Finite Group Databasebeta

← #13159532

Subgroups — #13159532

Order 1728.

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

normal subgroups only

This group has 792 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#11723431C211yesmaximal, characteristic
3#11722186C211yesmaximal, characteristic, complemented
4C3×#63647C211yesmaximal, characteristic
5C32×#817C211yesmaximal, characteristic, complemented
6#11722054C211yesmaximal, characteristic, complemented
7#11722041C211yesmaximal, characteristic, complemented
8#10592561—31nomaximal
9#10590674—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88807—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20#88803—21no—
21#64106—31no—
22C4⋊C4×C3⋊D3—31no—
23D3×C4⋊Dic3—34no—
24#63857—34no—
25C32×#106—31no—
26#63647D311yescharacteristic
27#63643—31no—
28#63643—34no—
29#63643—64no—
30#63637—31no—
31#63635—31no—
32C3×(C2×D3)⋊C8—34no—
33C3×#817D314yescomplemented
34#63573—34no—
35#63567—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40#6308—21no—
41#6308—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3—21no—
44C3×C4×C3⋊D3—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—21no—
47Dic3×C3⋊D3—42no—
48C33×C8—21no—
49C3×C3⋊(C3⋊C8)—21no—
50#4862—94no—
51#4721—91no—
52C2×C4×C3⋊D3—61no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634—31no—
55C2×#634—34no—
56C2×#634—64no—
57C2×C3×C4×D3—34no—
58C2×D3×Dic3—64no—
59C2×C32×C8—24no—
60C2×C32×C8C2×D314yes—
61C2×C32×C8C2×D311yescharacteristic
62C32×C4⋊C4—31no—
63C32×C4⋊C4—31no—
64#3752—61no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)—64no—
69#3750—61no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C4⋊Dic3C2×D314yes—
72C3×C4⋊Dic3C2×D314yes—
73C2×C3×C3⋊C8—34no—
74Dic3⋊Dic3—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—
88C2×C4⋊Dic3—91no—
89#924—91no—
90#897—94no—
91D3×C4⋊C4—94no—
92C3×#106—34no—
93C3×C22⋊C8—91no—
94(C2×D3)⋊C8—34no—
95#817—91no—
96#817C3⋊D311yescharacteristic, complemented
97#816—94no—
98#816—91no—
99#816—91no—
100#816—184no—
101#810—94no—
102#808—94no—
103C22×C3⋊D3D1211yescharacteristic
104C22×C3×D3—34no—
105C2×C32×C4—24no—
106C2×C32×C4—61no—
107C2×C32×C4—61no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—124no—
115C2×C3⋊Dic3—124no—
116C2×C3⋊Dic3D1211yescharacteristic
117C4×C3⋊D3—21no—
118C4×C3⋊D3—21no—
119C4×C3⋊D3—122no—
120#634—68no—
121#634—61no—
122#634—61no—
123#634—64no—
124#634—64no—
125#634—124no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—34no—
129C3×C4×D3—64no—
130C3×C4×D3—64no—
131D3×Dic3—128no—
132C32×C8—21no—
133C32×C8—24no—
134C32×C8—44no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—64no—
137#483—271noSylow
138C2×C33#12311yescharacteristic
139C2×C33C2×SD1611yescharacteristic
140C2×C33#13711yescharacteristic
141C3×C3⋊D3—22no—
142C22×C3×C4—91no—
143C22×Dic3—181no—
144C2×C4×D3—34no—
145C2×C4×D3—184no—
146C2×Dic6—94no—
147C2×Dic6—91no—
148C2×Dic6—184no—
149C2×C3×C8—24no—
150C2×C3×C8—91no—
151C2×C3×C8D3214yes—
152C2×C3×C8C2×C3⋊D311yescharacteristic
153C3×C4⋊C4—34no—
154C3×C4⋊C4—34no—
155#217—181no—
156(C2×D3)⋊C4—184no—
157C4⋊Dic3—94no—
158C4⋊Dic3—91no—
159C4⋊Dic3—184no—
160C4⋊Dic3—181no—
161C4⋊Dic3—184no—
162C4⋊Dic3C2×C3⋊D311yescharacteristic
163C4⋊Dic3C2×C3⋊D311yescharacteristic
164Dic3⋊C4—181no—
165Dic3⋊C4—184no—
166C2×C3⋊C8—34no—
167C22×C32—24no—
168C22×C32C2×D1211yescharacteristic
169C22×C32D3×D414yes—
170C2×C3⋊D3—21no—
171C2×C3⋊D3—21no—
172C2×C3⋊D3#20412yes—
173C2×C3×D3—38no—
174C2×C3×D3—64no—
175C2×C3×D3—64no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—121no—
179C32×C4—121no—
180C32×C4#23514yes—
181C32×C4#23711yescharacteristic
182C32×C4#23711yescharacteristic
183C32×C4#23914yes—
184C3⋊Dic3—21no—
185C3⋊Dic3—128no—
186C3⋊Dic3—121no—
187C3⋊Dic3—124no—
188C3⋊Dic3—121no—
189C3⋊Dic3—124no—
190C3⋊Dic3—244no—
191C3×Dic3—44no—
192C3×Dic3—44no—
193C3×Dic3—64no—
194C22⋊Q8—271no—
195C2×C4⋊C4—271no—
196#106—31no—
197#106—271no—
198#103—271no—
199#103—271no—
200C22⋊C8—91no—
201C33#48311yescharacteristic, complemented, Sylow
202C23×C3—91no—
203C22×D3—34no—
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—188no—
226Dic6—184no—
227Dic6—184no—
228Dic6—181no—
229Dic6—181no—
230Dic6—364no—
231C3×C8—24no—
232C3×C8—21no—
233C3×C8—44no—
234C3×C8—181no—
235C3⋊C8—64no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32#88611yescharacteristic
240C2×C32#89414yes—
241C2×C32C2×#20411yescharacteristic
242C2×C32#90911yescharacteristic
243C2×C32D3×SD1614yes—
244C2×C32#91514yes—
245C3⋊D3—22no—
246C3×D3—68no—
247C2×Q8—271no—
248C22×C4—91no—
249C22×C4—541no—
250C2×C8—91no—
251C2×C8D3×C3⋊D311yescharacteristic
252C4⋊C4—31no—
253C4⋊C4—31no—
254C4⋊C4—271no—
255C4⋊C4—541no—
256C4⋊C4—541no—
257C22⋊C4—541no—
258C22×C3—24no—
259C22×C3—92no—
260C22×C3—181no—
261C22×C3—181no—
262C22×C3D3×D1214yes—
263C22×C3D4×C3⋊D311yescharacteristic
264C2×D3—38no—
265C2×D3—64no—
266C2×D3—64no—
267C3×C4—24no—
268C3×C4—24no—
269C3×C4—124no—
270C3×C4—124no—
271C3×C4—181no—
272C3×C4#379614yes—
273C3×C4#380414yes—
274C3×C4#382811yescharacteristic
275C3×C4#383211yescharacteristic
276Dic3—41no—
277Dic3—41no—
278Dic3—64no—
279Dic3—368no—
280Dic3—364no—
281Dic3—361no—
282Dic3—361no—
283Dic3—364no—
284Dic3—724no—
285C32—24no—
286C32#472111yescharacteristic, complemented
287C32#486214yescomplemented
288C23—91no—
289Q8—541no—
290Q8—541no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—91no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—541no—
297C2×C4—541no—
298C2×C4—1082no—
299C2×C4C2×D3×C3⋊D311yescharacteristic
300C8—21no—
301C8—181no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—182no—
306C2×C3D3×#20414yes—
307C2×C3#6380014yes—
308C2×C3#6389014yes—
309C2×C3#6410311yescharacteristic
310C2×C3SD16×C3⋊D311yescharacteristic
311C2×C3#6412411yescharacteristic
312D3—68no—
313C22—92no—
314C22—181no—
315C22—181no—
316C22C3⋊D3×D1211yescharacteristic
317C4—121no—
318C4—121no—
319C4—181no—
320C4—1081no—
321C4—1081no—
322C4#8901711yescharacteristic
323C4#8903111yescharacteristic
324C3—24no—
325C3#1059067414yescomplemented
326C3#1059256111yescharacteristic, complemented
327C2—182no—
328C2C3⋊D3×#20411yescharacteristic
329C2#1172334011yescharacteristic
330C2#1172349311yescharacteristic
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. #11722186C2semidirect
5. C32×#817C2semidirect
6. #11722054C2semidirect
7. #11722041C2semidirect
33. C3×#817D3semidirect
96. #817C3⋊D3semidirect
201. C33#483semidirect
286. C32#4721semidirect
287. C32#4862semidirect
325. C3#10590674semidirect
326. C3#10592561semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×C4⋊Dic32
#117234312
#117221862
C3×#636472
C32×#8172
#117220542
#117220412
#105925613
#105906743