Finite Group Databasebeta

← #13137402

Subgroups — #13137402

Order 1728.

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

normal subgroups only

This group has 416 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
3C2×#88433C212yesmaximal
4C4⋊#6180C211yesmaximal, characteristic
5#11720313C212yesmaximal
6#10590388—31nomaximal
7#10590385—31nomaximal
8C2×#634⋊C3C2212yes—
9C2×#634⋊C3C2211yescharacteristic
10C42×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2212yes—
12C4⋊((C3⋊Dic3)⋊C3)—21no—
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15#3750⋊C3—21no—
16C4×(C3⋊Dic3)⋊C3—21no—
17#88433—24no—
18#64078D311yescharacteristic
19C4×#634D311yescharacteristic
20C3×#869—31no—
21C3×C4×Dic6—31no—
22C2×#3719—32no—
23C2×#3719—32no—
24C3×C4⋊(C3⋊C8)—31no—
25C3×C4⋊(C3⋊C8)—31no—
26C3×C4⋊(C3⋊C8)—61no—
27#63567—32no—
28#63566—32no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—21no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—21no—
35#634⋊C3—21no—
36#634⋊C3—22no—
37#634⋊C3—22no—
38#634⋊C3D412yes—
39#6180—22no—
40#5090—91no—
41#5031—91no—
42#4849—91no—
43C2×#634C2×D312yes—
44C2×#634C2×D311yescharacteristic
45C2×C3×Dic6—31no—
46C2×C3×Dic6—32no—
47C32×C42—21no—
48C32×C42C2×D311yescharacteristic
49C32×C42C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)C2×D311yescharacteristic
53#3750—21no—
54C4×C3⋊Dic3—21no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—61no—
57C3×C4⋊Dic3—61no—
58C3×Dic3⋊C4—61no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—62no—
63#3719—64no—
64#3719—64no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)Q1612yescomplemented
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)#4111yescharacteristic
70(C3⋊Dic3)⋊C3—22no—
71(C3⋊Dic3)⋊C3—42no—
72(C3⋊Dic3)⋊C3—41no—
73C3×C4⋊Q8—91no—
74C3×C4×Q8—91no—
75C2×#216—92no—
76C2×#216—92no—
77C2×Dic12—92no—
78#869—31no—
79#869—91no—
80C4×Dic6—31no—
81C4×Dic6—91no—
82C3×C4⋊C8—91no—
83#835—92no—
84#816—92no—
85#810—92no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—61no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4#6711yescharacteristic
93C2×C32×C4#6711yescharacteristic
94C2×C32×C4#6711yescharacteristic
95C2×C32×C4#6711yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C22×D311yescharacteristic
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—22no—
101#634—21no—
102#634—22no—
103#634—22no—
104#634D1212yes—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—62no—
108C3×Dic6—32no—
109C3×Dic6—61no—
110C3×Dic6—62no—
111C3×Dic6—62no—
112C3×C3⋊C8—62no—
113C3×C3⋊C8—62no—
114C3×C3⋊C8—122no—
115#462—271noSylow
116C2×H(3,𝔽3)#12111yescharacteristic
117C2×H(3,𝔽3)C2×Q1611yescharacteristic
118C2×H(3,𝔽3)#13711yescharacteristic
119C2×C3×Q8—91no—
120C2×C3×Q8—92no—
121C2×Dic6—31no—
122C2×Dic6—32no—
123C2×Dic6—91no—
124C2×Dic6—92no—
125C2×C3×C8—92no—
126C3×C4⋊C4—91no—
127C3×C4⋊C4—181no—
128C3×C4⋊C4—181no—
129C3×C4⋊C4—181no—
130C3×C42—31no—
131C3×C42—31no—
132C3×C42—61no—
133C3×C42—181no—
134C3×C42D3211yescharacteristic
135#216—184no—
136#216—184no—
137C4⋊Dic3—31no—
138C4⋊Dic3—61no—
139C4⋊Dic3—61no—
140C4⋊Dic3—91no—
141C4⋊Dic3—181no—
142C4⋊Dic3—181no—
143Dic3⋊C4—61no—
144Dic3⋊C4—181no—
145C4×Dic3—61no—
146C4×Dic3—181no—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—62no—
150Dic12—184no—
151C22×C32—21no—
152C22×C32C2×#6711yescharacteristic
153C22×C32C2×#6711yescharacteristic
154C32×C4—21no—
155C32×C4—21no—
156C32×C4—22no—
157C32×C4—21no—
158C32×C4—21no—
159C32×C4—41no—
160C32×C4#21612yescomplemented
161C32×C4#21612yescomplemented
162C32×C4C2×D1211yescharacteristic
163C32×C4#23511yescharacteristic
164C32×C4D3×D411yescharacteristic
165C32×C4#23711yescharacteristic
166C3⋊Dic3—22no—
167C3⋊Dic3—42no—
168C3⋊Dic3—41no—
169C3×Dic3—62no—
170C3×Dic3—122no—
171C3×Dic3—121no—
172C2×Q16—272no—
173C4⋊Q8—91no—
174C4×Q8—91no—
175C4⋊C8—91no—
176#103—272no—
177H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
178C3×Q8—92no—
179C3×Q8—181no—
180C3×Q8—182no—
181C3×Q8—182no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193C2×C3×C4—182no—
194C2×C3×C4#62511yescharacteristic
195C2×C3×C4#62511yescharacteristic
196C2×C3×C4C2×D3211yescharacteristic
197C2×Dic3—61no—
198C2×Dic3—61no—
199C2×Dic3—62no—
200C2×Dic3—181no—
201C2×Dic3—181no—
202C2×Dic3—182no—
203Dic6—32no—
204Dic6—61no—
205Dic6—62no—
206Dic6—62no—
207Dic6—92no—
208Dic6—181no—
209Dic6—182no—
210Dic6—182no—
211C3×C8—182no—
212C3⋊C8—62no—
213C3⋊C8—62no—
214C3⋊C8—122no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32#93011yescharacteristic
219C2×C32#93811yescharacteristic
220C2×C32#94211yescharacteristic
221C2×C32C2×#21611yescharacteristic
222C2×C32C2×#21611yescharacteristic
223C2×C32#95111yescharacteristic
224C2×Q8—91no—
225C2×Q8—92no—
226Q16—544no—
227C2×C8—92no—
228C4⋊C4—91no—
229C4⋊C4—181no—
230C4⋊C4—181no—
231C4⋊C4—181no—
232C42—181no—
233C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—61no—
237C22×C3C2×#62511yescharacteristic
238C3×C4—21no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—32no—
244C3×C4—32no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—62no—
250C3×C4—121no—
251C3×C4—182no—
252C3×C4—362no—
253C3×C4—361no—
254C3×C4#371812yes—
255C3×C4#379511yescharacteristic
256C3×C4D3×D1211yescharacteristic
257Dic3—62no—
258Dic3—121no—
259Dic3—122no—
260Dic3—182no—
261Dic3—362no—
262Dic3—361no—
263C32—21no—
264C32#503111yescharacteristic, complemented
265C32#509011yescharacteristic, complemented
266Q8—92no—
267Q8—181no—
268Q8—182no—
269Q8—182no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—182no—
273C2×C4#619811yescharacteristic
274C2×C4#619811yescharacteristic
275C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
276C8—182no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—61no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3#6382711yescharacteristic
287C2×C3C2×#371811yescharacteristic
288C2×C3#6390811yescharacteristic
289C22C2×#619811yescharacteristic
290C4—21no—
291C4—182no—
292C4—361no—
293C4—362no—
294C4#8843312yescomplemented
295C4#8864811yescharacteristic
296C4#8865411yescharacteristic
297C3—31no—
298C3—31no—
299C3—61no—
300C3#1059038411yescharacteristic
301C2#1172105811yescharacteristic
302C2C2×#8843311yescharacteristic
303C2#1172114111yescharacteristic
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)#462
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
67. C4×H(3,𝔽3)Q16semidirect
160. C32×C4#216semidirect
161. C32×C4#216semidirect
177. H(3,𝔽3)#462semidirect
264. C32#5031semidirect
265. C32#5090semidirect
294. C4#88433semidirect

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
C2×#884332
C4⋊#61802
#117203132
#105903883
#105903853