Finite Group Databasebeta

← #13159533

Subgroups — #13159533

Order 1728.

768 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 148 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⋊D3×C4⋊Dic3C211yesmaximal, characteristic
2#11723435C211yesmaximal, characteristic
3#11722187C211yesmaximal, characteristic, complemented
4C3×#63647C211yesmaximal, characteristic
5C32×#816C211yesmaximal, characteristic, complemented
6#11722054C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592504—31nomaximal
9#10590675—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88807—21no—
19#88806—21no—
20#88803—21no—
21#64155—31no—
22C4⋊C4×C3⋊D3—31no—
23D3×C4⋊Dic3—34no—
24#63856—34no—
25C32×#103—31no—
26#63662—31no—
27#63647D311yescharacteristic
28#63644—31no—
29#63644—34no—
30#63644—64no—
31#63635—31no—
32C3×(C2×D3)⋊C8—34no—
33C3×#816D314yescomplemented
34#63573—34no—
35#63566—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×C4×C3⋊D3—21no—
42C3×C4×C3⋊D3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45C32×Dic6—21no—
46C32×Dic6—21no—
47Dic3×C3⋊D3—42no—
48C33×C8—21no—
49C3×C3⋊(C3⋊C8)—21no—
50#4805—94no—
51#4721—91no—
52C2×C32×Q8—31no—
53C2×C4×C3⋊D3—61no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×C4×D3—34no—
56C2×C3×Dic6C2×D314yes—
57C2×D3×Dic3—64no—
58C2×C32×C8—24no—
59C2×C32×C8C2×D314yes—
60C2×C32×C8C2×D311yescharacteristic
61C32×C4⋊C4—31no—
62#3752—61no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—64no—
68C4⋊(C3⋊Dic3)—64no—
69#3750—61no—
70#3750—61no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×C4⋊Dic3C2×D314yes—
73C2×C3×C3⋊C8—34no—
74#3724—64no—
75Dic3⋊Dic3—64no—
76(C2×D3)⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3SD1612yes—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—21no—
86C32×Dic3—41no—
87C32×Dic3—41no—
88#946—94no—
89C2×C4⋊Dic3—91no—
90#924—91no—
91D3×C4⋊C4—94no—
92C3×#103—34no—
93C3×C22⋊C8—91no—
94#835—94no—
95(C2×D3)⋊C8—34no—
96#817—94no—
97#817—91no—
98#817—91no—
99#817—184no—
100#816—91no—
101#816C3⋊D311yescharacteristic, complemented
102#808—94no—
103C22×C3⋊D3D1211yescharacteristic
104C22×C3×D3—34no—
105C32×Q8—61no—
106C32×Q8—61no—
107C2×C32×C4—24no—
108C2×C32×C4—61no—
109C2×C32×C4—61no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—124no—
117C2×C3⋊Dic3—124no—
118C2×C3⋊Dic3D1211yescharacteristic
119C4×C3⋊D3—21no—
120C4×C3⋊D3—21no—
121C4×C3⋊D3—122no—
122C2×C3×Dic3—24no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—34no—
125C3×C4×D3—64no—
126C3×C4×D3—64no—
127C3×Dic6—24no—
128C3×Dic6—24no—
129D3×Dic3—128no—
130C32×C8—28no—
131C32×C8—24no—
132C32×C8—21no—
133C3⋊(C3⋊C8)—21no—
134C3×C3⋊C8—64no—
135#483—271noSylow
136C2×C33#12311yescharacteristic
137C2×C33C2×SD1611yescharacteristic
138C2×C33#13711yescharacteristic
139C3×C3⋊D3—22no—
140C2×C3×Q8—34no—
141C22×C3×C4—91no—
142C22×Dic3—181no—
143C2×C4×D3—34no—
144C2×C4×D3—184no—
145C2×Dic6C2×C3⋊D311yescharacteristic
146C2×C3×C8—24no—
147C2×C3×C8—91no—
148C2×C3×C8D3214yes—
149C2×C3×C8C2×C3⋊D311yescharacteristic
150C3×C4⋊C4—34no—
151#217—181no—
152(C2×D3)⋊C4—184no—
153C4⋊Dic3—94no—
154C4⋊Dic3—94no—
155C4⋊Dic3—91no—
156C4⋊Dic3—91no—
157C4⋊Dic3—184no—
158C4⋊Dic3—184no—
159C4⋊Dic3—181no—
160C4⋊Dic3C2×C3⋊D311yescharacteristic
161Dic3⋊C4—181no—
162Dic3⋊C4—184no—
163Dic3⋊C4—184no—
164C2×C3⋊C8—34no—
165C22×C32—24no—
166C22×C32C2×D1211yescharacteristic
167C22×C32D3×D414yes—
168C2×C3⋊D3—21no—
169C2×C3⋊D3—21no—
170C2×C3⋊D3#20412yes—
171C2×C3×D3—38no—
172C2×C3×D3—64no—
173C2×C3×D3—64no—
174C32×C4—24no—
175C32×C4—24no—
176C32×C4—121no—
177C32×C4—121no—
178C32×C4#23514yes—
179C32×C4#23714yes—
180C32×C4#23711yescharacteristic
181C32×C4#23711yescharacteristic
182C3⋊Dic3—21no—
183C3⋊Dic3—121no—
184C3⋊Dic3—121no—
185C3⋊Dic3—124no—
186C3⋊Dic3—124no—
187C3⋊Dic3—244no—
188C3⋊Dic3—244no—
189C3×Dic3—44no—
190C3×Dic3—44no—
191C3×Dic3—64no—
192C22⋊Q8—271no—
193C2×C4⋊C4—271no—
194#106—271no—
195#106—271no—
196#103—31no—
197#103—271no—
198C22⋊C8—91no—
199C33#48311yescharacteristic, complemented, Sylow
200C23×C3—91no—
201C22×D3—34no—
202C3×Q8—64no—
203C3×Q8—64no—
204C2×C3×C4—24no—
205C2×C3×C4—64no—
206C2×C3×C4—64no—
207C2×C3×C4—91no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4C2×D3214yes—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—34no—
215C2×Dic3—184no—
216C2×Dic3—184no—
217C2×Dic3—181no—
218C2×Dic3—181no—
219C2×Dic3—364no—
220C2×Dic3—364no—
221C2×Dic3—362no—
222C4×D3—64no—
223C4×D3—64no—
224C4×D3—368no—
225Dic6—21no—
226Dic6—21no—
227C3×C8—28no—
228C3×C8—24no—
229C3×C8—21no—
230C3×C8—181no—
231C3⋊C8—64no—
232C2×C32—24no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32#88314yes—
236C2×C32#88611yescharacteristic
237C2×C32C2×#20411yescharacteristic
238C2×C32#90911yescharacteristic
239C2×C32D3×SD1614yes—
240C2×C32#91814yes—
241C3⋊D3—22no—
242C3×D3—68no—
243C2×Q8—31no—
244C22×C4—91no—
245C22×C4—541no—
246C2×C8—91no—
247C2×C8D3×C3⋊D311yescharacteristic
248C4⋊C4—31no—
249C4⋊C4—271no—
250C4⋊C4—271no—
251C4⋊C4—541no—
252C4⋊C4—541no—
253C22⋊C4—541no—
254C22×C3—24no—
255C22×C3—92no—
256C22×C3—181no—
257C22×C3—181no—
258C22×C3D3×D1214yes—
259C22×C3D4×C3⋊D311yescharacteristic
260C2×D3—38no—
261C2×D3—64no—
262C2×D3—64no—
263C3×C4—24no—
264C3×C4—24no—
265C3×C4—124no—
266C3×C4—124no—
267C3×C4—181no—
268C3×C4#379514yes—
269C3×C4#380514yes—
270C3×C4#382811yescharacteristic
271C3×C4#383011yescharacteristic
272Dic3—41no—
273Dic3—41no—
274Dic3—64no—
275Dic3—364no—
276Dic3—364no—
277Dic3—361no—
278Dic3—361no—
279Dic3—724no—
280Dic3—724no—
281C32—24no—
282C32#472111yescharacteristic, complemented
283C32#480514yescomplemented
284C23—91no—
285Q8—61no—
286Q8—61no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—91no—
290C2×C4—181no—
291C2×C4—181no—
292C2×C4—541no—
293C2×C4—541no—
294C2×C4—1082no—
295C2×C4C2×D3×C3⋊D311yescharacteristic
296C8—21no—
297C8—181no—
298C2×C3—24no—
299C2×C3—24no—
300C2×C3—24no—
301C2×C3—182no—
302C2×C3D3×#20414yes—
303C2×C3#6380114yes—
304C2×C3#6389114yes—
305C2×C3#6409211yescharacteristic
306C2×C3SD16×C3⋊D311yescharacteristic
307C2×C3#6412711yescharacteristic
308D3—68no—
309C22—92no—
310C22—181no—
311C22—181no—
312C22C3⋊D3×D1211yescharacteristic
313C4—121no—
314C4—121no—
315C4—181no—
316C4—1081no—
317C4—1081no—
318C4#8902111yescharacteristic
319C4#8902911yescharacteristic
320C3—24no—
321C3#1059067514yescomplemented
322C3#1059250411yescharacteristic, complemented
323C2—182no—
324C2C3⋊D3×#20411yescharacteristic
325C2#1172332911yescharacteristic
326C2#1172352311yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#483
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11722187C2semidirect
5. C32×#816C2semidirect
6. #11722054C2semidirect
7. #11722038C2semidirect
33. C3×#816D3semidirect
101. #816C3⋊D3semidirect
199. C33#483semidirect
282. C32#4721semidirect
283. C32#4805semidirect
321. C3#10590675semidirect
322. C3#10592504semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×C4⋊Dic32
#117234352
#117221872
C3×#636472
C32×#8162
#117220542
#117220382
#105925043
#105906753