Finite Group Databasebeta

← #13137358

Subgroups — #13137358

Order 1728.

390 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 75 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#64084⋊C3C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3#11720520C211yesmaximal, characteristic, complemented
4#11720318C212yesmaximal
5#11720313C212yesmaximal, complemented
6#10590294—31nomaximal
7#10590276—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×#634⋊C3—21no—
10C2×#634⋊C3C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2212yes—
13#3752⋊C3—21no—
14#3752⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17#64084D311yescharacteristic
18#64078D311yescharacteristic, complemented
19C3×#875—31no—
20C3×#869—31no—
21C3×#803—31no—
22C3×#803—31no—
23C3×#803—61no—
24#63567—32no—
25#63566—32no—
26#63564—32no—
27#63563—32no—
28C22×(C3⋊D3)⋊C3—21no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—21no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35#634⋊C3—22no—
36#634⋊C3—42no—
37#6180—22no—
38#5089—91no—
39#5055—91no—
40#4714—91no—
41C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
42C2×#634—21no—
43C2×#634C2×D311yescharacteristic
44C2×C3×D12—31no—
45C2×C3×Dic6—31no—
46C2×C3×Dic6—61no—
47C32×C42—21no—
48C32×C42C2×D311yescharacteristic
49C32×C42C2×D311yescharacteristic
50#3752—21no—
51#3752—21no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)C2×D312yes—
54C3×(C2×D3)⋊C4—61no—
55C3×(C2×D3)⋊C4—61no—
56C3×C4⋊Dic3—32no—
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
62C2×(C3⋊D3)⋊C3—41no—
63C2×(C3⋊D3)⋊C3—41no—
64C2×(C3⋊D3)⋊C3—41no—
65C4×H(3,𝔽3)—21no—
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)#4112yes—
68(C3⋊Dic3)⋊C3—41no—
69(C3⋊Dic3)⋊C3—42no—
70C3×C4⋊Q8—91no—
71C3×#124—91no—
72#875—31no—
73#875—91no—
74#869—31no—
75#869—91no—
76C3×#97—91no—
77#835—92no—
78#832—92no—
79#821—92no—
80#816—92no—
81#810—92no—
82#809—92no—
83#803—31no—
84#803—31no—
85#803—61no—
86C22×C3⋊D3—21no—
87C22×C3×D3—61no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4#6711yescharacteristic
92C2×C32×C4#6711yescharacteristic
93C2×C32×C4#6711yescharacteristic
94C2×C32×C4#6711yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—21no—
99C4⋊(C3⋊D3)—22no—
100#634—22no—
101#634—42no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—61no—
104C3×D12—62no—
105C3×Dic6—62no—
106C3×Dic6—122no—
107C3×C3⋊C8—64no—
108C3×C3⋊C8—62no—
109C3×C3⋊C8—62no—
110#489—271noSylow
111C2×H(3,𝔽3)#12411yescharacteristic
112C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
113C2×H(3,𝔽3)#13711yescharacteristic
114(C3⋊D3)⋊C3—81no—
115C2×C3×Q8—91no—
116C2×C3×Q8—181no—
117C2×C3×D4—91no—
118C2×D12—31no—
119C2×D12—91no—
120C2×Dic6—31no—
121C2×Dic6—61no—
122C2×Dic6—91no—
123C2×Dic6—181no—
124C2×C3×C8—92no—
125C3×C4⋊C4—92no—
126C3×C4⋊C4—181no—
127C3×C22⋊C4—181no—
128C3×C22⋊C4—181no—
129C3×C42—31no—
130C3×C42—31no—
131C3×C42—61no—
132C3×C42D3211yescharacteristic
133(C2×D3)⋊C4—61no—
134(C2×D3)⋊C4—61no—
135(C2×D3)⋊C4—181no—
136(C2×D3)⋊C4—181no—
137C4⋊Dic3—32no—
138C4⋊Dic3—61no—
139C4⋊Dic3—92no—
140C4⋊Dic3—181no—
141C2×C3⋊C8—32no—
142C2×C3⋊C8—32no—
143C2×C3⋊C8—62no—
144C22×C32—21no—
145C22×C32C2×#6711yescharacteristic
146C22×C32C2×#6711yescharacteristic
147C2×C3⋊D3—41no—
148C2×C3⋊D3—41no—
149C2×C3⋊D3—41no—
150C2×C3×D3—121no—
151C2×C3×D3—121no—
152C2×C3×D3—121no—
153C32×C4—21no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—22no—
158C32×C4—41no—
159C32×C4—41no—
160C32×C4#23712yes—
161C32×C4#23912yes—
162C3⋊Dic3—42no—
163C3⋊Dic3—41no—
164C3×Dic3—121no—
165C3×Dic3—122no—
166C4⋊Q8—91no—
167#124—91no—
168#103—272no—
169#102—272no—
170#97—91no—
171H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
172C23×C3—181no—
173C22×D3—61no—
174C22×D3—181no—
175C3×Q8—182no—
176C3×Q8—362no—
177C3×D4—182no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—181no—
188C2×C3×C4—182no—
189C2×C3×C4#62511yescharacteristic
190C2×C3×C4#62511yescharacteristic
191C2×C3×C4C2×D3211yescharacteristic
192C2×Dic3—62no—
193C2×Dic3—61no—
194C2×Dic3—182no—
195C2×Dic3—181no—
196D12—62no—
197D12—182no—
198Dic6—62no—
199Dic6—122no—
200Dic6—182no—
201Dic6—362no—
202C3×C8—182no—
203C3⋊C8—62no—
204C3⋊C8—62no—
205C3⋊C8—64no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32#93211yescharacteristic
210C2×C32#93611yescharacteristic
211C2×C32#94211yescharacteristic
212C2×C32#94711yescharacteristic
213C2×C32#94911yescharacteristic
214C2×C32#95111yescharacteristic
215C3⋊D3—81no—
216C3×D3—241no—
217C2×Q8—91no—
218C2×Q8—181no—
219C2×D4—91no—
220C2×C8—92no—
221C4⋊C4—92no—
222C4⋊C4—181no—
223C22⋊C4—181no—
224C22⋊C4—181no—
225C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
226C22×C3—31no—
227C22×C3—31no—
228C22×C3—61no—
229C22×C3—361no—
230C22×C3—361no—
231C22×C3—361no—
232C22×C3C2×#62511yescharacteristic
233C2×D3—121no—
234C2×D3—121no—
235C2×D3—121no—
236C2×D3—361no—
237C2×D3—361no—
238C2×D3—361no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—32no—
242C3×C4—32no—
243C3×C4—61no—
244C3×C4—61no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—62no—
248C3×C4—121no—
249C3×C4—121no—
250C3×C4—361no—
251C3×C4—362no—
252C3×C4#379612yes—
253Dic3—122no—
254Dic3—121no—
255Dic3—361no—
256Dic3—362no—
257C32—21no—
258C32#505511yescharacteristic, complemented
259C32#508911yescharacteristic, complemented
260C23—181no—
261Q8—182no—
262Q8—362no—
263D4—182no—
264C2×C4—181no—
265C2×C4—182no—
266C2×C4#619811yescharacteristic
267C2×C4#619811yescharacteristic
268C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C8—182no—
270C2×C3—31no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—61no—
277C2×C3—61no—
278C2×C3—61no—
279C2×C3—721no—
280C2×C3#6382311yescharacteristic
281C2×C3#6382711yescharacteristic
282C2×C3#6386311yescharacteristic
283D3—241no—
284D3—721no—
285C22—361no—
286C22—361no—
287C22—361no—
288C22C2×#619811yescharacteristic
289C4—21no—
290C4—21no—
291C4—362no—
292C4—361no—
293C4#8864912yes—
294C3—31no—
295C3—31no—
296C3—61no—
297C3#1059028011yescharacteristic
298C2—721no—
299C2#1172105811yescharacteristic
300C2#1172106111yescharacteristic
301C2#1172111311yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#489
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
2. #64078⋊C3C2semidirect
3. #11720520C2semidirect
5. #11720313C2semidirect
18. #64078D3semidirect
171. H(3,𝔽3)#489semidirect
258. C32#5055semidirect
259. C32#5089semidirect

Maximal subgroups

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

subgroupindex
#64084⋊C32
#64078⋊C32
#117205202
#117203182
#117203132
#105902943
#105902763