Finite Group Databasebeta

← #13159222

Subgroups — #13159222

Order 1728.

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

normal subgroups only

This group has 744 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, complemented
2#11723429C211yesmaximal, characteristic
3#11722180C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic
5#11722042C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592467—31nomaximal
9#10590533—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
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(C2×D3)⋊(C3⋊Dic3)—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#64145—31no—
22#64099D311yescharacteristic, complemented
23C3×#890—34no—
24#63860—34no—
25C32×#102—31no—
26#63659—31no—
27#63643—31no—
28C8×C3⋊Dic3—31no—
29#63637—31no—
30#63637—34no—
31#63637—64no—
32C3×#809D314yes—
33#63567—34no—
34#63562—34no—
35Dic3×C3⋊C8—34no—
36C22×C32×D3—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40#6308—21no—
41#6308—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C3×C3⋊Dic3D411yescharacteristic
45C32×D12—21no—
46C32×D12—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C32×C3⋊C8—21no—
49#5068—91no—
50#4768—94no—
51C2×C32×D4—31no—
52C2×#634—31no—
53C2×#634—34no—
54C2×#634—64no—
55C2×C3×D12C2×D314yes—
56C2×C32×C8—31no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60#3757—61no—
61#3757—61no—
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—21no—
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—
74(C2×D3)⋊Dic3—64no—
75(C2×D3)⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C33×C4—41no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C3×#125—91no—
88#936—94no—
89#890—34no—
90#875—91no—
91C3×#102—34no—
92#832—94no—
93#816—94no—
94C8×Dic3—94no—
95#810—94no—
96#810—91no—
97#810—91no—
98#810—184no—
99#809—91no—
100#809C3⋊D311yescharacteristic
101C4×C3⋊C8—91no—
102C23×C32—61no—
103C22×C3×D3—24no—
104C32×D4—61no—
105C32×D4—61no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—44no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—124no—
116C2×C3⋊Dic3#6711yescharacteristic
117C2×C3⋊Dic3#6711yescharacteristic
118#634—68no—
119#634—64no—
120#634—61no—
121#634—61no—
122#634—64no—
123#634—124no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—64no—
127C3×D12—24no—
128C3×D12—24no—
129C32×C8—61no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133C3×C3⋊C8—24no—
134#488—271noSylow
135C2×C33#12411yescharacteristic
136C2×C33#13511yescharacteristic
137C2×C33#13511yescharacteristic
138C32×D3—81no—
139C2×C3×D4—34no—
140C2×D12C2×C3⋊D311yescharacteristic
141C2×Dic6—94no—
142C2×Dic6—91no—
143C2×Dic6—184no—
144C2×C3×C8—34no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—181no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4D3214yes—
150C3×C4⋊C4C2×C3⋊D311yescharacteristic
151C3×C42—91no—
152#217—184no—
153#217—184no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—181no—
156C4⋊Dic3—34no—
157C4⋊Dic3—64no—
158Dic3⋊C4—64no—
159C4×Dic3—34no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—184no—
163C2×C3⋊C8C2×C3⋊D311yescharacteristic
164C22×C32—24no—
165C22×C32—121no—
166C22×C32—121no—
167C22×C32—121no—
168C22×C32D3×D414yes—
169C22×C32C2×#6711yescharacteristic
170C2×C3×D3—44no—
171C2×C3×D3—44no—
172C2×C3×D3—44no—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—41no—
176C32×C4—44no—
177C32×C4—84no—
178C32×C4#23514yes—
179C32×C4#23714yes—
180C32×C4#23911yescharacteristic
181C32×C4#23911yescharacteristic
182C3⋊Dic3—21no—
183C3⋊Dic3—21no—
184C3⋊Dic3—41no—
185C3⋊Dic3—128no—
186C3⋊Dic3—124no—
187C3⋊Dic3—121no—
188C3×Dic3—64no—
189C3×Dic3—64no—
190C3×Dic3—124no—
191#125—91no—
192#124—271no—
193#103—271no—
194#103—271no—
195#102—31no—
196#102—271no—
197C4×C8—271no—
198C33#48811yescharacteristic, complemented, Sylow
199C23×C3—64no—
200C22×D3—21no—
201C3×D4—64no—
202C3×D4—64no—
203C2×C3×C4—24no—
204C2×C3×C4—21no—
205C2×C3×C4—24no—
206C2×C3×C4—44no—
207C2×C3×C4—91no—
208C2×C3×C4—91no—
209C2×C3×C4—181no—
210C2×C3×C4C2×D3214yes—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212C2×Dic3—34no—
213C2×Dic3—34no—
214C2×Dic3—64no—
215C2×Dic3—184no—
216C2×Dic3—181no—
217C2×Dic3—364no—
218D12—21no—
219D12—21no—
220Dic6—188no—
221Dic6—184no—
222Dic6—184no—
223Dic6—181no—
224Dic6—181no—
225Dic6—364no—
226C3×C8—64no—
227C3⋊C8—21no—
228C3⋊C8—184no—
229C3⋊C8—181no—
230C3⋊C8—364no—
231C2×C32—24no—
232C2×C32—24no—
233C2×C32—24no—
234C2×C32—241no—
235C2×C32#88514yes—
236C2×C32#91214yes—
237C2×C32#91614yes—
238C2×C32#94711yescharacteristic
239C2×C32#95011yescharacteristic
240C2×C32#95011yescharacteristic
241C3×D3—84no—
242C2×Q8—271no—
243C2×D4—31no—
244C2×C8—31no—
245C2×C8—271no—
246C4⋊C4—91no—
247C4⋊C4—181no—
248C4⋊C4—181no—
249C4⋊C4D3×C3⋊D311yescharacteristic
250C22⋊C4—541no—
251C22⋊C4—541no—
252C42—91no—
253C22×C3—24no—
254C22×C3—124no—
255C22×C3—124no—
256C22×C3—124no—
257C22×C3D3×#6714yes—
258C22×C3D4×C3⋊D311yescharacteristic
259C2×D3—41no—
260C2×D3—41no—
261C2×D3—41no—
262C3×C4—24no—
263C3×C4—24no—
264C3×C4—41no—
265C3×C4—44no—
266C3×C4—84no—
267C3×C4—181no—
268C3×C4—181no—
269C3×C4—361no—
270C3×C4#379614yes—
271C3×C4#379914yes—
272C3×C4#382811yescharacteristic
273C3×C4#383011yescharacteristic
274Dic3—64no—
275Dic3—64no—
276Dic3—124no—
277Dic3—368no—
278Dic3—361no—
279Dic3—364no—
280C32—24no—
281C32#476814yescomplemented
282C32#506811yescharacteristic, complemented
283C23—61no—
284Q8—541no—
285Q8—541no—
286D4—61no—
287D4—61no—
288C2×C4—21no—
289C2×C4—91no—
290C2×C4—91no—
291C2×C4—181no—
292C2×C4—541no—
293C2×C4C2×D3×C3⋊D311yescharacteristic
294C8—61no—
295C8—541no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—244no—
300C2×C3#6390714yes—
301C2×C3#6393314yes—
302C2×C3#6394714yes—
303C2×C3#6409411yescharacteristic
304C2×C3#6412111yescharacteristic
305C2×C3#6412511yescharacteristic
306D3—81no—
307C22—121no—
308C22—121no—
309C22—121no—
310C22C3⋊D3×#6711yescharacteristic
311C4—41no—
312C4—181no—
313C4—181no—
314C4—361no—
315C4—1081no—
316C4#8901511yescharacteristic
317C4#8902211yescharacteristic
318C3—24no—
319C3#1059053314yescomplemented
320C3#1059246711yescharacteristic, complemented
321C2—241no—
322C2#1172352111yescharacteristic
323C2#1172355911yescharacteristic
324C2#1172358311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#488
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
1. C3×#64099C2semidirect
3. #11722180C2semidirect
5. #11722042C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
22. #64099D3semidirect
198. C33#488semidirect
281. C32#4768semidirect
282. C32#5068semidirect
319. C3#10590533semidirect
320. C3#10592467semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
#117234292
#117221802
C32×#8092
#117220422
#117220312
C3⋊C8×C3⋊Dic32
#105924673
#105905333