Finite Group Databasebeta

← #13150515

Subgroups — #13150515

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#11721819C211yesmaximal, characteristic, complemented
2C9×#930C211yesmaximal, characteristic, complemented
3C3×#63488C211yesmaximal, characteristic, complemented
4#11721282C212yesmaximal, complemented
5#11721262C212yesmaximal, complemented
6#11721040C211yesmaximal, characteristic, complemented
7D12×Dic9C211yesmaximal, characteristic, complemented
8#11721020C212yesmaximal, complemented
9#11720992C211yesmaximal, characteristic, complemented
10#11720958C212yesmaximal, complemented
11#11720935C211yesmaximal, characteristic, complemented
12#10594985—31nomaximal
13#10589721—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C9×#67C2212yes—
18C2×C9×D12C2211yescharacteristic
19C2×#6195C2212yes—
20C2×#6195C2212yes—
21C2×D3×Dic9C2212yes—
22C2×(C3⋊D9)⋊C4C2212yes—
23#88536C2212yes—
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25C3×#3676C2211yescharacteristic
26C3×#3676C2211yescharacteristic
27C9×(C2×D3)⋊C4C2212yes—
28C9×C4⋊Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37#64139—31no—
38C3×#930D311yescharacteristic
39C3×#926—31no—
40#63971—32no—
41#63958—32no—
42#63911—31no—
43#63894—31no—
44Dic3×D12—31no—
45#63890—32no—
46#63869—31no—
47#63854—32no—
48C9×#121—31no—
49#63502—31no—
50#63501—32no—
51#63500—32no—
52D4×Dic9—31no—
53#63492—31no—
54#63492—61no—
55#63488D311yescharacteristic, complemented
56#63455—31no—
57#63452—31no—
58#63448—32no—
59#63445—32no—
60C23×C3×C9C2311yescharacteristic
61C22×C3⋊D9C2312yes—
62C22×D3×C9C2312yes—
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66#6233—42no—
67C2×C3⋊Dic9C2312yes—
68C4⋊(C3⋊D9)—22no—
69C9×#67—42no—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C9×D12—22no—
76#6195—42no—
77#6195—42no—
78D3×Dic9—42no—
79(C3⋊D9)⋊C4—42no—
80#5619—91no—
81#5535—91no—
82C22×C32×C4C2×D311yescharacteristic
83C2×#638—32no—
84C2×C4⋊(C3⋊D3)—31no—
85C2×C3×#67C2×D312yes—
86C2×C3×D12C2×D311yescharacteristic
87C2×#626—32no—
88C2×#626—32no—
89C2×#624—32no—
90C2×D3×Dic3—32no—
91#3752—32no—
92C4⋊(C3⋊Dic3)—31no—
93C3×#217—31no—
94C3×#217—31no—
95C3×(C2×D3)⋊C4C2×D312yes—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4×Dic3—31no—
100C3×C4×Dic3—31no—
101#3724—32no—
102Dic3⋊Dic3—32no—
103#3722—32no—
104(C2×D3)⋊Dic3—32no—
105C2×D4×C9—32no—
106C2×D4×C9—31no—
107C22×C4×C9—21no—
108C22×C4×C9C2×D311yescharacteristic
109C2×#611—32no—
110C2×#611—32no—
111C2×#611—32no—
112C2×#611—62no—
113C22×Dic9—32no—
114C2×D36—31no—
115C2×D36—61no—
116C2×C4×D9—32no—
117C9×C4⋊C4—31no—
118C9×C22⋊C4—32no—
119#3676—32no—
120#3676C2×D311yescharacteristic
121#3676C2×D311yescharacteristic
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—62no—
125C4⋊Dic9—31no—
126C4⋊Dic9—32no—
127C4⋊Dic9—61no—
128C4⋊Dic9C2×D311yescharacteristic
129C4⋊Dic9C2×D311yescharacteristic
130Dic9⋊C4—32no—
131C4×Dic9C2×D311yescharacteristic
132C4×Dic9C2×D311yescharacteristic
133C22×C3×C9—21no—
134C22×C3×C9—21no—
135C22×C3×C9—21no—
136C22×C3×C9C2411yescharacteristic
137C2×C3⋊D9—22no—
138C2×C3⋊D9—22no—
139C2×C3⋊D9—22no—
140C2×D3×C9—22no—
141C2×D3×C9—22no—
142C2×D3×C9—22no—
143C3×C4×C9—21no—
144C3×C4×C9#4112yes—
145C3⋊Dic9—22no—
146C9×Dic3—22no—
147C3×Dic9—21no—
148C3×Dic9—21no—
149C3×Dic9—21no—
150C3×Dic9—21no—
151C3×#121—31no—
152C3×#117—91no—
153#940—91no—
154#939—92no—
155#938—92no—
156D4×Dic3—91no—
157#930—91no—
158#930—91no—
159#930—181no—
160#930D911yescharacteristic, complemented
161#926—31no—
162#894—92no—
163#894—92no—
164#893—91no—
165#890—91no—
166#886—92no—
167#884—92no—
168#884—92no—
169#883—92no—
170#874—91no—
171C4×D12—91no—
172C4×D12—91no—
173#870—91no—
174C23×C32C22×D311yescharacteristic
175C22×C3⋊D3—32no—
176C22×C3×D3C22×D312yes—
177C2×C32×C4—22no—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180#638—122no—
181C2×C3⋊Dic3—32no—
182C4⋊(C3⋊D3)—62no—
183C3×#67—42no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3C22×D312yes—
189C3×D12—22no—
190#626—122no—
191#626—122no—
192#624—122no—
193D3×Dic3—122no—
194C23×C9—21no—
195C23×C9—32no—
196C23×C9C22×D311yescharacteristic
197C22×D9—32no—
198C22×D9—62no—
199D4×C9—62no—
200D4×C9—122no—
201C2×C4×C9—22no—
202C2×C4×C9—21no—
203C2×C4×C9—21no—
204C2×C4×C9—32no—
205C2×C4×C9—42no—
206C2×C4×C9C22×D311yescharacteristic
207C2×C4×C9C22×D311yescharacteristic
208#611—122no—
209#611—122no—
210#611—122no—
211#611—124no—
212C2×Dic9—32no—
213C2×Dic9—62no—
214C2×Dic9—122no—
215C2×Dic9C22×D311yescharacteristic
216C2×Dic9C22×D311yescharacteristic
217C2×Dic9C22×D311yescharacteristic
218C2×Dic9C22×D311yescharacteristic
219D36—62no—
220D36—122no—
221C4×D9—122no—
222#540—271noSylow
223C2×C3×C9—41no—
224C2×C3×C9C2×#4111yescharacteristic
225C2×C3×C9H(5,𝔽2)11yescharacteristic
226C2×C3×C9H(5,𝔽2)11yescharacteristic
227C3⋊D9—42no—
228D3×C9—42no—
229C2×C3×D4—32no—
230C2×C3×D4—31no—
231C22×C3×C4—21no—
232C22×C3×C4C2×D911yescharacteristic
233C22×C3×C4D3211yescharacteristic
234C2×#67—92no—
235C2×#67—92no—
236C2×#67—92no—
237C2×#67—92no—
238C2×#67—182no—
239C2×#67C2×D912yes—
240C22×Dic3—92no—
241C2×D12—91no—
242C2×D12—91no—
243C2×D12—92no—
244C2×D12—92no—
245C2×D12—181no—
246C2×D12C2×D911yescharacteristic
247C2×C4×D3—92no—
248C2×C4×D3—92no—
249C2×C4×D3—92no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—91no—
253C3×C22⋊C4—32no—
254C3×C22⋊C4—91no—
255C3×C22⋊C4—91no—
256C3×C42—91no—
257C3×C42—91no—
258#217—31no—
259#217—31no—
260#217—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—182no—
267(C2×D3)⋊C4C2×D912yes—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—92no—
271C4⋊Dic3—91no—
272C4⋊Dic3—91no—
273C4⋊Dic3—181no—
274C4⋊Dic3C2×D911yescharacteristic
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278C4×Dic3—31no—
279C4×Dic3—31no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32C23×D311yescharacteristic
284C2×C3⋊D3—62no—
285C2×C3⋊D3—62no—
286C2×C3⋊D3—62no—
287C2×C3×D3—22no—
288C2×C3×D3—22no—
289C2×C3×D3—22no—
290C32×C4—21no—
291C32×C4#23712yes—
292C3⋊Dic3—62no—
293C3×Dic3—22no—
294C3×Dic3—61no—
295C3×Dic3—61no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C22×C9—22no—
299C22×C9—21no—
300C22×C9—21no—
301C22×C9—21no—
302C22×C9—21no—
303C22×C9—41no—
304C22×C9—41no—
305C22×C9—62no—
306C22×C9—62no—
307C22×C9—62no—
308C22×C9C23×D311yescharacteristic
309C2×D9—62no—
310C2×D9—62no—
311C2×D9—62no—
312C2×D9—122no—
313C2×D9—122no—
314C2×D9—122no—
315C4×C9—21no—
316C4×C9—22no—
317C4×C9—41no—
318C4×C9—62no—
319C4×C9#23512yes—
320Dic9—21no—
321Dic9—21no—
322Dic9—21no—
323Dic9—21no—
324Dic9—62no—
325Dic9—122no—
326#127—271no—
327#125—271no—
328#123—272no—
329#123—272no—
330#121—31no—
331#121—272no—
332#121—272no—
333#121—271no—
334C4×D4—271no—
335C4×D4—271no—
336#117—91no—
337C3×C9#54011yescharacteristic, complemented, Sylow
338C23×C3—21no—
339C23×C3—32no—
340C23×C3C22×D911yescharacteristic
341C23×C3C2×D3211yescharacteristic
342C22×D3—92no—
343C22×D3—92no—
344C22×D3—182no—
345C22×D3C22×D912yes—
346C3×D4—62no—
347C3×D4—122no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—32no—
353C2×C3×C4—42no—
354C2×C3×C4—91no—
355C2×C3×C4—91no—
356C2×C3×C4—91no—
357C2×C3×C4—91no—
358C2×C3×C4C22×D911yescharacteristic
359C2×C3×C4C22×D911yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362#67—42no—
363#67—362no—
364#67—362no—
365#67—362no—
366#67—362no—
367#67—364no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—31no—
371C2×Dic3—31no—
372C2×Dic3—92no—
373C2×Dic3—92no—
374C2×Dic3—182no—
375C2×Dic3—362no—
376C2×Dic3C22×D912yes—
377D12—22no—
378D12—182no—
379D12—182no—
380D12—362no—
381D12—362no—
382D12—362no—
383C4×D3—362no—
384C4×D3—362no—
385C4×D3—362no—
386C2×C32—41no—
387C2×C32C2×#23711yescharacteristic
388C2×C32#100411yescharacteristic
389C2×C32#100911yescharacteristic
390C3⋊D3—122no—
391C3×D3—42no—
392C2×C9—21no—
393C2×C9—21no—
394C2×C9—21no—
395C2×C9—42no—
396C2×C9—41no—
397C2×C9—122no—
398C2×C9C2×#23511yescharacteristic
399C2×C9#100911yescharacteristic
400C2×C9#100911yescharacteristic
401D9—122no—
402D9—242no—
403C2×D4—32no—
404C2×D4—31no—
405C2×D4—272no—
406C2×D4—272no—
407C2×D4—272no—
408C2×D4—271no—
409C22×C4—272no—
410C22×C4—272no—
411C22×C4D3×D911yescharacteristic
412C4⋊C4—31no—
413C4⋊C4—91no—
414C4⋊C4—91no—
415C4⋊C4—272no—
416C4⋊C4—272no—
417C4⋊C4—271no—
418C22⋊C4—32no—
419C22⋊C4—91no—
420C22⋊C4—91no—
421C22⋊C4—272no—
422C22⋊C4—272no—
423C22⋊C4—272no—
424C42—91no—
425C42—91no—
426C22×C3—22no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—62no—
437C22×C3—62no—
438C22×C3—62no—
439C22×C3C23×D911yescharacteristic
440C22×C3C22×D3211yescharacteristic
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—182no—
445C2×D3—182no—
446C2×D3—182no—
447C2×D3—182no—
448C2×D3—182no—
449C2×D3—182no—
450C2×D3—362no—
451C2×D3—362no—
452C2×D3—362no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—22no—
456C3×C4—41no—
457C3×C4—62no—
458C3×C4—181no—
459C3×C4—181no—
460C3×C4—181no—
461C3×C4—181no—
462C3×C4#369912yes—
463C3×C4#380412yes—
464Dic3—22no—
465Dic3—61no—
466Dic3—61no—
467Dic3—61no—
468Dic3—61no—
469Dic3—182no—
470Dic3—182no—
471Dic3—362no—
472C32#561911yescharacteristic
473C9—21no—
474C9#553511yescharacteristic, complemented
475C23—32no—
476C23—272no—
477C23C2×D3×D911yescharacteristic
478D4—62no—
479D4—122no—
480D4—542no—
481D4—1082no—
482D4—1082no—
483D4—1082no—
484C2×C4—22no—
485C2×C4—32no—
486C2×C4—91no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—272no—
491C2×C4—1082no—
492C2×C4—1082no—
493C2×C4C2×D3×D911yescharacteristic
494C2×C4C2×D3×D911yescharacteristic
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—42no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—122no—
502C2×C3C2×#369911yescharacteristic
503C2×C3#6371011yescharacteristic
504C2×C3#6371511yescharacteristic
505C2×C3#6430711yescharacteristic
506C2×C3#6430811yescharacteristic
507C2×C3C2×#380411yescharacteristic
508D3—42no—
509D3—362no—
510D3—362no—
511D3—722no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—62no—
516C22—62no—
517C22—62no—
518C22—542no—
519C22—542no—
520C22—542no—
521C22C22×D3×D911yescharacteristic
522C4—21no—
523C4—62no—
524C4—181no—
525C4—181no—
526C4—181no—
527C4—181no—
528C4—542no—
529C4#8866212yes—
530C3—21no—
531C3#1058972111yescharacteristic, complemented
532C3#1059498511yescharacteristic
533C2—41no—
534C2—122no—
535C2—1082no—
536C2#1172247611yescharacteristic
537C2#1172247811yescharacteristic
538C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#540
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. #11721819C2semidirect
2. C9×#930C2semidirect
3. C3×#63488C2semidirect
4. #11721282C2semidirect
5. #11721262C2semidirect
6. #11721040C2semidirect
7. D12×Dic9C2semidirect
8. #11721020C2semidirect
9. #11720992C2semidirect
10. #11720958C2semidirect
11. #11720935C2semidirect
55. #63488D3semidirect
160. #930D9semidirect
337. C3×C9#540semidirect
474. C9#5535semidirect
531. C3#10589721semidirect

Maximal subgroups

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

subgroupindex
#117218192
C9×#9302
C3×#634882
#117212822
#117212622
#117210402
D12×Dic92
#117210202
#117209922
#117209582
#117209352
#105949853
#105897213