Finite Group Databasebeta

← #13137709

Subgroups — #13137709

Order 1728.

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

normal subgroups only

This group has 432 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
1C3×#63455C211yesmaximal, characteristic, complemented
2D12×Dic9C211yesmaximal, characteristic, complemented
3C2×#88421C211yesmaximal, characteristic
4#11720526C211yesmaximal, characteristic, complemented
5C9×#809C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720274C211yesmaximal, characteristic, complemented
8#10590455—31nomaximal
9#10588465—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C3×C4×D9—21no—
13C2×D3×Dic9—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C2×C3⋊(C9⋊C8)C2211yescharacteristic
17C3×(C2×D9)⋊C4—21no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic9—21no—
21#88421C414yescomplemented
22C3×#893—31no—
23Dic3×D12—31no—
24C2×#3713—31no—
25#63635—31no—
26C3×#809D311yescharacteristic
27#63563—31no—
28#63554—31no—
29D4×Dic9—31no—
30C2×#3673—31no—
31#63455D311yescharacteristic, complemented
32C9×#102—31no—
33#63380—31no—
34#63375—31no—
35#63367—31no—
36#63367—61no—
37C22×D3×C9—21no—
38C22×C3×D9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3×Dic9D411yescharacteristic
43C2×C3×Dic9D411yescharacteristic
44C9×D12—21no—
45C9×D12C2×C412yescomplemented
46C3×D36—21no—
47C3×D36C2×C412yescomplemented
48C3×C4×D9—42no—
49D3×Dic9—42no—
50C3⋊(C9⋊C8)C2×C412yescomplemented
51C9×C3⋊C8—21no—
52#5014—91no—
53#4785—91no—
54C2×C3×D12—31no—
55C2×C3×D12C2×D311yescharacteristic
56C2×C3×C4×D3—61no—
57C2×D3×Dic3—61no—
58C32×C4⋊C4C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)—31no—
60C2×C3⋊(C3⋊C8)—31no—
61C3×(C2×D3)⋊C4—61no—
62C3×C4×Dic3—31no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64(C2×D3)⋊Dic3—61no—
65#3713—34no—
66C2×D4×C9—31no—
67C22×Dic9—61no—
68C2×D36C2×D311yescharacteristic
69C2×C4×D9—21no—
70C2×C8×C9—31no—
71C9×C4⋊C4—21no—
72C9×C4⋊C4C2×D311yescharacteristic
73#3676—61no—
74#3673—34no—
75(C2×D9)⋊C4—21no—
76C4⋊Dic9—31no—
77C4⋊Dic9—61no—
78C4×Dic9C2×D311yescharacteristic
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×C9—41no—
89C3×C4×C9C22×C411yescharacteristic
90C3×C4×C9#4111yescharacteristic
91C3⋊Dic9—41no—
92C3×Dic9—21no—
93C3×Dic9—21no—
94C3×C4×D4—91no—
95D4×Dic3—91no—
96C2×#213—91no—
97C2×#213—91no—
98#893—31no—
99C4×D12—91no—
100C3×#102—31no—
101#832—91no—
102#821—91no—
103#815—91no—
104#809D911yescharacteristic, complemented
105#808—91no—
106#808—91no—
107#808—181no—
108#803—91no—
109C22×C3×D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—61no—
114C2×C3×Dic3—31no—
115C2×C3×Dic3—31no—
116C3×D12—21no—
117C3×D12—32no—
118C3×D12—61no—
119C3×D12C4×D312yes—
120C3×C4×D3—122no—
121D3×Dic3—122no—
122C23×C9—61no—
123C22×D9—21no—
124C3⋊(C3⋊C8)—32no—
125C3×C3⋊C8—21no—
126D4×C9—32no—
127D4×C9—61no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—41no—
131C2×C4×C9C22×D311yescharacteristic
132C2×Dic9—61no—
133C2×Dic9—121no—
134C2×Dic9—122no—
135C2×Dic9#6711yescharacteristic
136C2×Dic9#6711yescharacteristic
137D36—21no—
138D36C4×D312yescomplemented
139C4×D9—42no—
140C8×C9—61no—
141C9⋊C8—32no—
142C9⋊C8—62no—
143#442—271noSylow
144C2×C3×C9C4×D411yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
147D3×C9—42no—
148C3×D9—42no—
149C2×C3×D4—31no—
150C2×C3×D4—91no—
151C22×C3×C4—181no—
152C22×Dic3—181no—
153C2×D12—31no—
154C2×D12C2×D911yescharacteristic
155C2×C4×D3—61no—
156C2×C4×D3—181no—
157C2×C3×C8—31no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4C2×D911yescharacteristic
160C3×C4⋊C4D3211yescharacteristic
161C3×C22⋊C4—181no—
162C3×C42—91no—
163#217—181no—
164#213—94no—
165#213—94no—
166(C2×D3)⋊C4—61no—
167(C2×D3)⋊C4—181no—
168C4⋊Dic3—91no—
169C4⋊Dic3—91no—
170C4⋊Dic3—181no—
171C4×Dic3—31no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—181no—
175C2×C3⋊C8C2×D911yescharacteristic
176C22×C32D3×D411yescharacteristic
177C2×C3×D3—22no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C2×C3×D3—62no—
181C2×C3×D3—121no—
182C2×C3×D3—121no—
183C32×C4—41no—
184C32×C4C2×C4×D311yescharacteristic
185C32×C4#23711yescharacteristic
186C3⋊Dic3—121no—
187C3×Dic3—61no—
188C3×Dic3—61no—
189C22×C9—21no—
190C22×C9—62no—
191C22×C9—121no—
192C22×C9—121no—
193C22×C9C2×#6711yescharacteristic
194C2×D9—22no—
195C2×D9—41no—
196C2×D9—41no—
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—41no—
200C4×C9—81no—
201C4×C9C2×C4×D311yescharacteristic
202C4×C9#23511yescharacteristic
203Dic9—21no—
204Dic9—21no—
205Dic9—121no—
206Dic9—241no—
207C2×D8—271no—
208C4×D4—91no—
209C4×D4—271no—
210#106—271no—
211#102—31no—
212#102—271no—
213#97—271no—
214C3×C9#44211yescharacteristic, complemented, Sylow
215C23×C3—61no—
216C23×C3—181no—
217C22×D3—21no—
218C22×D3—61no—
219C3×D4—32no—
220C3×D4—61no—
221C3×D4—92no—
222C3×D4—181no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—41no—
227C2×C3×C4—91no—
228C2×C3×C4—91no—
229C2×C3×C4—362no—
230C2×C3×C4C22×D911yescharacteristic
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—31no—
233C2×Dic3—31no—
234C2×Dic3—181no—
235C2×Dic3—181no—
236C2×Dic3—361no—
237C2×Dic3—362no—
238D12—21no—
239D12—32no—
240D12—61no—
241D12C4×D912yescomplemented
242C4×D3—122no—
243C4×D3—362no—
244C3×C8—61no—
245C3⋊C8—21no—
246C3⋊C8—92no—
247C3⋊C8—92no—
248C3⋊C8—182no—
249C2×C32#67⋊C411yescharacteristic
250C2×C32#91111yescharacteristic
251C2×C32#91411yescharacteristic
252C3×D3—42no—
253C3×D3—122no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—122no—
258C2×C9C4×#6711yescharacteristic
259C2×C9#93211yescharacteristic
260C2×C9#94911yescharacteristic
261D9—42no—
262C2×D4—31no—
263C2×D4—91no—
264C22×C4—181no—
265C22×C4—541no—
266D8—274no—
267C2×C8—31no—
268C2×C8—271no—
269C4⋊C4—271no—
270C4⋊C4D3×D911yescharacteristic
271C22⋊C4—181no—
272C22⋊C4—541no—
273C42—91no—
274C22×C3—21no—
275C22×C3—62no—
276C22×C3—121no—
277C22×C3—121no—
278C22×C3—182no—
279C22×C3—361no—
280C22×C3—361no—
281C22×C3D4×D911yescharacteristic
282C22×C3D3×#6711yescharacteristic
283C2×D3—22no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—62no—
287C2×D3—121no—
288C2×D3—121no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—81no—
294C3×C4—181no—
295C3×C4—181no—
296C3×C4C2×C4×D911yescharacteristic
297C3×C4#369911yescharacteristic
298C3×C4#379511yescharacteristic
299C3×C4C4×D3211yescharacteristic
300Dic3—61no—
301Dic3—61no—
302Dic3—361no—
303Dic3—361no—
304Dic3—721no—
305C32#478511yescharacteristic
306C9—21no—
307C9#501411yescharacteristic, complemented
308C23—61no—
309C23—181no—
310D4—32no—
311D4—61no—
312D4—92no—
313D4—181no—
314C2×C4—21no—
315C2×C4—91no—
316C2×C4—91no—
317C2×C4—362no—
318C2×C4—541no—
319C2×C4—1082no—
320C2×C4C2×D3×D911yescharacteristic
321C8—61no—
322C8—272no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—122no—
327C2×C3—362no—
328C2×C3#611⋊C411yescharacteristic
329C2×C3#6347311yescharacteristic
330C2×C3#6347611yescharacteristic
331C2×C3#6387911yescharacteristic
332C2×C3#6393211yescharacteristic
333C2×C3#6393911yescharacteristic
334D3—42no—
335D3—122no—
336C22—62no—
337C22—121no—
338C22—121no—
339C22—182no—
340C22—361no—
341C22—361no—
342C22D9×#6711yescharacteristic
343C4—41no—
344C4—181no—
345C4—181no—
346C4—1081no—
347C4#8863811yescharacteristic
348C4C4×D3×D911yescharacteristic
349C3—21no—
350C3#1058846511yescharacteristic, complemented
351C3#1059045511yescharacteristic
352C2—122no—
353C2—362no—
354C2#1172101211yescharacteristic
355C2#1172117611yescharacteristic
356C2#1172121111yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88421C2
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#442
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#63455C2semidirect
2. D12×Dic9C2semidirect
4. #11720526C2semidirect
5. C9×#809C2semidirect
6. #11720283C2semidirect
7. #11720274C2semidirect
21. #88421C4semidirect
31. #63455D3semidirect
45. C9×D12C2×C4semidirect
47. C3×D36C2×C4semidirect
50. C3⋊(C9⋊C8)C2×C4semidirect
104. #809D9semidirect
138. D36C4×D3semidirect
214. C3×C9#442semidirect
241. D12C4×D9semidirect
307. C9#5014semidirect
350. C3#10588465semidirect

Maximal subgroups

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

subgroupindex
C3×#634552
D12×Dic92
C2×#884212
#117205262
C9×#8092
#117202832
#117202742
#105904553
#105884653