Finite Group Databasebeta

← #10592446⋊C3 #13141675

Subgroups — #10592446⋊C3 #13141675

Order 1728.

634 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 90 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#64361⋊C3C211yesmaximal, characteristic, complemented
2#64149⋊C3C211yesmaximal, characteristic, complemented
3#64117⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
5#63652⋊C3C212yesmaximal, complemented
6#10592446C311yesmaximal, characteristic, complemented
7C3×#4747—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—22no—
10C2×#638⋊C3—22no—
11#3830⋊C3—22no—
12D4×(C3⋊D3)⋊C3—22no—
13#3828⋊C3C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
15M16×H(3,𝔽3)C2212yescomplemented
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
17#3745⋊C3—22no—
18#3744⋊C3—22no—
19#64361C2×C311yescharacteristic, complemented
20C3×#1004—31no—
21#64149C2×C311yescharacteristic, complemented
22#64117C2×C312yescomplemented
23C3×#940—31no—
24C3×#908—32no—
25C32×#100—21no—
26C32×#100D311yescharacteristic, complemented
27C32×#100C2×C311yescharacteristic, complemented
28#63652C2×C312yescomplemented
29C3×#825—32no—
30C23×H(3,𝔽3)D412yes—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3—21no—
33D4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#638⋊C3—24no—
36#638⋊C3—22no—
37#638⋊C3—44no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C4⋊((C3⋊D3)⋊C3)—22no—
41C4⋊((C3⋊D3)⋊C3)D412yescomplemented
42C4×(C3⋊D3)⋊C3—22no—
43#634⋊C3D412yescomplemented
44C8×H(3,𝔽3)—22no—
45C3×#453—91no—
46#4747—31no—
47#4747—91no—
48C2×C32×D4—21no—
49C2×C32×D4C2×D311yescharacteristic
50C2×C32×D4C22×C311yescharacteristic
51C2×#638—22no—
52C2×#638—22no—
53#3830—22no—
54D4×C3⋊D3—22no—
55#3828C22×C312yes—
56C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
57C2×C3×#67—62no—
58C2×C3×#67—62no—
59C3×#237—62no—
60C3×D3×D4—62no—
61C3×#235—32no—
62C2×C3×D12—31no—
63C32×M16—22no—
64C32×M16C2×D312yescomplemented
65C32×M16C22×C312yescomplemented
66C4×C3⋊Dic3C22×C311yescharacteristic, complemented
67#3745—22no—
68#3744—22no—
69C3×C4×Dic3—31no—
70C3×D24—62no—
71C3×#204—62no—
72C22×H(3,𝔽3)—22no—
73C22×H(3,𝔽3)—42no—
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C2×(C3⋊D3)⋊C3—22no—
76C2×(C3⋊D3)⋊C3—42no—
77C2×(C3⋊D3)⋊C3—41no—
78C2×(C3⋊D3)⋊C3—82no—
79C4×H(3,𝔽3)C2×D412yes—
80(C3⋊Dic3)⋊C3—22no—
81(C3⋊Dic3)⋊C3—22no—
82C3×H(5,𝔽2)—91no—
83#1004—31no—
84#1004—91no—
85C3×AGL(1,ℤ/8)—92no—
86C3×#127—91no—
87#940—31no—
88#940—91no—
89#908—32no—
90#908—92no—
91C3×#104—92no—
92C3×#100—31no—
93C3×#100—31no—
94C3×#100—61no—
95C3×#100C3×D311yescharacteristic
96#825—32no—
97#825—92no—
98C23×C32—22no—
99C23×C32D1212yes—
100C23×C32C3×D412yes—
101C22×C3⋊D3—22no—
102C22×C3⋊D3—21no—
103C22×C3×D3—62no—
104C22×C3×D3—61no—
105C32×D4—22no—
106C32×D4—22no—
107C32×D4—42no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C32×C4C23×C311yescharacteristic
111#638—24no—
112#638—22no—
113#638—44no—
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3—21no—
116C4⋊(C3⋊D3)—22no—
117C4⋊(C3⋊D3)C3×D412yescomplemented
118C4×C3⋊D3—22no—
119#634C3×D412yescomplemented
120C3×#67—64no—
121C3×#67—62no—
122C3×#67—124no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—61no—
125C3×D12—32no—
126C3×D12—62no—
127C3×C4×D3—62no—
128C3×Dic6—32no—
129C32×C8—22no—
130C32×C8—22no—
131C32×C8—42no—
132#453—91noSylow
133C2×H(3,𝔽3)—21no—
134C2×H(3,𝔽3)—42no—
135C2×H(3,𝔽3)#12011yescharacteristic
136(C3⋊D3)⋊C3—42no—
137(C3⋊D3)⋊C3—81no—
138C3×#41—92no—
139C3×#41—182no—
140C2×C3×D4—31no—
141C2×C3×D4—31no—
142C2×C3×D4—61no—
143C2×C3×D4—91no—
144C2×C3×D4—182no—
145C2×C3×D4—182no—
146C2×C3×D4—182no—
147C2×C3×D4C2×C3×D311yescharacteristic
148C2×#67—62no—
149C2×#67—62no—
150C2×#67—182no—
151C2×#67—182no—
152#237—62no—
153#237—182no—
154D3×D4—62no—
155D3×D4—182no—
156#235—32no—
157#235—92no—
158C2×D12—31no—
159C2×D12—91no—
160C3×SD16—182no—
161C3×D8—182no—
162C3×M16—32no—
163C3×M16—32no—
164C3×M16—62no—
165C3×M16C2×C3×D312yes—
166C3×C42—91no—
167C4×Dic3—31no—
168C4×Dic3—91no—
169D24—62no—
170D24—182no—
171#204—62no—
172#204—182no—
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32—42no—
177C22×C32—42no—
178C22×C32—42no—
179C22×C32—82no—
180C22×C32C2×D1211yescharacteristic
181C22×C32C2×C3×D411yescharacteristic
182C2×C3⋊D3—22no—
183C2×C3⋊D3—42no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—82no—
186C2×C3×D3—62no—
187C2×C3×D3—122no—
188C2×C3×D3—121no—
189C2×C3×D3—242no—
190C32×C4—22no—
191C32×C4D3×D412yes—
192C32×C4C2×C3×D412yes—
193C3⋊Dic3—22no—
194C3⋊Dic3—22no—
195C3×Dic3—62no—
196C3×Dic3—62no—
197H(5,𝔽2)—91no—
198AGL(1,ℤ/8)—92no—
199#127—91no—
200#104—92no—
201#100(C3⋊D3)⋊C311yescharacteristic, complemented
202H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
203C23×C3—32no—
204C23×C3—32no—
205C23×C3—62no—
206C23×C3—181no—
207C23×C3—182no—
208C23×C3C3×D1212yes—
209C22×D3—62no—
210C22×D3—61no—
211C22×D3—182no—
212C22×D3—181no—
213C3×Q8—92no—
214C3×D4—22no—
215C3×D4—62no—
216C3×D4—62no—
217C3×D4—92no—
218C3×D4—122no—
219C3×D4—182no—
220C3×D4—182no—
221C3×D4—184no—
222C3×D4—364no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—61no—
226C2×C3×C4—182no—
227C2×C3×C4—181no—
228C2×C3×C4—182no—
229C2×C3×C4C22×C3×D311yescharacteristic
230#67—64no—
231#67—62no—
232#67—124no—
233#67—182no—
234#67—184no—
235#67—364no—
236C2×Dic3—62no—
237C2×Dic3—61no—
238C2×Dic3—181no—
239C2×Dic3—182no—
240D12—32no—
241D12—62no—
242D12—92no—
243D12—182no—
244C4×D3—62no—
245C4×D3—182no—
246Dic6—32no—
247Dic6—92no—
248C3×C8—22no—
249C3×C8—62no—
250C3×C8—62no—
251C3×C8—122no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—41no—
256C2×C32—42no—
257C2×C32—42no—
258C2×C32—82no—
259C2×C32#88211yescharacteristic
260C2×C32C3×#12011yescharacteristic
261C3⋊D3—42no—
262C3⋊D3—81no—
263C3×D3—122no—
264C3×D3—241no—
265#41—92no—
266#41—182no—
267C2×D4—91no—
268C2×D4—182no—
269C2×D4—182no—
270C2×D4—182no—
271C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
272SD16—182no—
273D8—182no—
274M16C2×(C3⋊D3)⋊C312yescomplemented
275C42—91no—
276C22×C3—22no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—42no—
280C22×C3—62no—
281C22×C3—62no—
282C22×C3—61no—
283C22×C3—122no—
284C22×C3—122no—
285C22×C3—122no—
286C22×C3—182no—
287C22×C3—242no—
288C22×C3—361no—
289C22×C3—362no—
290C22×C3—722no—
291C22×C3C2×C3×D1211yescharacteristic
292C2×D3—62no—
293C2×D3—122no—
294C2×D3—121no—
295C2×D3—182no—
296C2×D3—242no—
297C2×D3—361no—
298C2×D3—362no—
299C2×D3—722no—
300C3×C4—32no—
301C3×C4—32no—
302C3×C4—62no—
303C3×C4—182no—
304C3×C4—182no—
305C3×C4C3×D3×D412yes—
306Dic3—62no—
307Dic3—62no—
308Dic3—182no—
309Dic3—182no—
310C32—21no—
311C32#474711yescharacteristic, complemented
312C32C3×#45311yescharacteristic, complemented
313C23—182no—
314C23—181no—
315C23C4⋊((C3⋊D3)⋊C3)12yes—
316Q8—92no—
317D4—22no—
318D4—92no—
319D4—184no—
320D4—182no—
321D4—182no—
322D4—364no—
323C2×C4—182no—
324C2×C4—181no—
325C2×C4—182no—
326C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
327C8—22no—
328C2×C3—21no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—42no—
332C2×C3—61no—
333C2×C3—61no—
334C2×C3—61no—
335C2×C3—121no—
336C2×C3—122no—
337C2×C3—122no—
338C2×C3—242no—
339C2×C3—362no—
340C2×C3—721no—
341C2×C3C3×#88211yescharacteristic
342D3—122no—
343D3—241no—
344D3—362no—
345D3—721no—
346C22—22no—
347C22—42no—
348C22—182no—
349C22—362no—
350C22—361no—
351C22—722no—
352C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
353C4—182no—
354C4—182no—
355C4D4×(C3⋊D3)⋊C312yes—
356C3—31no—
357C3—31no—
358C3—61no—
359C3C3×#474711yescharacteristic
360C2—21no—
361C2—42no—
362C2—362no—
363C2—721no—
364C2#64091⋊C311yescharacteristic
subgroupG/H
centerC2#64091⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#100C2
socleC2×C3C3×#882
hypercenter#100(C3⋊D3)⋊C3
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
1. #64361⋊C3C2semidirect
2. #64149⋊C3C2semidirect
3. #64117⋊C3C2semidirect
4. H(3,𝔽3)×#100C2semidirect
5. #63652⋊C3C2semidirect
6. #10592446C3semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. M16×H(3,𝔽3)C22semidirect
16. C4×(C3⋊Dic3)⋊C3C22semidirect
19. #64361C2×C3semidirect
21. #64149C2×C3semidirect
22. #64117C2×C3semidirect
26. C32×#100D3semidirect
27. C32×#100C2×C3semidirect
28. #63652C2×C3semidirect
41. C4⋊((C3⋊D3)⋊C3)D4semidirect
43. #634⋊C3D4semidirect
56. C2×C4⋊(C3⋊D3)C22×C3semidirect
64. C32×M16C2×D3semidirect
65. C32×M16C22×C3semidirect
66. C4×C3⋊Dic3C22×C3semidirect
117. C4⋊(C3⋊D3)C3×D4semidirect
119. #634C3×D4semidirect
201. #100(C3⋊D3)⋊C3semidirect
202. H(3,𝔽3)#453semidirect
274. M16C2×(C3⋊D3)⋊C3semidirect
311. C32#4747semidirect
312. C32C3×#453semidirect

Maximal subgroups

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

subgroupindex
#64361⋊C32
#64149⋊C32
#64117⋊C32
H(3,𝔽3)×#1002
#63652⋊C32
#105924463
C3×#47473