Finite Group Databasebeta

← #13137343

Subgroups — #13137343

Order 1728.

374 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 89 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#64078⋊C3C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic
3C4⋊#6180C211yesmaximal, characteristic
4#11720315C212yesmaximal
5#11720313C212yesmaximal
6#10590284—31nomaximal
7#10590283—31nomaximal
8C2×#634⋊C3—21no—
9C2×#634⋊C3C2211yescharacteristic
10C42×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2212yes—
12C4⋊((C3⋊Dic3)⋊C3)—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15#3750⋊C3—21no—
16C4×(C3⋊Dic3)⋊C3—21no—
17#64078D311yescharacteristic
18C4×#634D311yescharacteristic
19C3×#869—31no—
20C3×C4×Dic6—31no—
21C3×C4⋊(C3⋊C8)—31no—
22C3×C4⋊(C3⋊C8)—31no—
23C3×C4⋊(C3⋊C8)—61no—
24#63575—32no—
25#63575—32no—
26#63567—32no—
27#63566—32no—
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×(C3⋊Dic3)⋊C3—21no—
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—21no—
34#634⋊C3—21no—
35#634⋊C3—42no—
36#634⋊C3Q812yescomplemented
37#6180—22no—
38#5091—91no—
39#5021—91no—
40#4850—91no—
41C2×#634—21no—
42C2×#634C2×D311yescharacteristic
43C2×C3×Dic6—31no—
44C2×C3×Dic6—61no—
45C32×C42—21no—
46C32×C42C2×D311yescharacteristic
47C32×C42C2×D311yescharacteristic
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)C2×D312yes—
51#3750—21no—
52C4×C3⋊Dic3—21no—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3—61no—
56C3×Dic3⋊C4—61no—
57C3×C4×Dic3—61no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—62no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C4×H(3,𝔽3)—21no—
63C4×H(3,𝔽3)Q1612yes—
64C4×H(3,𝔽3)C2×Q811yescharacteristic
65C4×H(3,𝔽3)#4111yescharacteristic
66(C3⋊Dic3)⋊C3—22no—
67(C3⋊Dic3)⋊C3—42no—
68(C3⋊Dic3)⋊C3—41no—
69C3×C4⋊Q8—91no—
70C3×C4×Q8—91no—
71#869—31no—
72#869—91no—
73C4×Dic6—31no—
74C4×Dic6—91no—
75C3×C4⋊C8—91no—
76#835—92no—
77#818—92no—
78#816—92no—
79#810—92no—
80#807—92no—
81#807—92no—
82C4⋊(C3⋊C8)—31no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—61no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4#6711yescharacteristic
89C2×C32×C4#6711yescharacteristic
90C2×C32×C4#6711yescharacteristic
91C2×C32×C4#6711yescharacteristic
92C2×C32×C4C22×D311yescharacteristic
93C2×C32×C4C22×D311yescharacteristic
94C2×C3⋊Dic3—21no—
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3—22no—
97#634—21no—
98#634—42no—
99#634Dic612yescomplemented
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—62no—
103C3×Dic6—32no—
104C3×Dic6—61no—
105C3×Dic6—122no—
106C3×C3⋊C8—62no—
107C3×C3⋊C8—62no—
108C3×C3⋊C8—122no—
109#477—271noSylow
110C2×H(3,𝔽3)C22⋊Q811yescharacteristic
111C2×H(3,𝔽3)C2×Q1611yescharacteristic
112C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
113C2×C3×Q8—91no—
114C2×C3×Q8—181no—
115C2×Dic6—31no—
116C2×Dic6—61no—
117C2×Dic6—91no—
118C2×Dic6—181no—
119C2×C3×C8—92no—
120C3×C4⋊C4—91no—
121C3×C4⋊C4—92no—
122C3×C4⋊C4—181no—
123C3×C4⋊C4—181no—
124C3×C42—31no—
125C3×C42—31no—
126C3×C42—61no—
127C3×C42—181no—
128C3×C42D3211yescharacteristic
129C4⋊Dic3—32no—
130C4⋊Dic3—31no—
131C4⋊Dic3—61no—
132C4⋊Dic3—91no—
133C4⋊Dic3—92no—
134C4⋊Dic3—181no—
135Dic3⋊C4—61no—
136Dic3⋊C4—181no—
137C4×Dic3—61no—
138C4×Dic3—181no—
139C2×C3⋊C8—32no—
140C2×C3⋊C8—32no—
141C2×C3⋊C8—62no—
142C22×C32—21no—
143C22×C32C2×#6711yescharacteristic
144C22×C32C2×#6711yescharacteristic
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—22no—
148C32×C4—21no—
149C32×C4—21no—
150C32×C4—41no—
151C32×C4#21612yes—
152C32×C4#21612yes—
153C32×C4C2×Dic611yescharacteristic
154C32×C4#23511yescharacteristic
155C32×C4D3×Q811yescharacteristic
156C32×C4#23911yescharacteristic
157C3⋊Dic3—22no—
158C3⋊Dic3—42no—
159C3⋊Dic3—41no—
160C3×Dic3—62no—
161C3×Dic3—121no—
162C3×Dic3—122no—
163C4⋊Q8—91no—
164C4×Q8—91no—
165#107—272no—
166C4⋊C8—91no—
167#103—272no—
168H(3,𝔽3)#47711yescharacteristic, complemented, Sylow
169C3×Q8—92no—
170C3×Q8—181no—
171C3×Q8—362no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—181no—
182C2×C3×C4—181no—
183C2×C3×C4—182no—
184C2×C3×C4#62511yescharacteristic
185C2×C3×C4#62511yescharacteristic
186C2×C3×C4C2×D3211yescharacteristic
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—62no—
190C2×Dic3—181no—
191C2×Dic3—181no—
192C2×Dic3—182no—
193Dic6—32no—
194Dic6—61no—
195Dic6—92no—
196Dic6—122no—
197Dic6—181no—
198Dic6—362no—
199C3×C8—182no—
200C3⋊C8—62no—
201C3⋊C8—62no—
202C3⋊C8—122no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32#92411yescharacteristic
207C2×C32#93211yescharacteristic
208C2×C32C2×#21611yescharacteristic
209C2×C32C2×#21611yescharacteristic
210C2×C32#94611yescharacteristic
211C2×C32#94911yescharacteristic
212C2×Q8—91no—
213C2×Q8—181no—
214C2×C8—92no—
215C4⋊C4—92no—
216C4⋊C4—91no—
217C4⋊C4—181no—
218C4⋊C4—181no—
219C42—181no—
220C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—61no—
224C22×C3C2×#62511yescharacteristic
225C3×C4—21no—
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—32no—
231C3×C4—32no—
232C3×C4—61no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—61no—
236C3×C4—62no—
237C3×C4—121no—
238C3×C4—182no—
239C3×C4—362no—
240C3×C4—361no—
241C3×C4#371812yes—
242C3×C4D3×Dic611yescharacteristic
243C3×C4#379911yescharacteristic
244Dic3—62no—
245Dic3—122no—
246Dic3—121no—
247Dic3—182no—
248Dic3—362no—
249Dic3—361no—
250C32—21no—
251C32#502111yescharacteristic, complemented
252C32#509111yescharacteristic, complemented
253Q8—92no—
254Q8—181no—
255Q8—362no—
256C2×C4—181no—
257C2×C4—181no—
258C2×C4—182no—
259C2×C4#619811yescharacteristic
260C2×C4#619811yescharacteristic
261C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
262C8—182no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—61no—
270C2×C3—61no—
271C2×C3—61no—
272C2×C3#6382311yescharacteristic
273C2×C3C2×#371811yescharacteristic
274C2×C3#6385611yescharacteristic
275C22C2×#619811yescharacteristic
276C4—21no—
277C4—182no—
278C4—362no—
279C4—361no—
280C4#8843312yes—
281C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
282C4#8865211yescharacteristic
283C3—31no—
284C3—31no—
285C3—61no—
286C3#1059031011yescharacteristic
287C2#1172106111yescharacteristic
288C2C2×#8843311yescharacteristic
289C2#1172110611yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#477
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
36. #634⋊C3Q8semidirect
99. #634Dic6semidirect
168. H(3,𝔽3)#477semidirect
251. C32#5021semidirect
252. C32#5091semidirect

Maximal subgroups

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

subgroupindex
#64078⋊C32
C4×#634⋊C32
C4⋊#61802
#117203152
#117203132
#105902843
#105902833