Finite Group Databasebeta

← #10592776⋊C3 #13142987

Subgroups — #10592776⋊C3 #13142987

Order 1728.

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

normal subgroups only

This group has 696 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
2H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
3#64141⋊C3C212yesmaximal, complemented
4#63660⋊C3C211yesmaximal, characteristic, complemented
5#63632⋊C3C212yesmaximal, complemented
6#10592776C311yesmaximal, characteristic, complemented
7C3×#5077—31nomaximal
8C2×D4×H(3,𝔽3)—22no—
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×#638⋊C3—22no—
12#3830⋊C3—22no—
13D4×(C3⋊D3)⋊C3—22no—
14#3828⋊C3C2212yes—
15C42×H(3,𝔽3)C2211yescharacteristic, complemented
16#3753⋊C3—22no—
17#3754⋊C3—22no—
18#3748⋊C3C2212yescomplemented
19#64361C2×C311yescharacteristic, complemented
20C3×#1004—31no—
21C32×#127—21no—
22C32×#127D311yescharacteristic, complemented
23C32×#127C2×C311yescharacteristic, complemented
24#64141C2×C312yescomplemented
25C3×#932—32no—
26#63660C2×C311yescharacteristic, complemented
27#63632C2×C312yescomplemented
28C3×#833—31no—
29C3×#805—32no—
30C23×H(3,𝔽3)—21no—
31C23×H(3,𝔽3)D412yes—
32C22×(C3⋊D3)⋊C3—22no—
33D4×H(3,𝔽3)—22no—
34D4×H(3,𝔽3)—22no—
35D4×H(3,𝔽3)—44no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38#638⋊C3—24no—
39#638⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C4⋊((C3⋊D3)⋊C3)D412yescomplemented
42C4×(C3⋊D3)⋊C3—22no—
43#634⋊C3D412yescomplemented
44(C3⋊(C3⋊C8))⋊C3—22no—
45C3×#453—91no—
46#5077—31no—
47#5077—91no—
48C2×C32×D4—24no—
49C2×C32×D4—21no—
50C2×C32×D4—21no—
51C2×C32×D4—22no—
52C2×C32×D4—22no—
53C2×C32×D4C2×D311yescharacteristic
54C2×C32×D4C2×D311yescharacteristic, complemented
55C2×C32×D4C22×C311yescharacteristic
56C2×C32×D4C22×C311yescharacteristic, complemented
57C2×#638—22no—
58#3830—22no—
59D4×C3⋊D3—22no—
60#3828C22×C312yes—
61C2×C3×#67—62no—
62C3×#237—62no—
63C3×D3×D4—62no—
64C3×#235—32no—
65C32×C42—21no—
66C32×C42C2×D311yescharacteristic, complemented
67C32×C42C22×C311yescharacteristic, complemented
68#3754—22no—
69#3753—22no—
70#3748C22×C312yescomplemented
71C3×#214—62no—
72C3×#213—62no—
73C3×#208—32no—
74C22×H(3,𝔽3)—22no—
75C22×H(3,𝔽3)—42no—
76C22×H(3,𝔽3)—42no—
77C22×H(3,𝔽3)—41no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—22no—
80C2×(C3⋊D3)⋊C3—82no—
81C4×H(3,𝔽3)—22no—
82C4×H(3,𝔽3)C2×D412yes—
83(C3⋊Dic3)⋊C3—22no—
84C3×H(5,𝔽2)—91no—
85#1004—31no—
86#1004—91no—
87C3×AGL(1,ℤ/8)—92no—
88C3×#127—31no—
89C3×#127—31no—
90C3×#127—61no—
91C3×#127C3×D311yescharacteristic
92#932—32no—
93#932—92no—
94C3×#104—92no—
95C3×#100—91no—
96#833—31no—
97#833—91no—
98#805—32no—
99#805—92no—
100C23×C32—22no—
101C23×C32—21no—
102C23×C32—21no—
103C23×C32—22no—
104C23×C32#6712yes—
105C23×C32C3×D412yes—
106C22×C3⋊D3—22no—
107C22×C3×D3—62no—
108C32×D4—22no—
109C32×D4—22no—
110C32×D4—22no—
111C32×D4—22no—
112C32×D4—42no—
113C32×D4—42no—
114C32×D4—44no—
115C32×D4—44no—
116C32×D4—48no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C23×C311yescharacteristic
123#638—24no—
124#638—22no—
125C2×C3⋊Dic3—22no—
126C4⋊(C3⋊D3)C3×D412yescomplemented
127C4×C3⋊D3—22no—
128#634C3×D412yescomplemented
129C3×#67—64no—
130C3×#67—62no—
131C2×C3×Dic3—62no—
132C3×D12—32no—
133C3×C4×D3—62no—
134C3×Dic6—32no—
135C3⋊(C3⋊C8)—22no—
136C3×C3⋊C8—62no—
137#453—91noSylow
138C2×H(3,𝔽3)—21no—
139C2×H(3,𝔽3)—42no—
140C2×H(3,𝔽3)—81no—
141C2×H(3,𝔽3)#12011yescharacteristic
142(C3⋊D3)⋊C3—42no—
143C3×#41—92no—
144C3×#41—182no—
145C2×C3×D4—22no—
146C2×C3×D4—31no—
147C2×C3×D4—31no—
148C2×C3×D4—31no—
149C2×C3×D4—31no—
150C2×C3×D4—62no—
151C2×C3×D4—62no—
152C2×C3×D4—64no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C2×C3×D4—182no—
156C2×C3×D4—182no—
157C2×C3×D4C2×C3×D311yescharacteristic
158C2×C3×D4C2×C3×D311yescharacteristic
159C2×#67—62no—
160C2×#67—182no—
161#237—62no—
162#237—182no—
163D3×D4—62no—
164D3×D4—182no—
165#235—32no—
166#235—92no—
167C3×SD16—182no—
168C3×D8—182no—
169C3×M16—92no—
170C3×C42—31no—
171C3×C42—31no—
172C3×C42—61no—
173C3×C42C2×C3×D311yescharacteristic
174#214—62no—
175#214—182no—
176#213—62no—
177#213—182no—
178#208—32no—
179#208—92no—
180C22×C32—22no—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—42no—
184C22×C32—42no—
185C22×C32—42no—
186C22×C32—44no—
187C22×C32—44no—
188C22×C32—42no—
189C22×C32—42no—
190C22×C32—42no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32C2×#6711yescharacteristic
194C22×C32C2×C3×D411yescharacteristic
195C2×C3⋊D3—22no—
196C2×C3⋊D3—82no—
197C2×C3×D3—62no—
198C2×C3×D3—242no—
199C32×C4—24no—
200C32×C4—22no—
201C32×C4—22no—
202C32×C4—22no—
203C32×C4D3×D412yes—
204C32×C4C2×C3×D412yes—
205C3⋊Dic3—22no—
206C3×Dic3—62no—
207H(5,𝔽2)—91no—
208AGL(1,ℤ/8)—92no—
209#127(C3⋊D3)⋊C311yescharacteristic, complemented
210#104—92no—
211#100—91no—
212H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
213C23×C3—21no—
214C23×C3—32no—
215C23×C3—32no—
216C23×C3—61no—
217C23×C3—61no—
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—182no—
221C23×C3C3×#6712yes—
222C22×D3—62no—
223C22×D3—182no—
224C3×Q8—92no—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—44no—
228C3×D4—62no—
229C3×D4—62no—
230C3×D4—62no—
231C3×D4—62no—
232C3×D4—92no—
233C3×D4—122no—
234C3×D4—122no—
235C3×D4—124no—
236C3×D4—124no—
237C3×D4—128no—
238C3×D4—182no—
239C3×D4—184no—
240C2×C3×C4—21no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—62no—
246C2×C3×C4—61no—
247C2×C3×C4—182no—
248C2×C3×C4—182no—
249C2×C3×C4C22×C3×D311yescharacteristic
250#67—64no—
251#67—62no—
252#67—182no—
253#67—184no—
254C2×Dic3—62no—
255C2×Dic3—182no—
256D12—32no—
257D12—92no—
258C4×D3—62no—
259C4×D3—182no—
260Dic6—32no—
261Dic6—92no—
262C3×C8—182no—
263C3⋊C8—62no—
264C3⋊C8—182no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—42no—
270C2×C32—42no—
271C2×C32—82no—
272C2×C32—82no—
273C2×C32—81no—
274C2×C32—81no—
275C2×C32#93711yescharacteristic
276C2×C32C3×#12011yescharacteristic
277C3⋊D3—42no—
278C3×D3—122no—
279#41—92no—
280#41—182no—
281C2×D4—22no—
282C2×D4—182no—
283C2×D4—182no—
284C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
285C2×D4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
286SD16—182no—
287D8—182no—
288M16—92no—
289C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
290C22×C3—22no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—41no—
296C22×C3—62no—
297C22×C3—62no—
298C22×C3—61no—
299C22×C3—122no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3—122no—
303C22×C3—122no—
304C22×C3—124no—
305C22×C3—122no—
306C22×C3—122no—
307C22×C3—124no—
308C22×C3—122no—
309C22×C3—182no—
310C22×C3—722no—
311C22×C3C2×C3×#6711yescharacteristic
312C2×D3—62no—
313C2×D3—182no—
314C2×D3—242no—
315C2×D3—722no—
316C3×C4—22no—
317C3×C4—32no—
318C3×C4—32no—
319C3×C4—62no—
320C3×C4—62no—
321C3×C4—64no—
322C3×C4—62no—
323C3×C4—182no—
324C3×C4C3×D3×D412yes—
325Dic3—62no—
326Dic3—182no—
327C32—21no—
328C32#507711yescharacteristic, complemented
329C32C3×#45311yescharacteristic, complemented
330C23—21no—
331C23—182no—
332C23#638⋊C312yes—
333Q8—92no—
334D4—22no—
335D4—22no—
336D4—44no—
337D4—92no—
338D4—184no—
339D4—182no—
340C2×C4—21no—
341C2×C4—182no—
342C2×C4—182no—
343C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
344C8—182no—
345C2×C3—21no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—42no—
349C2×C3—61no—
350C2×C3—61no—
351C2×C3—62no—
352C2×C3—61no—
353C2×C3—81no—
354C2×C3—122no—
355C2×C3—122no—
356C2×C3—242no—
357C2×C3—241no—
358C2×C3—241no—
359C2×C3—242no—
360C2×C3—362no—
361C2×C3C3×#93711yescharacteristic
362D3—122no—
363D3—362no—
364C22—22no—
365C22—42no—
366C22—42no—
367C22—41no—
368C22—182no—
369C22—722no—
370C22C2×#638⋊C311yescharacteristic
371C4—22no—
372C4—182no—
373C4D4×(C3⋊D3)⋊C312yes—
374C3—31no—
375C3—31no—
376C3—61no—
377C3C3×#507711yescharacteristic
378C2—21no—
379C2—42no—
380C2—81no—
381C2—362no—
382C2#64146⋊C311yescharacteristic
subgroupG/H
centerC2#64146⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#127C2
socleC2×C3C3×#937
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #64361⋊C3C2semidirect
2. H(3,𝔽3)×#127C2semidirect
3. #64141⋊C3C2semidirect
4. #63660⋊C3C2semidirect
5. #63632⋊C3C2semidirect
6. #10592776C3semidirect
10. C2×D4×H(3,𝔽3)C22semidirect
15. C42×H(3,𝔽3)C22semidirect
18. #3748⋊C3C22semidirect
19. #64361C2×C3semidirect
22. C32×#127D3semidirect
23. C32×#127C2×C3semidirect
24. #64141C2×C3semidirect
26. #63660C2×C3semidirect
27. #63632C2×C3semidirect
41. C4⋊((C3⋊D3)⋊C3)D4semidirect
43. #634⋊C3D4semidirect
54. C2×C32×D4C2×D3semidirect
56. C2×C32×D4C22×C3semidirect
66. C32×C42C2×D3semidirect
67. C32×C42C22×C3semidirect
70. #3748C22×C3semidirect
126. C4⋊(C3⋊D3)C3×D4semidirect
128. #634C3×D4semidirect
209. #127(C3⋊D3)⋊C3semidirect
212. H(3,𝔽3)#453semidirect
285. C2×D4C2×(C3⋊D3)⋊C3semidirect
289. C42C2×(C3⋊D3)⋊C3semidirect
328. C32#5077semidirect
329. C32C3×#453semidirect

Maximal subgroups

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

subgroupindex
#64361⋊C32
H(3,𝔽3)×#1272
#64141⋊C32
#63660⋊C32
#63632⋊C32
#105927763
C3×#50773