Finite Group Databasebeta

← #13161227

Subgroups — #13161227

Order 1728.

708 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 144 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×#64153C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic
3#11722205C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic
6#11722039C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic
8#10592496—31nomaximal
9#10591613—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×#3750—21no—
15C3×#3750—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#64153D311yescharacteristic
22#64099—31no—
23C3×#944—34no—
24#63869—34no—
25C32×#103—31no—
26#63662—31no—
27#63662—34no—
28#63662—64no—
29#63643—31no—
30#63642—31no—
31#63637—31no—
32C3×#835D314yes—
33#63566—34no—
34#63565—34no—
35#63554—34no—
36C33×Q8—21no—
37C33×Q8—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×#634—21no—
45C3×#634—21no—
46C2×C32×Dic3—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C32×C3⋊C8—21no—
49#5082—91no—
50#4797—94no—
51C2×C32×Q8—24no—
52C2×C32×Q8C2×D314yes—
53C2×C32×Q8C2×D311yescharacteristic
54C2×#634C2×D311yescharacteristic
55C2×C3×Dic6—34no—
56C2×C32×C8—31no—
57C32×C4⋊C4—31no—
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—34no—
60C4⋊(C3⋊Dic3)—61no—
61C4⋊(C3⋊Dic3)—64no—
62#3750—21no—
63#3750—21no—
64#3750—61no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×C4⋊Dic3C2×D314yes—
70C3×Dic3⋊C4—64no—
71C3×Dic3⋊C4—64no—
72C3×C4×Dic3—34no—
73C2×C3×C3⋊C8C2×D314yes—
74#3724—64no—
75Dic3⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C33×C4—41no—
78C33×C4#4111yescharacteristic
79C33×C4#4111yescharacteristic
80C3⋊(C3⋊Dic3)—41no—
81C3×C3⋊Dic3—21no—
82C3×C3⋊Dic3—21no—
83C3×C3⋊Dic3—41no—
84C32×Dic3—41no—
85C3×C4⋊Q8—91no—
86#944—34no—
87#890—94no—
88#870—91no—
89C3×#103—34no—
90#835—94no—
91#835—91no—
92#835—91no—
93#835—91no—
94#835—184no—
95#835C3⋊D311yescharacteristic
96#816—94no—
97#815—94no—
98#810—94no—
99#803—91no—
100C32×Q8—28no—
101C32×Q8—24no—
102C32×Q8—21no—
103C32×Q8—21no—
104C32×Q8—24no—
105C32×Q8—44no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—44no—
110C2×C32×C4—61no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—124no—
117C2×C3⋊Dic3#6711yescharacteristic
118C2×C3⋊Dic3#6711yescharacteristic
119#634—21no—
120#634—21no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—64no—
125C3×Dic6—64no—
126C3×Dic6—64no—
127C32×C8—61no—
128C3⋊(C3⋊C8)—61no—
129C3⋊(C3⋊C8)—64no—
130C3⋊(C3⋊C8)—124no—
131C3×C3⋊C8—24no—
132#491—271noSylow
133C2×C33#12411yescharacteristic
134C2×C33#13711yescharacteristic
135C2×C33#13711yescharacteristic
136C2×C3×Q8—24no—
137C2×C3×Q8—91no—
138C2×C3×Q8D3214yes—
139C2×C3×Q8C2×C3⋊D311yescharacteristic
140C2×Dic6—34no—
141C2×C3×C8—34no—
142C3×C4⋊C4—34no—
143C3×C4⋊C4—181no—
144C3×C4⋊C4—181no—
145C3×C42—91no—
146C4⋊Dic3—94no—
147C4⋊Dic3—91no—
148C4⋊Dic3—184no—
149C4⋊Dic3—184no—
150C4⋊Dic3C2×C3⋊D311yescharacteristic
151Dic3⋊C4—64no—
152Dic3⋊C4—64no—
153Dic3⋊C4—181no—
154Dic3⋊C4—181no—
155Dic3⋊C4—184no—
156C4×Dic3—34no—
157C2×C3⋊C8—94no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—184no—
160C2×C3⋊C8C2×C3⋊D311yescharacteristic
161C22×C32—24no—
162C22×C32D3×D414yes—
163C22×C32C2×#6711yescharacteristic
164C32×C4—24no—
165C32×C4—24no—
166C32×C4—48no—
167C32×C4—44no—
168C32×C4—41no—
169C32×C4—121no—
170C32×C4#23514yes—
171C32×C4#23714yes—
172C32×C4#23711yescharacteristic
173C32×C4#23711yescharacteristic
174C3⋊Dic3—21no—
175C3⋊Dic3—21no—
176C3⋊Dic3—41no—
177C3⋊Dic3—121no—
178C3⋊Dic3—124no—
179C3⋊Dic3—244no—
180C3×Dic3—44no—
181C3×Dic3—64no—
182C3×Dic3—64no—
183C3×Dic3—124no—
184C4⋊Q8—91no—
185#125—271no—
186#103—31no—
187#103—271no—
188#103—271no—
189#103—271no—
190#97—271no—
191C33#49111yescharacteristic, complemented, Sylow
192C3×Q8—28no—
193C3×Q8—24no—
194C3×Q8—24no—
195C3×Q8—21no—
196C3×Q8—21no—
197C3×Q8—44no—
198C3×Q8—181no—
199C3×Q8—181no—
200C2×C3×C4—24no—
201C2×C3×C4—21no—
202C2×C3×C4—24no—
203C2×C3×C4—44no—
204C2×C3×C4—64no—
205C2×C3×C4—91no—
206C2×C3×C4—91no—
207C2×C3×C4—181no—
208C2×C3×C4C2×D3214yes—
209C2×C3×C4C22×C3⋊D311yescharacteristic
210C2×Dic3—21no—
211C2×Dic3—34no—
212C2×Dic3—34no—
213C2×Dic3—64no—
214C2×Dic3—184no—
215C2×Dic3—181no—
216C2×Dic3—364no—
217Dic6—64no—
218Dic6—64no—
219C3×C8—64no—
220C3⋊C8—21no—
221C3⋊C8—184no—
222C3⋊C8—181no—
223C3⋊C8—364no—
224C2×C32—24no—
225C2×C32—24no—
226C2×C32—24no—
227C2×C32#88514yes—
228C2×C32#91514yes—
229C2×C32#91814yes—
230C2×C32#93611yescharacteristic
231C2×C32#94211yescharacteristic
232C2×C32#94211yescharacteristic
233C2×Q8—91no—
234C2×Q8D3×C3⋊D311yescharacteristic
235C2×C8—31no—
236C2×C8—271no—
237C4⋊C4—31no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C4⋊C4—271no—
241C4⋊C4—541no—
242C4⋊C4—541no—
243C42—91no—
244C22×C3—24no—
245C22×C3D3×#6714yes—
246C22×C3D4×C3⋊D311yescharacteristic
247C3×C4—24no—
248C3×C4—24no—
249C3×C4—48no—
250C3×C4—41no—
251C3×C4—44no—
252C3×C4—124no—
253C3×C4—181no—
254C3×C4—181no—
255C3×C4—361no—
256C3×C4#380414yes—
257C3×C4#380514yes—
258C3×C4#382811yescharacteristic
259C3×C4#383011yescharacteristic
260Dic3—41no—
261Dic3—64no—
262Dic3—64no—
263Dic3—124no—
264Dic3—361no—
265Dic3—364no—
266Dic3—724no—
267C32—24no—
268C32#479714yescomplemented
269C32#508211yescharacteristic, complemented
270Q8—21no—
271Q8—21no—
272Q8—181no—
273Q8—181no—
274C2×C4—21no—
275C2×C4—61no—
276C2×C4—91no—
277C2×C4—91no—
278C2×C4—181no—
279C2×C4—541no—
280C2×C4C2×D3×C3⋊D311yescharacteristic
281C8—61no—
282C8—541no—
283C2×C3—24no—
284C2×C3—24no—
285C2×C3—24no—
286C2×C3#6394614yes—
287C2×C3#6394814yes—
288C2×C3#6395914yes—
289C2×C3#6409411yescharacteristic
290C2×C3#6412411yescharacteristic
291C2×C3#6412711yescharacteristic
292C22C3⋊D3×#6711yescharacteristic
293C4—41no—
294C4—121no—
295C4—181no—
296C4—181no—
297C4—361no—
298C4—1081no—
299C4#8902911yescharacteristic
300C4#8903111yescharacteristic
301C3—24no—
302C3#1059161314yescomplemented
303C3#1059249611yescharacteristic, complemented
304C2#1172358411yescharacteristic
305C2#1172358611yescharacteristic
306C2#1172359111yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#491
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
191. C33#491semidirect
268. C32#4797semidirect
269. C32#5082semidirect
302. C3#10591613semidirect
303. C3#10592496semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
#117234472
#117222052
C32×#8352
#117220402
#117220392
#117220202
#105924963
#105916133