Finite Group Databasebeta

← #13147074

Subgroups — #13147074

Order 1728.

430 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C2×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88611C211yesmaximal, characteristic, complemented
3#11720738C211yesmaximal, characteristic, complemented
4C3×#5427—33nomaximal
5C3×#5238—31nomaximal
6#41⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8#88611—21no—
9#88542—41nomaximal
10C2×C32×#41D311yescharacteristic, complemented
11C2×C3×#41⋊C3D313yescomplemented
12C2×C3×GL(2,𝔽3)—33no—
13C2×C3×#215—31no—
14C2×C3×#213—31no—
15C3×#930—31no—
16C3×#859—33no—
17C3×#835—31no—
18C3×#832—31no—
19C3×#830—31no—
20C22×#272—41no—
21C2×C4×H(3,𝔽3)—41no—
22C2×#1084—41no—
23Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
24#5427—33no—
25C3×#449—91no—
26#5238—31no—
27C32×#41—21no—
28C32×#41—61no—
29C2×C32×Q8C2×D311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×C3×#67—61no—
33C2×C3×D12—31no—
34C3×#41⋊C3—23no—
35C2×C3×SL(2,𝔽3)C2×D313yes—
36C3×GL(2,𝔽3)—63no—
37C3×#215—61no—
38C3×#215—61no—
39C3×#213—62no—
40C3×(C2×D3)⋊C4—61no—
41C3×(C2×D3)⋊C4—123no—
42C3×C4⋊Dic3—31no—
43C2×C3×C3⋊C8—31no—
44C2×C3×C3⋊C8—31no—
45C22×H(3,𝔽3)—41no—
46C2×#272—42no—
47C2×#272—81no—
48C2×#272—81no—
49C4×H(3,𝔽3)—81no—
50#1084—81no—
51C2×C3×#41—31no—
52C2×C3×#41C3⋊D311yescharacteristic
53C2×#41⋊C3—33no—
54C2×GL(2,𝔽3)—33no—
55C2×C3×SD16—91no—
56C2×C3×D8—91no—
57C3×#121—91no—
58C2×#215—31no—
59C2×#213—31no—
60#930—31no—
61#859—33no—
62C3×#103—91no—
63C3×#102—91no—
64C3×C22⋊C8—91no—
65#835—31no—
66#832—31no—
67#830—31no—
68C23×C32—31no—
69C22×C3×D3—61no—
70C22×C3×D3—123no—
71C32×Q8—31no—
72C32×Q8#6711yescharacteristic
73C32×D4—32no—
74C32×D4—61no—
75C2×C32×C4—31no—
76C2×C32×C4—43no—
77C2×C32×C4—61no—
78C2×C32×C4—61no—
79C2×C32×C4S411yescharacteristic
80C3×#67—122no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—123no—
83C3×D12—61no—
84C3×D12—61no—
85C3×SL(2,𝔽3)#6713yescomplemented
86C3×C3⋊C8—61no—
87C3×C3⋊C8—61no—
88#449—91noSylow
89C2×H(3,𝔽3)—41no—
90C2×H(3,𝔽3)—41no—
91C2×H(3,𝔽3)—41no—
92#272—82no—
93C3×#41—21no—
94C3×#41—61no—
95C3×#41—62no—
96C3×#41—61no—
97C3×#41—61no—
98C2×C3×Q8—31no—
99C2×C3×Q8C2×C3⋊D311yescharacteristic
100C2×C3×D4—31no—
101C2×C3×D4—31no—
102C2×C3×D4—61no—
103C2×C3×D4—91no—
104C2×C3×D4—181no—
105C22×C3×C4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—61no—
108C2×#67—61no—
109C2×D12—31no—
110#41⋊C3—63no—
111C2×SL(2,𝔽3)—33no—
112GL(2,𝔽3)—63no—
113C3×SD16—181no—
114C3×SD16—181no—
115C3×D8—182no—
116C2×C3×C8—91no—
117C2×C3×C8—91no—
118C3×C4⋊C4—91no—
119C3×C22⋊C4—181no—
120#215—61no—
121#215—61no—
122#213—62no—
123(C2×D3)⋊C4—61no—
124(C2×D3)⋊C4—123no—
125C4⋊Dic3—31no—
126C2×C3⋊C8—31no—
127C2×C3⋊C8—31no—
128C22×C32—43no—
129C22×C32—61no—
130C22×C32—61no—
131C22×C32—61no—
132C22×C32C2×S411yescharacteristic
133C2×C3×D3—126no—
134C2×C3×D3—121no—
135C2×C3×D3—121no—
136C2×C3×D3—121no—
137C2×C3×D3—243no—
138C2×C3×D3—243no—
139C32×C4—21no—
140C32×C4—31no—
141C32×C4—31no—
142C32×C4—83no—
143C3×Dic3—121no—
144C3×Dic3—243no—
145C2×#41#27211yescharacteristic, complemented
146C2×SD16—91no—
147C2×D8—91no—
148#121—91no—
149#103—91no—
150#102—91no—
151C22⋊C8—91no—
152H(3,𝔽3)—41noSylow
153C23×C3—31no—
154C23×C3—31no—
155C23×C3—61no—
156C23×C3—181no—
157C22×D3—61no—
158C22×D3—123no—
159C3×Q8—31no—
160C3×Q8—31no—
161C3×Q8—31no—
162C3×Q8—61no—
163C3×Q8#63811yescharacteristic
164C3×D4—32no—
165C3×D4—32no—
166C3×D4—62no—
167C3×D4—61no—
168C3×D4—61no—
169C3×D4—62no—
170C3×D4—181no—
171C3×D4—181no—
172C3×D4—362no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—62no—
182C2×C3×C4—62no—
183C2×C3×C4—123no—
184C2×C3×C4—181no—
185C2×C3×C4C3⋊S411yescharacteristic
186#67—122no—
187C2×Dic3—61no—
188C2×Dic3—123no—
189D12—61no—
190D12—61no—
191SL(2,𝔽3)—33no—
192C3×C8—181no—
193C3×C8—181no—
194C3⋊C8—61no—
195C3⋊C8—61no—
196C2×C32—43no—
197C2×C32—43no—
198C2×C32—43no—
199C2×C32—121no—
200C2×C32#98511yescharacteristic
201C2×C32#98611yescharacteristic
202C2×C32GL(2,ℤ/4)11yescharacteristic
203C3×D3—246no—
204C3×D3—241no—
205#41—21no—
206#41—61no—
207C2×Q8C2×#27211yescharacteristic
208C2×D4—31no—
209C2×D4—91no—
210C2×D4—181no—
211C22×C4—31no—
212SD16—181no—
213SD16—181no—
214D8—182no—
215C2×C8—91no—
216C2×C8—91no—
217C4⋊C4—91no—
218C22⋊C4—181no—
219C22×C3—31no—
220C22×C3—61no—
221C22×C3—62no—
222C22×C3—61no—
223C22×C3—61no—
224C22×C3—61no—
225C22×C3—61no—
226C22×C3—61no—
227C22×C3—121no—
228C22×C3—121no—
229C22×C3—123no—
230C22×C3—361no—
231C22×C3—361no—
232C22×C3—361no—
233C22×C3C2×C3⋊S411yescharacteristic
234C2×D3—121no—
235C2×D3—121no—
236C2×D3—121no—
237C2×D3—126no—
238C2×D3—243no—
239C2×D3—243no—
240C3×C4—21no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—243no—
249C3×C4—361no—
250Dic3—121no—
251Dic3—243no—
252C32—43no—
253C32#542711yescharacteristic, complemented
254C23—31no—
255C23—181no—
256Q8—31no—
257Q8#623511yescharacteristic
258D4—32no—
259D4—61no—
260D4—181no—
261D4—181no—
262D4—362no—
263C2×C4—31no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—181no—
267C2×C4#625811yescharacteristic
268C8—181no—
269C8—181no—
270C2×C3—31no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—121no—
274C2×C3—122no—
275C2×C3—121no—
276C2×C3—123no—
277C2×C3—123no—
278C2×C3—123no—
279C2×C3—721no—
280C2×C3#6426611yescharacteristic
281C2×C3#6426711yescharacteristic
282C2×C3#6426911yescharacteristic
283D3—241no—
284D3—246no—
285C22—61no—
286C22—61no—
287C22—61no—
288C22—361no—
289C22—361no—
290C22—361no—
291C22C2×#625811yescharacteristic
292C4—21no—
293C4—31no—
294C4—31no—
295C4—361no—
296C3—31no—
297C3—123no—
298C3#1059366111yescharacteristic
299C2—121no—
300C2—721no—
301C2#1172239311yescharacteristic
302C2#1172239411yescharacteristic
303C2#1172239811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C2×C3⋊S4
commutator, cosocleC2×Q8⋊H(3,𝔽3)C22
FittingC2×C32×#41D3
hypercenterC2×C3×C4C3⋊S4
hypocenterQ8⋊H(3,𝔽3)D4
LCSG ▷ C2×Q8⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. C2×#88611C2semidirect
3. #11720738C2semidirect
10. C2×C32×#41D3semidirect
11. C2×C3×#41⋊C3D3semidirect
23. Q8⋊H(3,𝔽3)D4semidirect
85. C3×SL(2,𝔽3)#67semidirect
145. C2×#41#272semidirect
253. C32#5427semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
C2×#886112
#117207382
C3×#54273
C3×#52383
#885424