Finite Group Databasebeta

← #13149009

Subgroups — #13149009

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
2C9×#893C211yesmaximal, characteristic, complemented
3C3×#63454C211yesmaximal, characteristic, complemented
4#11721045C211yesmaximal, characteristic, complemented
5#11721044C212yesmaximal, complemented
6#11721036C211yesmaximal, characteristic, complemented
7#11721034C212yesmaximal, complemented
8C4×#6192C211yesmaximal, characteristic, complemented
9#11720970C211yesmaximal, characteristic, complemented
10#11720959C212yesmaximal, complemented
11#11720956C212yesmaximal, complemented
12#10594555—31nomaximal
13#10589665—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×#6195C2212yes—
19C2×#6194C2212yes—
20C2×#6192C2211yescharacteristic
21C2×#6189C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23#88536C2212yes—
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25C4×C3⋊Dic9C2212yes—
26C9×(C2×D3)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C4×C9×Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×C4×Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33#88445C2211yescharacteristic
34(C2×D9)⋊Dic3C2211yescharacteristic
35Dic3⋊Dic9C2212yes—
36Dic3⋊Dic9C2211yescharacteristic
37#64101—31no—
38C3×#893D311yescharacteristic
39C3×#892—31no—
40#63918—31no—
41#63916—31no—
42#63915—32no—
43#63908—31no—
44#63906—32no—
45C4×#626—31no—
46#63855—32no—
47#63852—32no—
48C4×D4×C9—31no—
49#63500—31no—
50#63500—31no—
51#63492—32no—
52#63488—32no—
53#63454—31no—
54#63454—61no—
55#63454D311yescharacteristic, complemented
56#63447—32no—
57#63447—32no—
58C4×D36—31no—
59#63431—31no—
60C22×C3⋊D9C2311yescharacteristic
61C22×D3×C9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67C4×C3⋊D9—22no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C9×D12—22no—
73C4×D3×C9—24no—
74C3×C4×D9—22no—
75#6195—42no—
76#6194—42no—
77#6192—22no—
78#6189—42no—
79#5563—91no—
80#5532—91no—
81C2×C4×C3⋊D3—31no—
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3C2×D312yes—
85C2×#627—32no—
86C2×#626—31no—
87C2×#626—32no—
88C2×#625—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90#3752—32no—
91C4⋊(C3⋊Dic3)—31no—
92C4×C3⋊Dic3—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4C2×D312yes—
95C3×C4⋊Dic3—31no—
96C3×C4×Dic3—32no—
97C3×C4×Dic3C2×D311yescharacteristic
98Dic3⋊Dic3—31no—
99Dic3⋊Dic3—32no—
100#3722—31no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103(C2×D3)⋊Dic3—31no—
104C2×D4×C9—31no—
105C22×C4×C9—32no—
106C2×#611—32no—
107C2×#611—32no—
108C2×D36—31no—
109C2×C4×D9—31no—
110C2×C4×D9—61no—
111C2×C4×D9C2×D311yescharacteristic
112C2×Dic18—32no—
113C9×C4⋊C4—21no—
114C9×C4⋊C4C2×D311yescharacteristic
115C9×C22⋊C4—32no—
116C42×C9—31no—
117#3676—32no—
118#3676—32no—
119(C2×D9)⋊C4—31no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—31no—
122(C2×D9)⋊C4—62no—
123(C2×D9)⋊C4C2×D312yes—
124C4⋊Dic9—31no—
125C4⋊Dic9—31no—
126C4⋊Dic9—32no—
127C4⋊Dic9—61no—
128C4⋊Dic9C2×D311yescharacteristic
129C4×Dic9—32no—
130C4×Dic9—62no—
131C4×Dic9C2×D312yes—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—21no—
134C2×C3⋊D9—21no—
135C2×C3⋊D9—21no—
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C2×C3×D9—21no—
140C2×C3×D9—21no—
141C2×C3×D9—21no—
142C3×C4×C9—22no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—21no—
145C3⋊Dic9—22no—
146C9×Dic3—21no—
147C9×Dic3#4112yes—
148C3×Dic9—21no—
149C3×Dic9—22no—
150C3×C4×D4—31no—
151C3×#117—91no—
152#938—91no—
153#938—91no—
154#930—92no—
155#930—91no—
156C4×#67—91no—
157#926—92no—
158#895—91no—
159#893D911yescharacteristic, complemented
160#892—31no—
161#892—91no—
162#892—91no—
163#892—181no—
164#888—91no—
165#885—92no—
166#885—92no—
167#885—92no—
168#884—92no—
169#875—92no—
170C4×D12—91no—
171#872—92no—
172#869—91no—
173C22×C3⋊D3—31no—
174C22×C3×D3—31no—
175C22×C3×D3C22×D312yes—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—32no—
180C4×C3⋊D3—62no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3C22×D311yescharacteristic
184C2×C3×Dic3C22×D311yescharacteristic
185C3×D12—22no—
186C3×C4×D3—24no—
187C3×C4×D3—62no—
188#627—122no—
189#626—62no—
190#626—122no—
191#625—122no—
192C23×C9—32no—
193C22×D9—31no—
194C22×D9—61no—
195C22×D9C22×D311yescharacteristic
196D4×C9—62no—
197C2×C4×C9—21no—
198C2×C4×C9—22no—
199C2×C4×C9—31no—
200C2×C4×C9—31no—
201C2×C4×C9—64no—
202C2×C4×C9C22×D312yes—
203C2×C4×C9C22×D311yescharacteristic
204#611—122no—
205#611—122no—
206C2×Dic9—31no—
207C2×Dic9—32no—
208C2×Dic9—61no—
209C2×Dic9—62no—
210C2×Dic9C22×D311yescharacteristic
211C2×Dic9C22×D312yes—
212D36—62no—
213C4×D9—22no—
214C4×D9—64no—
215C4×D9—62no—
216Dic18—122no—
217#555—271noSylow
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9C2×#4111yescharacteristic
220C2×C3×C9H(5,𝔽2)11yescharacteristic
221C3⋊D9—41no—
222D3×C9—42no—
223C3×D9—41no—
224C2×C3×D4—31no—
225C22×C3×C4—32no—
226C22×C3×C4—91no—
227C2×#67—92no—
228C2×#67—92no—
229C2×#67—92no—
230C2×#67—91no—
231C2×D12—91no—
232C2×D12—92no—
233C2×D12C2×D911yescharacteristic
234C2×C4×D3—31no—
235C2×C4×D3—91no—
236C2×C4×D3—91no—
237C2×C4×D3—181no—
238C2×C4×D3C2×D912yes—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4C2×D911yescharacteristic
244C3×C4⋊C4D3211yescharacteristic
245C3×C22⋊C4—32no—
246C3×C22⋊C4—92no—
247C3×C42—31no—
248C3×C42—92no—
249#217—92no—
250#217—92no—
251#217—91no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—91no—
254(C2×D3)⋊C4—91no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—91no—
260(C2×D3)⋊C4—182no—
261(C2×D3)⋊C4C2×D912yes—
262C4⋊Dic3—31no—
263C4⋊Dic3—92no—
264C4⋊Dic3—91no—
265C4⋊Dic3—91no—
266C4⋊Dic3—91no—
267C4⋊Dic3—181no—
268Dic3⋊C4—91no—
269Dic3⋊C4—92no—
270C4×Dic3—32no—
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—22no—
280C2×C3×D3—22no—
281C2×C3×D3—22no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—61no—
285C32×C4—22no—
286C32×C4#23512yes—
287C3⋊Dic3—61no—
288C3⋊Dic3—62no—
289C3×Dic3—21no—
290C3×Dic3—61no—
291C3×Dic3—62no—
292C3×Dic3#23712yes—
293C22×C9—21no—
294C22×C9—62no—
295C22×C9—62no—
296C22×C9—62no—
297C22×C9C23×D311yescharacteristic
298C2×D9—21no—
299C2×D9—21no—
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—61no—
312C4×C9#23512yes—
313Dic9—21no—
314Dic9—22no—
315Dic9—62no—
316Dic9—61no—
317Dic9—62no—
318Dic9—122no—
319C4⋊Q8—271no—
320#124—272no—
321#124—272no—
322#121—272no—
323#121—271no—
324#121—271no—
325C4×D4—31no—
326C4×D4—271no—
327#117—91no—
328#117—272no—
329#117—271no—
330C3×C9#55511yescharacteristic, complemented, Sylow
331C23×C3—32no—
332C23×C3—91no—
333C22×D3—31no—
334C22×D3—91no—
335C22×D3—91no—
336C22×D3—181no—
337C22×D3C22×D912yes—
338C3×D4—62no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—64no—
344C2×C3×C4—91no—
345C2×C3×C4—92no—
346C2×C3×C4—182no—
347C2×C3×C4C22×D912yes—
348C2×C3×C4C22×D911yescharacteristic
349C2×C3×C4C2×D3212yes—
350C2×C3×C4C2×D3211yescharacteristic
351#67—182no—
352#67—362no—
353#67—362no—
354#67—362no—
355C2×Dic3—31no—
356C2×Dic3—32no—
357C2×Dic3—91no—
358C2×Dic3—92no—
359C2×Dic3—91no—
360C2×Dic3—92no—
361C2×Dic3—181no—
362C2×Dic3—182no—
363C2×Dic3C22×D911yescharacteristic
364C2×Dic3C22×D911yescharacteristic
365D12—22no—
366D12—182no—
367D12—362no—
368C4×D3—24no—
369C4×D3—62no—
370C4×D3—184no—
371C4×D3—182no—
372C4×D3—182no—
373Dic6—362no—
374Dic6—362no—
375C2×C32C2×#23511yescharacteristic
376C2×C32C2×#23711yescharacteristic
377C2×C32#100911yescharacteristic
378C3⋊D3—121no—
379C3×D3—42no—
380C3×D3—121no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—122no—
385C2×C9C2×#23511yescharacteristic
386C2×C9D3×#4111yescharacteristic
387C2×C9#100911yescharacteristic
388D9—41no—
389D9—122no—
390D9—121no—
391C2×Q8—272no—
392C2×D4—31no—
393C2×D4—271no—
394C2×D4—272no—
395C2×D4—272no—
396C22×C4—32no—
397C22×C4—91no—
398C22×C4—271no—
399C4⋊C4—91no—
400C4⋊C4—271no—
401C4⋊C4—272no—
402C4⋊C4—271no—
403C4⋊C4D3×D911yescharacteristic
404C22⋊C4—32no—
405C22⋊C4—92no—
406C22⋊C4—271no—
407C22⋊C4—271no—
408C22⋊C4—272no—
409C22⋊C4—272no—
410C22⋊C4—272no—
411C42—31no—
412C42—92no—
413C42—272no—
414C22×C3—21no—
415C22×C3—62no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—181no—
419C22×C3—181no—
420C22×C3—181no—
421C22×C3C23×D911yescharacteristic
422C22×C3C22×D3211yescharacteristic
423C2×D3—22no—
424C2×D3—22no—
425C2×D3—22no—
426C2×D3—61no—
427C2×D3—61no—
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—61no—
444C3×C4—181no—
445C3×C4—182no—
446C3×C4#369712yes—
447C3×C4#379812yes—
448Dic3—21no—
449Dic3—61no—
450Dic3—62no—
451Dic3—182no—
452Dic3—181no—
453Dic3—182no—
454Dic3—181no—
455Dic3—182no—
456Dic3—362no—
457Dic3#369912yes—
458C32#556311yescharacteristic
459C9—21no—
460C9#553211yescharacteristic, complemented
461C23—32no—
462C23—91no—
463C23—271no—
464Q8—1082no—
465D4—62no—
466D4—542no—
467D4—1082no—
468D4—1082no—
469C2×C4—31no—
470C2×C4—31no—
471C2×C4—64no—
472C2×C4—91no—
473C2×C4—92no—
474C2×C4—182no—
475C2×C4—271no—
476C2×C4—272no—
477C2×C4—542no—
478C2×C4C2×D3×D912yes—
479C2×C4C2×D3×D911yescharacteristic
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—122no—
484C2×C3—361no—
485C2×C3C2×#369711yescharacteristic
486C2×C3C2×#369911yescharacteristic
487C2×C3#6371511yescharacteristic
488C2×C3C2×#379811yescharacteristic
489C2×C3D3×#23711yescharacteristic
490C2×C3#6432011yescharacteristic
491D3—42no—
492D3—121no—
493D3—362no—
494D3—361no—
495D3—361no—
496C22—62no—
497C22—62no—
498C22—62no—
499C22—181no—
500C22—181no—
501C22—181no—
502C22—541no—
503C22—541no—
504C22—541no—
505C22C22×D3×D911yescharacteristic
506C4—22no—
507C4—32no—
508C4—61no—
509C4—182no—
510C4—181no—
511C4—542no—
512C4—541no—
513C4#8864012yes—
514C3—21no—
515C3#1058966511yescharacteristic, complemented
516C3#1059455511yescharacteristic
517C2—122no—
518C2—361no—
519C2—1081no—
520C2C2×#8864011yescharacteristic
521C2D3×#369911yescharacteristic
522C2#1172251511yescharacteristic
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. C9×#893C2semidirect
3. C3×#63454C2semidirect
4. #11721045C2semidirect
5. #11721044C2semidirect
6. #11721036C2semidirect
7. #11721034C2semidirect
8. C4×#6192C2semidirect
9. #11720970C2semidirect
10. #11720959C2semidirect
11. #11720956C2semidirect
55. #63454D3semidirect
159. #893D9semidirect
330. C3×C9#555semidirect
460. C9#5532semidirect
515. C3#10589665semidirect

Maximal subgroups

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

subgroupindex
#117217482
C9×#8932
C3×#634542
#117210452
#117210442
#117210362
#117210342
C4×#61922
#117209702
#117209592
#117209562
#105945553
#105896653