Finite Group Databasebeta

← #13139529

Subgroups — #13139529

Order 1728.

434 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88532C211yesmaximal, characteristic, complemented
2C4×C9×#67C211yesmaximal, characteristic
3C2×#88420C211yesmaximal, characteristic, complemented
4C3×#63389C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
6Dic3⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
7Dic3×C9⋊C8C211yesmaximal, characteristic, complemented
8#10591067—31nomaximal
9#10588689—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×C9×#67C411yescharacteristic
12C2×C4×D3×C9C2211yescharacteristic
13#88532—22no—
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×#217C411yescharacteristic
17C9×(C2×D3)⋊C4C411yescharacteristic
18C9×Dic3⋊C4C411yescharacteristic
19C4×C9×Dic3C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22#88420—22no—
23C2×#3748—31no—
24C3×C4×#67D311yescharacteristic
25C2×#3711—31no—
26C3×#830—31no—
27Dic3⋊(C3⋊C8)—31no—
28(C2×D3)⋊(C3⋊C8)—31no—
29Dic3×C3⋊C8—31no—
30C4×D4×C9—31no—
31C2×#3667—31no—
32C2×#3667—31no—
33C2×#3667—61no—
34#63389—31no—
35#63389D311yescharacteristic, complemented
36C4⋊(C9⋊C8)—31no—
37C4×C9⋊C8—31no—
38C23×C3×C9C2×C411yescharacteristic
39C22×D3×C9C2×C411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2×C411yescharacteristic
43C2×C3×C4×C9C2311yescharacteristic
44C9×#67—22no—
45C2×C9×Dic3C2×C411yescharacteristic
46C2×C9×Dic3C2×C411yescharacteristic
47C2×C9×Dic3C2×C411yescharacteristic
48C4×D3×C9—21no—
49C4×D3×C9—21no—
50C3⋊(C9⋊C8)—21no—
51C3⋊(C9⋊C8)D412yescomplemented
52C3×C9⋊C8—21no—
53C3×C9⋊C8—21no—
54#5012—91no—
55#4729—91no—
56C22×C32×C4C2×D311yescharacteristic
57C2×C3×#67Dic311yescharacteristic
58C2×C3×C4×D3C2×D311yescharacteristic
59#3748—62no—
60C2×C3⋊(C3⋊C8)—31no—
61C2×C3⋊(C3⋊C8)—31no—
62C3×#217Dic311yescharacteristic
63C3×(C2×D3)⋊C4Dic311yescharacteristic
64C3×Dic3⋊C4Dic311yescharacteristic
65C3×C4×Dic3C2×D311yescharacteristic
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68#3711—62no—
69C2×D4×C9—31no—
70C22×C4×C9—21no—
71C22×C4×C9—31no—
72C22×C4×C9C2×D311yescharacteristic
73C9×C4⋊C4—31no—
74C9×C22⋊C4—31no—
75C9×C22⋊C4—31no—
76C42×C9—31no—
77#3667—62no—
78#3667—62no—
79#3667—64no—
80C2×C9⋊C8—31no—
81C2×C9⋊C8—31no—
82C2×C9⋊C8—61no—
83C2×C9⋊C8—61no—
84C2×C9⋊C8C2×D311yescharacteristic
85C2×C9⋊C8C2×D311yescharacteristic
86C22×C3×C9—21no—
87C22×C3×C9—21no—
88C22×C3×C9—21no—
89C22×C3×C9C22×C411yescharacteristic
90C2×D3×C9—21no—
91C2×D3×C9—21no—
92C2×D3×C9—21no—
93C3×C4×C9—21no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C9×Dic3—21no—
97C9×Dic3—21no—
98C9×Dic3M1612yescomplemented
99C3×C4×D4—31no—
100C4×#67D911yescharacteristic
101C2×#208—91no—
102C2×#208—91no—
103C2×#208—91no—
104C2×#208—181no—
105C2×ΓL(1,𝔽25)—91no—
106C3×C22⋊C8—91no—
107#830—31no—
108#830—91no—
109(C2×D3)⋊C8—91no—
110Dic3⋊C8—91no—
111C8×Dic3—91no—
112C4⋊(C3⋊C8)—91no—
113C4×C3⋊C8—91no—
114C23×C32C2×Dic311yescharacteristic
115C22×C3×D3C2×Dic311yescharacteristic
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4C2×Dic311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C3×#67—22no—
121C2×C3×Dic3C2×Dic311yescharacteristic
122C2×C3×Dic3C2×Dic311yescharacteristic
123C2×C3×Dic3C2×Dic311yescharacteristic
124C3×C4×D3—21no—
125C3×C4×D3—21no—
126C23×C9—21no—
127C23×C9—31no—
128C23×C9C4×D311yescharacteristic
129C3⋊(C3⋊C8)—32no—
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8—61no—
132C3×C3⋊C8—61no—
133D4×C9—62no—
134C2×C4×C9—22no—
135C2×C4×C9—22no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—31no—
141C2×C4×C9—31no—
142C2×C4×C9—31no—
143C2×C4×C9—61no—
144C2×C4×C9—61no—
145C2×C4×C9C4×D311yescharacteristic
146C2×C4×C9C22×D311yescharacteristic
147C9⋊C8—21no—
148C9⋊C8—21no—
149C9⋊C8—32no—
150C9⋊C8—62no—
151C9⋊C8—62no—
152C9⋊C8—61no—
153#436—271noSylow
154C2×C3×C9—41no—
155C2×C3×C9C4×D411yescharacteristic
156C2×C3×C9C2×M1611yescharacteristic
157C2×C3×C9#13111yescharacteristic
158D3×C9—41no—
159C2×C3×D4—31no—
160C22×C3×C4—21no—
161C22×C3×C4—31no—
162C22×C3×C4C2×D911yescharacteristic
163C22×C3×C4D3211yescharacteristic
164C2×#67Dic911yescharacteristic
165C2×C4×D3C2×D911yescharacteristic
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—31no—
169C3×C22⋊C4—31no—
170C3×C22⋊C4—31no—
171C3×C42—31no—
172#217Dic911yescharacteristic
173(C2×D3)⋊C4Dic911yescharacteristic
174Dic3⋊C4Dic911yescharacteristic
175C4×Dic3C2×D911yescharacteristic
176#208—182no—
177#208—182no—
178#208—182no—
179#208—184no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—91no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—181no—
187C2×C3⋊C8—181no—
188ΓL(1,𝔽25)—182no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32C22×Dic311yescharacteristic
193C2×C3×D3—21no—
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C32×C4—21no—
197C32×C4D3×D411yescharacteristic
198C32×C4#23711yescharacteristic
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3#20812yes—
202C22×C9—22no—
203C22×C9—21no—
204C22×C9—21no—
205C22×C9—21no—
206C22×C9—21no—
207C22×C9—41no—
208C22×C9—41no—
209C22×C9—61no—
210C22×C9—61no—
211C22×C9—61no—
212C22×C9C2×C4×D311yescharacteristic
213C4×C9—22no—
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—21no—
217C4×C9—32no—
218C4×C9—61no—
219C4×C9—61no—
220C4×C9D3×D411yescharacteristic
221C4×C9#23711yescharacteristic
222C2×M16—271no—
223C2×M16—271no—
224C4×D4—31no—
225C4⋊C8—271no—
226C22⋊C8—91no—
227C22⋊C8—271no—
228C4×C8—271no—
229C3×C9#43611yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—31no—
232C23×C3C2×Dic911yescharacteristic
233C23×C3D3×Dic311yescharacteristic
234C22×D3C2×Dic911yescharacteristic
235C3×D4—62no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4C2×Dic911yescharacteristic
250C2×C3×C4C22×D911yescharacteristic
251C2×C3×C4D3×Dic311yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253#67—22no—
254C2×Dic3C2×Dic911yescharacteristic
255C2×Dic3C2×Dic911yescharacteristic
256C2×Dic3C2×Dic911yescharacteristic
257C4×D3—21no—
258C4×D3—21no—
259C3×C8—181no—
260C3×C8—181no—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C3⋊C8—92no—
264C3⋊C8—92no—
265C3⋊C8—182no—
266C3⋊C8—182no—
267C3⋊C8—181no—
268C3⋊C8—181no—
269C2×C32—41no—
270C2×C32C2×#20811yescharacteristic
271C2×C32D4×Dic311yescharacteristic
272C2×C32#94811yescharacteristic
273C3×D3—41no—
274C2×C9—21no—
275C2×C9—21no—
276C2×C9—21no—
277C2×C9—42no—
278C2×C9—41no—
279C2×C9—121no—
280C2×C9#67⋊C411yescharacteristic
281C2×C9#90111yescharacteristic
282C2×C9D3×M1611yescharacteristic
283C2×D4—31no—
284C22×C4—31no—
285C22×C4D3×D911yescharacteristic
286M16—542no—
287M16—542no—
288C2×C8—91no—
289C2×C8—91no—
290C2×C8—271no—
291C2×C8—271no—
292C4⋊C4—31no—
293C22⋊C4—31no—
294C22⋊C4—31no—
295C42—31no—
296C22×C3—22no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—61no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3C22×Dic911yescharacteristic
310C22×C3C2×D3×Dic311yescharacteristic
311C2×D3—21no—
312C2×D3—21no—
313C2×D3—21no—
314C3×C4—22no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—32no—
320C3×C4—61no—
321C3×C4—61no—
322C3×C4D4×D911yescharacteristic
323C3×C4#369911yescharacteristic
324C3×C4#380511yescharacteristic
325C3×C4#381111yescharacteristic
326Dic3—21no—
327Dic3—21no—
328Dic3#366712yescomplemented
329C32#501211yescharacteristic
330C9—21no—
331C9#472911yescharacteristic, complemented
332C23—31no—
333C23D3×Dic911yescharacteristic
334D4—62no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4D3×Dic911yescharacteristic
343C2×C4C2×D3×D911yescharacteristic
344C8—181no—
345C8—181no—
346C8—272no—
347C8—541no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—42no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—121no—
355C2×C3C2×#366711yescharacteristic
356C2×C3D4×Dic911yescharacteristic
357C2×C3#6351011yescharacteristic
358C2×C3D3×#20811yescharacteristic
359C2×C3#6381411yescharacteristic
360C2×C3#67⋊Dic311yescharacteristic
361D3—41no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22C2×D3×Dic911yescharacteristic
369C4—21no—
370C4—32no—
371C4—61no—
372C4—61no—
373C4#8866311yescharacteristic
374C4#8866811yescharacteristic
375C3—21no—
376C3#1058868911yescharacteristic, complemented
377C3#1059106711yescharacteristic
378C2—41no—
379C2—121no—
380C2D3×#366711yescharacteristic
381C2#1172087711yescharacteristic
382C2#67⋊Dic911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C22×Dic3
prime radicalC2×C3×#67Dic3
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#436
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88532C2semidirect
3. C2×#88420C2semidirect
4. C3×#63389C2semidirect
5. (C2×D3)⋊(C9⋊C8)C2semidirect
6. Dic3⋊(C9⋊C8)C2semidirect
7. Dic3×C9⋊C8C2semidirect
35. #63389D3semidirect
51. C3⋊(C9⋊C8)D4semidirect
98. C9×Dic3M16semidirect
229. C3×C9#436semidirect
328. Dic3#3667semidirect
331. C9#4729semidirect
376. C3#10588689semidirect

Maximal subgroups

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

subgroupindex
C2×#885322
C4×C9×#672
C2×#884202
C3×#633892
(C2×D3)⋊(C9⋊C8)2
Dic3⋊(C9⋊C8)2
Dic3×C9⋊C82
#105910673
#105886893