Finite Group Databasebeta

← #13159902

Subgroups — #13159902

Order 1728.

700 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 168 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#11723450C211yesmaximal, characteristic
3#11722185C211yesmaximal, characteristic
4C3×#63645C211yesmaximal, characteristic
5C32×#817C211yesmaximal, characteristic
6#11722056C211yesmaximal, characteristic
7#11722053C211yesmaximal, characteristic
8#10592555—31nomaximal
9#10590903—34nomaximal
10C2×#6293—21no—
11C2×C33×C8C2211yescharacteristic
12C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88807—21no—
19#88806—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#64099—31no—
22#64097—31no—
23#63919—34no—
24#63872—34no—
25C32×#106—31no—
26#63645D311yescharacteristic
27#63642—31no—
28#63642—34no—
29#63642—64no—
30#63635—31no—
31#63634—31no—
32C3×#818—34no—
33C3×#817D314yes—
34#63576—34no—
35#63574—34no—
36C2×C33×C4C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)D411yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—21no—
43#6293—42no—
44C33×C8Q812yescomplemented
45C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
46#4876—91no—
47#4856—94no—
48C2×#634—61no—
49C2×#627—64no—
50C2×C32×C8—24no—
51C2×C32×C8C2×D314yes—
52C2×C32×C8C2×D311yescharacteristic
53C32×C4⋊C4—31no—
54C32×C4⋊C4—31no—
55C4⋊(C3⋊Dic3)—61no—
56C4⋊(C3⋊Dic3)C2×D311yescharacteristic
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58#3750—61no—
59#3750—61no—
60C4×C3⋊Dic3—31no—
61C4×C3⋊Dic3—34no—
62C4×C3⋊Dic3—64no—
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—34no—
65C2×C3⋊(C3⋊C8)—64no—
66C3×C4⋊Dic3—34no—
67C3×C4⋊Dic3—34no—
68C3×C4⋊Dic3C2×D314yes—
69C3×C4⋊Dic3C2×D314yes—
70#3724—64no—
71Dic3⋊Dic3—64no—
72Dic3⋊Dic3—64no—
73C22×C33C2×D411yescharacteristic
74C33×C4C2×Q811yescharacteristic
75C33×C4C2×Q811yescharacteristic
76C3⋊(C3⋊Dic3)—21no—
77C3⋊(C3⋊Dic3)—21no—
78C3×C3⋊Dic3—41no—
79C3×C3⋊Dic3—41no—
80C32×Dic3—41no—
81C32×Dic3—41no—
82#890—94no—
83#890—91no—
84#888—94no—
85#888—91no—
86C3×#107—91no—
87C3×#106—34no—
88#818—34no—
89#817C3⋊D311yescharacteristic
90#815—94no—
91#815—91no—
92#815—184no—
93#808—94no—
94#808—91no—
95#807—94no—
96#807—91no—
97C2×C32×C4—24no—
98C2×C32×C4—61no—
99C2×C32×C4—61no—
100C2×C32×C4C22×D314yes—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—31no—
106C2×C3⋊Dic3—34no—
107C2×C3⋊Dic3—34no—
108C2×C3⋊Dic3—64no—
109C2×C3⋊Dic3—64no—
110#634—122no—
111C2×C3×Dic3—24no—
112C2×C3×Dic3—24no—
113C2×C3×Dic3—64no—
114C2×C3×Dic3—64no—
115#627—128no—
116C32×C8—28no—
117C32×C8Dic618yescomplemented
118C32×C8Dic612yescomplemented
119C3⋊(C3⋊C8)—32no—
120C3⋊(C3⋊C8)—38no—
121C3⋊(C3⋊C8)—68no—
122#501—271noSylow
123C2×C33C4⋊Q811yescharacteristic
124C2×C33AGL(1,ℤ/8)11yescharacteristic
125C2×C33#13711yescharacteristic
126C2×Dic6—181no—
127C2×Dic6—184no—
128C2×C3×C8—24no—
129C2×C3×C8D3214yes—
130C2×C3×C8C2×C3⋊D311yescharacteristic
131C3×C4⋊C4—34no—
132C3×C4⋊C4—34no—
133C3×C4⋊C4—91no—
134C3×C4⋊C4—91no—
135C4⋊Dic3—34no—
136C4⋊Dic3—34no—
137C4⋊Dic3—181no—
138C4⋊Dic3—184no—
139C4⋊Dic3C2×C3⋊D311yescharacteristic
140C4⋊Dic3C2×C3⋊D311yescharacteristic
141Dic3⋊C4—181no—
142Dic3⋊C4—181no—
143Dic3⋊C4—184no—
144Dic3⋊C4—184no—
145C4×Dic3—94no—
146C4×Dic3—91no—
147C4×Dic3—184no—
148C2×C3⋊C8—94no—
149C2×C3⋊C8—91no—
150C2×C3⋊C8—184no—
151C22×C32—24no—
152C22×C32D3×D414yes—
153C22×C32D3×D411yescharacteristic
154C32×C4—24no—
155C32×C4—24no—
156C32×C4—121no—
157C32×C4—121no—
158C32×C4C2×Dic614yes—
159C32×C4C2×Dic611yescharacteristic
160C32×C4D3×Q814yes—
161C32×C4D3×Q811yescharacteristic
162C3⋊Dic3—41no—
163C3⋊Dic3—41no—
164C3⋊Dic3—61no—
165C3⋊Dic3—61no—
166C3⋊Dic3—64no—
167C3⋊Dic3—64no—
168C3⋊Dic3—124no—
169C3⋊Dic3—124no—
170C3×Dic3—44no—
171C3×Dic3—44no—
172C3×Dic3—124no—
173C3×Dic3—124no—
174C4⋊Q8—271no—
175#125—271no—
176#107—91no—
177#107—271no—
178#106—31no—
179#106—271no—
180#97—271no—
181C33#50111yescharacteristic, complemented, Sylow
182C2×C3×C4—24no—
183C2×C3×C4—64no—
184C2×C3×C4—64no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4C2×D3214yes—
188C2×C3×C4C22×C3⋊D311yescharacteristic
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—64no—
192C2×Dic3—64no—
193C2×Dic3—94no—
194C2×Dic3—94no—
195C2×Dic3—91no—
196C2×Dic3—91no—
197C2×Dic3—184no—
198C2×Dic3—184no—
199Dic6—362no—
200Dic6—368no—
201C3×C8—28no—
202C3×C8#62718yescomplemented
203C3×C8#63412yescomplemented
204C3⋊C8—98no—
205C3⋊C8—92no—
206C3⋊C8—188no—
207C2×C32—24no—
208C2×C32—24no—
209C2×C32—24no—
210C2×C32#88814yes—
211C2×C32#88811yescharacteristic
212C2×C32#91111yescharacteristic
213C2×C32#91414yes—
214C2×C32#91514yes—
215C2×C32#91811yescharacteristic
216C2×Q8—541no—
217C2×C8—271no—
218C2×C8D3×C3⋊D311yescharacteristic
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—91no—
222C4⋊C4—91no—
223C4⋊C4—541no—
224C4⋊C4—541no—
225C4⋊C4—541no—
226C42—271no—
227C22×C3—24no—
228C22×C3#380214yes—
229C22×C3D4×C3⋊D311yescharacteristic
230C3×C4—24no—
231C3×C4—24no—
232C3×C4—124no—
233C3×C4—124no—
234C3×C4—361no—
235C3×C4—361no—
236C3×C4#379714yes—
237C3×C4C2×#62714yes—
238C3×C4C2×#63411yescharacteristic
239C3×C4Q8×C3⋊D311yescharacteristic
240Dic3—41no—
241Dic3—41no—
242Dic3—124no—
243Dic3—124no—
244Dic3—184no—
245Dic3—184no—
246Dic3—181no—
247Dic3—181no—
248Dic3—364no—
249Dic3—364no—
250C32—24no—
251C32#485614yescomplemented
252C32#487611yescharacteristic, complemented
253Q8—1082no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—181no—
257C2×C4—181no—
258C2×C4—271no—
259C2×C4—271no—
260C2×C4C2×D3×C3⋊D311yescharacteristic
261C8—272no—
262C8#629312yescomplemented
263C2×C3—24no—
264C2×C3—24no—
265C2×C3—24no—
266C2×C3#6379814yes—
267C2×C3#6381114yes—
268C2×C3#6391914yes—
269C2×C3#6409711yescharacteristic
270C2×C3#6412311yescharacteristic
271C2×C3#6412411yescharacteristic
272C22#8902611yescharacteristic
273C4—121no—
274C4—121no—
275C4—361no—
276C4—361no—
277C4—541no—
278C4—541no—
279C4#8901911yescharacteristic
280C4C2×#629311yescharacteristic
281C3—24no—
282C3#1059090314yescomplemented
283C3#1059255511yescharacteristic, complemented
284C2#1172332311yescharacteristic
285C2#1172334411yescharacteristic
286C2#1172353611yescharacteristic
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#501
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
44. C33×C8Q8semidirect
45. C3⋊(C3⋊(C3⋊C8))Q8semidirect
117. C32×C8Dic6semidirect
118. C32×C8Dic6semidirect
181. C33#501semidirect
202. C3×C8#627semidirect
203. C3×C8#634semidirect
251. C32#4856semidirect
252. C32#4876semidirect
262. C8#6293semidirect
282. C3#10590903semidirect
283. C3#10592555semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117234502
#117221852
C3×#636452
C32×#8172
#117220562
#117220532
#105925553
#105909033