Finite Group Databasebeta

← #13167097

Subgroups — #13167097

Order 1728.

412 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 47 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#11723660C211yesmaximal, characteristic, complemented
2C2×C3×#3713C211yesmaximal, characteristic, complemented
3#11723098C211yesmaximal, characteristic, complemented
4#11722767C212yesmaximal, complemented
5#11722765C212yesmaximal, complemented
6#10593334—31nomaximal
7C3×#3828—21no—
8C2×C32×D12—21no—
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10#88812C2211yescharacteristic, complemented
11#88811—21no—
12#88810C2211yescharacteristic, complemented
13C3×#3713—21no—
14C3×#3713C2212yes—
15C3⋊#3708C2212yescomplemented
16C2×C3×#213—61no—
17#63829—61no—
18#63822—31no—
19C2×#3713D311yescharacteristic, complemented
20#63773—31no—
21#63735—32no—
22#63734—32no—
23C22×C32×D3—41no—
24C2×C33×C4D411yescharacteristic
25C3×#638—41no—
26C3×C4⋊(C3⋊D3)—21no—
27C3×C4×C3⋊D3—41no—
28C3×#634—21no—
29C32×D12—22no—
30C32×D12—41no—
31C3×C3⋊(C3⋊C8)D411yescharacteristic
32C3×C3⋊(C3⋊C8)C2311yescharacteristic
33#5147—91nomaximal
34C2×C32×D4—61no—
35#3828—21no—
36C3×#235—61no—
37C3×#235—61no—
38C3×#235—121no—
39C2×C3×D12—21no—
40C2×C3×D12—21no—
41C2×C3×D12—41no—
42C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
43C3×#213—62no—
44C3×#213—62no—
45C2×C3×C3⋊C8—61no—
46C2×C3×C3⋊C8—61no—
47#3719—61no—
48#3718—31no—
49#3717—61no—
50#3716—61no—
51#3715—61no—
52#3714—61no—
53#3713—21no—
54#3713—31no—
55#3713C2×D312yes—
56#3708—32no—
57C22×C33D811yescharacteristic
58C2×C3×C3⋊D3—41no—
59C2×C32×D3—42no—
60C2×C32×D3—82no—
61C33×C4D811yescharacteristic
62C33×C4C2×D411yescharacteristic
63C3×C3⋊Dic3—41no—
64C2×C3×D8—91no—
65#950—181no—
66C2×#213—61no—
67#904—91no—
68#827—92no—
69#826—92no—
70#812—91no—
71C23×C32—121no—
72C22×C3×D3—41no—
73C22×C3×D3—42no—
74C22×C3×D3—41no—
75C32×D4—62no—
76C32×D4—121no—
77C2×C32×C4—21no—
78C2×C32×C4—21no—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C32×C4#6711yescharacteristic
82#638—41no—
83C4⋊(C3⋊D3)—21no—
84C4×C3⋊D3—41no—
85#634—21no—
86C3×#67—121no—
87C3×#67—122no—
88C3×#67—121no—
89C3×D12—22no—
90C3×D12—22no—
91C3×D12—42no—
92C3×D12—41no—
93C3×D12—42no—
94C3×D12—41no—
95C3×D12—61no—
96C3×D12—61no—
97C3×D12—121no—
98C3×C4×D3—121no—
99C3×C4×D3—122no—
100C3×C4×D3—121no—
101C3×Dic6—61no—
102C3×Dic6—61no—
103C3×Dic6—121no—
104C3⋊(C3⋊C8)#6711yescharacteristic
105C3⋊(C3⋊C8)C22×D311yescharacteristic
106C3×C3⋊C8—61no—
107C3×C3⋊C8—61no—
108C3×C3⋊C8—61no—
109C3×C3⋊C8—61no—
110#509—271noSylow
111C2×C33—21no—
112C2×C33C2×D811yescharacteristic
113C3×C3⋊D3—81no—
114C32×D3—82no—
115C3×#41—181no—
116C2×C3×D4—61no—
117C2×C3×D4—61no—
118C2×C3×D4—121no—
119#235—61no—
120#235—61no—
121#235—121no—
122C2×D12—21no—
123C3×D8—92no—
124C3×D8—181no—
125C2×C3×C8—91no—
126#216—181no—
127#215—181no—
128#214—181no—
129#213—62no—
130#213—62no—
131#213—181no—
132C2×C3⋊C8—61no—
133C2×C3⋊C8—61no—
134Dic12—91no—
135D24—91no—
136#204—181no—
137C3⋊C16—92no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—41no—
141C22×C32—41no—
142C22×C32—122no—
143C22×C32—242no—
144C22×C32#21311yescharacteristic
145C2×C3⋊D3—41no—
146C2×C3×D3—42no—
147C2×C3×D3—44no—
148C2×C3×D3—42no—
149C2×C3×D3—82no—
150C2×C3×D3—82no—
151C2×C3×D3—84no—
152C2×C3×D3—121no—
153C2×C3×D3—122no—
154C2×C3×D3—121no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—21no—
158C32×C4—21no—
159C32×C4—41no—
160C32×C4—41no—
161C32×C4—41no—
162C32×C4—41no—
163C32×C4#21311yescharacteristic
164C32×C4C2×#6711yescharacteristic
165C3⋊Dic3—41no—
166C3×Dic3—121no—
167C3×Dic3—122no—
168C3×Dic3—121no—
169#135—271no—
170C2×D8—91no—
171SD32—272no—
172D16—272no—
173M32—271no—
174C33#50911yescharacteristic, complemented, Sylow
175C23×C3—121no—
176C23×C3—122no—
177C23×C3—121no—
178C22×D3—41no—
179C3×Q8—181no—
180C3×D4—62no—
181C3×D4—62no—
182C3×D4—122no—
183C3×D4—121no—
184C3×D4—122no—
185C3×D4—121no—
186C3×D4—181no—
187C3×D4—361no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—361no—
193C2×C3×C4GO+(4,𝔽2)11yescharacteristic
194#67—121no—
195#67—122no—
196#67—121no—
197D12—22no—
198D12—41no—
199D12—61no—
200D12—61no—
201D12—121no—
202C4×D3—121no—
203C4×D3—122no—
204C4×D3—121no—
205Dic6—61no—
206Dic6—61no—
207Dic6—121no—
208C3×C8—91no—
209C3×C8—91no—
210C3⋊C8—61no—
211C3⋊C8—61no—
212C3⋊C8—61no—
213C3⋊C8—61no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—42no—
222C2×C32—81no—
223C2×C32—242no—
224C2×C32C2×#21311yescharacteristic
225C3⋊D3—81no—
226C3×D3—82no—
227C3×D3—84no—
228C3×D3—82no—
229C3×D3—241no—
230C3×D3—242no—
231C3×D3—241no—
232#41—181no—
233C2×D4—61no—
234Q16—271no—
235SD16—541no—
236D8—92no—
237D8—181no—
238D8—271no—
239C2×C8—91no—
240C16—272no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—122no—
246C22×C3—124no—
247C22×C3—122no—
248C22×C3—242no—
249C22×C3—242no—
250C22×C3—244no—
251C22×C3—361no—
252C22×C3#377411yescharacteristic
253C2×D3—42no—
254C2×D3—82no—
255C2×D3—121no—
256C2×D3—122no—
257C2×D3—121no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—361no—
267C3×C4#377411yescharacteristic
268C3×C4C2×GO+(4,𝔽2)11yescharacteristic
269Dic3—121no—
270Dic3—122no—
271Dic3—121no—
272C32—21no—
273C32—21no—
274C32—41no—
275C32—41no—
276C32#514711yescharacteristic, complemented
277C23—121no—
278Q8—181no—
279D4—62no—
280D4—121no—
281D4—181no—
282D4—361no—
283C2×C4—361no—
284C2×C4D32⋊D311yescharacteristic
285C8—91no—
286C8—91no—
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—41no—
292C2×C3—42no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—81no—
296C2×C3—242no—
297C2×C3—244no—
298C2×C3—242no—
299C2×C3—721no—
300C2×C3C2×#377411yescharacteristic
301D3—82no—
302D3—241no—
303D3—242no—
304D3—241no—
305C22—122no—
306C22—242no—
307C22—361no—
308C22#8893511yescharacteristic
309C4—361no—
310C4#8893511yescharacteristic
311C4C2×D32⋊D311yescharacteristic
312C3—21no—
313C3—21no—
314C3—41no—
315C3—41no—
316C3#1059333411yescharacteristic, complemented
317C2—21no—
318C2—242no—
319C2—721no—
320C2C2×#8893511yescharacteristic
subgroupG/H
centerC2C2×#88935
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×D8
hypercenterC2×C4D32⋊D3
hypocenterC33#509
LCSG ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723660C2semidirect
2. C2×C3×#3713C2semidirect
3. #11723098C2semidirect
4. #11722767C2semidirect
5. #11722765C2semidirect
10. #88812C22semidirect
12. #88810C22semidirect
15. C3⋊#3708C22semidirect
19. C2×#3713D3semidirect
174. C33#509semidirect
276. C32#5147semidirect
316. C3#10593334semidirect

Maximal subgroups

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

subgroupindex
#117236602
C2×C3×#37132
#117230982
#117227672
#117227652
#105933343
#51479