×C3D3×#23511yescharacteristic
525C2×C3C2×D3×#6711yescharacteristic
526D3—42no—
527D3—62no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—62no—
532C22—62no—
533C22—62no—
534C22—92no—
535C22—181no—
536C22—181no—
537C22C22×D3×D911yescharacteristic
538C4—21no—
539C4—34no—
540C4—181no—
541C4—181no—
542C4—181no—
543C4—542no—
544C4—542no—
545C4D3×#61112yes—
546C3—21no—
547C3#1058993311yescharacteristic, complemented
548C3#1059489911yescharacteristic
549C2—41no—
550C2—122no—
551C2—182no—
552C2#1172246511yescharacteristic
553C2D9×#23511yescharacteristic
554C2C2×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. C2×C9×#235C2semidirect
2. C2×D9×Dic6C2semidirect
3. #11721813C2semidirect
4. C3×C4×#611C2semidirect
5. #11721270C2semidirect
6. #11721263C2semidirect
7. #11721036C2semidirect
8. #11720976C2semidirect
9. #11720964C2semidirect
10. #11720955C2semidirect
11. Dic6×Dic9C2semidirect
58. C4×#611D3semidirect
80. C9×Dic6D4semidirect
150. C2×C3×D9#41semidirect
161. C2×#235D9semidirect
326. C2×D9#235semidirect
350. C3×C9#548semidirect
401. Dic6#611semidirect
489. C9#5552semidirect
547. C3#10589933semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C2×D9×Dic62
#117218132
C3×C4×#6112
#117212702
#117212632
#117210362
#117209762
#117209642
#117209552
Dic6×Dic92
#105948993
#105899333