Finite Group Databasebeta

← #13159313

Subgroups — #13159313

Order 1728.

860 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723929C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic
3C3×#63637C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic
6#11722038C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic
8#10592491—31nomaximal
9#10590567—34nomaximal
10C2×#6308—21no—
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15#88857—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20#88806—21no—
21#64153—31no—
22#64097—31no—
23#64097—34no—
24#64097—64no—
25#63873—34no—
26C32×#103—31no—
27#63662—31no—
28#63643—31no—
29C8×C3⋊Dic3—31no—
30#63637D311yescharacteristic
31C3×#810—34no—
32C3×#810D314yes—
33#63566—34no—
34#63566—34no—
35#63553—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41#6308—42no—
42C2×C3×C3⋊Dic3—21no—
43C3×#634—21no—
44C3×#634—21no—
45C2×C32×Dic3—21no—
46C32×Dic6—21no—
47C32×Dic6—21no—
48C3×C3⋊(C3⋊C8)—21no—
49C32×C3⋊C8—21no—
50#4792—94no—
51#4792—91no—
52C2×C32×Q8—31no—
53C2×#634—61no—
54C2×#634—64no—
55C2×#634—68no—
56C2×#634C2×D311yescharacteristic
57C2×C3×Dic6—34no—
58C2×C3×Dic6C2×D314yes—
59C2×C32×C8—31no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66#3750—61no—
67#3750—61no—
68#3750—61no—
69#3750—64no—
70#3750—68no—
71C4×C3⋊Dic3—31no—
72C4×C3⋊Dic3—34no—
73C4×C3⋊Dic3—64no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C2×C3×C3⋊C8—34no—
76C2×C3×C3⋊C8C2×D314yes—
77#3724—64no—
78#3724—64no—
79C22×C33C2×D411yescharacteristic
80C33×C4—41no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3⋊(C3⋊Dic3)SD1612yes—
85C3⋊(C3⋊Dic3)Q1612yes—
86C3×C3⋊Dic3—41no—
87C32×Dic3—41no—
88#944—94no—
89#944—91no—
90#888—94no—
91#888—91no—
92#888—184no—
93C3×#103—34no—
94C3×#103—91no—
95#835—94no—
96#835—91no—
97#816—94no—
98#816—91no—
99C8×Dic3—94no—
100C8×Dic3—91no—
101#810—34no—
102#810C3⋊D311yescharacteristic
103C32×Q8—61no—
104C32×Q8—61no—
105C2×C32×C4—28no—
106C2×C32×C4—24no—
107C2×C32×C4—21no—
108C2×C32×C4—24no—
109C2×C32×C4—61no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—34no—
116C2×C3⋊Dic3—34no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—68no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—61no—
122#634—21no—
123#634—21no—
124#634—122no—
125#634—128no—
126#634—1216no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—64no—
129C3×Dic6—24no—
130C3×Dic6—24no—
131C3×Dic6—64no—
132C3×Dic6—64no—
133C32×C8—61no—
134C3⋊(C3⋊C8)—21no—
135C3×C3⋊C8—24no—
136C3×C3⋊C8—64no—
137#487—271noSylow
138C2×C33#12411yescharacteristic
139C2×C33C2×SD1611yescharacteristic
140C2×C33C2×Q1611yescharacteristic
141C2×C3×Q8—34no—
142C2×C3×Q8—91no—
143C2×Dic6—34no—
144C2×Dic6—184no—
145C2×Dic6—181no—
146C2×Dic6—188no—
147C2×Dic6C2×C3⋊D311yescharacteristic
148C2×C3×C8—34no—
149C2×C3×C8—91no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4D3214yes—
152C3×C4⋊C4C2×C3⋊D311yescharacteristic
153C4⋊Dic3—94no—
154C4⋊Dic3—91no—
155C4⋊Dic3—184no—
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158Dic3⋊C4—184no—
159Dic3⋊C4—184no—
160Dic3⋊C4—184no—
161Dic3⋊C4—181no—
162Dic3⋊C4—188no—
163C4×Dic3—94no—
164C4×Dic3—91no—
165C4×Dic3—184no—
166C2×C3⋊C8—34no—
167C2×C3⋊C8C2×C3⋊D311yescharacteristic
168C22×C32—24no—
169C22×C32D3×D414yes—
170C22×C32D3×D411yescharacteristic
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—48no—
174C32×C4—44no—
175C32×C4—41no—
176C32×C4—121no—
177C32×C4#23514yes—
178C32×C4#23511yescharacteristic
179C32×C4#23714yes—
180C32×C4#23711yescharacteristic
181C3⋊Dic3—32no—
182C3⋊Dic3—32no—
183C3⋊Dic3—38no—
184C3⋊Dic3—38no—
185C3⋊Dic3—41no—
186C3⋊Dic3—68no—
187C3⋊Dic3—68no—
188C3⋊Dic3—128no—
189C3⋊Dic3—124no—
190C3⋊Dic3—121no—
191C3×Dic3—44no—
192C3×Dic3—124no—
193C4⋊Q8—271no—
194C4⋊Q8—271no—
195#103—31no—
196#103—91no—
197#103—271no—
198#103—271no—
199C4×C8—271no—
200C33#48711yescharacteristic, complemented, Sylow
201C3×Q8—64no—
202C3×Q8—64no—
203C3×Q8—181no—
204C3×Q8—181no—
205C2×C3×C4—28no—
206C2×C3×C4—24no—
207C2×C3×C4—21no—
208C2×C3×C4—24no—
209C2×C3×C4—64no—
210C2×C3×C4—181no—
211C2×C3×C4C2×D3214yes—
212C2×C3×C4C22×C3⋊D311yescharacteristic
213C2×Dic3—21no—
214C2×Dic3—64no—
215C2×Dic3—94no—
216C2×Dic3—94no—
217C2×Dic3—91no—
218C2×Dic3—91no—
219C2×Dic3—184no—
220C2×Dic3—184no—
221C2×Dic3—188no—
222C2×Dic3—184no—
223C2×Dic3—181no—
224Dic6—21no—
225Dic6—21no—
226Dic6—64no—
227Dic6—64no—
228Dic6—362no—
229Dic6—368no—
230Dic6—3616no—
231C3×C8—64no—
232C3×C8—181no—
233C3⋊C8—21no—
234C3⋊C8—64no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32#88514yes—
239C2×C32#88511yescharacteristic
240C2×C32D3×SD1614yes—
241C2×C32D3×SD1611yescharacteristic
242C2×C32D3×Q1614yes—
243C2×C32D3×Q1611yescharacteristic
244C2×Q8—31no—
245C2×Q8—91no—
246C2×Q8—541no—
247C2×C8—31no—
248C2×C8—91no—
249C4⋊C4—271no—
250C4⋊C4—541no—
251C4⋊C4—541no—
252C4⋊C4—541no—
253C4⋊C4D3×C3⋊D311yescharacteristic
254C42—271no—
255C22×C3—24no—
256C22×C3#381114yes—
257C22×C3D4×C3⋊D311yescharacteristic
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—48no—
261C3×C4—41no—
262C3×C4—44no—
263C3×C4—124no—
264C3×C4—361no—
265C3×C4#379514yes—
266C3×C4#379514yes—
267C3×C4#382811yescharacteristic
268C3×C4#383011yescharacteristic
269Dic3—41no—
270Dic3—92no—
271Dic3—92no—
272Dic3—98no—
273Dic3—98no—
274Dic3—124no—
275Dic3—188no—
276Dic3—188no—
277Dic3—368no—
278Dic3—361no—
279Dic3—364no—
280C32—24no—
281C32#479214yescomplemented
282C32#479211yescharacteristic, complemented
283Q8—61no—
284Q8—61no—
285Q8—181no—
286Q8—181no—
287Q8—1082no—
288C2×C4—21no—
289C2×C4—61no—
290C2×C4—181no—
291C2×C4—271no—
292C2×C4—271no—
293C2×C4—541no—
294C2×C4C2×D3×C3⋊D311yescharacteristic
295C8—61no—
296C8—181no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—24no—
300C2×C3#6391314yes—
301C2×C3#6393014yes—
302C2×C3#6394414yes—
303C2×C3#6409411yescharacteristic
304C2×C3SD16×C3⋊D311yescharacteristic
305C2×C3Q16×C3⋊D311yescharacteristic
306C22#8903911yescharacteristic
307C4—41no—
308C4—121no—
309C4—272no—
310C4—272no—
311C4—361no—
312C4—1081no—
313C4#8901411yescharacteristic
314C4#8902111yescharacteristic
315C3—24no—
316C3#1059056714yescomplemented
317C3#1059249111yescharacteristic, complemented
318C2#1172352811yescharacteristic
319C2#1172355311yescharacteristic
320C2#1172357711yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#487
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
200. C33#487semidirect
281. C32#4792semidirect
282. C32#4792semidirect
316. C3#10590567semidirect
317. C3#10592491semidirect

Maximal subgroups

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

subgroupindex
#117239292
#117234512
C3×#636372
C32×#8102
#117220402
#117220382
#117220182
#105924913
#105905673