Finite Group Databasebeta

← #13154235

Subgroups — #13154235

Order 1728.

960 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#126C211yesmaximal, characteristic
2#11721746C211yesmaximal, characteristic, complemented
3#11721745C211yesmaximal, characteristic, complemented
4#11721745C211yesmaximal, characteristic, complemented
5#11721745C211yesmaximal, characteristic, complemented
6#11721744C211yesmaximal, characteristic, complemented
7#11721743C211yesmaximal, characteristic, complemented
8#11721725C211yesmaximal, characteristic
9#11721725C211yesmaximal, characteristic
10#11721725C211yesmaximal, characteristic
11#11721724C211yesmaximal, characteristic
12#11721724C211yesmaximal, characteristic
13#11721723C211yesmaximal, characteristic
14#11721707C211yesmaximal, characteristic, complemented
15C4×#6226C211yesmaximal, characteristic, complemented
16#10596172—31nomaximal
17#10589803—33nomaximal
18C22×C3⋊Dic9C2211yescharacteristic
19C2×#6226C2211yescharacteristic
20C2×#6226C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C4⋊C4C2211yescharacteristic
23C3×C9×C4⋊C4C2211yescharacteristic
24C3×C9×C22⋊C4C2211yescharacteristic
25C3×C9×C22⋊C4C2211yescharacteristic
26C3×C9×C22⋊C4C2211yescharacteristic
27C3×C42×C9C2211yescharacteristic
28#88551C2211yescharacteristic
29#88551C2211yescharacteristic
30#88551C2211yescharacteristic
31C4⋊(C3⋊Dic9)C2211yescharacteristic
32C4⋊(C3⋊Dic9)C2211yescharacteristic
33C4⋊(C3⋊Dic9)C2211yescharacteristic
34C4⋊(C3⋊Dic9)C2211yescharacteristic
35C4⋊(C3⋊Dic9)C2211yescharacteristic
36#88534C2211yescharacteristic
37#88534C2211yescharacteristic
38#88534C2211yescharacteristic
39#88534C2211yescharacteristic
40#88534C2211yescharacteristic
41#88534C2211yescharacteristic
42#88534C2211yescharacteristic
43#88534C2211yescharacteristic
44#88534C2211yescharacteristic
45#88534C2211yescharacteristic
46#88534C2211yescharacteristic
47#88534C2211yescharacteristic
48C4×C3⋊Dic9C2211yescharacteristic
49C4×C3⋊Dic9C2211yescharacteristic
50C4×C3⋊Dic9C2211yescharacteristic
51C4×C3⋊Dic9C2211yescharacteristic
52C4×C3⋊Dic9C2211yescharacteristic
53C32×#126D311yescharacteristic
54#64099—31no—
55#64098—31no—
56#64098—31no—
57#64098—31no—
58#64097—31no—
59#64096—31no—
60#64088—31no—
61#64088—31no—
62#64088—31no—
63#64087—31no—
64#64087—31no—
65#64086—31no—
66#64079—31no—
67C4×#634—31no—
68C9×#126D313yes—
69#63452—33no—
70#63451—33no—
71#63451—33no—
72#63451—33no—
73#63450—33no—
74#63449—33no—
75#63441—33no—
76#63441—33no—
77#63441—33no—
78#63440—33no—
79#63440—33no—
80#63439—33no—
81#63432—33no—
82C4×Dic18—33no—
83C23×C3×C9C2311yescharacteristic
84C2×C3×C4×C9C2311yescharacteristic
85C2×C3×C4×C9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3×C4×C9C2311yescharacteristic
90C2×C3⋊Dic9—22no—
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C2×C3⋊Dic9C2311yescharacteristic
94C2×C3⋊Dic9C2311yescharacteristic
95C2×C3⋊Dic9C2311yescharacteristic
96C2×C3⋊Dic9C2311yescharacteristic
97C2×C3⋊Dic9C2311yescharacteristic
98C2×C3⋊Dic9C2311yescharacteristic
99#6226—22no—
100#6226—22no—
101#5701—93no—
102#5701—91no—
103C22×C3⋊Dic3—31no—
104C2×#634—31no—
105C2×#634—31no—
106C32×C4⋊C4C2×D311yescharacteristic
107C32×C4⋊C4C2×D311yescharacteristic
108C32×C4⋊C4C2×D311yescharacteristic
109C32×C22⋊C4C2×D311yescharacteristic
110C32×C22⋊C4C2×D311yescharacteristic
111C32×C22⋊C4C2×D311yescharacteristic
112C32×C42C2×D311yescharacteristic
113#3757—31no—
114#3757—31no—
115#3757—31no—
116C4⋊(C3⋊Dic3)—31no—
117C4⋊(C3⋊Dic3)—31no—
118C4⋊(C3⋊Dic3)—31no—
119C4⋊(C3⋊Dic3)—31no—
120C4⋊(C3⋊Dic3)—31no—
121#3750—31no—
122#3750—31no—
123#3750—31no—
124#3750—31no—
125#3750—31no—
126#3750—31no—
127#3750—31no—
128#3750—31no—
129#3750—31no—
130#3750—31no—
131#3750—31no—
132#3750—31no—
133C4×C3⋊Dic3—31no—
134C4×C3⋊Dic3—31no—
135C4×C3⋊Dic3—31no—
136C4×C3⋊Dic3—31no—
137C4×C3⋊Dic3—31no—
138C22×Dic9—33no—
139C2×Dic18—33no—
140C2×Dic18—33no—
141C9×C4⋊C4C2×D313yes—
142C9×C4⋊C4C2×D313yes—
143C9×C4⋊C4C2×D313yes—
144C9×C22⋊C4C2×D313yes—
145C9×C22⋊C4C2×D313yes—
146C9×C22⋊C4C2×D313yes—
147C42×C9C2×D313yes—
148#3676—33no—
149#3676—33no—
150#3676—33no—
151C4⋊Dic9—33no—
152C4⋊Dic9—33no—
153C4⋊Dic9—33no—
154C4⋊Dic9—33no—
155C4⋊Dic9—33no—
156Dic9⋊C4—33no—
157Dic9⋊C4—33no—
158Dic9⋊C4—33no—
159Dic9⋊C4—33no—
160Dic9⋊C4—33no—
161Dic9⋊C4—33no—
162Dic9⋊C4—33no—
163Dic9⋊C4—33no—
164Dic9⋊C4—33no—
165Dic9⋊C4—33no—
166Dic9⋊C4—33no—
167Dic9⋊C4—33no—
168C4×Dic9—33no—
169C4×Dic9—33no—
170C4×Dic9—33no—
171C4×Dic9—33no—
172C4×Dic9—33no—
173C22×C3×C9—21no—
174C22×C3×C9—21no—
175C22×C3×C9—21no—
176C22×C3×C9C2411yescharacteristic
177C3×C4×C9—21no—
178C3×C4×C9—21no—
179C3×C4×C9—21no—
180C3×C4×C9—21no—
181C3×C4×C9—21no—
182C3×C4×C9—21no—
183C3⋊Dic9—21no—
184C3⋊Dic9—21no—
185C3⋊Dic9—21no—
186C3⋊Dic9—21no—
187C3⋊Dic9—21no—
188C3⋊Dic9—21no—
189C3⋊Dic9—21no—
190C3⋊Dic9#4112yes—
191C3×#126D913yes—
192C3×#126C3⋊D311yescharacteristic
193#890—93no—
194#890—91no—
195#889—93no—
196#889—93no—
197#889—93no—
198#889—91no—
199#889—91no—
200#889—91no—
201#888—93no—
202#888—91no—
203#887—93no—
204#887—91no—
205#879—93no—
206#879—93no—
207#879—93no—
208#879—91no—
209#879—91no—
210#879—91no—
211#878—93no—
212#878—93no—
213#878—91no—
214#878—91no—
215#877—93no—
216#877—91no—
217#870—93no—
218#870—91no—
219C4×Dic6—93no—
220C4×Dic6—91no—
221C23×C32C22×D311yescharacteristic
222C2×C32×C4C22×D311yescharacteristic
223C2×C32×C4C22×D311yescharacteristic
224C2×C32×C4C22×D311yescharacteristic
225C2×C32×C4C22×D311yescharacteristic
226C2×C32×C4C22×D311yescharacteristic
227C2×C32×C4C22×D311yescharacteristic
228C2×C3⋊Dic3—31no—
229C2×C3⋊Dic3—31no—
230C2×C3⋊Dic3—31no—
231C2×C3⋊Dic3—31no—
232C2×C3⋊Dic3—31no—
233C2×C3⋊Dic3—31no—
234C2×C3⋊Dic3—31no—
235C2×C3⋊Dic3—31no—
236C2×C3⋊Dic3—62no—
237#634—62no—
238#634—62no—
239C23×C9C22×D313yes—
240C2×C4×C9C22×D313yes—
241C2×C4×C9C22×D313yes—
242C2×C4×C9C22×D313yes—
243C2×C4×C9C22×D313yes—
244C2×C4×C9C22×D313yes—
245C2×C4×C9C22×D313yes—
246C2×Dic9—33no—
247C2×Dic9—33no—
248C2×Dic9—33no—
249C2×Dic9—33no—
250C2×Dic9—33no—
251C2×Dic9—33no—
252C2×Dic9—33no—
253C2×Dic9—33no—
254C2×Dic9—66no—
255Dic18—66no—
256Dic18—66no—
257#541—271noSylow
258C2×C3×C9—41no—
259C2×C3×C9C2×#4111yescharacteristic
260C2×C3×C92−1+411yescharacteristic
261C2×C3×C92−1+411yescharacteristic
262C22×Dic3—93no—
263C22×Dic3—91no—
264C2×Dic6—93no—
265C2×Dic6—93no—
266C2×Dic6—91no—
267C2×Dic6—91no—
268C3×C4⋊C4C2×D913yes—
269C3×C4⋊C4C2×D913yes—
270C3×C4⋊C4C2×D913yes—
271C3×C4⋊C4C2×C3⋊D311yescharacteristic
272C3×C4⋊C4C2×C3⋊D311yescharacteristic
273C3×C4⋊C4C2×C3⋊D311yescharacteristic
274C3×C22⋊C4C2×D913yes—
275C3×C22⋊C4C2×D913yes—
276C3×C22⋊C4C2×D913yes—
277C3×C22⋊C4C2×C3⋊D311yescharacteristic
278C3×C22⋊C4C2×C3⋊D311yescharacteristic
279C3×C22⋊C4C2×C3⋊D311yescharacteristic
280C3×C42C2×D913yes—
281C3×C42C2×C3⋊D311yescharacteristic
282#217—93no—
283#217—93no—
284#217—93no—
285#217—91no—
286#217—91no—
287#217—91no—
288C4⋊Dic3—93no—
289C4⋊Dic3—93no—
290C4⋊Dic3—93no—
291C4⋊Dic3—93no—
292C4⋊Dic3—93no—
293C4⋊Dic3—91no—
294C4⋊Dic3—91no—
295C4⋊Dic3—91no—
296C4⋊Dic3—91no—
297C4⋊Dic3—91no—
298Dic3⋊C4—93no—
299Dic3⋊C4—93no—
300Dic3⋊C4—93no—
301Dic3⋊C4—93no—
302Dic3⋊C4—93no—
303Dic3⋊C4—93no—
304Dic3⋊C4—93no—
305Dic3⋊C4—93no—
306Dic3⋊C4—93no—
307Dic3⋊C4—93no—
308Dic3⋊C4—93no—
309Dic3⋊C4—93no—
310Dic3⋊C4—91no—
311Dic3⋊C4—91no—
312Dic3⋊C4—91no—
313Dic3⋊C4—91no—
314Dic3⋊C4—91no—
315Dic3⋊C4—91no—
316Dic3⋊C4—91no—
317Dic3⋊C4—91no—
318Dic3⋊C4—91no—
319Dic3⋊C4—91no—
320Dic3⋊C4—91no—
321Dic3⋊C4—91no—
322C4×Dic3—93no—
323C4×Dic3—93no—
324C4×Dic3—93no—
325C4×Dic3—93no—
326C4×Dic3—93no—
327C4×Dic3—91no—
328C4×Dic3—91no—
329C4×Dic3—91no—
330C4×Dic3—91no—
331C4×Dic3—91no—
332C22×C32—21no—
333C22×C32—21no—
334C22×C32—21no—
335C22×C32C23×D311yescharacteristic
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—21no—
342C3⋊Dic3—32no—
343C3⋊Dic3—61no—
344C3⋊Dic3—61no—
345C3⋊Dic3—61no—
346C3⋊Dic3—61no—
347C3⋊Dic3—61no—
348C3⋊Dic3—61no—
349C3⋊Dic3—61no—
350C22×C9—23no—
351C22×C9—23no—
352C22×C9—23no—
353C22×C9C23×D313yes—
354C4×C9—23no—
355C4×C9—23no—
356C4×C9—23no—
357C4×C9—23no—
358C4×C9—23no—
359C4×C9—23no—
360Dic9—36no—
361Dic9—63no—
362Dic9—63no—
363Dic9—63no—
364Dic9—63no—
365Dic9—63no—
366Dic9—63no—
367Dic9—63no—
368C4⋊Q8—271no—
369#126—271no—
370#126—271no—
371#126—271no—
372#126C3⋊D911yescharacteristic
373#125—271no—
374#125—271no—
375#125—271no—
376#125—271no—
377#125—271no—
378C22⋊Q8—271no—
379C22⋊Q8—271no—
380C4×Q8—271no—
381C4×Q8—271no—
382#117—271no—
383C3×C9#54111yescharacteristic, complemented, Sylow
384C23×C3C22×D913yes—
385C23×C3C22×C3⋊D311yescharacteristic
386C2×C3×C4C22×D913yes—
387C2×C3×C4C22×D913yes—
388C2×C3×C4C22×D913yes—
389C2×C3×C4C22×D913yes—
390C2×C3×C4C22×D913yes—
391C2×C3×C4C22×D913yes—
392C2×C3×C4C22×C3⋊D311yescharacteristic
393C2×C3×C4C22×C3⋊D311yescharacteristic
394C2×C3×C4C22×C3⋊D311yescharacteristic
395C2×C3×C4C22×C3⋊D311yescharacteristic
396C2×C3×C4C22×C3⋊D311yescharacteristic
397C2×C3×C4C22×C3⋊D311yescharacteristic
398C2×Dic3—91no—
399C2×Dic3—93no—
400C2×Dic3—91no—
401C2×Dic3—91no—
402C2×Dic3—91no—
403C2×Dic3—91no—
404C2×Dic3—91no—
405C2×Dic3—91no—
406C2×Dic3—91no—
407C2×Dic3—93no—
408C2×Dic3—93no—
409C2×Dic3—93no—
410C2×Dic3—93no—
411C2×Dic3—93no—
412C2×Dic3—93no—
413C2×Dic3—93no—
414C2×Dic3—182no—
415C2×Dic3—186no—
416Dic6—182no—
417Dic6—182no—
418Dic6—186no—
419Dic6—186no—
420C2×C32—41no—
421C2×C32D3×#4111yescharacteristic
422C2×C32#101011yescharacteristic
423C2×C32#101011yescharacteristic
424C2×C9—43no—
425C2×C9D3×#4113yes—
426C2×C9#101013yes—
427C2×C9#101013yes—
428C2×Q8—271no—
429C2×Q8—271no—
430C22×C4—271no—
431C4⋊C4—271no—
432C4⋊C4—271no—
433C4⋊C4—271no—
434C4⋊C4—271no—
435C4⋊C4—271no—
436C4⋊C4—271no—
437C4⋊C4—271no—
438C4⋊C4—271no—
439C4⋊C4—271no—
440C4⋊C4—271no—
441C4⋊C4—271no—
442C4⋊C4—271no—
443C4⋊C4—271no—
444C4⋊C4—271no—
445C4⋊C4—271no—
446C4⋊C4—271no—
447C4⋊C4—271no—
448C4⋊C4C2×C3⋊D911yescharacteristic
449C4⋊C4C2×C3⋊D911yescharacteristic
450C4⋊C4C2×C3⋊D911yescharacteristic
451C22⋊C4—271no—
452C22⋊C4—271no—
453C22⋊C4—271no—
454C22⋊C4C2×C3⋊D911yescharacteristic
455C22⋊C4C2×C3⋊D911yescharacteristic
456C22⋊C4C2×C3⋊D911yescharacteristic
457C42—271no—
458C42—271no—
459C42—271no—
460C42—271no—
461C42—271no—
462C42C2×C3⋊D911yescharacteristic
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—23no—
467C22×C3—23no—
468C22×C3—23no—
469C22×C3C23×D913yes—
470C22×C3C23×C3⋊D311yescharacteristic
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—23no—
478C3×C4—23no—
479C3×C4—23no—
480C3×C4—23no—
481C3×C4—23no—
482C3×C4—23no—
483Dic3—92no—
484Dic3—96no—
485Dic3—181no—
486Dic3—181no—
487Dic3—181no—
488Dic3—181no—
489Dic3—181no—
490Dic3—181no—
491Dic3—181no—
492Dic3—183no—
493Dic3—183no—
494Dic3—183no—
495Dic3—183no—
496Dic3—183no—
497Dic3—183no—
498Dic3—183no—
499C32#570111yescharacteristic
500C9#570113yescomplemented
501C23C22×C3⋊D911yescharacteristic
502Q8—542no—
503Q8—542no—
504C2×C4—271no—
505C2×C4—271no—
506C2×C4—271no—
507C2×C4—271no—
508C2×C4—271no—
509C2×C4—271no—
510C2×C4—271no—
511C2×C4—271no—
512C2×C4—542no—
513C2×C4C22×C3⋊D911yescharacteristic
514C2×C4C22×C3⋊D911yescharacteristic
515C2×C4C22×C3⋊D911yescharacteristic
516C2×C4C22×C3⋊D911yescharacteristic
517C2×C4C22×C3⋊D911yescharacteristic
518C2×C4C22×C3⋊D911yescharacteristic
519C2×C3—41no—
520C2×C3—43no—
521C2×C3#41×D913yes—
522C2×C3#6371613yes—
523C2×C3#6371613yes—
524C2×C3#41×C3⋊D311yescharacteristic
525C2×C3#6436711yescharacteristic
526C2×C3#6436711yescharacteristic
527C22—21no—
528C22—21no—
529C22—21no—
530C22C23×C3⋊D911yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—272no—
538C4—541no—
539C4—541no—
540C4—541no—
541C4—541no—
542C4—541no—
543C4—541no—
544C4—541no—
545C3#1058980313yescomplemented
546C3#1059617211yescharacteristic
547C2—41no—
548C2#41×C3⋊D911yescharacteristic
549C2#1172267311yescharacteristic
550C2#1172267311yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#126C2
socleC22×C32C23×D3
prime radicalC23×C32C22×D3
hypercenter#126C3⋊D9
hypocenterC3×C9#541
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
2. #11721746C2semidirect
3. #11721745C2semidirect
4. #11721745C2semidirect
5. #11721745C2semidirect
6. #11721744C2semidirect
7. #11721743C2semidirect
14. #11721707C2semidirect
15. C4×#6226C2semidirect
383. C3×C9#541semidirect
500. C9#5701semidirect
545. C3#10589803semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1262
#117217462
#117217452
#117217442
#117217432
#117217252
#117217242
#117217232
#117217072
C4×#62262
#105961723
#105898033