Finite Group Databasebeta

← #13148915

Subgroups — #13148915

Order 1728.

784 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721748C211yesmaximal, characteristic, complemented
2C3×#63457C211yesmaximal, characteristic, complemented
3C9×#888C211yesmaximal, characteristic, complemented
4#11721007C211yesmaximal, characteristic, complemented
5#11721003C212yesmaximal, complemented
6C4×#6192C211yesmaximal, characteristic, complemented
7Dic3×D36C211yesmaximal, characteristic, complemented
8#11720983C212yesmaximal, complemented
9#11720975C212yesmaximal, complemented
10#11720951C212yesmaximal, complemented
11#11720907C211yesmaximal, characteristic, complemented
12#10594463—31nomaximal
13#10589836—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×D36C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2212yes—
19C2×#6192C2211yescharacteristic
20C2×#6192C2212yes—
21C2×Dic3×D9C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23#88536C2212yes—
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25C4×C3⋊Dic9C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C9×C4⋊Dic3C2211yescharacteristic
28C9×Dic3⋊C4C2212yes—
29C4×C9×Dic3C2211yescharacteristic
30#88445C2211yescharacteristic
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34(C2×D9)⋊Dic3C2211yescharacteristic
35Dic3⋊Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64101—31no—
38C3×#895—31no—
39C3×#888D311yescharacteristic
40#63916—31no—
41#63907—32no—
42C4×#626—31no—
43Dic3×D12—31no—
44#63886—32no—
45#63878—32no—
46#63863—32no—
47#63841—31no—
48C9×C4⋊Q8—31no—
49#63509—32no—
50#63509—32no—
51#63457—32no—
52#63457—31no—
53#63457D311yescharacteristic, complemented
54#63454—31no—
55#63454—31no—
56#63454—32no—
57#63454—61no—
58C4×D36—31no—
59C4×D36—31no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67C4×C3⋊D9—22no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C3×D36—22no—
73C3×D36—42no—
74C3×C4×D9—22no—
75C9×Dic6—42no—
76#6192—22no—
77#6192—42no—
78Dic3×D9—24no—
79#5734—91no—
80#5616—91no—
81C2×C4×C3⋊D3—31no—
82C2×C3×D12—31no—
83C2×C3×D12—32no—
84C2×C3×C4×D3—31no—
85C2×C3×Dic6C2×D312yes—
86C2×#626—31no—
87C2×#626—32no—
88C2×D3×Dic3—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90#3752—32no—
91C4⋊(C3⋊Dic3)—31no—
92C4×C3⋊Dic3—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×Dic3⋊C4C2×D312yes—
96C3×C4×Dic3C2×D311yescharacteristic
97Dic3⋊Dic3—31no—
98#3722—31no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—31no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×Q8×C9—32no—
105C2×D36—31no—
106C2×D36—32no—
107C2×D36C2×D311yescharacteristic
108C2×D36C2×D312yes—
109C2×C4×D9—32no—
110C2×C4×D9—31no—
111C2×C4×D9—61no—
112C2×C4×D9C2×D311yescharacteristic
113C9×C4⋊C4—21no—
114C9×C4⋊C4—32no—
115C9×C4⋊C4—31no—
116C9×C4⋊C4C2×D311yescharacteristic
117C42×C9—31no—
118(C2×D9)⋊C4—31no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—31no—
124(C2×D9)⋊C4—62no—
125(C2×D9)⋊C4C2×D312yes—
126C4⋊Dic9—31no—
127C4⋊Dic9—31no—
128C4⋊Dic9—61no—
129C4×Dic9—32no—
130C4×Dic9—32no—
131C4×Dic9—62no—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—21no—
134C2×C3⋊D9—21no—
135C2×C3⋊D9—21no—
136C2×C3×D9—22no—
137C2×C3×D9—21no—
138C2×C3×D9—21no—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C2×C3×D9—21no—
142C3×C4×C9—22no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—21no—
145C3⋊Dic9—22no—
146C9×Dic3—22no—
147C9×Dic3—21no—
148C9×Dic3#4112yes—
149C3×Dic9—21no—
150C3×C4⋊Q8—31no—
151C3×#121—91no—
152#947—92no—
153#947—92no—
154#939—92no—
155#938—91no—
156#936—92no—
157D4×Dic3—91no—
158C4×#67—91no—
159#926—91no—
160#895—31no—
161#895—92no—
162#895—91no—
163#892—92no—
164#892—91no—
165#892—91no—
166#892—91no—
167#892—181no—
168#888D911yescharacteristic, complemented
169#885—92no—
170#877—92no—
171C4×D12—91no—
172C4×D12—91no—
173C22×C3⋊D3—31no—
174C22×C3×D3—31no—
175C22×C3×D3—32no—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—32no—
179C2×C3⋊Dic3—31no—
180C4×C3⋊D3—62no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3C22×D312yes—
183C2×C3×Dic3C22×D311yescharacteristic
184C2×C3×Dic3C22×D311yescharacteristic
185C3×D12—62no—
186C3×D12—122no—
187C3×C4×D3—62no—
188C3×Dic6—42no—
189#626—62no—
190#626—122no—
191D3×Dic3—64no—
192C22×D9—31no—
193C22×D9—61no—
194C22×D9C22×D311yescharacteristic
195C22×D9C22×D312yes—
196Q8×C9—122no—
197C2×C4×C9—21no—
198C2×C4×C9—22no—
199C2×C4×C9—31no—
200C2×C4×C9—32no—
201C2×C4×C9—31no—
202C2×C4×C9C22×D312yes—
203C2×C4×C9C22×D311yescharacteristic
204C2×Dic9—31no—
205C2×Dic9—32no—
206C2×Dic9—61no—
207C2×Dic9—62no—
208C2×Dic9C22×D311yescharacteristic
209D36—22no—
210D36—42no—
211D36—62no—
212D36—122no—
213C4×D9—22no—
214C4×D9—64no—
215C4×D9—64no—
216C4×D9—62no—
217#555—271noSylow
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9C2×#4111yescharacteristic
220C2×C3×C9H(5,𝔽2)11yescharacteristic
221C3⋊D9—41no—
222C3×D9—42no—
223C3×D9—41no—
224C2×C3×Q8—32no—
225C2×C3×D4—91no—
226C2×C3×D4—92no—
227C22×C3×C4—91no—
228C2×#67—91no—
229C2×#67—92no—
230C22×Dic3—92no—
231C2×D12—31no—
232C2×D12—32no—
233C2×D12—91no—
234C2×D12—92no—
235C2×C4×D3—31no—
236C2×C4×D3—92no—
237C2×C4×D3—91no—
238C2×C4×D3—91no—
239C2×C4×D3—181no—
240C2×Dic6C2×D912yes—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—32no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4C2×D911yescharacteristic
245C3×C4⋊C4D3211yescharacteristic
246C3×C22⋊C4—92no—
247C3×C42—31no—
248#217—92no—
249#217—92no—
250#217—92no—
251#217—91no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—91no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—91no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—91no—
261(C2×D3)⋊C4—182no—
262C4⋊Dic3—91no—
263C4⋊Dic3—91no—
264C4⋊Dic3—91no—
265C4⋊Dic3—181no—
266C4⋊Dic3C2×D911yescharacteristic
267Dic3⋊C4—91no—
268Dic3⋊C4C2×D912yes—
269C4×Dic3—92no—
270C4×Dic3—92no—
271C4×Dic3—92no—
272C4×Dic3—92no—
273C4×Dic3—182no—
274C4×Dic3C2×D911yescharacteristic
275C22×C32C23×D311yescharacteristic
276C2×C3⋊D3—61no—
277C2×C3⋊D3—61no—
278C2×C3⋊D3—61no—
279C2×C3×D3—62no—
280C2×C3×D3—61no—
281C2×C3×D3—62no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—62no—
285C32×C4—22no—
286C32×C4#23912yes—
287C3⋊Dic3—61no—
288C3⋊Dic3—62no—
289C3×Dic3—22no—
290C3×Dic3—21no—
291C3×Dic3—61no—
292C3×Dic3#23912yes—
293C22×C9—21no—
294C22×C9C23×D311yescharacteristic
295C2×D9—21no—
296C2×D9—22no—
297C2×D9—21no—
298C2×D9—22no—
299C2×D9—22no—
300C2×D9—21no—
301C2×D9—62no—
302C2×D9—61no—
303C2×D9—61no—
304C2×D9—61no—
305C2×D9—121no—
306C2×D9—121no—
307C4×C9—22no—
308C4×C9—22no—
309C4×C9—32no—
310C4×C9—42no—
311C4×C9—62no—
312C4×C9—61no—
313C4×C9#23712yes—
314Dic9—21no—
315Dic9—62no—
316Dic9—61no—
317Dic9—62no—
318Dic9—122no—
319C4⋊Q8—31no—
320#124—272no—
321#124—272no—
322#121—91no—
323#121—271no—
324#121—272no—
325C4×D4—271no—
326C4×D4—271no—
327#117—272no—
328#117—271no—
329#117—271no—
330C3×C9#55511yescharacteristic, complemented, Sylow
331C23×C3—91no—
332C23×C3—92no—
333C22×D3—32no—
334C22×D3—31no—
335C22×D3—91no—
336C22×D3—91no—
337C22×D3—181no—
338C3×Q8—122no—
339C3×D4—182no—
340C3×D4—362no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—31no—
344C2×C3×C4—32no—
345C2×C3×C4—31no—
346C2×C3×C4—91no—
347C2×C3×C4—182no—
348C2×C3×C4C22×D912yes—
349C2×C3×C4C22×D911yescharacteristic
350C2×C3×C4C2×D3212yes—
351C2×C3×C4C2×D3211yescharacteristic
352#67—182no—
353#67—362no—
354C2×Dic3—31no—
355C2×Dic3—91no—
356C2×Dic3—92no—
357C2×Dic3—91no—
358C2×Dic3—92no—
359C2×Dic3—181no—
360C2×Dic3—182no—
361C2×Dic3—184no—
362C2×Dic3C22×D912yes—
363C2×Dic3C22×D911yescharacteristic
364C2×Dic3C22×D911yescharacteristic
365D12—62no—
366D12—122no—
367D12—182no—
368D12—362no—
369C4×D3—62no—
370C4×D3—184no—
371C4×D3—184no—
372C4×D3—182no—
373C4×D3—182no—
374Dic6—42no—
375C2×C32#100411yescharacteristic
376C2×C32C2×#23911yescharacteristic
377C2×C32C2×#23911yescharacteristic
378C3⋊D3—121no—
379C3×D3—122no—
380C3×D3—121no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9C2×#23711yescharacteristic
385C2×C9#100411yescharacteristic
386C2×C9D3×#4111yescharacteristic
387D9—41no—
388D9—42no—
389D9—122no—
390D9—121no—
391C2×Q8—32no—
392C2×D4—91no—
393C2×D4—92no—
394C2×D4—271no—
395C2×D4—272no—
396C22×C4—91no—
397C22×C4—272no—
398C22×C4—271no—
399C4⋊C4—32no—
400C4⋊C4—31no—
401C4⋊C4—271no—
402C4⋊C4—271no—
403C4⋊C4D3×D911yescharacteristic
404C22⋊C4—92no—
405C22⋊C4—271no—
406C22⋊C4—272no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C22⋊C4—271no—
410C22⋊C4—272no—
411C42—31no—
412C42—272no—
413C42—272no—
414C22×C3—21no—
415C22×C3—182no—
416C22×C3—181no—
417C22×C3—182no—
418C22×C3—181no—
419C22×C3—181no—
420C22×C3—182no—
421C22×C3C23×D911yescharacteristic
422C22×C3C22×D3211yescharacteristic
423C2×D3—61no—
424C2×D3—62no—
425C2×D3—61no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—61no—
429C2×D3—182no—
430C2×D3—181no—
431C2×D3—181no—
432C2×D3—181no—
433C2×D3—181no—
434C2×D3—181no—
435C2×D3—181no—
436C2×D3—361no—
437C2×D3—361no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—32no—
442C3×C4—42no—
443C3×C4—62no—
444C3×C4—61no—
445C3×C4—181no—
446C3×C4#370112yes—
447C3×C4#379612yes—
448Dic3—21no—
449Dic3—22no—
450Dic3—61no—
451Dic3—182no—
452Dic3—181no—
453Dic3—182no—
454Dic3—181no—
455Dic3—182no—
456Dic3—362no—
457Dic3#370112yes—
458C32#573411yescharacteristic
459C9—21no—
460C9#561611yescharacteristic, complemented
461C23—92no—
462C23—91no—
463C23—271no—
464Q8—122no—
465D4—182no—
466D4—362no—
467D4—542no—
468D4—1082no—
469C2×C4—31no—
470C2×C4—31no—
471C2×C4—32no—
472C2×C4—91no—
473C2×C4—182no—
474C2×C4—272no—
475C2×C4—271no—
476C2×C4—544no—
477C2×C4—542no—
478C2×C4C2×D3×D912yes—
479C2×C4C2×D3×D911yescharacteristic
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—362no—
484C2×C3—361no—
485C2×C3#6371011yescharacteristic
486C2×C3C2×#370111yescharacteristic
487C2×C3C2×#370111yescharacteristic
488C2×C3C2×#379611yescharacteristic
489C2×C3#6431311yescharacteristic
490C2×C3D3×#23911yescharacteristic
491D3—121no—
492D3—122no—
493D3—362no—
494D3—361no—
495D3—361no—
496C22—181no—
497C22—182no—
498C22—182no—
499C22—181no—
500C22—181no—
501C22—182no—
502C22—541no—
503C22—541no—
504C22—541no—
505C22C22×D3×D911yescharacteristic
506C4—22no—
507C4—32no—
508C4—62no—
509C4—61no—
510C4—181no—
511C4—542no—
512C4—541no—
513C4#8863412yes—
514C3—21no—
515C3#1058983611yescharacteristic, complemented
516C3#1059446311yescharacteristic
517C2—361no—
518C2—362no—
519C2—1081no—
520C2C2×#8863411yescharacteristic
521C2#1172249911yescharacteristic
522C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#555
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11721748C2semidirect
2. C3×#63457C2semidirect
3. C9×#888C2semidirect
4. #11721007C2semidirect
5. #11721003C2semidirect
6. C4×#6192C2semidirect
7. Dic3×D36C2semidirect
8. #11720983C2semidirect
9. #11720975C2semidirect
10. #11720951C2semidirect
11. #11720907C2semidirect
53. #63457D3semidirect
168. #888D9semidirect
330. C3×C9#555semidirect
460. C9#5616semidirect
515. C3#10589836semidirect

Maximal subgroups

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

subgroupindex
#117217482
C3×#634572
C9×#8882
#117210072
#117210032
C4×#61922
Dic3×D362
#117209832
#117209752
#117209512
#117209072
#105944633
#105898363