Finite Group Databasebeta

← #13147062

Subgroups — #13147062

Order 1728.

504 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88611C211yesmaximal, characteristic, complemented
3#11720738C211yesmaximal, characteristic, complemented
4C3×#5421—33nomaximal
5C3×#5227—31nomaximal
6C2×#6235—41nomaximal
7C2×Q8⋊H(3,𝔽3)—21no—
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9#88611—21no—
10C22×C32×Q8D311yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)D313yescomplemented
12C2×C3×GL(2,𝔽3)—33no—
13C2×C3×#215—31no—
14C2×C3×#215—31no—
15C3×#930—31no—
16C3×#859—33no—
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20C23×H(3,𝔽3)—41no—
21C22×#272—41no—
22#6235—44no—
23C2×#1084—41no—
24Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
25#5421—33no—
26C3×#448—91no—
27#5227—31no—
28C2×C32×Q8—21no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C2×C32×Q8C2×D311yescharacteristic
32C22×C32×C4—31no—
33C2×C3×#67—61no—
34C2×C3×#67—123no—
35C2×C3×D12—31no—
36C2×C3×SL(2,𝔽3)—23no—
37C2×C3×SL(2,𝔽3)C2×D313yes—
38C3×GL(2,𝔽3)—63no—
39C3×#215—61no—
40C3×#215—62no—
41C3×#215—61no—
42C3×(C2×D3)⋊C4—61no—
43C3×C4⋊Dic3—31no—
44C2×C3×C3⋊C8—31no—
45C2×C3×C3⋊C8—31no—
46C22×H(3,𝔽3)—41no—
47C22×H(3,𝔽3)—42no—
48C22×H(3,𝔽3)—81no—
49C22×H(3,𝔽3)—81no—
50C2×#272—42no—
51C2×#272—81no—
52C2×#272—81no—
53#1084—42no—
54C22×C3×Q8—31no—
55C22×C3×Q8C3⋊D311yescharacteristic
56C22×SL(2,𝔽3)—33no—
57C2×GL(2,𝔽3)—33no—
58C2×C3×SD16—91no—
59C2×C3×SD16—91no—
60C3×#121—91no—
61C2×#215—31no—
62C2×#215—31no—
63#930—31no—
64#859—33no—
65C3×#103—91no—
66C3×#103—91no—
67C3×C22⋊C8—91no—
68#835—31no—
69#835—31no—
70#830—31no—
71C23×C32—43no—
72C23×C32S411yescharacteristic
73C22×C3×D3—61no—
74C22×C3×D3—123no—
75C32×Q8—31no—
76C32×Q8—32no—
77C32×Q8—61no—
78C32×Q8#6711yescharacteristic, complemented
79C2×C32×C4—31no—
80C2×C32×C4—31no—
81C2×C32×C4—61no—
82C2×C32×C4—61no—
83C3×#67—1212no—
84C3×#67—122no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—123no—
87C3×D12—61no—
88C3×D12—61no—
89C3×SL(2,𝔽3)#6713yescomplemented
90C3×C3⋊C8—61no—
91C3×C3⋊C8—61no—
92#448—91noSylow
93C2×H(3,𝔽3)—41no—
94C2×H(3,𝔽3)—41no—
95C2×H(3,𝔽3)—41no—
96C2×H(3,𝔽3)—82no—
97#272—82no—
98C2×C3×Q8—21no—
99C2×C3×Q8—31no—
100C2×C3×Q8—31no—
101C2×C3×Q8—31no—
102C2×C3×Q8—61no—
103C2×C3×Q8—61no—
104C2×C3×Q8—62no—
105C2×C3×Q8—61no—
106C2×C3×Q8—61no—
107C2×C3×Q8C2×C3⋊D311yescharacteristic
108C2×C3×D4—91no—
109C2×C3×D4—181no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—61no—
113C2×#67—61no—
114C2×#67—123no—
115C2×D12—31no—
116C2×SL(2,𝔽3)—33no—
117C2×SL(2,𝔽3)—63no—
118GL(2,𝔽3)—63no—
119C3×SD16—181no—
120C3×SD16—182no—
121C3×SD16—181no—
122C2×C3×C8—91no—
123C2×C3×C8—91no—
124C3×C4⋊C4—91no—
125C3×C22⋊C4—181no—
126#215—61no—
127#215—62no—
128#215—61no—
129(C2×D3)⋊C4—61no—
130C4⋊Dic3—31no—
131C2×C3⋊C8—31no—
132C2×C3⋊C8—31no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—43no—
136C22×C32—46no—
137C22×C32—83no—
138C22×C32—83no—
139C22×C32GL(2,𝔽3)12yescomplemented
140C22×C32C2×S411yescharacteristic
141C2×C3×D3—126no—
142C2×C3×D3—121no—
143C2×C3×D3—121no—
144C2×C3×D3—121no—
145C2×C3×D3—243no—
146C2×C3×D3—243no—
147C32×C4—31no—
148C32×C4—31no—
149C32×C4—61no—
150C3×Dic3—126no—
151C3×Dic3—121no—
152C22×Q8#27211yescharacteristic, complemented
153C2×SD16—91no—
154C2×SD16—91no—
155#121—91no—
156#103—91no—
157#103—91no—
158C22⋊C8—91no—
159H(3,𝔽3)—41noSylow
160C23×C3—31no—
161C23×C3—123no—
162C23×C3—181no—
163C23×C3C3⋊S411yescharacteristic
164C22×D3—61no—
165C22×D3—123no—
166C3×Q8—31no—
167C3×Q8—31no—
168C3×Q8—32no—
169C3×Q8—32no—
170C3×Q8—31no—
171C3×Q8—61no—
172C3×Q8—62no—
173C3×Q8—61no—
174C3×Q8—61no—
175C3×Q8—62no—
176C3×Q8#63811yescharacteristic
177C3×D4—181no—
178C3×D4—181no—
179C3×D4—362no—
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—61no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—181no—
193#67—122no—
194#67—1212no—
195C2×Dic3—61no—
196C2×Dic3—123no—
197D12—61no—
198D12—61no—
199SL(2,𝔽3)—33no—
200C3×C8—181no—
201C3×C8—181no—
202C3⋊C8—61no—
203C3⋊C8—61no—
204C2×C32—22no—
205C2×C32—43no—
206C2×C32—43no—
207C2×C32—43no—
208C2×C32—86no—
209C2×C32C2×GL(2,𝔽3)11yescharacteristic
210C2×C32#98311yescharacteristic
211C2×C32GL(2,ℤ/4)11yescharacteristic
212C3×D3—246no—
213C3×D3—241no—
214C2×Q8—21no—
215C2×Q8—31no—
216C2×Q8—61no—
217C2×Q8C2×#27211yescharacteristic
218C2×D4—91no—
219C2×D4—181no—
220C22×C4—31no—
221SD16—181no—
222SD16—182no—
223SD16—181no—
224C2×C8—91no—
225C2×C8—91no—
226C4⋊C4—91no—
227C22⋊C4—181no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—32no—
231C22×C3—31no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3—126no—
235C22×C3—123no—
236C22×C3—243no—
237C22×C3—243no—
238C22×C3—361no—
239C22×C3—361no—
240C22×C3—361no—
241C22×C3C3⋊GL(2,𝔽3)12yes—
242C22×C3C2×C3⋊S411yescharacteristic
243C2×D3—121no—
244C2×D3—121no—
245C2×D3—121no—
246C2×D3—126no—
247C2×D3—243no—
248C2×D3—243no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—62no—
257C3×C4—61no—
258C3×C4—361no—
259Dic3—121no—
260Dic3—126no—
261C32—43no—
262C32#542111yescharacteristic, complemented
263C23—181no—
264C23#625811yescharacteristic
265Q8—31no—
266Q8—32no—
267Q8—61no—
268Q8#623511yescharacteristic, complemented
269D4—181no—
270D4—181no—
271D4—362no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—181no—
277C8—181no—
278C8—181no—
279C2×C3—22no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—62no—
284C2×C3—123no—
285C2×C3—123no—
286C2×C3—123no—
287C2×C3—246no—
288C2×C3—721no—
289C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
290C2×C3#6426411yescharacteristic
291C2×C3#6426911yescharacteristic
292D3—241no—
293D3—246no—
294C22—21no—
295C22—21no—
296C22—361no—
297C22—361no—
298C22—361no—
299C22#8861112yescomplemented
300C22C2×#625811yescharacteristic
301C4—31no—
302C4—31no—
303C4—61no—
304C4—361no—
305C3—31no—
306C3—123no—
307C3#1059365511yescharacteristic
308C2—22no—
309C2—721no—
310C2C2×#8861111yescharacteristic
311C2#1172238811yescharacteristic
312C2#1172239811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C2×C3⋊S4
commutator, cosocleC2×Q8⋊H(3,𝔽3)C22
FittingC22×C32×Q8D3
hypercenterC23×C3C3⋊S4
hypocenterQ8⋊H(3,𝔽3)D4
LCSG ▷ C2×Q8⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×Q8⋊H(3,𝔽3)C2semidirect
2. C2×#88611C2semidirect
3. #11720738C2semidirect
10. C22×C32×Q8D3semidirect
11. C22×C3×SL(2,𝔽3)D3semidirect
24. Q8⋊H(3,𝔽3)D4semidirect
78. C32×Q8#67semidirect
89. C3×SL(2,𝔽3)#67semidirect
139. C22×C32GL(2,𝔽3)semidirect
152. C22×Q8#272semidirect
262. C32#5421semidirect
268. Q8#6235semidirect
299. C22#88611semidirect

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C2×#886112
#117207382
C3×#54213
C3×#52273
C2×#62354