Finite Group Databasebeta

← #13140697

Subgroups — #13140697

Order 1728.

358 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 93 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
1Q8×C3⋊Dic9C211yesmaximal, characteristic
2#11720971C211yesmaximal, characteristic
3C9×#835C211yesmaximal, characteristic
4C3×#63395C211yesmaximal, characteristic
5#11720278C211yesmaximal, characteristic
6#11720273C211yesmaximal, characteristic
7#11720264C211yesmaximal, characteristic
8#10591630—31nomaximal
9#10588471—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6189—21no—
12C4⋊(C3⋊Dic9)—21no—
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C4×C3⋊Dic9—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21Q8×C3⋊Dic3—31no—
22#63919—31no—
23C3×#835—31no—
24C3×#835D311yescharacteristic
25#63573—31no—
26#63573—31no—
27#63572—31no—
28Q8×Dic9—31no—
29Q8×Dic9—61no—
30#63450—31no—
31C9×#103—31no—
32#63395D311yescharacteristic
33#63377—31no—
34Dic9⋊C8—31no—
35#63367—31no—
36C3×Q8×C9—21no—
37C3×Q8×C9Q812yescomplemented
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C2×C9×Dic3—21no—
44C2×C3×Dic9—21no—
45#6189—42no—
46C9×C3⋊C8—21no—
47C3×C9⋊C8—21no—
48#4791—91no—
49#4791—91no—
50C2×C32×Q8C2×D311yescharacteristic
51C2×#627—61no—
52C4⋊(C3⋊Dic3)—31no—
53C4⋊(C3⋊Dic3)—61no—
54C4×C3⋊Dic3—31no—
55C4×C3⋊Dic3—61no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60#3724—61no—
61C2×Q8×C9—21no—
62C2×Q8×C9C2×D311yescharacteristic
63C2×Dic18—61no—
64C2×C8×C9—31no—
65C9×C4⋊C4—31no—
66C4⋊Dic9—31no—
67C4⋊Dic9—61no—
68C4⋊Dic9—61no—
69C4⋊Dic9—62no—
70C4⋊Dic9C2×D311yescharacteristic
71Dic9⋊C4—61no—
72C4×Dic9—31no—
73C4×Dic9—61no—
74C4×Dic9—61no—
75C4×Dic9—62no—
76C2×C9⋊C8C2×D311yescharacteristic
77C22×C3×C9C2×D411yescharacteristic
78C3×C4×C9—22no—
79C3×C4×C9C2×Q811yescharacteristic
80C3×C4×C9#4111yescharacteristic
81C3⋊Dic9—21no—
82C3⋊Dic9—41no—
83C3⋊Dic9SD1612yes—
84C9×Dic3—41no—
85C3×Dic9—41no—
86Q8×Dic3—91no—
87Q8×Dic3—91no—
88Q8×Dic3—181no—
89#888—91no—
90#888—91no—
91C3×#103—31no—
92C3×#103—91no—
93#835—31no—
94#835D911yescharacteristic
95#817—91no—
96#817—91no—
97Dic3⋊C8—91no—
98Dic3⋊C8—91no—
99#808—91no—
100#808—91no—
101C32×Q8—21no—
102C32×Q8Dic612yes—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—31no—
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—61no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—61no—
110#627—122no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—61no—
113Q8×C9—22no—
114Q8×C9—21no—
115Q8×C9—22no—
116Q8×C9Dic612yescomplemented
117C2×C4×C9—22no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—61no—
121C2×C4×C9C22×D311yescharacteristic
122C2×Dic9—21no—
123C2×Dic9—31no—
124C2×Dic9—31no—
125C2×Dic9—61no—
126C2×Dic9—61no—
127C2×Dic9—62no—
128C2×Dic9—61no—
129Dic18—122no—
130C8×C9—61no—
131C9⋊C8—21no—
132#475—271noSylow
133C2×C3×C9C22⋊Q811yescharacteristic
134C2×C3×C9C2×SD1611yescharacteristic
135C2×C3×C9#13711yescharacteristic
136C2×C3×Q8—21no—
137C2×C3×Q8C2×D911yescharacteristic
138C2×C3×Q8D3211yescharacteristic
139C2×Dic6—181no—
140C2×Dic6—181no—
141C2×C3×C8—31no—
142C2×C3×C8—91no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—91no—
145C4⋊Dic3—31no—
146C4⋊Dic3—91no—
147C4⋊Dic3—91no—
148C4⋊Dic3—181no—
149C4⋊Dic3—181no—
150C4⋊Dic3—181no—
151C4⋊Dic3—182no—
152C4⋊Dic3C2×D911yescharacteristic
153Dic3⋊C4—181no—
154Dic3⋊C4—181no—
155C4×Dic3—91no—
156C4×Dic3—91no—
157C4×Dic3—181no—
158C4×Dic3—181no—
159C4×Dic3—181no—
160C4×Dic3—182no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8C2×D911yescharacteristic
163C22×C32D3×D411yescharacteristic
164C32×C4—22no—
165C32×C4C2×Dic611yescharacteristic
166C32×C4#23711yescharacteristic
167C3⋊Dic3—32no—
168C3⋊Dic3—61no—
169C3⋊Dic3—121no—
170C3×Dic3—41no—
171C3×Dic3—121no—
172C22×C9—21no—
173C22×C9D3×D411yescharacteristic
174C4×C9—24no—
175C4×C9—22no—
176C4×C9—21no—
177C4×C9—21no—
178C4×C9—121no—
179C4×C9C2×Dic611yescharacteristic
180C4×C9#23711yescharacteristic
181Dic9—32no—
182Dic9—41no—
183Dic9—62no—
184Dic9—61no—
185Dic9—121no—
186Dic9—122no—
187Dic9—121no—
188C4⋊Q8—271no—
189C4×Q8—271no—
190#106—271no—
191#106—271no—
192C4⋊C8—271no—
193#103—31no—
194#103—91no—
195C3×C9#47511yescharacteristic, complemented, Sylow
196C3×Q8—22no—
197C3×Q8—21no—
198C3×Q8—21no—
199C3×Q8—22no—
200C3×Q8Dic1812yescomplemented
201C3×Q8#62712yes—
202C2×C3×C4—22no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—61no—
207C2×C3×C4—181no—
208C2×C3×C4C22×D911yescharacteristic
209C2×C3×C4C2×D3211yescharacteristic
210C2×Dic3—21no—
211C2×Dic3—61no—
212C2×Dic3—91no—
213C2×Dic3—91no—
214C2×Dic3—91no—
215C2×Dic3—91no—
216C2×Dic3—181no—
217C2×Dic3—181no—
218C2×Dic3—182no—
219C2×Dic3—181no—
220C2×Dic3—181no—
221Dic6—362no—
222Dic6—362no—
223C3×C8—61no—
224C3×C8—181no—
225C3⋊C8—21no—
226C3⋊C8—61no—
227C2×C32#87811yescharacteristic
228C2×C32D3×SD1611yescharacteristic
229C2×C32#91811yescharacteristic
230C2×C9—21no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9#87811yescharacteristic
234C2×C9D3×SD1611yescharacteristic
235C2×C9#91811yescharacteristic
236C2×Q8—541no—
237C2×Q8D3×D911yescharacteristic
238C2×C8—31no—
239C2×C8—91no—
240C4⋊C4—31no—
241C4⋊C4—91no—
242C4⋊C4—271no—
243C4⋊C4—541no—
244C4⋊C4—541no—
245C42—271no—
246C42—541no—
247C22×C3—21no—
248C22×C3D4×D911yescharacteristic
249C22×C3#381111yescharacteristic
250C3×C4—24no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—121no—
256C3×C4—361no—
257C3×C4C2×Dic1811yescharacteristic
258C3×C4#369911yescharacteristic
259C3×C4#380511yescharacteristic
260C3×C4C2×#62711yescharacteristic
261Dic3—41no—
262Dic3—92no—
263Dic3—92no—
264Dic3—121no—
265Dic3—182no—
266Dic3—181no—
267Dic3—181no—
268Dic3—361no—
269Dic3—362no—
270Dic3—361no—
271Dic3—361no—
272C32#479111yescharacteristic
273C9—21no—
274C9#479111yescharacteristic, complemented
275Q8—21no—
276Q8—1082no—
277Q8#618912yescomplemented
278C2×C4—21no—
279C2×C4—61no—
280C2×C4—181no—
281C2×C4—271no—
282C2×C4—271no—
283C2×C4—541no—
284C2×C4C2×D3×D911yescharacteristic
285C8—61no—
286C8—181no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3#6344011yescharacteristic
291C2×C3SD16×D911yescharacteristic
292C2×C3#6348011yescharacteristic
293C2×C3#6394011yescharacteristic
294C2×C3#6394511yescharacteristic
295C2×C3#6398211yescharacteristic
296C22#8866811yescharacteristic
297C4—22no—
298C4—121no—
299C4—272no—
300C4—361no—
301C4—541no—
302C4—1081no—
303C4C2×#618911yescharacteristic
304C4#8866311yescharacteristic
305C3—21no—
306C3#1058847111yescharacteristic, complemented
307C3#1059163011yescharacteristic
308C2#1172121211yescharacteristic
309C2#1172122111yescharacteristic
310C2#1172126711yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#475
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
37. C3×Q8×C9Q8semidirect
116. Q8×C9Dic6semidirect
195. C3×C9#475semidirect
200. C3×Q8Dic18semidirect
274. C9#4791semidirect
277. Q8#6189semidirect
306. C3#10588471semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic92
#117209712
C9×#8352
C3×#633952
#117202782
#117202732
#117202642
#105916303
#105884713