Finite Group Databasebeta

← #13159290

Subgroups — #13159290

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×#64099C211yesmaximal, characteristic
2#11723455C211yesmaximal, characteristic
3#11722180C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic
5#11722042C211yesmaximal, characteristic
6#11722037C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic
8#10592496—31nomaximal
9#10590562—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20#88807—21no—
21#64153—31no—
22#64099D311yescharacteristic
23C3×#890—34no—
24#63866—34no—
25C32×#103—31no—
26#63662—31no—
27#63643—31no—
28#63642—31no—
29#63637—31no—
30#63637—34no—
31#63637—64no—
32C3×#810D314yes—
33#63567—34no—
34#63565—34no—
35#63554—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39#6308—21no—
40#6308—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C32×Dic3—21no—
45C32×Dic6—21no—
46C32×Dic6—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C32×C3⋊C8—21no—
49#5069—91no—
50#4797—94no—
51C2×C32×Q8—31no—
52C2×#634—31no—
53C2×#634—34no—
54C2×#634—64no—
55C2×C3×Dic6C2×D314yes—
56C2×C32×C8—31no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—21no—
63#3750—61no—
64#3750—61no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3—64no—
71C3×Dic3⋊C4—64no—
72C3×C4×Dic3—34no—
73C2×C3×C3⋊C8C2×D314yes—
74Dic3⋊Dic3—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×#125—91no—
86#944—94no—
87#890—34no—
88#869—91no—
89C3×#103—34no—
90#835—94no—
91#816—94no—
92#815—94no—
93#810—91no—
94#810—94no—
95#810—91no—
96#810—91no—
97#810—184no—
98#810C3⋊D311yescharacteristic
99#803—91no—
100C32×Q8—61no—
101C32×Q8—61no—
102C2×C32×C4—21no—
103C2×C32×C4—24no—
104C2×C32×C4—24no—
105C2×C32×C4—44no—
106C2×C32×C4—61no—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—64no—
112C2×C3⋊Dic3—124no—
113C2×C3⋊Dic3#6711yescharacteristic
114C2×C3⋊Dic3#6711yescharacteristic
115#634—68no—
116#634—64no—
117#634—61no—
118#634—61no—
119#634—64no—
120#634—124no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—64no—
125C3×Dic6—24no—
126C3×Dic6—24no—
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—34no—
137C2×Dic6—94no—
138C2×Dic6—91no—
139C2×Dic6—184no—
140C2×Dic6C2×C3⋊D311yescharacteristic
141C2×C3×C8—34no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—181no—
145C3×C4⋊C4—181no—
146C3×C4⋊C4D3214yes—
147C3×C4⋊C4C2×C3⋊D311yescharacteristic
148C3×C42—91no—
149C4⋊Dic3—34no—
150C4⋊Dic3—64no—
151C4⋊Dic3—181no—
152C4⋊Dic3—181no—
153Dic3⋊C4—64no—
154Dic3⋊C4—184no—
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—41no—
167C32×C4—44no—
168C32×C4—84no—
169C32×C4—121no—
170C32×C4#23514yes—
171C32×C4#23714yes—
172C32×C4#23911yescharacteristic
173C32×C4#23911yescharacteristic
174C3⋊Dic3—21no—
175C3⋊Dic3—21no—
176C3⋊Dic3—41no—
177C3⋊Dic3—128no—
178C3⋊Dic3—124no—
179C3⋊Dic3—121no—
180C3×Dic3—44no—
181C3×Dic3—64no—
182C3×Dic3—64no—
183C3×Dic3—124no—
184C4⋊Q8—271no—
185#125—91no—
186#103—31no—
187#103—271no—
188#103—271no—
189#103—271no—
190#97—271no—
191C33#49111yescharacteristic, complemented, Sylow
192C3×Q8—64no—
193C3×Q8—64no—
194C2×C3×C4—24no—
195C2×C3×C4—21no—
196C2×C3×C4—24no—
197C2×C3×C4—44no—
198C2×C3×C4—64no—
199C2×C3×C4—91no—
200C2×C3×C4—91no—
201C2×C3×C4—181no—
202C2×C3×C4C2×D3214yes—
203C2×C3×C4C22×C3⋊D311yescharacteristic
204C2×Dic3—21no—
205C2×Dic3—34no—
206C2×Dic3—34no—
207C2×Dic3—64no—
208C2×Dic3—184no—
209C2×Dic3—181no—
210C2×Dic3—364no—
211Dic6—21no—
212Dic6—21no—
213Dic6—188no—
214Dic6—184no—
215Dic6—184no—
216Dic6—181no—
217Dic6—181no—
218Dic6—364no—
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#94711yescharacteristic
231C2×C32#95111yescharacteristic
232C2×C32#95111yescharacteristic
233C2×Q8—31no—
234C2×Q8—271no—
235C2×C8—31no—
236C2×C8—271no—
237C4⋊C4—91no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C4⋊C4—541no—
241C4⋊C4—541no—
242C4⋊C4D3×C3⋊D311yescharacteristic
243C42—91no—
244C22×C3—24no—
245C22×C3D3×#6714yes—
246C22×C3D4×C3⋊D311yescharacteristic
247C3×C4—24no—
248C3×C4—24no—
249C3×C4—41no—
250C3×C4—44no—
251C3×C4—84no—
252C3×C4—124no—
253C3×C4—181no—
254C3×C4—181no—
255C3×C4—361no—
256C3×C4#379614yes—
257C3×C4#379914yes—
258C3×C4#382811yescharacteristic
259C3×C4#383011yescharacteristic
260Dic3—41no—
261Dic3—64no—
262Dic3—64no—
263Dic3—124no—
264Dic3—368no—
265Dic3—361no—
266Dic3—364no—
267C32—24no—
268C32#479714yescomplemented
269C32#506911yescharacteristic, complemented
270Q8—61no—
271Q8—61no—
272Q8—541no—
273Q8—541no—
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#6390714yes—
287C2×C3#6392914yes—
288C2×C3#6394314yes—
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#8901511yescharacteristic
300C4#8902211yescharacteristic
301C3—24no—
302C3#1059056214yescomplemented
303C3#1059249611yescharacteristic, complemented
304C2#1172352111yescharacteristic
305C2#1172355211yescharacteristic
306C2#1172357611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#491
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
191. C33#491semidirect
268. C32#4797semidirect
269. C32#5069semidirect
302. C3#10590562semidirect
303. C3#10592496semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
#117234552
#117221802
C32×#8102
#117220422
#117220372
#117220202
#105924963
#105905623