Finite Group Databasebeta

← #13150996

Subgroups — #13150996

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
1C2×C9×#237C211yesmaximal, characteristic, complemented
2C2×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
3#11721838C211yesmaximal, characteristic, complemented
4C3×#63498C211yesmaximal, characteristic, complemented
5#11721263C212yesmaximal, complemented
6#11721262C212yesmaximal, complemented
7#11721249C212yesmaximal, complemented
8#11721240C212yesmaximal, complemented
9C4×#6195C211yesmaximal, characteristic, complemented
10#11720960C211yesmaximal, characteristic, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10595237—31nomaximal
13#10589933—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×#67C2212yes—
18C22×C9×Dic3C2212yes—
19C9×#237—24no—
20C2×C4×D3×C9C2211yescharacteristic
21C2×C9×Dic6C2211yescharacteristic
22C2×C3×Dic18C2211yescharacteristic
23C2×#6195C2211yescharacteristic
24C2×(C3⋊D9)⋊C4C2212yes—
25C2×#6189C2212yes—
26(C3⋊D9)⋊Q8—24no—
27#88551C2212yes—
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C3×#3676C2212yes—
30C3×#3676C2212yes—
31C3×C4×Dic9C2211yescharacteristic
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2211yescharacteristic
34#88445C2211yescharacteristic
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37Dic9⋊Dic3C2211yescharacteristic
38Dic3×Dic9C2212yes—
39C2×C3×#237D311yescharacteristic
40C2×#3797—31no—
41#64147—31no—
42C3×#936—31no—
43#63964—32no—
44#63959—32no—
45#63958—32no—
46#63952—32no—
47C4×#626—31no—
48#63856—31no—
49Dic3×Dic6—31no—
50C2×C9×#41—31no—
51C2×Q8×D9—31no—
52Q8×Dic9—31no—
53#63500—31no—
54#63500—32no—
55#63500—61no—
56#63498—32no—
57#63498D311yescharacteristic, complemented
58C4×#611—32no—
59C4×#611—31no—
60#63486—31no—
61#63486—32no—
62C23×C3×C9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C3×D4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67#6233—24no—
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2312yes—
70C4×C3⋊D9—22no—
71C9×#67—24no—
72C2×C9×Dic3—24no—
73C2×C9×Dic3C2311yescharacteristic
74C2×C9×Dic3C2312yes—
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C4×D3×C9—22no—
79C9×Dic6D414yescomplemented
80C3×Dic18—22no—
81#6195—22no—
82(C3⋊D9)⋊C4—24no—
83#6189—24no—
84#5831—91no—
85#5677—91no—
86C2×C32×D4C2×D311yescharacteristic
87C2×#638—32no—
88C2×C4×C3⋊D3—31no—
89C2×C3×#67C2×D312yes—
90C22×C3×Dic3C2×D312yes—
91C3×#237—24no—
92C2×C3×C4×D3C2×D311yescharacteristic
93C2×C3×Dic6—31no—
94C2×C3×Dic6C2×D311yescharacteristic
95C2×#627—32no—
96C2×#626—31no—
97C2×#624—32no—
98#3797—64no—
99#3757—32no—
100C4⋊(C3⋊Dic3)—31no—
101C3×#217—32no—
102C3×#217—32no—
103C3×C4×Dic3—31no—
104#3724—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—31no—
107#3722—31no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—31no—
110Dic32—32no—
111C9×#41—64no—
112C2×Q8×C9—31no—
113C2×D4×C9—21no—
114C2×D4×C9—32no—
115C2×D4×C9C2×D311yescharacteristic
116C22×C4×C9—32no—
117C22×C4×C9—31no—
118C2×#611—32no—
119C2×#611—31no—
120C2×#611—62no—
121Q8×D9—64no—
122C2×C4×D9—31no—
123C2×C4×D9—32no—
124C2×C4×D9—61no—
125C2×Dic18—32no—
126C2×Dic18C2×D311yescharacteristic
127#3676—32no—
128#3676—31no—
129#3676—32no—
130#3676—62no—
131#3676C2×D312yes—
132#3676C2×D312yes—
133(C2×D9)⋊C4—31no—
134C4⋊Dic9—31no—
135C4⋊Dic9—32no—
136C4⋊Dic9—61no—
137Dic9⋊C4—31no—
138Dic9⋊C4—32no—
139C4×Dic9—32no—
140C4×Dic9C2×D311yescharacteristic
141C22×C3×C9—22no—
142C22×C3×C9—22no—
143C22×C3×C9—22no—
144C22×C3×C9C2411yescharacteristic
145C2×C3⋊D9—21no—
146C2×C3⋊D9—21no—
147C2×C3⋊D9#4112yescomplemented
148C2×D3×C9—21no—
149C2×D3×C9—21no—
150C2×D3×C9—21no—
151C3×C4×C9C2×D412yes—
152C3⋊Dic9—21no—
153C3⋊Dic9—22no—
154C9×Dic3—21no—
155C9×Dic3C2×D414yes—
156C3×Dic9—22no—
157C3×Dic9—21no—
158C3×Dic9—21no—
159C2×C3×#41—31no—
160C2×D3×Q8—91no—
161C2×D3×Q8—91no—
162C2×#237D911yescharacteristic, complemented
163C3×#124—91no—
164#946—91no—
165Q8×Dic3—91no—
166#938—92no—
167#938—91no—
168#938—91no—
169#938—181no—
170#936—31no—
171#936—92no—
172C4×#67—91no—
173C4×#67—92no—
174#924—92no—
175#924—91no—
176#885—92no—
177#884—92no—
178#67⋊C4—92no—
179#878—92no—
180C4×D12—91no—
181C4×Dic6—91no—
182C23×C32C22×D312yes—
183C22×C3⋊D3—31no—
184C22×C3×D3C22×D311yescharacteristic
185C32×D4—22no—
186C2×C32×C4C22×D311yescharacteristic
187#638—64no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—31no—
190C4×C3⋊D3—62no—
191C3×#67—24no—
192C2×C3×Dic3—24no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3C22×D311yescharacteristic
197C2×C3×Dic3C22×D312yes—
198C3×C4×D3—22no—
199C3×Dic6—62no—
200C3×Dic6#6714yes—
201#627—64no—
202#626—62no—
203#624—64no—
204C23×C9—22no—
205C23×C9—31no—
206C23×C9C22×D312yes—
207C22×D9—31no—
208C22×D9—61no—
209Q8×C9—34no—
210D4×C9—22no—
211D4×C9—42no—
212D4×C9—64no—
213C2×C4×C9—21no—
214C2×C4×C9—32no—
215C2×C4×C9—31no—
216C2×C4×C9—64no—
217C2×C4×C9—62no—
218C2×C4×C9C22×D311yescharacteristic
219#611—62no—
220#611—64no—
221#611—124no—
222C2×Dic9—32no—
223C2×Dic9—31no—
224C2×Dic9—62no—
225C2×Dic9—61no—
226C2×Dic9C22×D311yescharacteristic
227C2×Dic9C22×D311yescharacteristic
228C2×Dic9C22×D312yes—
229C4×D9—64no—
230C4×D9—62no—
231C4×D9—122no—
232Dic18—22no—
233Dic18—64no—
234#548—271noSylow
235C2×C3×C9—42no—
236C2×C3×C9C22×D411yescharacteristic
237C2×C3×C9C2×#4111yescharacteristic
238C2×C3×C92−1+411yescharacteristic
239C3⋊D9—22no—
240D3×C9—41no—
241C3×#41—64no—
242C2×C3×Q8—31no—
243C2×C3×Q8—91no—
244C2×C3×D4—21no—
245C2×C3×D4—32no—
246C2×C3×D4C2×D911yescharacteristic
247C2×C3×D4D3211yescharacteristic
248C22×C3×C4—32no—
249C22×C3×C4—31no—
250C2×#67—91no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—182no—
254C2×#67C2×D912yes—
255C22×Dic3C2×D912yes—
256D3×Q8—184no—
257D3×Q8—184no—
258#237—24no—
259C2×D12—91no—
260C2×C4×D3—92no—
261C2×C4×D3—91no—
262C2×C4×D3—91no—
263C2×C4×D3—92no—
264C2×C4×D3—181no—
265C2×C4×D3C2×D911yescharacteristic
266C2×Dic6—31no—
267C2×Dic6—92no—
268C2×Dic6—92no—
269C2×Dic6C2×D911yescharacteristic
270C3×C22⋊C4—92no—
271C3×C22⋊C4—92no—
272C3×C42—91no—
273#217—32no—
274#217—32no—
275#217—92no—
276#217—91no—
277#217—92no—
278#217—92no—
279#217—182no—
280(C2×D3)⋊C4—91no—
281(C2×D3)⋊C4—91no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—91no—
284C4⋊Dic3—92no—
285C4⋊Dic3—91no—
286C4⋊Dic3—91no—
287C4⋊Dic3—91no—
288C4⋊Dic3—181no—
289Dic3⋊C4—91no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293C4×Dic3—31no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32—22no—
299C22×C32C23×D311yescharacteristic
300C2×C3⋊D3—32no—
301C2×C3⋊D3—61no—
302C2×C3⋊D3—61no—
303C2×C3×D3—21no—
304C2×C3×D3—21no—
305C2×C3×D3—21no—
306C32×C4C2×#6712yes—
307C3⋊Dic3—62no—
308C3⋊Dic3—61no—
309C3×Dic3—21no—
310C3×Dic3—61no—
311C3×Dic3—61no—
312C3×Dic3—62no—
313C3×Dic3C2×#6714yes—
314C22×C9—22no—
315C22×C9—22no—
316C22×C9—22no—
317C22×C9—21no—
318C22×C9—42no—
319C22×C9—42no—
320C22×C9—42no—
321C22×C9—61no—
322C22×C9—61no—
323C22×C9—61no—
324C22×C9C23×D311yescharacteristic
325C2×D9—32no—
326C2×D9—62no—
327C2×D9—61no—
328C2×D9—61no—
329C2×D9—121no—
330C2×D9—121no—
331C4×C9—22no—
332C4×C9—34no—
333C4×C9—61no—
334C4×C9D3×D412yes—
335Dic9—22no—
336Dic9—21no—
337Dic9—21no—
338Dic9—61no—
339Dic9—62no—
340Dic9—121no—
341Dic9—122no—
342C2×#41—31no—
343C22×Q8—271no—
344#124—91no—
345#124—272no—
346C22⋊Q8—272no—
347C22⋊Q8—271no—
348#121—272no—
349#121—271no—
350C4×Q8—271no—
351C4×D4—271no—
352C4×D4—272no—
353C3×C9#54811yescharacteristic, complemented, Sylow
354C23×C3—22no—
355C23×C3—31no—
356C23×C3C22×D912yes—
357C23×C3C2×D3212yes—
358C22×D3—91no—
359C22×D3—91no—
360C22×D3—181no—
361C22×D3C22×D911yescharacteristic
362C3×Q8—34no—
363C3×Q8—182no—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—42no—
367C3×D4—64no—
368C2×C3×C4—21no—
369C2×C3×C4—32no—
370C2×C3×C4—31no—
371C2×C3×C4—64no—
372C2×C3×C4—62no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—92no—
376C2×C3×C4C22×D911yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378#67—24no—
379#67—182no—
380#67—184no—
381#67—184no—
382#67—364no—
383C2×Dic3—24no—
384C2×Dic3—31no—
385C2×Dic3—31no—
386C2×Dic3—32no—
387C2×Dic3—92no—
388C2×Dic3—91no—
389C2×Dic3—92no—
390C2×Dic3—91no—
391C2×Dic3—182no—
392C2×Dic3—181no—
393C2×Dic3C22×D911yescharacteristic
394C2×Dic3C22×D912yes—
395D12—182no—
396C4×D3—22no—
397C4×D3—184no—
398C4×D3—182no—
399C4×D3—182no—
400C4×D3—184no—
401C4×D3—362no—
402Dic6—62no—
403Dic6—184no—
404Dic6—184no—
405Dic6#61114yescomplemented
406C2×C32—42no—
407C2×C32D3×#4111yescharacteristic
408C2×C32#101011yescharacteristic
409C2×C32C22×#6711yescharacteristic
410C3⋊D3—62no—
411C3×D3—41no—
412C2×C9—21no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—42no—
416C2×C9—82no—
417C2×C9—121no—
418C2×C9C2×D3×D411yescharacteristic
419C2×C9D3×#4111yescharacteristic
420C2×C9#101011yescharacteristic
421D9—62no—
422D9—122no—
423#41—64no—
424C2×Q8—31no—
425C2×Q8—91no—
426C2×Q8—272no—
427C2×Q8—544no—
428C2×D4—32no—
429C2×D4—271no—
430C2×D4—272no—
431C2×D4D3×D911yescharacteristic
432C22×C4—32no—
433C22×C4—31no—
434C22×C4—272no—
435C22×C4—271no—
436C4⋊C4—272no—
437C4⋊C4—271no—
438C4⋊C4—272no—
439C4⋊C4—271no—
440C22⋊C4—92no—
441C22⋊C4—92no—
442C22⋊C4—271no—
443C22⋊C4—271no—
444C22⋊C4—272no—
445C22⋊C4—272no—
446C42—91no—
447C42—272no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—21no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—61no—
459C22×C3—61no—
460C22×C3—61no—
461C22×C3C23×D911yescharacteristic
462C22×C3C22×D3211yescharacteristic
463C2×D3—21no—
464C2×D3—21no—
465C2×D3—21no—
466C2×D3—92no—
467C2×D3—92no—
468C2×D3—182no—
469C2×D3—181no—
470C2×D3—181no—
471C2×D3—181no—
472C2×D3—181no—
473C2×D3—361no—
474C2×D3—361no—
475C3×C4—22no—
476C3×C4—34no—
477C3×C4—61no—
478C3×C4—181no—
479C3×C4—181no—
480C3×C4—182no—
481C3×C4C2×#61112yes—
482C3×C4D3×#6712yes—
483Dic3—21no—
484Dic3—61no—
485Dic3—61no—
486Dic3—62no—
487Dic3—182no—
488Dic3—181no—
489Dic3—182no—
490Dic3—181no—
491Dic3—361no—
492Dic3—362no—
493Dic3C2×#61114yes—
494C32#583111yescharacteristic
495C9—21no—
496C9#567711yescharacteristic, complemented
497C23—31no—
498C23—271no—
499C23C2×D3×D912yes—
500Q8—34no—
501Q8—182no—
502Q8—544no—
503D4—22no—
504D4—64no—
505D4—542no—
506D4—544no—
507C2×C4—32no—
508C2×C4—31no—
509C2×C4—64no—
510C2×C4—62no—
511C2×C4—91no—
512C2×C4—91no—
513C2×C4—92no—
514C2×C4—272no—
515C2×C4—271no—
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—121no—
526C2×C3#41×D911yescharacteristic
527C2×C3#6371611yescharacteristic
528C2×C3C22×#61111yescharacteristic
529C2×C3#6430911yescharacteristic
530C2×C3#6431211yescharacteristic
531C2×C3C2×D3×#6711yescharacteristic
532D3—41no—
533D3—182no—
534D3—182no—
535D3—362no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—61no—
540C22—61no—
541C22—61no—
542C22—272no—
543C22—541no—
544C22—541no—
545C22C22×D3×D911yescharacteristic
546C4—34no—
547C4—61no—
548C4—181no—
549C4—182no—
550C4—181no—
551C4—542no—
552C4—541no—
553C4D3×#61112yes—
554C3—21no—
555C3#1058993311yescharacteristic, complemented
556C3#1059523711yescharacteristic
557C2—42no—
558C2—121no—
559C2—542no—
560C2#1172249411yescharacteristic
561C2#1172249811yescharacteristic
562C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#548
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×C9×#237C2semidirect
2. C2×(C3⋊D9)⋊Q8C2semidirect
3. #11721838C2semidirect
4. C3×#63498C2semidirect
5. #11721263C2semidirect
6. #11721262C2semidirect
7. #11721249C2semidirect
8. #11721240C2semidirect
9. C4×#6195C2semidirect
10. #11720960C2semidirect
11. Dic6×Dic9C2semidirect
57. #63498D3semidirect
79. C9×Dic6D4semidirect
147. C2×C3⋊D9#41semidirect
162. C2×#237D9semidirect
353. C3×C9#548semidirect
405. Dic6#611semidirect
496. C9#5677semidirect
555. C3#10589933semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
C2×(C3⋊D9)⋊Q82
#117218382
C3×#634982
#117212632
#117212622
#117212492
#117212402
C4×#61952
#117209602
Dic6×Dic92
#105952373
#105899333