Finite Group Databasebeta

← #10592757⋊C3 #13142920

Subgroups — #10592757⋊C3 #13142920

Order 1728.

580 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 100 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
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C212yesmaximal, complemented
3C2×#3753⋊C3C212yesmaximal, complemented
4#11721385C211yesmaximal, characteristic, complemented
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#10592757C311yesmaximal, characteristic, complemented
7C3×#5058—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)—22no—
11C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
12C22⋊C4×H(3,𝔽3)—22no—
13C42×H(3,𝔽3)C2211yescharacteristic
14#3755⋊C3—24no—
15#3753⋊C3—24no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C32×#124—21no—
18C32×#124D311yescharacteristic, complemented
19C32×#124C2×C311yescharacteristic, complemented
20C2×#3755C2×C312yescomplemented
21C2×#3753C2×C312yescomplemented
22#64083C2×C311yescharacteristic, complemented
23C2×C3×#215—32no—
24C2×C3×#213—32no—
25C3×#874—31no—
26#63630C2×C311yescharacteristic, complemented
27C3×#803—31no—
28C23×H(3,𝔽3)—21no—
29C22×(C3⋊D3)⋊C3—22no—
30Q8×H(3,𝔽3)—22no—
31D4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)D412yes—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C4⋊((C3⋊D3)⋊C3)—22no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37C4⋊((C3⋊D3)⋊C3)—44no—
38(C3⋊(C3⋊C8))⋊C3D414yescomplemented
39C3×#496—91no—
40#5058—31no—
41#5058—91no—
42C2×C32×Q8—21no—
43C2×C32×Q8C2×D311yescharacteristic
44C2×C32×Q8C22×C311yescharacteristic
45C2×C32×D4—21no—
46C2×C32×D4C2×D311yescharacteristic
47C2×C32×D4C22×C311yescharacteristic
48C2×C4⋊(C3⋊D3)—22no—
49C2×C4⋊(C3⋊D3)C22×C312yes—
50C2×C3×D12—32no—
51C2×C3×D12—62no—
52C32×C22⋊C4—24no—
53C32×C22⋊C4—22no—
54C32×C22⋊C4—22no—
55C32×C42—21no—
56C32×C42C2×D311yescharacteristic
57C32×C42C22×C311yescharacteristic
58#3755—24no—
59#3753—24no—
60C2×C3⋊(C3⋊C8)C22×C312yes—
61C3×#215—64no—
62C3×#213—64no—
63C2×C3×C3⋊C8—32no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)—42no—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C2×(C3⋊D3)⋊C3—42no—
68C2×(C3⋊D3)⋊C3—44no—
69C4×H(3,𝔽3)—22no—
70C4×H(3,𝔽3)—41no—
71C4×H(3,𝔽3)C2×D412yes—
72C2×C3×SD16—92no—
73C2×C3×D8—92no—
74C3×#127—91no—
75C3×#124—31no—
76C3×#124—31no—
77C3×#124—61no—
78C3×#124C3×D311yescharacteristic
79C2×#215—32no—
80C2×#215—92no—
81C2×#213—32no—
82C2×#213—92no—
83#874—31no—
84#874—91no—
85C3×#97—91no—
86#803—31no—
87#803—91no—
88C23×C32—22no—
89C23×C32—21no—
90C23×C32—21no—
91C22×C3⋊D3—22no—
92C22×C3×D3—62no—
93C32×Q8—22no—
94C32×Q8—22no—
95C32×Q8—42no—
96C32×D4—22no—
97C32×D4—22no—
98C32×D4—42no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4#6712yes—
105C2×C32×C4C3×D412yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C32×C4C23×C311yescharacteristic
108C4⋊(C3⋊D3)—22no—
109C4⋊(C3⋊D3)—22no—
110C4⋊(C3⋊D3)—44no—
111C3×D12—62no—
112C3×D12—62no—
113C3×D12—124no—
114C3⋊(C3⋊C8)C3×D414yescomplemented
115C3×C3⋊C8—34no—
116#496—91noSylow
117C2×H(3,𝔽3)—81no—
118C2×H(3,𝔽3)#12711yescharacteristic
119C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
120(C3⋊D3)⋊C3—82no—
121C2×C3×Q8—31no—
122C2×C3×Q8—31no—
123C2×C3×Q8—61no—
124C2×C3×Q8C2×C3×D311yescharacteristic
125C2×C3×D4—31no—
126C2×C3×D4—31no—
127C2×C3×D4—61no—
128C2×C3×D4—92no—
129C2×C3×D4—182no—
130C2×C3×D4C2×C3×D311yescharacteristic
131C2×D12—32no—
132C2×D12—62no—
133C2×D12—92no—
134C2×D12—182no—
135C3×SD16—184no—
136C3×D8—184no—
137C2×C3×C8—92no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—62no—
140C3×C22⋊C4—62no—
141C3×C22⋊C4—64no—
142C3×C42—31no—
143C3×C42—31no—
144C3×C42—61no—
145C3×C42C2×C3×D311yescharacteristic
146#215—64no—
147#215—184no—
148#213—64no—
149#213—184no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—92no—
152C22×C32—21no—
153C22×C32—42no—
154C22×C32—44no—
155C22×C32—41no—
156C22×C32—42no—
157C22×C32—41no—
158C22×C32—42no—
159C22×C32C2×#6711yescharacteristic
160C22×C32C2×C3×D411yescharacteristic
161C2×C3⋊D3—42no—
162C2×C3⋊D3—44no—
163C2×C3×D3—122no—
164C2×C3×D3—124no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—42no—
169C32×C4—42no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4D3×D412yes—
173C32×C4C2×C3×D412yes—
174C2×SD16—92no—
175C2×D8—92no—
176#127—91no—
177#124(C3⋊D3)⋊C311yescharacteristic, complemented
178#97—91no—
179H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
180C23×C3—21no—
181C23×C3—61no—
182C23×C3—61no—
183C23×C3—62no—
184C23×C3—182no—
185C22×D3—62no—
186C22×D3—182no—
187C3×Q8—22no—
188C3×Q8—62no—
189C3×Q8—62no—
190C3×Q8—122no—
191C3×D4—22no—
192C3×D4—62no—
193C3×D4—62no—
194C3×D4—122no—
195C3×D4—182no—
196C3×D4—182no—
197C3×D4—364no—
198C2×C3×C4—21no—
199C2×C3×C4—32no—
200C2×C3×C4—32no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—61no—
208C2×C3×C4C3×#6712yes—
209C2×C3×C4C22×C3×D311yescharacteristic
210D12—62no—
211D12—62no—
212D12—124no—
213D12—182no—
214D12—182no—
215D12—364no—
216C3×C8—94no—
217C3⋊C8—34no—
218C3⋊C8—94no—
219C2×C32—21no—
220C2×C32—22no—
221C2×C32—82no—
222C2×C32—81no—
223C2×C32—81no—
224C2×C32#94011yescharacteristic
225C2×C32#94912yes—
226C2×C32C3×#12711yescharacteristic
227C2×C32C3×AGL(1,ℤ/8)12yes—
228C3⋊D3—82no—
229C3×D3—242no—
230C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
231C2×D4—92no—
232C2×D4—182no—
233C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
234SD16—184no—
235D8—184no—
236C2×C8—92no—
237C22⋊C4—22no—
238C42C2×(C3⋊D3)⋊C311yescharacteristic
239C22×C3—31no—
240C22×C3—31no—
241C22×C3—41no—
242C22×C3—42no—
243C22×C3—61no—
244C22×C3—121no—
245C22×C3—121no—
246C22×C3—122no—
247C22×C3—122no—
248C22×C3—122no—
249C22×C3—124no—
250C22×C3—362no—
251C22×C3—364no—
252C22×C3C2×C3×#6711yescharacteristic
253C2×D3—122no—
254C2×D3—124no—
255C2×D3—362no—
256C2×D3—364no—
257C3×C4—22no—
258C3×C4—32no—
259C3×C4—32no—
260C3×C4—41no—
261C3×C4—62no—
262C3×C4—62no—
263C3×C4—62no—
264C3×C4—122no—
265C3×C4—121no—
266C3×C4—121no—
267C3×C4—122no—
268C3×C4C3×D3×D412yes—
269C32—21no—
270C32#505811yescharacteristic, complemented
271C32C3×#49611yescharacteristic, complemented
272C23—21no—
273C23—182no—
274Q8—22no—
275D4—22no—
276D4—182no—
277D4—182no—
278D4—364no—
279C2×C4—21no—
280C2×C4#638⋊C312yes—
281C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
282C8—94no—
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—32no—
286C2×C3—32no—
287C2×C3—61no—
288C2×C3—62no—
289C2×C3—81no—
290C2×C3—241no—
291C2×C3—241no—
292C2×C3—242no—
293C2×C3—722no—
294C2×C3C3×#94011yescharacteristic
295C2×C3C3×#94912yes—
296D3—242no—
297D3—722no—
298C22—41no—
299C22—42no—
300C22—362no—
301C22—364no—
302C22C2×#638⋊C311yescharacteristic
303C4—22no—
304C4—41no—
305C4D4×(C3⋊D3)⋊C312yes—
306C3—31no—
307C3—31no—
308C3—61no—
309C3C3×#505811yescharacteristic
310C2—81no—
311C2—722no—
312C2#64149⋊C311yescharacteristic
313C2#64158⋊C312yes—
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#124C2
socleC22×C3C2×C3×#67
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#496
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#124C2semidirect
2. C2×#3755⋊C3C2semidirect
3. C2×#3753⋊C3C2semidirect
4. #11721385C2semidirect
5. #63630⋊C3C2semidirect
6. #10592757C3semidirect
18. C32×#124D3semidirect
19. C32×#124C2×C3semidirect
20. C2×#3755C2×C3semidirect
21. C2×#3753C2×C3semidirect
22. #64083C2×C3semidirect
26. #63630C2×C3semidirect
38. (C3⋊(C3⋊C8))⋊C3D4semidirect
114. C3⋊(C3⋊C8)C3×D4semidirect
177. #124(C3⋊D3)⋊C3semidirect
179. H(3,𝔽3)#496semidirect
270. C32#5058semidirect
271. C32C3×#496semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
C2×#3755⋊C32
C2×#3753⋊C32
#117213852
#63630⋊C32
#105927573
C3×#50583