Finite Group Databasebeta

← #13139304

Subgroups — #13139304

Order 1728.

530 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2C2×#88532C211yesmaximal, characteristic, complemented
3C9×#926C211yesmaximal, characteristic, complemented
4#11720260C214yesmaximal, complemented
5#10591111—31nomaximal
6#10588685—31nomaximal
7C22×C3×C4×C9C2211yescharacteristic
8C2×C3×#611—21no—
9C3×#3697—21no—
10C3×#3697—21no—
11C3×#3697C412yescomplemented
12C3×#3697C2212yes—
13C2×C3×D36C411yescharacteristic, complemented
14C2×C3×C4×D9—21no—
15C2×C3×Dic18C411yescharacteristic, complemented
16#88532—21no—
17#88532C2212yescomplemented
18C2×C3⋊(C9⋊C8)—21no—
19C9×#217—21no—
20C9×C4⋊Dic3—21no—
21C4×C9×Dic3C2212yescomplemented
22C2×C3×#235—31no—
23C2×#3748—31no—
24C3×#926D311yescharacteristic
25C2×#3697D311yescharacteristic
26#63569—34no—
27C9×#117—31no—
28C2×#3667—31no—
29C2×#3667—61no—
30#63364—34no—
31C23×C3×C9D411yescharacteristic
32C22×C3×D9—21no—
33C2×C3×C4×C9C2×C411yescharacteristic
34C2×C3×C4×C9C2×C411yescharacteristic
35C2×C3×C4×C9D411yescharacteristic
36C2×C3×C4×C9D411yescharacteristic
37C2×C3×C4×C9D411yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39C3×#611—22no—
40C3×#611—22no—
41C2×C9×Dic3—22no—
42C2×C3×Dic9—21no—
43C3×D36—21no—
44C3×D36C2×C412yes—
45C3×C4×D9—22no—
46C3×C4×D9—22no—
47C3×Dic18—21no—
48C3×Dic18C2×C412yes—
49C3⋊(C9⋊C8)—22no—
50#5236—91no—
51#5005—91no—
52C22×C32×C4C2×D311yescharacteristic
53C2×C3×#67—61no—
54C3×#235—32no—
55C3×#235—32no—
56C3×#235—61no—
57C3×#235—61no—
58C2×C3×D12—31no—
59C2×C3×C4×D3—61no—
60C2×C3×Dic6—31no—
61#3748—32no—
62#3748—61no—
63C2×C3⋊(C3⋊C8)—61no—
64C3×#217—21no—
65C3×C4⋊Dic3—21no—
66C3×C4×Dic3C2×D312yes—
67C22×C4×C9—21no—
68C22×C4×C9C2×D311yescharacteristic
69C2×#611—21no—
70#3697—21no—
71#3697—21no—
72#3697Dic312yescomplemented
73#3697C2×D312yes—
74C2×D36Dic311yescharacteristic, complemented
75C2×C4×D9—21no—
76C2×Dic18Dic311yescharacteristic, complemented
77C9×C4⋊C4—61no—
78C9×C22⋊C4—61no—
79C42×C9—32no—
80#3667—32no—
81#3667—62no—
82#3667—62no—
83#3667—61no—
84C2×C9⋊C8—62no—
85C2×C9⋊C8—61no—
86C22×C3×C9—41no—
87C22×C3×C9C22⋊C411yescharacteristic
88C22×C3×C9C22⋊C411yescharacteristic
89C22×C3×C9C2×D411yescharacteristic
90C2×C3×D9—22no—
91C2×C3×D9—42no—
92C3×C4×C9C22⋊C411yescharacteristic
93C3×C4×C9C22⋊C411yescharacteristic
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C9×Dic3—42no—
97C3×Dic9—22no—
98C2×C3×#41—91no—
99C2×#235—31no—
100C3×#117—31no—
101#926D911yescharacteristic, complemented
102C2×#208—91no—
103C2×#208—91no—
104C2×#208—181no—
105#837—94no—
106#805—94no—
107C23×C32#6711yescharacteristic
108C22×C3×D3—61no—
109C2×C32×C4C4×D311yescharacteristic
110C2×C32×C4C4×D311yescharacteristic
111C2×C32×C4D1211yescharacteristic
112C2×C32×C4D1211yescharacteristic
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C3×#67—62no—
116C3×#67—62no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—61no—
119C3×D12—32no—
120C3×D12—61no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×Dic6—32no—
124C3×Dic6—61no—
125C23×C9—21no—
126C23×C9#6711yescharacteristic
127C22×D9—21no—
128C3⋊(C3⋊C8)—62no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—21no—
135C2×C4×C9—62no—
136C2×C4×C9C2×Dic311yescharacteristic
137C2×C4×C9C2×Dic311yescharacteristic
138C2×C4×C9#6711yescharacteristic
139C2×C4×C9#6711yescharacteristic
140C2×C4×C9#6711yescharacteristic
141C2×C4×C9C22×D311yescharacteristic
142#611—22no—
143#611—22no—
144C2×Dic9—21no—
145D36—21no—
146D36C2×Dic312yes—
147C4×D9—22no—
148C4×D9—22no—
149Dic18—21no—
150Dic18C2×Dic312yes—
151C9⋊C8—64no—
152C9⋊C8—62no—
153#421—271noSylow
154C2×C3×C9—21no—
155C2×C3×C9—21no—
156C2×C3×C9—21no—
157C2×C3×C9C2×C22⋊C411yescharacteristic
158C3×D9—42no—
159C3×#41—92no—
160C3×#41—92no—
161C3×#41—181no—
162C3×#41—181no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C2×C3×D4—181no—
166C22×C3×C4—21no—
167C22×C3×C4—181no—
168C22×C3×C4C2×D911yescharacteristic
169C22×C3×C4D3211yescharacteristic
170C2×#67—61no—
171#235—32no—
172#235—32no—
173#235—61no—
174#235—61no—
175C2×D12—31no—
176C2×C4×D3—61no—
177C2×Dic6—31no—
178C3×C4⋊C4—61no—
179C3×C22⋊C4—61no—
180C3×C42—32no—
181#217—21no—
182C4⋊Dic3—21no—
183C4×Dic3C2×D912yescomplemented
184#208—92no—
185#208—92no—
186#208—182no—
187#208—182no—
188#208—181no—
189#208—181no—
190C2×C3⋊C8—182no—
191C2×C3⋊C8—181no—
192C2×C3⋊C8—181no—
193C22×C32—41no—
194C22×C32(C2×D3)⋊C411yescharacteristic
195C22×C32(C2×D3)⋊C411yescharacteristic
196C22×C32C2×#6711yescharacteristic
197C2×C3×D3—62no—
198C2×C3×D3—122no—
199C32×C4(C2×D3)⋊C411yescharacteristic
200C32×C4(C2×D3)⋊C411yescharacteristic
201C32×C4C2×C4×D311yescharacteristic
202C32×C4C2×D1211yescharacteristic
203C3×Dic3—42no—
204C3×Dic3—62no—
205C22×C9—21no—
206C22×C9—21no—
207C22×C9—21no—
208C22×C9—41no—
209C22×C9—42no—
210C22×C9#21711yescharacteristic
211C22×C9#21711yescharacteristic
212C22×C9C2×#6711yescharacteristic
213C2×D9—22no—
214C2×D9—42no—
215C4×C9—21no—
216C4×C9—21no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—122no—
220C4×C9#21711yescharacteristic
221C4×C9#21711yescharacteristic
222C4×C9C22×Dic311yescharacteristic
223C4×C9C2×#6711yescharacteristic
224Dic9—22no—
225C2×#41—91no—
226C2×M16—271no—
227#117—31no—
228#104—274no—
229C3×C9#42111yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—181no—
232C23×C3#61111yescharacteristic
233C23×C3#62511yescharacteristic
234C22×D3—61no—
235C3×Q8—92no—
236C3×Q8—181no—
237C3×D4—92no—
238C3×D4—182no—
239C3×D4—182no—
240C3×D4—181no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—62no—
248C2×C3×C4—181no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4C4×D911yescharacteristic
252C2×C3×C4C4×D911yescharacteristic
253C2×C3×C4D3611yescharacteristic
254C2×C3×C4D3611yescharacteristic
255C2×C3×C4#61111yescharacteristic
256C2×C3×C4C22×D911yescharacteristic
257C2×C3×C4D3×Dic311yescharacteristic
258C2×C3×C4D3×Dic311yescharacteristic
259C2×C3×C4#62511yescharacteristic
260C2×C3×C4#62611yescharacteristic
261C2×C3×C4#62611yescharacteristic
262C2×C3×C4C2×D3211yescharacteristic
263#67—62no—
264#67—62no—
265C2×Dic3—22no—
266C2×Dic3—61no—
267D12—32no—
268D12—61no—
269C4×D3—62no—
270C4×D3—62no—
271Dic6—32no—
272Dic6—61no—
273C3⋊C8—184no—
274C3⋊C8—182no—
275C3⋊C8—182no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32C2×(C2×D3)⋊C411yescharacteristic
280C3×D3—122no—
281C2×C9—22no—
282C2×C9—21no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9—41no—
287C2×C9—41no—
288C2×C9C2×#21711yescharacteristic
289D9—42no—
290#41—92no—
291#41—92no—
292#41—181no—
293#41—181no—
294C2×Q8—91no—
295C2×D4—91no—
296C2×D4—181no—
297C22×C4—181no—
298C22×C4D3×D911yescharacteristic
299M16—272no—
300M16—541no—
301C2×C8—541no—
302C4⋊C4—61no—
303C22⋊C4—61no—
304C42—32no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—42no—
311C22×C3—182no—
312C22×C3—362no—
313C22×C3(C2×D9)⋊C411yescharacteristic
314C22×C3(C2×D9)⋊C411yescharacteristic
315C22×C3C2×#61111yescharacteristic
316C22×C3(C2×D3)⋊Dic311yescharacteristic
317C22×C3(C2×D3)⋊Dic311yescharacteristic
318C22×C3C2×#62511yescharacteristic
319C2×D3—62no—
320C2×D3—122no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—122no—
326C3×C4—182no—
327C3×C4(C2×D9)⋊C411yescharacteristic
328C3×C4(C2×D9)⋊C411yescharacteristic
329C3×C4C2×C4×D911yescharacteristic
330C3×C4C2×D3611yescharacteristic
331C3×C4(C2×D3)⋊Dic311yescharacteristic
332C3×C4(C2×D3)⋊Dic311yescharacteristic
333C3×C4C2×D3×Dic311yescharacteristic
334C3×C4C2×#62611yescharacteristic
335Dic3—42no—
336Dic3—62no—
337C32#500511yescharacteristic
338C9—21no—
339C9#523611yescharacteristic, complemented
340C23—181no—
341C23#619411yescharacteristic
342Q8—92no—
343Q8—181no—
344D4—92no—
345D4—182no—
346D4—182no—
347D4—181no—
348C2×C4—62no—
349C2×C4—181no—
350C2×C4—182no—
351C2×C4—182no—
352C2×C4Dic3×D911yescharacteristic
353C2×C4Dic3×D911yescharacteristic
354C2×C4#619211yescharacteristic
355C2×C4#619211yescharacteristic
356C2×C4#619411yescharacteristic
357C2×C4C2×D3×D911yescharacteristic
358C8—542no—
359C2×C3—22no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—362no—
370C2×C3C2×(C2×D9)⋊C411yescharacteristic
371C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
372D3—122no—
373C22—41no—
374C22—182no—
375C22—362no—
376C22(C2×D9)⋊Dic311yescharacteristic
377C22(C2×D9)⋊Dic311yescharacteristic
378C22C2×#619411yescharacteristic
379C4—122no—
380C4—182no—
381C4(C2×D9)⋊Dic311yescharacteristic
382C4(C2×D9)⋊Dic311yescharacteristic
383C4C2×Dic3×D911yescharacteristic
384C4C2×#619211yescharacteristic
385C3—21no—
386C3#1058868511yescharacteristic, complemented
387C3#1059111111yescharacteristic
388C2—21no—
389C2—21no—
390C2—21no—
391C2—362no—
392C2C2×(C2×D9)⋊Dic311yescharacteristic
subgroupG/H
centerC4C2×#6192
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC2×C32C2×(C2×D3)⋊C4
prime radicalC2×C3×#3697C2
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#421
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88532C2semidirect
3. C9×#926C2semidirect
4. #11720260C2semidirect
11. C3×#3697C4semidirect
13. C2×C3×D36C4semidirect
15. C2×C3×Dic18C4semidirect
17. #88532C22semidirect
21. C4×C9×Dic3C22semidirect
72. #3697Dic3semidirect
74. C2×D36Dic3semidirect
76. C2×Dic18Dic3semidirect
101. #926D9semidirect
183. C4×Dic3C2×D9semidirect
229. C3×C9#421semidirect
339. C9#5236semidirect
386. C3#10588685semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×#885322
C9×#9262
#117202602
#105911113
#105886853