Finite Group Databasebeta

← #548⋊H(3,𝔽3) #13152192

Subgroups — #548⋊H(3,𝔽3) #13152192

Order 1728.

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

normal subgroups only

This group has 515 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
1C2×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C32×#548C311yesmaximal, characteristic, complemented
5C3×#548⋊C3C313yesmaximal, complemented
6#41⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)—21no—
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9C22⋊C4×H(3,𝔽3)—41nomaximal
10C2×C32×#41C2×C311yescharacteristic, complemented
11C22×C32×Q8C2×C311yescharacteristic
12C2×C3×#41⋊C3C2×C313yescomplemented
13C22×C3×SL(2,𝔽3)C2×C313yescomplemented
14C32×#124—31no—
15C32×C22⋊Q8—31no—
16C32×#121—31no—
17C32×C4×Q8C2×C311yescharacteristic, complemented
18C32×C4×D4—31no—
19C3×C4×SL(2,𝔽3)C2×C313yescomplemented
20C23×H(3,𝔽3)—41no—
21C2×C4×H(3,𝔽3)—41no—
22C2×C4×H(3,𝔽3)—41no—
23Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
24C3×#548—31no—
25C3×#548C3211yescharacteristic
26#548⋊C3—33no—
27C32×#41—21no—
28C32×#41—61no—
29C2×C32×Q8—21no—
30C2×C32×Q8—31no—
31C2×C32×Q8—61no—
32C2×C32×Q8C22×C311yescharacteristic
33C2×C32×D4—31no—
34C2×C32×D4—31no—
35C22×C32×C4—31no—
36C22×C32×C4—31no—
37C3×#41⋊C3—23no—
38C2×C3×SL(2,𝔽3)—23no—
39C2×C3×SL(2,𝔽3)C22×C313yes—
40C32×C4⋊C4—31no—
41C32×C4⋊C4—31no—
42C32×C22⋊C4—32no—
43C32×C22⋊C4—31no—
44C32×C22⋊C4—43no—
45C32×C22⋊C4A411yescharacteristic
46C32×C42—31no—
47C22×H(3,𝔽3)—41no—
48C22×H(3,𝔽3)—42no—
49C22×H(3,𝔽3)—81no—
50C22×H(3,𝔽3)—81no—
51C4×H(3,𝔽3)—81no—
52C4×H(3,𝔽3)—81no—
53C2×C3×#41—31no—
54C2×C3×#41C2×C3211yescharacteristic
55C22×C3×Q8—31no—
56C22×C3×Q8C2×C3211yescharacteristic
57C2×#41⋊C3—33no—
58C22×SL(2,𝔽3)—33no—
59C3×#124—33no—
60C3×#124—31no—
61C3×C22⋊Q8—33no—
62C3×C22⋊Q8—31no—
63C3×#121—31no—
64C3×#121—33no—
65C3×C4×Q8—31no—
66C3×C4×Q8C2×C3211yescharacteristic
67C3×C4×D4—33no—
68C3×C4×D4—31no—
69C4×SL(2,𝔽3)—33no—
70C23×C32—31no—
71C23×C32—43no—
72C23×C32C2×A411yescharacteristic
73C32×Q8—31no—
74C32×Q8—61no—
75C32×Q8—61no—
76C32×Q8C3×D411yescharacteristic
77C32×D4—61no—
78C32×D4—61no—
79C32×D4—62no—
80C2×C32×C4—31no—
81C2×C32×C4—31no—
82C2×C32×C4—31no—
83C2×C32×C4—43no—
84C2×C32×C4—43no—
85C2×C32×C4—61no—
86C2×C32×C4—61no—
87C2×C32×C4—61no—
88C2×C32×C4—61no—
89C2×C32×C4C2×A411yescharacteristic
90C2×C32×C4C2×A411yescharacteristic
91C3×SL(2,𝔽3)C3×D413yescomplemented
92#548H(3,𝔽3)11yescharacteristic, complemented, Sylow
93C2×H(3,𝔽3)—41no—
94C2×H(3,𝔽3)—41no—
95C2×H(3,𝔽3)—41no—
96C2×H(3,𝔽3)—82no—
97C3×#41—21no—
98C3×#41—61no—
99C3×#41—63no—
100C3×#41—61no—
101C2×C3×Q8—21no—
102C2×C3×Q8—33no—
103C2×C3×Q8—31no—
104C2×C3×Q8—31no—
105C2×C3×Q8—61no—
106C2×C3×Q8—63no—
107C2×C3×Q8—61no—
108C2×C3×Q8C22×C3211yescharacteristic
109C2×C3×D4—31no—
110C2×C3×D4—33no—
111C2×C3×D4—33no—
112C2×C3×D4—31no—
113C22×C3×C4—33no—
114C22×C3×C4—33no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117#41⋊C3—63no—
118C2×SL(2,𝔽3)—33no—
119C2×SL(2,𝔽3)—63no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—33no—
122C3×C4⋊C4—33no—
123C3×C4⋊C4—31no—
124C3×C22⋊C4—33no—
125C3×C22⋊C4—36no—
126C3×C22⋊C4—32no—
127C3×C22⋊C4—31no—
128C3×C22⋊C4—31no—
129C3×C22⋊C4—123no—
130C3×C22⋊C4C3×A411yescharacteristic
131C3×C42—33no—
132C3×C42—31no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—43no—
136C22×C32—46no—
137C22×C32—61no—
138C22×C32—61no—
139C22×C32—61no—
140C22×C32—83no—
141C22×C32—83no—
142C22×C32#41⋊C312yescomplemented
143C22×C32C22×A411yescharacteristic
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—31no—
147C32×C4—31no—
148C32×C4—61no—
149C32×C4—61no—
150C32×C4—83no—
151C32×C4—83no—
152C2×#41C2×H(3,𝔽3)11yescharacteristic, complemented
153C22×Q8C2×H(3,𝔽3)11yescharacteristic
154#124—31no—
155C22⋊Q8—31no—
156#121—31no—
157C4×Q8C2×H(3,𝔽3)11yescharacteristic, complemented
158C4×D4—31no—
159H(3,𝔽3)—41noSylow
160C23×C3—33no—
161C23×C3—31no—
162C23×C3—31no—
163C23×C3—123no—
164C23×C3C2×C3×A411yescharacteristic
165C3×Q8—31no—
166C3×Q8—33no—
167C3×Q8—31no—
168C3×Q8—61no—
169C3×Q8—61no—
170C3×Q8—63no—
171C3×Q8—63no—
172C3×Q8C32×D411yescharacteristic
173C3×D4—61no—
174C3×D4—61no—
175C3×D4—63no—
176C3×D4—63no—
177C3×D4—66no—
178C3×D4—62no—
179C2×C3×C4—33no—
180C2×C3×C4—31no—
181C2×C3×C4—33no—
182C2×C3×C4—33no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—63no—
192C2×C3×C4—63no—
193C2×C3×C4—63no—
194C2×C3×C4—63no—
195C2×C3×C4—123no—
196C2×C3×C4—123no—
197C2×C3×C4C2×C3×A411yescharacteristic
198C2×C3×C4C2×C3×A411yescharacteristic
199SL(2,𝔽3)—33no—
200C2×C32—22no—
201C2×C32—43no—
202C2×C32—43no—
203C2×C32—43no—
204C2×C32—86no—
205C2×C32—121no—
206C2×C32D4×A411yescharacteristic
207C2×C32C2×#41⋊C311yescharacteristic
208C2×C322−1+4⋊C311yescharacteristic
209#41—21no—
210#41—61no—
211C2×Q8—21no—
212C2×Q8—31no—
213C2×Q8—61no—
214C2×Q8C22×H(3,𝔽3)11yescharacteristic
215C2×D4—31no—
216C2×D4—31no—
217C22×C4—31no—
218C22×C4—31no—
219C4⋊C4—31no—
220C4⋊C4—31no—
221C22⋊C4—32no—
222C22⋊C4—31no—
223C22⋊C4C22⋊H(3,𝔽3)11yescharacteristic
224C42—31no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—31no—
228C22×C3—32no—
229C22×C3—61no—
230C22×C3—61no—
231C22×C3—63no—
232C22×C3—63no—
233C22×C3—63no—
234C22×C3—61no—
235C22×C3—61no—
236C22×C3—61no—
237C22×C3—123no—
238C22×C3—126no—
239C22×C3—243no—
240C22×C3—243no—
241C22×C3C3×#41⋊C312yes—
242C22×C3C22×C3×A411yescharacteristic
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—33no—
246C3×C4—33no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—61no—
250C3×C4—61no—
251C3×C4—63no—
252C3×C4—63no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—243no—
256C3×C4—243no—
257C32—43no—
258C32#548⋊C311yescharacteristic, complemented
259C23—31no—
260C23C2×C22⋊H(3,𝔽3)11yescharacteristic
261Q8—31no—
262Q8—61no—
263Q8—61no—
264Q8D4×H(3,𝔽3)11yescharacteristic
265D4—61no—
266D4—61no—
267D4—62no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4C2×C22⋊H(3,𝔽3)11yescharacteristic
276C2×C4C2×C22⋊H(3,𝔽3)11yescharacteristic
277C2×C3—22no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—62no—
282C2×C3—123no—
283C2×C3—121no—
284C2×C3—123no—
285C2×C3—123no—
286C2×C3—123no—
287C2×C3—246no—
288C2×C3C3×D4×A411yescharacteristic
289C2×C3C2×C3×#41⋊C311yescharacteristic
290C2×C3C3×2−1+4⋊C311yescharacteristic
291C22—21no—
292C22—21no—
293C22—61no—
294C22—61no—
295C22—61no—
296C22#41⋊H(3,𝔽3)12yescomplemented
297C22C22×C22⋊H(3,𝔽3)11yescharacteristic
298C4—21no—
299C4—21no—
300C4—31no—
301C4—31no—
302C4—61no—
303C4—61no—
304C3—31no—
305C3—123no—
306C3C3×#548⋊C311yescharacteristic
307C2—22no—
308C2—121no—
309C2D4×C22⋊H(3,𝔽3)11yescharacteristic
310C2C2×#41⋊H(3,𝔽3)11yescharacteristic
311C22−1+4⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×C3×A4
commutator, cosocleC2×C3×Q8C22×C32
FittingC32×#548C3
hypercenterC32×C22⋊C4A4
hypocenterQ8D4×H(3,𝔽3)
LCSG ▷ C2×C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C32×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. C22×Q8⋊H(3,𝔽3)C2semidirect
3. C4×Q8⋊H(3,𝔽3)C2semidirect
4. C32×#548C3semidirect
5. C3×#548⋊C3C3semidirect
10. C2×C32×#41C2×C3semidirect
12. C2×C3×#41⋊C3C2×C3semidirect
13. C22×C3×SL(2,𝔽3)C2×C3semidirect
17. C32×C4×Q8C2×C3semidirect
19. C3×C4×SL(2,𝔽3)C2×C3semidirect
23. Q8⋊H(3,𝔽3)D4semidirect
91. C3×SL(2,𝔽3)C3×D4semidirect
92. #548H(3,𝔽3)semidirect
142. C22×C32#41⋊C3semidirect
152. C2×#41C2×H(3,𝔽3)semidirect
157. C4×Q8C2×H(3,𝔽3)semidirect
258. C32#548⋊C3semidirect
296. C22#41⋊H(3,𝔽3)semidirect

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
C22×Q8⋊H(3,𝔽3)2
C4×Q8⋊H(3,𝔽3)2
C32×#5483
C3×#548⋊C33
C22⋊C4×H(3,𝔽3)4