Finite Group Databasebeta

← #13148954

Subgroups — #13148954

Order 1728.

718 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721743C211yesmaximal, characteristic, complemented
2C3×#63457C211yesmaximal, characteristic
3C9×#888C211yesmaximal, characteristic, complemented
4#11720998C211yesmaximal, characteristic, complemented
5#11720984C212yesmaximal, complemented
6#11720981C212yesmaximal, complemented
7#11720951C212yesmaximal, complemented
8#11720950C212yesmaximal, complemented
9#11720948C211yesmaximal, characteristic, complemented
10#11720945C211yesmaximal, characteristic, complemented
11#11720943C211yesmaximal, characteristic, complemented
12#10594494—31nomaximal
13#10589835—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×D36C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2212yes—
19C2×Dic3×D9C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21#88534C2212yes—
22C4×C3⋊Dic9C2211yescharacteristic
23C4×C3⋊Dic9C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28(C2×D9)⋊Dic3C2211yescharacteristic
29(C2×D9)⋊Dic3C2212yes—
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2211yescharacteristic
34Dic3⋊Dic9C2212yes—
35Dic9⋊Dic3C2211yescharacteristic
36Dic9⋊Dic3C2211yescharacteristic
37#64096—31no—
38C3×#895—31no—
39C3×#888D311yescharacteristic
40#63898—31no—
41#63887—32no—
42#63884—32no—
43#63863—32no—
44#63862—32no—
45#63860—31no—
46#63857—31no—
47#63855—31no—
48C9×C4⋊Q8—31no—
49#63509—32no—
50#63508—32no—
51#63460—32no—
52#63459—31no—
53#63457D311yescharacteristic
54#63456—32no—
55#63455—31no—
56#63449—31no—
57#63449—61no—
58#63437—31no—
59#63434—31no—
60C22×C3×D9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67#6226—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—
76Dic3×D9—42no—
77#5733—91no—
78#5603—91no—
79C2×#634—31no—
80C2×C3×D12—31no—
81C2×C3×D12—32no—
82C2×C3×C4×D3—31no—
83C2×C3×Dic6C2×D312yes—
84C2×D3×Dic3—32no—
85C32×C4⋊C4C2×D311yescharacteristic
86#3750—32no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3C2×D311yescharacteristic
93Dic3⋊Dic3—31no—
94Dic3⋊Dic3—31no—
95Dic3⋊Dic3—32no—
96(C2×D3)⋊Dic3—31no—
97(C2×D3)⋊Dic3—31no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×Q8×C9—32no—
103C2×D36C2×D311yescharacteristic
104C2×D36C2×D312yes—
105C2×C4×D9—32no—
106C2×C4×D9C2×D311yescharacteristic
107C2×Dic18—31no—
108C2×Dic18—61no—
109C9×C4⋊C4—21no—
110C9×C4⋊C4—32no—
111C9×C4⋊C4—31no—
112C9×C4⋊C4C2×D311yescharacteristic
113C42×C9—31no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—31no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—31no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4C2×D312yes—
121C4⋊Dic9—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—31no—
124Dic9⋊C4—31no—
125Dic9⋊C4—62no—
126C4×Dic9—32no—
127C4×Dic9—31no—
128C4×Dic9—62no—
129C4×Dic9—61no—
130C22×C3×C9C2411yescharacteristic
131C2×C3×D9—21no—
132C2×C3×D9—22no—
133C2×C3×D9—22no—
134C2×C3×D9—21no—
135C2×C3×D9—22no—
136C2×C3×D9—21no—
137C3×C4×C9—22no—
138C3×C4×C9#4112yes—
139C3⋊Dic9—22no—
140C3⋊Dic9—21no—
141C3⋊Dic9—21no—
142C9×Dic3—22no—
143C9×Dic3—21no—
144C9×Dic3—21no—
145C3×Dic9—21no—
146C3×C4⋊Q8—31no—
147C3×#121—91no—
148#947—92no—
149#946—92no—
150#936—92no—
151#936—91no—
152#935—92no—
153D4×Dic3—91no—
154#926—91no—
155#924—91no—
156#898—92no—
157#897—91no—
158#895—31no—
159#894—92no—
160#893—91no—
161#888D911yescharacteristic, complemented
162#887—91no—
163#887—91no—
164#887—181no—
165#879—92no—
166#878—92no—
167#875—91no—
168#872—91no—
169C22×C3×D3—31no—
170C22×C3×D3—32no—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—31no—
176#634—62no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3C22×D311yescharacteristic
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D312yes—
181C3×D12—62no—
182C3×D12—122no—
183C3×C4×D3—62no—
184C3×Dic6—42no—
185D3×Dic3—122no—
186C22×D9C22×D311yescharacteristic
187C22×D9C22×D312yes—
188Q8×C9—122no—
189C2×C4×C9—21no—
190C2×C4×C9—22no—
191C2×C4×C9—31no—
192C2×C4×C9—31no—
193C2×C4×C9—32no—
194C2×C4×C9C22×D312yes—
195C2×C4×C9C22×D311yescharacteristic
196C2×Dic9—31no—
197C2×Dic9—31no—
198C2×Dic9—32no—
199C2×Dic9—61no—
200C2×Dic9—62no—
201C2×Dic9—61no—
202C2×Dic9C22×D311yescharacteristic
203D36—22no—
204D36—42no—
205C4×D9—22no—
206C4×D9—122no—
207Dic18—64no—
208Dic18—62no—
209#542—271noSylow
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C9H(5,𝔽2)11yescharacteristic
212C2×C3×C92−1+411yescharacteristic
213C3×D9—41no—
214C3×D9—42no—
215C2×C3×Q8—32no—
216C2×C3×D4—91no—
217C2×C3×D4—92no—
218C22×C3×C4—91no—
219C22×Dic3—92no—
220C2×D12—31no—
221C2×D12—32no—
222C2×C4×D3—31no—
223C2×C4×D3—92no—
224C2×Dic6—91no—
225C2×Dic6—91no—
226C2×Dic6—181no—
227C2×Dic6C2×D912yes—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4C2×D911yescharacteristic
232C3×C4⋊C4D3211yescharacteristic
233C3×C22⋊C4—92no—
234C3×C42—31no—
235#217—92no—
236#217—92no—
237#217—92no—
238#217—92no—
239#217—91no—
240#217—91no—
241(C2×D3)⋊C4—32no—
242(C2×D3)⋊C4—91no—
243(C2×D3)⋊C4—91no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248C4⋊Dic3—92no—
249C4⋊Dic3—91no—
250C4⋊Dic3—91no—
251C4⋊Dic3C2×D911yescharacteristic
252Dic3⋊C4—91no—
253Dic3⋊C4—91no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—182no—
258Dic3⋊C4C2×D912yes—
259C4×Dic3—92no—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C4×Dic3—92no—
263C4×Dic3—181no—
264C4×Dic3—182no—
265C4×Dic3C2×D911yescharacteristic
266C22×C32C23×D311yescharacteristic
267C2×C3×D3—62no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—62no—
271C2×C3×D3—61no—
272C2×C3×D3—62no—
273C32×C4—22no—
274C32×C4#23912yes—
275C3⋊Dic3—62no—
276C3⋊Dic3—61no—
277C3⋊Dic3—61no—
278C3×Dic3—22no—
279C3×Dic3—21no—
280C3×Dic3—21no—
281C3×Dic3—61no—
282C22×C9—21no—
283C22×C9C23×D311yescharacteristic
284C2×D9—21no—
285C2×D9—22no—
286C2×D9—22no—
287C2×D9—21no—
288C2×D9—21no—
289C2×D9—22no—
290C4×C9—22no—
291C4×C9—22no—
292C4×C9—42no—
293C4×C9—62no—
294C4×C9—61no—
295C4×C9—61no—
296C4×C9#23712yes—
297Dic9—21no—
298Dic9—64no—
299Dic9—62no—
300Dic9—62no—
301Dic9—61no—
302Dic9—61no—
303Dic9—121no—
304C4⋊Q8—31no—
305#126—272no—
306#124—271no—
307#124—272no—
308#123—272no—
309C22⋊Q8—272no—
310C22⋊Q8—271no—
311#121—91no—
312C4×Q8—271no—
313C4×D4—271no—
314#117—271no—
315C3×C9#54211yescharacteristic, complemented, Sylow
316C23×C3—91no—
317C23×C3—92no—
318C22×D3—32no—
319C22×D3—31no—
320C3×Q8—122no—
321C3×D4—182no—
322C3×D4—362no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—31no—
326C2×C3×C4—31no—
327C2×C3×C4—32no—
328C2×C3×C4—91no—
329C2×C3×C4—182no—
330C2×C3×C4C22×D912yes—
331C2×C3×C4C22×D911yescharacteristic
332C2×C3×C4C2×D3212yes—
333C2×C3×C4C2×D3211yescharacteristic
334C2×Dic3—31no—
335C2×Dic3—91no—
336C2×Dic3—91no—
337C2×Dic3—92no—
338C2×Dic3—91no—
339C2×Dic3—91no—
340C2×Dic3—92no—
341C2×Dic3—181no—
342C2×Dic3—182no—
343C2×Dic3—181no—
344C2×Dic3—362no—
345C2×Dic3C22×D911yescharacteristic
346C2×Dic3C22×D911yescharacteristic
347C2×Dic3C22×D912yes—
348D12—62no—
349D12—122no—
350C4×D3—62no—
351C4×D3—362no—
352Dic6—42no—
353Dic6—184no—
354Dic6—182no—
355Dic6—182no—
356C2×C32#100411yescharacteristic
357C2×C32C2×#23911yescharacteristic
358C2×C32#100711yescharacteristic
359C3×D3—122no—
360C3×D3—121no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9C2×#23711yescharacteristic
365C2×C9#100411yescharacteristic
366C2×C9#101011yescharacteristic
367D9—41no—
368D9—42no—
369C2×Q8—32no—
370C2×Q8—271no—
371C2×D4—91no—
372C2×D4—92no—
373C22×C4—91no—
374C22×C4—272no—
375C4⋊C4—31no—
376C4⋊C4—32no—
377C4⋊C4—271no—
378C4⋊C4—271no—
379C4⋊C4—272no—
380C4⋊C4—272no—
381C4⋊C4D3×D911yescharacteristic
382C22⋊C4—92no—
383C22⋊C4—271no—
384C22⋊C4—271no—
385C22⋊C4—272no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C42—31no—
390C42—271no—
391C42—272no—
392C22×C3—21no—
393C22×C3—182no—
394C22×C3—181no—
395C22×C3—181no—
396C22×C3—182no—
397C22×C3—181no—
398C22×C3—182no—
399C22×C3C23×D911yescharacteristic
400C22×C3C22×D3211yescharacteristic
401C2×D3—61no—
402C2×D3—62no—
403C2×D3—62no—
404C2×D3—61no—
405C2×D3—62no—
406C2×D3—61no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—42no—
411C3×C4—62no—
412C3×C4—61no—
413C3×C4—61no—
414C3×C4—181no—
415C3×C4#370112yes—
416C3×C4#379612yes—
417Dic3—21no—
418Dic3—22no—
419Dic3—21no—
420Dic3—61no—
421Dic3—184no—
422Dic3—182no—
423Dic3—182no—
424Dic3—181no—
425Dic3—181no—
426Dic3—181no—
427Dic3—182no—
428Dic3—181no—
429Dic3—361no—
430C32#573311yescharacteristic
431C9—21no—
432C9#560311yescharacteristic, complemented
433C23—92no—
434C23—91no—
435Q8—122no—
436Q8—542no—
437D4—182no—
438D4—362no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—32no—
442C2×C4—91no—
443C2×C4—182no—
444C2×C4—271no—
445C2×C4—272no—
446C2×C4—271no—
447C2×C4—1082no—
448C2×C4C2×D3×D912yes—
449C2×C4C2×D3×D911yescharacteristic
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—362no—
454C2×C3—361no—
455C2×C3#6371011yescharacteristic
456C2×C3C2×#370111yescharacteristic
457C2×C3#6371311yescharacteristic
458C2×C3C2×#379611yescharacteristic
459C2×C3#6431311yescharacteristic
460C2×C3#6431511yescharacteristic
461D3—121no—
462D3—122no—
463C22—182no—
464C22—181no—
465C22—182no—
466C22—181no—
467C22—181no—
468C22—182no—
469C22C22×D3×D911yescharacteristic
470C4—22no—
471C4—62no—
472C4—61no—
473C4—61no—
474C4—181no—
475C4—541no—
476C4—541no—
477C4—542no—
478C4#8863412yes—
479C3—21no—
480C3#1058983511yescharacteristic, complemented
481C3#1059449411yescharacteristic
482C2—362no—
483C2—361no—
484C2C2×#8863411yescharacteristic
485C2#1172249911yescharacteristic
486C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC3×#63457C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#542
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. #11721743C2semidirect
3. C9×#888C2semidirect
4. #11720998C2semidirect
5. #11720984C2semidirect
6. #11720981C2semidirect
7. #11720951C2semidirect
8. #11720950C2semidirect
9. #11720948C2semidirect
10. #11720945C2semidirect
11. #11720943C2semidirect
161. #888D9semidirect
315. C3×C9#542semidirect
432. C9#5603semidirect
480. C3#10589835semidirect

Maximal subgroups

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

subgroupindex
#117217432
C3×#634572
C9×#8882
#117209982
#117209842
#117209812
#117209512
#117209502
#117209482
#117209452
#117209432
#105944943
#105898353