Finite Group Databasebeta

← #13138487

Subgroups — #13138487

Order 1728.

446 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721025C211yesmaximal, characteristic, complemented
2#11720998C211yesmaximal, characteristic, complemented
3C2×#88421C211yesmaximal, characteristic
4C8×C3⋊Dic9C211yesmaximal, characteristic, complemented
5C9×#821C211yesmaximal, characteristic, complemented
6C3×#63380C211yesmaximal, characteristic, complemented
7#11720263C211yesmaximal, characteristic, complemented
8#10590740—31nomaximal
9#10588436—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×D3×Dic9—21no—
13C2×Dic3×D9—21no—
14C2×C3×C8×C9C2211yescharacteristic
15C4×C3⋊Dic9C2211yescharacteristic
16C2×C3⋊(C9⋊C8)C2211yescharacteristic
17C9×C4⋊Dic3C2211yescharacteristic
18C3×C4⋊Dic9C2211yescharacteristic
19(C2×D3)⋊Dic9—21no—
20(C2×D9)⋊Dic3—21no—
21#88421C414yescomplemented
22#63898—31no—
23#63898—31no—
24C2×#3713—31no—
25C8×C3⋊Dic3—31no—
26C3×#821—31no—
27C3×#821D311yescharacteristic
28#63576—31no—
29D4×Dic9—31no—
30C2×#3673—31no—
31#63455—31no—
32C9×#102—31no—
33#63380D311yescharacteristic, complemented
34C8×Dic9—31no—
35C8×Dic9—61no—
36#63366—31no—
37C22×D3×C9—21no—
38C22×C3×D9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C9×D12—21no—
45C9×D12C2×C412yescomplemented
46C3×D36—21no—
47C3×D36C2×C412yescomplemented
48D3×Dic9—42no—
49Dic3×D9—42no—
50C3×C8×C9—21no—
51C3⋊(C9⋊C8)C2×C412yescomplemented
52#4756—91no—
53#4756—91no—
54C2×C3×D12—31no—
55C2×C3×D12C2×D311yescharacteristic
56C2×D3×Dic3—61no—
57C2×D3×Dic3—61no—
58C2×C32×C8C2×D311yescharacteristic
59C4×C3⋊Dic3—31no—
60C2×C3⋊(C3⋊C8)—31no—
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3C2×D311yescharacteristic
63(C2×D3)⋊Dic3—61no—
64(C2×D3)⋊Dic3—61no—
65#3713—34no—
66C2×D4×C9—31no—
67C22×Dic9—61no—
68C2×D36C2×D311yescharacteristic
69C2×C4×D9—61no—
70C2×C8×C9—21no—
71C2×C8×C9C2×D311yescharacteristic
72C9×C4⋊C4—31no—
73#3676—61no—
74#3673—34no—
75(C2×D9)⋊C4—61no—
76C4⋊Dic9C2×D311yescharacteristic
77C4×Dic9—31no—
78C4×Dic9—61no—
79C2×C9⋊C8—31no—
80C2×C9⋊C8—61no—
81C22×C3×C9C2×D411yescharacteristic
82C2×D3×C9—22no—
83C2×D3×C9—41no—
84C2×D3×C9—41no—
85C2×C3×D9—22no—
86C2×C3×D9—41no—
87C2×C3×D9—41no—
88C3×C4×C9C22×C411yescharacteristic
89C3×C4×C9#4111yescharacteristic
90C3⋊Dic9—21no—
91C3⋊Dic9D812yescomplemented
92C9×Dic3—41no—
93C3×Dic9—41no—
94D4×Dic3—91no—
95D4×Dic3—91no—
96C2×#213—91no—
97C2×#213—91no—
98#893—91no—
99#893—91no—
100C3×#102—31no—
101C3×#102—91no—
102#821—31no—
103#821D911yescharacteristic, complemented
104C8×Dic3—91no—
105C8×Dic3—91no—
106C8×Dic3—181no—
107#807—91no—
108#807—91no—
109C22×C3×D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—31no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—61no—
116C3×D12—21no—
117C3×D12—32no—
118C3×D12—61no—
119C3×D12C4×D312yes—
120D3×Dic3—122no—
121D3×Dic3—122no—
122C23×C9—61no—
123C22×D9—21no—
124C32×C8—21no—
125C3⋊(C3⋊C8)—32no—
126D4×C9—32no—
127D4×C9—61no—
128C2×C4×C9—21no—
129C2×C4×C9—61no—
130C2×C4×C9C22×D311yescharacteristic
131C2×Dic9—21no—
132C2×Dic9—31no—
133C2×Dic9—31no—
134C2×Dic9—61no—
135C2×Dic9—61no—
136C2×Dic9—122no—
137D36—21no—
138D36C4×D312yescomplemented
139C4×D9—122no—
140C8×C9—22no—
141C8×C9—21no—
142C9⋊C8—32no—
143C9⋊C8—62no—
144C4×D8—271noSylow
145C2×C3×C9C4×D411yescharacteristic
146C2×C3×C9C2×D811yescharacteristic
147C2×C3×C9#13511yescharacteristic
148D3×C9—42no—
149C3×D9—42no—
150C2×C3×D4—31no—
151C2×C3×D4—91no—
152C22×Dic3—181no—
153C22×Dic3—181no—
154C2×D12—31no—
155C2×D12C2×D911yescharacteristic
156C2×C4×D3—181no—
157C2×C4×D3—181no—
158C2×C3×C8—21no—
159C2×C3×C8C2×D911yescharacteristic
160C2×C3×C8D3211yescharacteristic
161C3×C4⋊C4—31no—
162C3×C4⋊C4—91no—
163#217—181no—
164#217—181no—
165#213—94no—
166#213—94no—
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—31no—
170C4⋊Dic3C2×D911yescharacteristic
171C4×Dic3—91no—
172C4×Dic3—91no—
173C4×Dic3—181no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—181no—
177C22×C32D3×D411yescharacteristic
178C2×C3×D3—22no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C2×C3×D3—62no—
182C2×C3×D3—121no—
183C2×C3×D3—121no—
184C32×C4C2×C4×D311yescharacteristic
185C32×C4#23711yescharacteristic
186C3⋊Dic3—32no—
187C3⋊Dic3—61no—
188C3×Dic3—41no—
189C3×Dic3—121no—
190C22×C9—21no—
191C22×C9—62no—
192C22×C9—121no—
193C22×C9—121no—
194C22×C9D3×D411yescharacteristic
195C2×D9—22no—
196C2×D9—41no—
197C2×D9—41no—
198C4×C9—21no—
199C4×C9—21no—
200C4×C9—121no—
201C4×C9C2×C4×D311yescharacteristic
202C4×C9#23711yescharacteristic
203Dic9—32no—
204Dic9—41no—
205Dic9—62no—
206Dic9—62no—
207Dic9—61no—
208C2×D8—271no—
209C4×D4—271no—
210C4×D4—271no—
211#107—271no—
212#102—31no—
213#102—91no—
214C4×C8—271no—
215C3×C9C4×D811yescharacteristic, complemented, Sylow
216C23×C3—61no—
217C23×C3—181no—
218C22×D3—21no—
219C22×D3—61no—
220C3×D4—32no—
221C3×D4—61no—
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—21no—
225C2×C3×C4—61no—
226C2×C3×C4—181no—
227C2×C3×C4C22×D911yescharacteristic
228C2×C3×C4C2×D3211yescharacteristic
229C2×Dic3—21no—
230C2×Dic3—61no—
231C2×Dic3—91no—
232C2×Dic3—91no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—181no—
236C2×Dic3—181no—
237C2×Dic3—362no—
238C2×Dic3—362no—
239D12—21no—
240D12—32no—
241D12—61no—
242D12C4×D912yescomplemented
243C4×D3—362no—
244C4×D3—362no—
245C3×C8—22no—
246C3×C8—21no—
247C3×C8—21no—
248C3⋊C8—92no—
249C3⋊C8—92no—
250C3⋊C8—182no—
251C2×C32#67⋊C411yescharacteristic
252C2×C32D3×D811yescharacteristic
253C2×C32#91611yescharacteristic
254C3×D3—42no—
255C3×D3—122no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—21no—
259C2×C9—122no—
260C2×C9#67⋊C411yescharacteristic
261C2×C9D3×D811yescharacteristic
262C2×C9#91611yescharacteristic
263D9—42no—
264C2×D4—31no—
265C2×D4—91no—
266C22×C4—541no—
267C22×C4—541no—
268D8—274no—
269C2×C8—271no—
270C2×C8D3×D911yescharacteristic
271C4⋊C4—31no—
272C4⋊C4—91no—
273C22⋊C4—541no—
274C22⋊C4—541no—
275C42—271no—
276C22×C3—21no—
277C22×C3—62no—
278C22×C3—121no—
279C22×C3—121no—
280C22×C3—182no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3D4×D911yescharacteristic
284C22×C3#380211yescharacteristic
285C2×D3—22no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—62no—
289C2×D3—121no—
290C2×D3—121no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—121no—
294C3×C4—361no—
295C3×C4C2×C4×D911yescharacteristic
296C3×C4#369911yescharacteristic
297C3×C4C4×D3211yescharacteristic
298C3×C4#380511yescharacteristic
299Dic3—41no—
300Dic3—92no—
301Dic3—92no—
302Dic3—121no—
303Dic3—182no—
304Dic3—182no—
305Dic3—181no—
306Dic3—181no—
307C32#475611yescharacteristic
308C9—21no—
309C9#475611yescharacteristic, complemented
310C23—61no—
311C23—181no—
312D4—32no—
313D4—61no—
314D4—92no—
315D4—181no—
316C2×C4—61no—
317C2×C4—181no—
318C2×C4—271no—
319C2×C4—271no—
320C2×C4—1082no—
321C2×C4—1082no—
322C2×C4C2×D3×D911yescharacteristic
323C8—21no—
324C8—272no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—122no—
329C2×C3—362no—
330C2×C3#611⋊C411yescharacteristic
331C2×C3D8×D911yescharacteristic
332C2×C3#6347811yescharacteristic
333C2×C3#6379711yescharacteristic
334C2×C3#6380911yescharacteristic
335C2×C3#6390211yescharacteristic
336D3—42no—
337D3—122no—
338C22—62no—
339C22—121no—
340C22—121no—
341C22—182no—
342C22—361no—
343C22—361no—
344C22#8864611yescharacteristic
345C4—121no—
346C4—272no—
347C4—361no—
348C4—541no—
349C4C4×D3×D911yescharacteristic
350C4#8866311yescharacteristic
351C3—21no—
352C3#1058843611yescharacteristic, complemented
353C3#1059074011yescharacteristic
354C2—122no—
355C2—362no—
356C2#1172081811yescharacteristic
357C2#1172082311yescharacteristic
358C2#1172102811yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88421C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9C4×D8
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11721025C2semidirect
2. #11720998C2semidirect
4. C8×C3⋊Dic9C2semidirect
5. C9×#821C2semidirect
6. C3×#63380C2semidirect
7. #11720263C2semidirect
21. #88421C4semidirect
33. #63380D3semidirect
45. C9×D12C2×C4semidirect
47. C3×D36C2×C4semidirect
51. C3⋊(C9⋊C8)C2×C4semidirect
91. C3⋊Dic9D8semidirect
103. #821D9semidirect
138. D36C4×D3semidirect
215. C3×C9C4×D8semidirect
242. D12C4×D9semidirect
309. C9#4756semidirect
352. C3#10588436semidirect

Maximal subgroups

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

subgroupindex
#117210252
#117209982
C2×#884212
C8×C3⋊Dic92
C9×#8212
C3×#633802
#117202632
#105907403
#105884363