Finite Group Databasebeta

← #13159256

Subgroups — #13159256

Order 1728.

684 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C32×#888C211yesmaximal, characteristic
2#11723454C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic
5#11722056C211yesmaximal, characteristic
6#11722054C211yesmaximal, characteristic
7#11722019C211yesmaximal, characteristic
8#10592780—31nomaximal
9#10590420—34nomaximal
10C2×C32×Dic6—21no—
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88807—21no—
20#88806—21no—
21C32×C4⋊Q8—31no—
22#64079—31no—
23C3×#888D314yes—
24#63869—34no—
25#63635—31no—
26#63635—31no—
27#63635—34no—
28#63635—64no—
29#63634—31no—
30#63634D311yescharacteristic
31#63630—31no—
32C3×#807—34no—
33#63576—34no—
34#63573—34no—
35#63554—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3D411yescharacteristic
42C2×C32×Dic3D411yescharacteristic
43C32×Dic6—42no—
44C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
45C3×C3⋊(C3⋊C8)Q812yescomplemented
46#5081—94no—
47#4876—91no—
48C2×C32×Q8—61no—
49C2×C3×Dic6—24no—
50C32×C4⋊C4—24no—
51C32×C4⋊C4—31no—
52C32×C4⋊C4—61no—
53C32×C4⋊C4C2×D314yes—
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C42—31no—
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—34no—
58C4⋊(C3⋊Dic3)—64no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60#3750—61no—
61#3750—61no—
62C2×C3⋊(C3⋊C8)—31no—
63C2×C3⋊(C3⋊C8)—34no—
64C2×C3⋊(C3⋊C8)—64no—
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×C4⋊Dic3—34no—
67C3×C4⋊Dic3C2×D314yes—
68C3×Dic3⋊C4—24no—
69C3×C4×Dic3C2×D314yes—
70C2×C3×C3⋊C8—34no—
71#3724—64no—
72Dic3⋊Dic3—64no—
73C22×C33C2×D411yescharacteristic
74C33×C4—41no—
75C33×C4C2×Q811yescharacteristic
76C33×C4C2×Q811yescharacteristic
77C3⋊(C3⋊Dic3)—41no—
78C3×C3⋊Dic3—41no—
79C32×Dic3—21no—
80C32×Dic3—21no—
81C32×Dic3—41no—
82C3×C4⋊Q8—34no—
83#890—91no—
84#888C3⋊D311yescharacteristic
85#870—94no—
86C3×#107—91no—
87#817—91no—
88#815—91no—
89#808—94no—
90#808—94no—
91#808—91no—
92#808—184no—
93#807—34no—
94#807—94no—
95#807—91no—
96#803—94no—
97C32×Q8—122no—
98C2×C32×C4—21no—
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4—31no—
102C2×C32×C4—31no—
103C2×C32×C4—44no—
104C2×C32×C4—61no—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—64no—
110C2×C3⋊Dic3—124no—
111C2×C3×Dic3—24no—
112C2×C3×Dic3—64no—
113C2×C3×Dic3#6714yes—
114C2×C3×Dic3#6714yes—
115C3×Dic6—48no—
116C3⋊(C3⋊C8)—32no—
117C3⋊(C3⋊C8)—38no—
118C3⋊(C3⋊C8)—68no—
119C3⋊(C3⋊C8)Dic612yescomplemented
120C3×C3⋊C8—38no—
121#501—271noSylow
122C2×C33C4⋊Q811yescharacteristic
123C2×C33AGL(1,ℤ/8)11yescharacteristic
124C2×C33#13711yescharacteristic
125C2×C3×Q8—64no—
126C2×Dic6—21no—
127C2×C3×C8—91no—
128C3×C4⋊C4—24no—
129C3×C4⋊C4—34no—
130C3×C4⋊C4—64no—
131C3×C4⋊C4—91no—
132C3×C4⋊C4D3214yes—
133C3×C4⋊C4C2×C3⋊D311yescharacteristic
134C3×C42—34no—
135C4⋊Dic3—34no—
136C4⋊Dic3—94no—
137C4⋊Dic3—91no—
138C4⋊Dic3—184no—
139C4⋊Dic3—181no—
140C4⋊Dic3C2×C3⋊D311yescharacteristic
141Dic3⋊C4—21no—
142Dic3⋊C4—184no—
143Dic3⋊C4—184no—
144Dic3⋊C4—181no—
145C4×Dic3C2×C3⋊D311yescharacteristic
146C2×C3⋊C8—34no—
147C2×C3⋊C8—94no—
148C2×C3⋊C8—91no—
149C2×C3⋊C8—184no—
150C22×C32—24no—
151C22×C32D3×D411yescharacteristic
152C22×C32C2×#6714yes—
153C32×C4—24no—
154C32×C4—24no—
155C32×C4—41no—
156C32×C4—44no—
157C32×C4—61no—
158C32×C4—61no—
159C32×C4—84no—
160C32×C4—121no—
161C32×C4C2×Dic611yescharacteristic
162C32×C4D3×Q814yes—
163C32×C4D3×Q814yes—
164C32×C4D3×Q811yescharacteristic
165C3⋊Dic3—41no—
166C3⋊Dic3—121no—
167C3⋊Dic3—124no—
168C3⋊Dic3—244no—
169C3×Dic3—24no—
170C3×Dic3—24no—
171C3×Dic3—44no—
172C3×Dic3—124no—
173C4⋊Q8—31no—
174#125—271no—
175#107—91no—
176#107—271no—
177#106—271no—
178#106—271no—
179#97—271no—
180C33#50111yescharacteristic, complemented, Sylow
181C3×Q8—128no—
182C2×C3×C4—24no—
183C2×C3×C4—21no—
184C2×C3×C4—24no—
185C2×C3×C4—34no—
186C2×C3×C4—34no—
187C2×C3×C4—44no—
188C2×C3×C4—64no—
189C2×C3×C4—181no—
190C2×C3×C4C2×D3214yes—
191C2×C3×C4C22×C3⋊D311yescharacteristic
192C2×Dic3—21no—
193C2×Dic3—64no—
194C2×Dic3—184no—
195C2×Dic3—181no—
196C2×Dic3—364no—
197C2×Dic3#63811yescharacteristic
198C2×Dic3#63811yescharacteristic
199Dic6—42no—
200C3×C8—92no—
201C3⋊C8—38no—
202C3⋊C8—98no—
203C3⋊C8—92no—
204C3⋊C8—188no—
205C2×C32—24no—
206C2×C32—24no—
207C2×C32—24no—
208C2×C32#88811yescharacteristic
209C2×C32#91111yescharacteristic
210C2×C32#91811yescharacteristic
211C2×C32#93214yes—
212C2×C32#94214yes—
213C2×C32#94414yes—
214C2×Q8—61no—
215C2×C8—91no—
216C2×C8—271no—
217C4⋊C4—31no—
218C4⋊C4—61no—
219C4⋊C4—91no—
220C4⋊C4—271no—
221C4⋊C4—541no—
222C4⋊C4—541no—
223C4⋊C4D3×C3⋊D311yescharacteristic
224C42—31no—
225C22×C3—24no—
226C22×C3D3×#6714yes—
227C22×C3C2×#63811yescharacteristic
228C3×C4—24no—
229C3×C4—24no—
230C3×C4—41no—
231C3×C4—44no—
232C3×C4—64no—
233C3×C4—64no—
234C3×C4—84no—
235C3×C4—124no—
236C3×C4—361no—
237C3×C4D3×Dic614yes—
238C3×C4#379714yes—
239C3×C4Q8×C3⋊D311yescharacteristic
240C3×C4Q8×C3⋊D311yescharacteristic
241Dic3—21no—
242Dic3—21no—
243Dic3—41no—
244Dic3—124no—
245Dic3—361no—
246Dic3—364no—
247Dic3—724no—
248C32—24no—
249C32#487611yescharacteristic, complemented
250C32#508114yescomplemented
251Q8—122no—
252C2×C4—21no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—61no—
256C2×C4—181no—
257C2×C4—541no—
258C2×C4C2×D3×C3⋊D311yescharacteristic
259C8—92no—
260C8—272no—
261C2×C3—24no—
262C2×C3—24no—
263C2×C3—24no—
264C2×C3#6383914yes—
265C2×C3#6393214yes—
266C2×C3#6394814yes—
267C2×C3#6414111yescharacteristic
268C2×C3#6415111yescharacteristic
269C2×C3#6415311yescharacteristic
270C22D3×#63811yescharacteristic
271C4—41no—
272C4—61no—
273C4—61no—
274C4—121no—
275C4—361no—
276C4—1081no—
277C4C3⋊D3×Dic611yescharacteristic
278C4#8901911yescharacteristic
279C3—24no—
280C3#1059042014yescomplemented
281C3#1059278011yescharacteristic, complemented
282C2#1172339311yescharacteristic
283C2#1172355511yescharacteristic
284C2#1172358111yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#501
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
44. C3⋊(C3⋊(C3⋊C8))Q8semidirect
45. C3×C3⋊(C3⋊C8)Q8semidirect
119. C3⋊(C3⋊C8)Dic6semidirect
180. C33#501semidirect
249. C32#4876semidirect
250. C32#5081semidirect
280. C3#10590420semidirect
281. C3#10592780semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
#117234542
#117221782
C3×#636342
#117220562
#117220542
#117220192
#105927803
#105904203