Finite Group Databasebeta

← #13159898

Subgroups — #13159898

Order 1728.

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

normal subgroups only

This group has 764 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#11723536C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic
3C8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
4C3×#63643C211yesmaximal, characteristic
5C32×#816C211yesmaximal, characteristic
6#11722039C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592491—31nomaximal
9#10590901—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×#6293—21no—
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88806—21no—
19#88806—21no—
20#88806—21no—
21#64153—31no—
22#64097—31no—
23#63919—34no—
24#63873—34no—
25C32×#103—31no—
26#63662—31no—
27#63643D311yescharacteristic
28C8×C3⋊Dic3—31no—
29C8×C3⋊Dic3—34no—
30C8×C3⋊Dic3—64no—
31#63637—31no—
32C3×#816—34no—
33C3×#816D314yes—
34#63565—34no—
35#63565—34no—
36C2×C33×C4C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)D411yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C3×#634—21no—
42C3×#634—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45C32×Dic6—21no—
46C32×Dic6—21no—
47#6293—42no—
48C33×C8—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50#4792—94no—
51#4792—91no—
52C2×C32×Q8—31no—
53C2×#634—61no—
54C2×#634C2×D311yescharacteristic
55C2×C3×Dic6—34no—
56C2×C3×Dic6C2×D314yes—
57C2×#627—64no—
58C2×C32×C8—24no—
59C2×C32×C8C2×D314yes—
60C2×C32×C8C2×D311yescharacteristic
61C32×C4⋊C4—31no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63#3750—61no—
64#3750—61no—
65#3750—61no—
66C4×C3⋊Dic3—31no—
67C4×C3⋊Dic3—34no—
68C4×C3⋊Dic3—64no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74#3724—64no—
75#3724—64no—
76#3724—64no—
77C22×C33C2×D411yescharacteristic
78C33×C4#4111yescharacteristic
79C33×C4#4111yescharacteristic
80C3⋊(C3⋊Dic3)SD1612yes—
81C3⋊(C3⋊Dic3)Q1612yes—
82C3×C3⋊Dic3—41no—
83C3×C3⋊Dic3—41no—
84C32×Dic3—41no—
85C32×Dic3—41no—
86#944—94no—
87#944—91no—
88#888—94no—
89#888—91no—
90C3×#103—34no—
91C3×#103—91no—
92#835—94no—
93#835—91no—
94#816—34no—
95#816C3⋊D311yescharacteristic
96C8×Dic3—94no—
97C8×Dic3—91no—
98C8×Dic3—184no—
99#810—94no—
100#810—91no—
101C32×Q8—61no—
102C32×Q8—61no—
103C2×C32×C4—24no—
104C2×C32×C4—61no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—64no—
116#634—21no—
117#634—21no—
118#634—122no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C3×Dic6—24no—
124C3×Dic6—24no—
125C3×Dic6—64no—
126C3×Dic6—64no—
127#627—128no—
128C32×C8—28no—
129C32×C8—24no—
130C32×C8—21no—
131C3⋊(C3⋊C8)—68no—
132C3⋊(C3⋊C8)—64no—
133C3⋊(C3⋊C8)—61no—
134#487—271noSylow
135C2×C33#12411yescharacteristic
136C2×C33C2×SD1611yescharacteristic
137C2×C33C2×Q1611yescharacteristic
138C2×C3×Q8—34no—
139C2×C3×Q8—91no—
140C2×Dic6—34no—
141C2×Dic6—181no—
142C2×Dic6—184no—
143C2×Dic6C2×C3⋊D311yescharacteristic
144C2×C3×C8—24no—
145C2×C3×C8D3214yes—
146C2×C3×C8C2×C3⋊D311yescharacteristic
147C3×C4⋊C4—34no—
148C3×C4⋊C4—91no—
149C4⋊Dic3—34no—
150C4⋊Dic3C2×C3⋊D311yescharacteristic
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154Dic3⋊C4—184no—
155Dic3⋊C4—184no—
156Dic3⋊C4—184no—
157C4×Dic3—94no—
158C4×Dic3—91no—
159C4×Dic3—184no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—184no—
163C22×C32—24no—
164C22×C32D3×D414yes—
165C22×C32D3×D411yescharacteristic
166C32×C4—24no—
167C32×C4—24no—
168C32×C4—121no—
169C32×C4—121no—
170C32×C4#23514yes—
171C32×C4#23511yescharacteristic
172C32×C4#23714yes—
173C32×C4#23711yescharacteristic
174C3⋊Dic3—32no—
175C3⋊Dic3—32no—
176C3⋊Dic3—38no—
177C3⋊Dic3—38no—
178C3⋊Dic3—41no—
179C3⋊Dic3—41no—
180C3⋊Dic3—68no—
181C3⋊Dic3—68no—
182C3×Dic3—44no—
183C3×Dic3—44no—
184C3×Dic3—124no—
185C3×Dic3—124no—
186C4⋊Q8—271no—
187C4⋊Q8—271no—
188#103—31no—
189#103—91no—
190#103—271no—
191#103—271no—
192C4×C8—271no—
193C33#48711yescharacteristic, complemented, Sylow
194C3×Q8—64no—
195C3×Q8—64no—
196C3×Q8—181no—
197C3×Q8—181no—
198C2×C3×C4—24no—
199C2×C3×C4—64no—
200C2×C3×C4—64no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4C2×D3214yes—
204C2×C3×C4C22×C3⋊D311yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—64no—
208C2×Dic3—64no—
209C2×Dic3—94no—
210C2×Dic3—94no—
211C2×Dic3—91no—
212C2×Dic3—91no—
213C2×Dic3—184no—
214C2×Dic3—184no—
215Dic6—21no—
216Dic6—21no—
217Dic6—64no—
218Dic6—64no—
219Dic6—362no—
220Dic6—368no—
221C3×C8—28no—
222C3×C8—24no—
223C3×C8—21no—
224C3⋊C8—188no—
225C3⋊C8—184no—
226C3⋊C8—181no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32#88514yes—
231C2×C32#88511yescharacteristic
232C2×C32D3×SD1614yes—
233C2×C32D3×SD1611yescharacteristic
234C2×C32D3×Q1614yes—
235C2×C32D3×Q1611yescharacteristic
236C2×Q8—31no—
237C2×Q8—91no—
238C2×Q8—541no—
239C2×C8—271no—
240C2×C8D3×C3⋊D311yescharacteristic
241C4⋊C4—31no—
242C4⋊C4—91no—
243C4⋊C4—541no—
244C4⋊C4—541no—
245C4⋊C4—541no—
246C42—271no—
247C22×C3—24no—
248C22×C3#380214yes—
249C22×C3D4×C3⋊D311yescharacteristic
250C3×C4—24no—
251C3×C4—24no—
252C3×C4—124no—
253C3×C4—124no—
254C3×C4—361no—
255C3×C4—361no—
256C3×C4#379814yes—
257C3×C4#380514yes—
258C3×C4#382811yescharacteristic
259C3×C4#383011yescharacteristic
260Dic3—41no—
261Dic3—41no—
262Dic3—98no—
263Dic3—98no—
264Dic3—92no—
265Dic3—92no—
266Dic3—124no—
267Dic3—124no—
268Dic3—188no—
269Dic3—188no—
270C32—24no—
271C32#479214yescomplemented
272C32#479211yescharacteristic, complemented
273Q8—61no—
274Q8—61no—
275Q8—181no—
276Q8—181no—
277Q8—1082no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—271no—
283C2×C4—271no—
284C2×C4C2×D3×C3⋊D311yescharacteristic
285C8—21no—
286C8—541no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—24no—
290C2×C3#6379614yes—
291C2×C3#6380214yes—
292C2×C3#6391514yes—
293C2×C3#6409411yescharacteristic
294C2×C3SD16×C3⋊D311yescharacteristic
295C2×C3Q16×C3⋊D311yescharacteristic
296C22#8902611yescharacteristic
297C4—121no—
298C4—121no—
299C4—272no—
300C4—272no—
301C4—361no—
302C4—361no—
303C4#8902011yescharacteristic
304C4#8902911yescharacteristic
305C3—24no—
306C3#1059090114yescomplemented
307C3#1059249111yescharacteristic, complemented
308C2#1172332111yescharacteristic
309C2#1172333011yescharacteristic
310C2#1172353011yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#487
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
193. C33#487semidirect
271. C32#4792semidirect
272. C32#4792semidirect
306. C3#10590901semidirect
307. C3#10592491semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117234512
C8×C3⋊(C3⋊Dic3)2
C3×#636432
C32×#8162
#117220392
#117220372
#105924913
#105909013