Finite Group Databasebeta

← #13151096

Subgroups — #13151096

Order 1728.

814 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721838C211yesmaximal, characteristic, complemented
2C3×#63502C211yesmaximal, characteristic, complemented
3D4×C9×Dic3C211yesmaximal, characteristic, complemented
4#11721277C212yesmaximal, complemented
5#11721271C212yesmaximal, complemented
6#11721269C212yesmaximal, complemented
7#11721259C212yesmaximal, complemented
8#11721007C211yesmaximal, characteristic, complemented
9Dic3×D36C211yesmaximal, characteristic, complemented
10#11720935C211yesmaximal, characteristic, complemented
11#11720931C211yesmaximal, characteristic, complemented
12#10595218—31nomaximal
13#10589664—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×#611C2212yes—
18C2×C3×#611C2212yes—
19C22×C9×Dic3C2212yes—
20C2×C3×D36C2211yescharacteristic
21C2×#6192C2212yes—
22C2×Dic3×D9C2212yes—
23#88551C2212yes—
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25C9×#217C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C4×C9×Dic3C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29#88445C2211yescharacteristic
30#88445C2211yescharacteristic
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33Dic3⋊Dic9C2212yes—
34#88442C2211yescharacteristic
35#88442C2212yes—
36#88442C2211yescharacteristic
37#64147—31no—
38C3×#940—31no—
39C3×D4×Dic3D311yescharacteristic
40#63978—32no—
41#63971—32no—
42#63969—32no—
43#63961—32no—
44#63916—31no—
45Dic3×D12—31no—
46#63874—31no—
47#63869—31no—
48C4×D4×C9—31no—
49#63502D311yescharacteristic, complemented
50#63500—31no—
51#63500—61no—
52#63492—32no—
53#63491—32no—
54#63457—31no—
55#63452—31no—
56#63446—32no—
57#63445—32no—
58C4×D36—31no—
59#63434—31no—
60C23×C3×C9C2312yes—
61C22×C3⋊D9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C3×D4×C9—22no—
64C2×C3×C4×C9C2311yescharacteristic
65#6233—42no—
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2311yescharacteristic
68C4×C3⋊D9—22no—
69C3×#611—42no—
70C3×#611—42no—
71C2×C9×Dic3—42no—
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2312yes—
74C2×C9×Dic3C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C3×D36—22no—
78#6192—42no—
79Dic3×D9—42no—
80#5719—91no—
81#5562—91no—
82C2×C32×D4C2×D311yescharacteristic
83C2×#638—32no—
84C2×C4×C3⋊D3—31no—
85C2×C3×#67—32no—
86C2×C3×#67—32no—
87C22×C3×Dic3C2×D312yes—
88C2×C3×D12—31no—
89C2×#626—32no—
90C2×D3×Dic3—32no—
91#3757—32no—
92C4⋊(C3⋊Dic3)—31no—
93C3×#217C2×D312yes—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4×Dic3—31no—
96C3×C4×Dic3C2×D311yescharacteristic
97#3724—32no—
98#3724—31no—
99#3724—31no—
100Dic3⋊Dic3—32no—
101#3722—31no—
102#3722—31no—
103(C2×D3)⋊Dic3—32no—
104(C2×D3)⋊Dic3—32no—
105C2×D4×C9—21no—
106C2×D4×C9C2×D311yescharacteristic
107C22×C4×C9—32no—
108C2×#611—32no—
109C2×#611—62no—
110C2×#611C2×D312yes—
111C2×#611C2×D312yes—
112C2×D36—32no—
113C2×D36C2×D311yescharacteristic
114C2×C4×D9—31no—
115C2×C4×D9—32no—
116C2×C4×D9—61no—
117C9×C4⋊C4—31no—
118C9×C22⋊C4—32no—
119C42×C9—31no—
120#3676—32no—
121#3676—62no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—31no—
125(C2×D9)⋊C4—31no—
126C4⋊Dic9—31no—
127C4⋊Dic9—32no—
128C4⋊Dic9—61no—
129Dic9⋊C4—31no—
130Dic9⋊C4—32no—
131Dic9⋊C4—31no—
132C4×Dic9C2×D311yescharacteristic
133C22×C3×C9—22no—
134C22×C3×C9—22no—
135C22×C3×C9—22no—
136C22×C3×C9C2411yescharacteristic
137C2×C3⋊D9—21no—
138C2×C3⋊D9—21no—
139C2×C3⋊D9—21no—
140C2×C3×D9—22no—
141C2×C3×D9—22no—
142C2×C3×D9—22no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—21no—
145C3⋊Dic9—22no—
146C9×Dic3—21no—
147C9×Dic3—22no—
148C9×Dic3—21no—
149C3×Dic9—21no—
150C3×Dic9—21no—
151C3×#127—91no—
152C3×C4×D4—31no—
153#940—31no—
154#939—92no—
155#939—92no—
156#938—91no—
157#938—91no—
158#938—91no—
159#938—181no—
160#935—92no—
161#935—92no—
162D4×Dic3—91no—
163D4×Dic3D911yescharacteristic, complemented
164#930—92no—
165#929—92no—
166#895—91no—
167#890—91no—
168#884—92no—
169#883—92no—
170C4×D12—91no—
171#872—91no—
172#872—91no—
173#870—91no—
174C23×C32C22×D312yes—
175C22×C3⋊D3—31no—
176C22×C3×D3—32no—
177C32×D4—22no—
178C2×C32×C4C22×D311yescharacteristic
179#638—122no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—32no—
182C4×C3⋊D3—62no—
183C3×#67—122no—
184C3×#67—122no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—42no—
188C2×C3×Dic3C22×D311yescharacteristic
189C2×C3×Dic3C22×D312yes—
190C2×C3×Dic3C22×D311yescharacteristic
191C3×D12—62no—
192#626—122no—
193D3×Dic3—122no—
194C23×C9—22no—
195C23×C9C22×D312yes—
196C22×D9—31no—
197C22×D9—61no—
198C22×D9C22×D312yes—
199D4×C9—22no—
200D4×C9—42no—
201C2×C4×C9—21no—
202C2×C4×C9—32no—
203C2×C4×C9—31no—
204C2×C4×C9—31no—
205C2×C4×C9—122no—
206C2×C4×C9C22×D311yescharacteristic
207#611—42no—
208#611—42no—
209#611—122no—
210#611—124no—
211C2×Dic9—31no—
212C2×Dic9—32no—
213C2×Dic9—62no—
214C2×Dic9—61no—
215C2×Dic9C22×D311yescharacteristic
216C2×Dic9C22×D311yescharacteristic
217D36—22no—
218D36—122no—
219C4×D9—62no—
220C4×D9—122no—
221C4×D9—122no—
222#540—271noSylow
223C2×C3×C9—42no—
224C2×C3×C9C2×#4111yescharacteristic
225C2×C3×C9H(5,𝔽2)11yescharacteristic
226C2×C3×C9H(5,𝔽2)11yescharacteristic
227C3⋊D9—41no—
228C3×D9—42no—
229C2×C3×D4—21no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—91no—
233C2×C3×D4C2×D911yescharacteristic
234C2×C3×D4D3211yescharacteristic
235C22×C3×C4—32no—
236C2×#67—32no—
237C2×#67—32no—
238C2×#67—92no—
239C2×#67—92no—
240C2×#67—92no—
241C2×#67—182no—
242C22×Dic3—92no—
243C22×Dic3C2×D912yes—
244C2×D12—31no—
245C2×D12—92no—
246C2×C4×D3—91no—
247C2×C4×D3—91no—
248C2×C4×D3—92no—
249C2×C4×D3—181no—
250C3×C4⋊C4—31no—
251C3×C22⋊C4—32no—
252C3×C42—31no—
253C3×C42—91no—
254#217—92no—
255#217—92no—
256#217—92no—
257#217—92no—
258#217—182no—
259#217C2×D912yes—
260(C2×D3)⋊C4—91no—
261(C2×D3)⋊C4—91no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—91no—
266C4⋊Dic3—92no—
267C4⋊Dic3—91no—
268C4⋊Dic3—91no—
269C4⋊Dic3—181no—
270C4⋊Dic3C2×D911yescharacteristic
271Dic3⋊C4—91no—
272Dic3⋊C4—91no—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—91no—
277Dic3⋊C4—91no—
278C4×Dic3—31no—
279C4×Dic3C2×D911yescharacteristic
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32C23×D311yescharacteristic
284C2×C3⋊D3—61no—
285C2×C3⋊D3—61no—
286C2×C3⋊D3—61no—
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C32×C4#23512yes—
291C3⋊Dic3—61no—
292C3⋊Dic3—62no—
293C3×Dic3—22no—
294C3×Dic3—21no—
295C3×Dic3—21no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C22×C9—22no—
299C22×C9—22no—
300C22×C9—22no—
301C22×C9—21no—
302C22×C9—42no—
303C22×C9—42no—
304C22×C9—42no—
305C22×C9C23×D311yescharacteristic
306C2×D9—22no—
307C2×D9—22no—
308C2×D9—22no—
309C2×D9—62no—
310C2×D9—61no—
311C2×D9—61no—
312C2×D9—61no—
313C2×D9—121no—
314C2×D9—121no—
315C4×C9—22no—
316C4×C9—61no—
317C4×C9—62no—
318C4×C9—61no—
319C4×C9#23712yes—
320Dic9—21no—
321Dic9—21no—
322Dic9—61no—
323Dic9—62no—
324Dic9—121no—
325Dic9—122no—
326#127—91no—
327#125—271no—
328#123—272no—
329#123—272no—
330#121—272no—
331#121—271no—
332#121—272no—
333#121—271no—
334C4×D4—31no—
335C4×D4—271no—
336#117—271no—
337C3×C9#54011yescharacteristic, complemented, Sylow
338C23×C3—22no—
339C23×C3—92no—
340C23×C3C22×D912yes—
341C23×C3C2×D3212yes—
342C22×D3—32no—
343C22×D3—91no—
344C22×D3—91no—
345C22×D3—181no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—42no—
349C3×D4—182no—
350C3×D4—362no—
351C3×D4—362no—
352C2×C3×C4—21no—
353C2×C3×C4—32no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—91no—
357C2×C3×C4—91no—
358C2×C3×C4—122no—
359C2×C3×C4C22×D911yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361#67—122no—
362#67—122no—
363#67—362no—
364#67—362no—
365#67—362no—
366#67—364no—
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—42no—
370C2×Dic3—91no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—91no—
374C2×Dic3—182no—
375C2×Dic3—181no—
376C2×Dic3—362no—
377C2×Dic3C22×D911yescharacteristic
378C2×Dic3C22×D911yescharacteristic
379C2×Dic3C22×D912yes—
380D12—62no—
381D12—362no—
382C4×D3—182no—
383C4×D3—182no—
384C4×D3—362no—
385C4×D3—362no—
386C2×C32—42no—
387C2×C32C2×#23511yescharacteristic
388C2×C32#100411yescharacteristic
389C2×C32#100911yescharacteristic
390C3⋊D3—121no—
391C3×D3—122no—
392C2×C9—21no—
393C2×C9—21no—
394C2×C9—21no—
395C2×C9—42no—
396C2×C9—82no—
397C2×C9C2×#23711yescharacteristic
398C2×C9#100411yescharacteristic
399C2×C9#100411yescharacteristic
400D9—42no—
401D9—122no—
402D9—121no—
403C2×D4—92no—
404C2×D4—92no—
405C2×D4—91no—
406C2×D4—272no—
407C2×D4—272no—
408C2×D4D3×D911yescharacteristic
409C22×C4—32no—
410C22×C4—272no—
411C22×C4—271no—
412C4⋊C4—31no—
413C4⋊C4—271no—
414C4⋊C4—271no—
415C4⋊C4—272no—
416C4⋊C4—272no—
417C4⋊C4—271no—
418C22⋊C4—32no—
419C22⋊C4—271no—
420C22⋊C4—271no—
421C22⋊C4—272no—
422C22⋊C4—272no—
423C22⋊C4—272no—
424C42—31no—
425C42—91no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—21no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—182no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3C23×D911yescharacteristic
440C22×C3C22×D3211yescharacteristic
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—182no—
445C2×D3—181no—
446C2×D3—181no—
447C2×D3—181no—
448C2×D3—181no—
449C2×D3—181no—
450C2×D3—181no—
451C2×D3—361no—
452C2×D3—361no—
453C3×C4—22no—
454C3×C4—62no—
455C3×C4—61no—
456C3×C4—61no—
457C3×C4—181no—
458C3×C4—181no—
459C3×C4#369712yes—
460C3×C4#380412yes—
461Dic3—21no—
462Dic3—22no—
463Dic3—21no—
464Dic3—61no—
465Dic3—61no—
466Dic3—182no—
467Dic3—181no—
468Dic3—181no—
469Dic3—182no—
470Dic3—361no—
471Dic3—362no—
472C32#556211yescharacteristic
473C9—21no—
474C9#571911yescharacteristic, complemented
475C23—92no—
476C23—271no—
477C23C2×D3×D912yes—
478D4—22no—
479D4—182no—
480D4—362no—
481D4—362no—
482D4—1082no—
483D4—1082no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—32no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—122no—
490C2×C4—271no—
491C2×C4—272no—
492C2×C4—542no—
493C2×C4—1082no—
494C2×C4C2×D3×D911yescharacteristic
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—42no—
499C2×C3—42no—
500C2×C3—82no—
501C2×C3—362no—
502C2×C3C2×#369711yescharacteristic
503C2×C3#6371011yescharacteristic
504C2×C3#6371511yescharacteristic
505C2×C3#6431311yescharacteristic
506C2×C3#6431411yescharacteristic
507C2×C3C2×#380411yescharacteristic
508D3—122no—
509D3—362no—
510D3—361no—
511D3—361no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—182no—
516C22—182no—
517C22—182no—
518C22—541no—
519C22—541no—
520C22—541no—
521C22C22×D3×D911yescharacteristic
522C4—61no—
523C4—62no—
524C4—61no—
525C4—181no—
526C4—181no—
527C4—542no—
528C4—541no—
529C4#8865812yes—
530C3—21no—
531C3#1058966411yescharacteristic, complemented
532C3#1059521811yescharacteristic
533C2—42no—
534C2—362no—
535C2—1081no—
536C2#1172249911yescharacteristic
537C2#1172250111yescharacteristic
538C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#540
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. #11721838C2semidirect
2. C3×#63502C2semidirect
3. D4×C9×Dic3C2semidirect
4. #11721277C2semidirect
5. #11721271C2semidirect
6. #11721269C2semidirect
7. #11721259C2semidirect
8. #11721007C2semidirect
9. Dic3×D36C2semidirect
10. #11720935C2semidirect
11. #11720931C2semidirect
49. #63502D3semidirect
163. D4×Dic3D9semidirect
337. C3×C9#540semidirect
474. C9#5719semidirect
531. C3#10589664semidirect

Maximal subgroups

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

subgroupindex
#117218382
C3×#635022
D4×C9×Dic32
#117212772
#117212712
#117212692
#117212592
#117210072
Dic3×D362
#117209352
#117209312
#105952183
#105896643