Finite Group Databasebeta

← #10592760⋊C3 #13142925

Subgroups — #10592760⋊C3 #13142925

Order 1728.

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

normal subgroups only

This group has 608 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
2H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
3#64151⋊C3C212yesmaximal, complemented
4#63660⋊C3C211yesmaximal, characteristic, complemented
5#63632⋊C3C212yesmaximal, complemented
6#10592760C311yesmaximal, characteristic, complemented
7C3×#5061—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3—22no—
11#3830⋊C3—22no—
12D4×(C3⋊D3)⋊C3—22no—
13#3828⋊C3C2212yes—
14C22⋊C4×H(3,𝔽3)—22no—
15C42×H(3,𝔽3)C2211yescharacteristic, complemented
16#3755⋊C3—22no—
17#3756⋊C3—22no—
18#3748⋊C3C2212yescomplemented
19#64361C2×C311yescharacteristic
20C3×#1004—31no—
21C32×#124—21no—
22C32×#124D311yescharacteristic, complemented
23C32×#124C2×C311yescharacteristic, complemented
24#64151C2×C312yescomplemented
25C3×#942—32no—
26#63660C2×C311yescharacteristic, complemented
27#63632C2×C312yescomplemented
28C3×#833—31no—
29C3×#805—32no—
30C23×H(3,𝔽3)D412yes—
31C22×(C3⋊D3)⋊C3—22no—
32Q8×H(3,𝔽3)—22no—
33D4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37#638⋊C3—24no—
38#638⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C4⋊((C3⋊D3)⋊C3)D412yes—
41C4×(C3⋊D3)⋊C3—22no—
42#634⋊C3D412yes—
43(C3⋊(C3⋊C8))⋊C3—22no—
44C3×#455—91no—
45#5061—31no—
46#5061—91no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×D311yescharacteristic, complemented
49C2×C32×Q8C22×C311yescharacteristic, complemented
50C2×C32×D4—21no—
51C2×C32×D4C2×D311yescharacteristic
52C2×C32×D4C22×C311yescharacteristic
53C2×#638—22no—
54#3830—22no—
55D4×C3⋊D3—22no—
56#3828C22×C312yes—
57C2×C3×#67—62no—
58C3×#237—62no—
59C3×D3×D4—62no—
60C3×#235—32no—
61C32×C22⋊C4—24no—
62C32×C22⋊C4—22no—
63C32×C22⋊C4—22no—
64C32×C42—21no—
65C32×C42C2×D311yescharacteristic, complemented
66C32×C42C22×C311yescharacteristic, complemented
67#3756—22no—
68#3755—22no—
69#3748C22×C312yescomplemented
70C3×#216—62no—
71C3×#215—62no—
72C3×#208—32no—
73C22×H(3,𝔽3)—22no—
74C22×H(3,𝔽3)—42no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C2×(C3⋊D3)⋊C3—22no—
77C2×(C3⋊D3)⋊C3—82no—
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)C2×D412yes—
81(C3⋊Dic3)⋊C3—22no—
82C3×H(5,𝔽2)—91no—
83#1004—31no—
84#1004—91no—
85C3×#137—92no—
86C3×#124—31no—
87C3×#124—31no—
88C3×#124—61no—
89C3×#124C3×D311yescharacteristic
90#942—32no—
91#942—92no—
92C3×#104—92no—
93C3×#100—91no—
94#833—31no—
95#833—91no—
96#805—32no—
97#805—92no—
98C23×C32—22no—
99C23×C32#6712yes—
100C23×C32C3×D412yes—
101C22×C3⋊D3—22no—
102C22×C3×D3—62no—
103C32×Q8—22no—
104C32×Q8—22no—
105C32×Q8—42no—
106C32×D4—22no—
107C32×D4—22no—
108C32×D4—42no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C23×C311yescharacteristic
118#638—24no—
119#638—22no—
120C2×C3⋊Dic3—22no—
121C4⋊(C3⋊D3)C3×D412yes—
122C4×C3⋊D3—22no—
123#634C3×D412yes—
124C3×#67—64no—
125C3×#67—62no—
126C2×C3×Dic3—62no—
127C3×D12—32no—
128C3×C4×D3—62no—
129C3×Dic6—32no—
130C3⋊(C3⋊C8)—22no—
131C3×C3⋊C8—62no—
132#455—91noSylow
133C2×H(3,𝔽3)—21no—
134C2×H(3,𝔽3)—42no—
135C2×H(3,𝔽3)#12011yescharacteristic
136(C3⋊D3)⋊C3—42no—
137C3×#41—92no—
138C3×#41—182no—
139C2×C3×Q8—31no—
140C2×C3×Q8—31no—
141C2×C3×Q8—61no—
142C2×C3×Q8C2×C3×D311yescharacteristic
143C2×C3×D4—31no—
144C2×C3×D4—31no—
145C2×C3×D4—61no—
146C2×C3×D4—182no—
147C2×C3×D4—182no—
148C2×C3×D4C2×C3×D311yescharacteristic
149C2×#67—62no—
150C2×#67—182no—
151#237—62no—
152#237—182no—
153D3×D4—62no—
154D3×D4—182no—
155#235—32no—
156#235—92no—
157C3×Q16—182no—
158C3×SD16—182no—
159C3×M16—92no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—62no—
162C3×C22⋊C4—62no—
163C3×C22⋊C4—64no—
164C3×C42—31no—
165C3×C42—31no—
166C3×C42—61no—
167C3×C42C2×C3×D311yescharacteristic
168#216—62no—
169#216—182no—
170#215—62no—
171#215—182no—
172#208—32no—
173#208—92no—
174C22×C32—22no—
175C22×C32—22no—
176C22×C32—21no—
177C22×C32—42no—
178C22×C32—42no—
179C22×C32—42no—
180C22×C32—44no—
181C22×C32C2×#6711yescharacteristic
182C22×C32C2×C3×D411yescharacteristic
183C2×C3⋊D3—22no—
184C2×C3⋊D3—82no—
185C2×C3×D3—62no—
186C2×C3×D3—242no—
187C32×C4—22no—
188C32×C4—42no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4—42no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4D3×D412yes—
195C32×C4C2×C3×D412yes—
196C3⋊Dic3—22no—
197C3×Dic3—62no—
198H(5,𝔽2)—91no—
199#137—92no—
200#124(C3⋊D3)⋊C311yescharacteristic, complemented
201#104—92no—
202#100—91no—
203H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
204C23×C3—32no—
205C23×C3—32no—
206C23×C3—62no—
207C23×C3—182no—
208C23×C3C3×#6712yes—
209C22×D3—62no—
210C22×D3—182no—
211C3×Q8—22no—
212C3×Q8—62no—
213C3×Q8—62no—
214C3×Q8—92no—
215C3×Q8—122no—
216C3×D4—22no—
217C3×D4—62no—
218C3×D4—62no—
219C3×D4—92no—
220C3×D4—122no—
221C3×D4—182no—
222C3×D4—184no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—61no—
234C2×C3×C4—182no—
235C2×C3×C4—182no—
236C2×C3×C4C22×C3×D311yescharacteristic
237#67—64no—
238#67—62no—
239#67—182no—
240#67—184no—
241C2×Dic3—62no—
242C2×Dic3—182no—
243D12—32no—
244D12—92no—
245C4×D3—62no—
246C4×D3—182no—
247Dic6—32no—
248Dic6—92no—
249C3×C8—182no—
250C3⋊C8—62no—
251C3⋊C8—182no—
252C2×C32—22no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—42no—
257C2×C32—42no—
258C2×C32—82no—
259C2×C32#93711yescharacteristic
260C2×C32C3×#12011yescharacteristic
261C3⋊D3—42no—
262C3×D3—122no—
263#41—92no—
264#41—182no—
265C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
266C2×D4—182no—
267C2×D4—182no—
268C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
269Q16—182no—
270SD16—182no—
271M16—92no—
272C22⋊C4—22no—
273C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
274C22×C3—22no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—42no—
278C22×C3—62no—
279C22×C3—62no—
280C22×C3—61no—
281C22×C3—122no—
282C22×C3—122no—
283C22×C3—122no—
284C22×C3—124no—
285C22×C3—182no—
286C22×C3—722no—
287C22×C3C2×C3×#6711yescharacteristic
288C2×D3—62no—
289C2×D3—182no—
290C2×D3—242no—
291C2×D3—722no—
292C3×C4—32no—
293C3×C4—32no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—62no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—122no—
300C3×C4—121no—
301C3×C4—121no—
302C3×C4—122no—
303C3×C4—182no—
304C3×C4C3×D3×D412yes—
305Dic3—62no—
306Dic3—182no—
307C32—21no—
308C32#506111yescharacteristic, complemented
309C32C3×#45511yescharacteristic, complemented
310C23—182no—
311C23#638⋊C312yes—
312Q8—22no—
313Q8—92no—
314D4—22no—
315D4—92no—
316D4—184no—
317D4—182no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—182no—
321C2×C4—182no—
322C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
323C8—182no—
324C2×C3—21no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—42no—
328C2×C3—61no—
329C2×C3—61no—
330C2×C3—62no—
331C2×C3—61no—
332C2×C3—122no—
333C2×C3—122no—
334C2×C3—242no—
335C2×C3—362no—
336C2×C3C3×#93711yescharacteristic
337D3—122no—
338D3—362no—
339C22—22no—
340C22—42no—
341C22—182no—
342C22—722no—
343C22C2×#638⋊C311yescharacteristic
344C4—41no—
345C4—41no—
346C4—182no—
347C4D4×(C3⋊D3)⋊C312yes—
348C3—31no—
349C3—31no—
350C3—61no—
351C3C3×#506111yescharacteristic
352C2—21no—
353C2—42no—
354C2—362no—
355C2#64146⋊C311yescharacteristic
subgroupG/H
centerC2#64146⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#124C2
socleC2×C3C3×#937
prime radical#64361⋊C3C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#124C2semidirect
3. #64151⋊C3C2semidirect
4. #63660⋊C3C2semidirect
5. #63632⋊C3C2semidirect
6. #10592760C3semidirect
8. C2×Q8×H(3,𝔽3)C22semidirect
15. C42×H(3,𝔽3)C22semidirect
18. #3748⋊C3C22semidirect
22. C32×#124D3semidirect
23. C32×#124C2×C3semidirect
24. #64151C2×C3semidirect
26. #63660C2×C3semidirect
27. #63632C2×C3semidirect
48. C2×C32×Q8C2×D3semidirect
49. C2×C32×Q8C22×C3semidirect
65. C32×C42C2×D3semidirect
66. C32×C42C22×C3semidirect
69. #3748C22×C3semidirect
200. #124(C3⋊D3)⋊C3semidirect
203. H(3,𝔽3)#455semidirect
265. C2×Q8C2×(C3⋊D3)⋊C3semidirect
273. C42C2×(C3⋊D3)⋊C3semidirect
308. C32#5061semidirect
309. C32C3×#455semidirect

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)×#1242
#64151⋊C32
#63660⋊C32
#63632⋊C32
#105927603
C3×#50613