Finite Group Databasebeta

← #10592522⋊C3 #13141983

Subgroups — #10592522⋊C3 #13141983

Order 1728.

408 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64159⋊C3C211yesmaximal, characteristic
2#11721689C211yesmaximal, characteristic, complemented
3#64116⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
5#63649⋊C3C211yesmaximal, characteristic, complemented
6#63632⋊C3C211yesmaximal, characteristic, complemented
7#63630⋊C3C211yesmaximal, characteristic, complemented
8#10592522C311yesmaximal, characteristic, complemented
9C3×#4823—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12M16×H(3,𝔽3)C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14#3755⋊C3C411yescharacteristic, complemented
15#3756⋊C3C411yescharacteristic, complemented
16#3753⋊C3C411yescharacteristic, complemented
17#3754⋊C3C411yescharacteristic, complemented
18#3748⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#3743⋊C3—21no—
24C8×(C3⋊D3)⋊C3—21no—
25#64159C2×C311yescharacteristic
26#64157C2×C311yescharacteristic, complemented
27#64116C2×C311yescharacteristic, complemented
28C3×#950—31no—
29C3×#948—31no—
30C3×#907—31no—
31C32×#104—21no—
32C32×#104D311yescharacteristic, complemented
33C32×#104C2×C311yescharacteristic, complemented
34#63649C2×C311yescharacteristic, complemented
35#63632C2×C311yescharacteristic, complemented
36#63630C2×C311yescharacteristic, complemented
37C3×#822—31no—
38C3×#805—31no—
39C3×#803—31no—
40Q8×H(3,𝔽3)C2×C411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46#638⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49#634⋊C3C2×C411yescharacteristic
50C8×H(3,𝔽3)—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
53(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
56C3×#444—91no—
57#4823—31no—
58#4823—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C22×C311yescharacteristic
62#3828C22×C311yescharacteristic
63C3×#235—31no—
64C32×M16—21no—
65C32×M16C2×D311yescharacteristic
66C32×M16C22×C311yescharacteristic
67C32×C42—21no—
68C32×C42C2×D311yescharacteristic
69C32×C42C22×C311yescharacteristic
70#3756C3×C411yescharacteristic, complemented
71#3755C3×C411yescharacteristic, complemented
72#3754C3×C411yescharacteristic, complemented
73#3753C3×C411yescharacteristic, complemented
74#3748—21no—
75#3748C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)—21no—
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79#3743—21no—
80C8×C3⋊D3—21no—
81C3×#216—31no—
82C3×#215—31no—
83C3×#214—31no—
84C3×#213—31no—
85C3×#208—31no—
86C3×#208—61no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—61no—
90C3×ΓL(1,𝔽25)—61no—
91C3×D3×C8—61no—
92C22×H(3,𝔽3)—21no—
93C22×H(3,𝔽3)#4111yescharacteristic
94C2×(C3⋊D3)⋊C3—21no—
95C4×H(3,𝔽3)—21no—
96C4×H(3,𝔽3)—41no—
97C4×H(3,𝔽3)C22×C411yescharacteristic
98C4×H(3,𝔽3)C2×D411yescharacteristic
99(C3⋊Dic3)⋊C3—21no—
100C3×#135—91no—
101C3×#131—91no—
102C3×#131—91no—
103#950—31no—
104#950—91no—
105#948—31no—
106#948—91no—
107#907—31no—
108#907—91no—
109C3×#108—91no—
110C3×#104—31no—
111C3×#104—31no—
112C3×#104—61no—
113C3×#104—91no—
114C3×#104C3×D311yescharacteristic
115C3×#97—91no—
116#822—31no—
117#822—91no—
118#805—31no—
119#805—91no—
120#803—31no—
121#803—91no—
122C32×Q8—21no—
123C32×Q8C4×D311yescharacteristic
124C32×Q8C2×C3×C411yescharacteristic
125C32×D4—21no—
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—41no—
129C32×D4C4×D311yescharacteristic
130C32×D4C2×C3×C411yescharacteristic
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4—41no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C23×C311yescharacteristic
140#638—21no—
141C4⋊(C3⋊D3)C2×C3×C411yescharacteristic
142C4×C3⋊D3—21no—
143#634C2×C3×C411yescharacteristic
144C3×#67—61no—
145C3×D12—31no—
146C3×C4×D3—61no—
147C3×Dic6—31no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—41no—
151C3⋊(C3⋊C8)—21no—
152C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
153C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
154C3⋊(C3⋊C8)C3×D411yescharacteristic
155C3⋊(C3⋊C8)C3×D411yescharacteristic
156C3×C3⋊C8—31no—
157C3×C3⋊C8—31no—
158C3×C3⋊C8—31no—
159C3×C3⋊C8—31no—
160C3×C3⋊C8—61no—
161#444—91noSylow
162C2×H(3,𝔽3)—21no—
163C2×H(3,𝔽3)—41no—
164C2×H(3,𝔽3)C4×D411yescharacteristic
165(C3⋊D3)⋊C3—41no—
166C3×#41—31no—
167C3×#41—31no—
168C3×#41—61no—
169C3×#41—91no—
170C3×#41C2×C3×D311yescharacteristic
171#235—31no—
172#235—91no—
173C3×Q16—91no—
174C3×SD16—91no—
175C3×SD16—91no—
176C3×D8—91no—
177C3×M16—31no—
178C3×M16—31no—
179C3×M16—61no—
180C3×M16—91no—
181C3×M16—181no—
182C3×M16—181no—
183C3×M16C2×C3×D311yescharacteristic
184C2×C3×C8—91no—
185C2×C3×C8—91no—
186C2×C3×C8—181no—
187C2×C3×C8—181no—
188C3×C42—31no—
189C3×C42—31no—
190C3×C42—61no—
191C3×C42C2×C3×D311yescharacteristic
192#216—31no—
193#216—91no—
194#215—31no—
195#215—91no—
196#214—31no—
197#214—91no—
198#213—31no—
199#213—91no—
200#208—31no—
201#208—61no—
202#208—91no—
203#208—181no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—61no—
207C2×C3⋊C8—91no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—181no—
210ΓL(1,𝔽25)—61no—
211ΓL(1,𝔽25)—181no—
212D3×C8—61no—
213D3×C8—181no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32#23711yescharacteristic
219C22×C32C3×#4111yescharacteristic
220C2×C3⋊D3—21no—
221C2×C3×D3—61no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4—42no—
230C32×C4C2×C4×D311yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4C22×C3×C411yescharacteristic
233C32×C4C2×C3×D411yescharacteristic
234C3⋊Dic3—21no—
235C3×Dic3—61no—
236#135—91no—
237#131—91no—
238#131—91no—
239#108—91no—
240#104—91no—
241#104(C3⋊D3)⋊C311yescharacteristic, complemented
242#97—91no—
243H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
244C3×Q8—31no—
245C3×Q8—31no—
246C3×Q8—61no—
247C3×Q8—91no—
248C3×Q8C3×C4×D311yescharacteristic
249C3×D4—21no—
250C3×D4—31no—
251C3×D4—31no—
252C3×D4—61no—
253C3×D4—61no—
254C3×D4—61no—
255C3×D4—91no—
256C3×D4—121no—
257C3×D4—181no—
258C3×D4C3×C4×D311yescharacteristic
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—121no—
269C2×C3×C4—121no—
270C2×C3×C4—181no—
271C2×C3×C4C22×C3×D311yescharacteristic
272#67—61no—
273#67—181no—
274D12—31no—
275D12—91no—
276C4×D3—61no—
277C4×D3—181no—
278Dic6—31no—
279Dic6—91no—
280C3×C8—21no—
281C3×C8—61no—
282C3×C8—61no—
283C3×C8—91no—
284C3×C8—91no—
285C3×C8—91no—
286C3×C8—91no—
287C3×C8—121no—
288C3×C8—181no—
289C3⋊C8—31no—
290C3⋊C8—31no—
291C3⋊C8—31no—
292C3⋊C8—31no—
293C3⋊C8—61no—
294C3⋊C8—91no—
295C3⋊C8—91no—
296C3⋊C8—91no—
297C3⋊C8—91no—
298C3⋊C8—181no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—41no—
303C2×C32—41no—
304C2×C32—41no—
305C2×C32—81no—
306C2×C32#67⋊C411yescharacteristic
307C2×C32C3×C4×D411yescharacteristic
308C3⋊D3—41no—
309C3×D3—121no—
310#41—91no—
311#41C2×(C3⋊D3)⋊C311yescharacteristic
312Q16—91no—
313SD16—91no—
314SD16—91no—
315D8—91no—
316M16—91no—
317M16—181no—
318M16—181no—
319M16C2×(C3⋊D3)⋊C311yescharacteristic
320C2×C8—91no—
321C2×C8—91no—
322C2×C8—181no—
323C2×C8—181no—
324C42C2×(C3⋊D3)⋊C311yescharacteristic
325C22×C3—21no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—61no—
330C22×C3—61no—
331C22×C3—121no—
332C22×C3—181no—
333C22×C3C3×#23711yescharacteristic
334C2×D3—61no—
335C2×D3—181no—
336C3×C4—21no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—41no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—121no—
347C3×C4—121no—
348C3×C4—121no—
349C3×C4—122no—
350C3×C4—181no—
351C3×C4C2×C3×C4×D311yescharacteristic
352C3×C4C3×D3×D411yescharacteristic
353Dic3—61no—
354Dic3—181no—
355C32—21no—
356C32#482311yescharacteristic, complemented
357C32C3×#44411yescharacteristic, complemented
358Q8—91no—
359Q8C4×(C3⋊D3)⋊C311yescharacteristic
360D4—21no—
361D4—91no—
362D4—181no—
363D4C4×(C3⋊D3)⋊C311yescharacteristic
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—181no—
367C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
368C8—21no—
369C8—91no—
370C8—91no—
371C8—91no—
372C8—91no—
373C8—181no—
374C2×C3—21no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—41no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—121no—
382C2×C3—121no—
383C2×C3—121no—
384C2×C3—241no—
385C2×C3—361no—
386C2×C3C3×#67⋊C411yescharacteristic
387D3—121no—
388D3—361no—
389C22—21no—
390C22—181no—
391C22#3830⋊C311yescharacteristic
392C4—21no—
393C4—41no—
394C4—181no—
395C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
396C4D4×(C3⋊D3)⋊C311yescharacteristic
397C3—31no—
398C3—31no—
399C3—61no—
400C3C3×#482311yescharacteristic
401C2—21no—
402C2—41no—
403C2—361no—
404C2#64090⋊C311yescharacteristic
subgroupG/H
centerC4D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#104C2
socleC2×C3C3×#67⋊C4
prime radical#64159⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#104(C3⋊D3)⋊C3
hypocenterC32C3×#444
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #11721689C2semidirect
3. #64116⋊C3C2semidirect
4. H(3,𝔽3)×#104C2semidirect
5. #63649⋊C3C2semidirect
6. #63632⋊C3C2semidirect
7. #63630⋊C3C2semidirect
8. #10592522C3semidirect
14. #3755⋊C3C4semidirect
15. #3756⋊C3C4semidirect
16. #3753⋊C3C4semidirect
17. #3754⋊C3C4semidirect
26. #64157C2×C3semidirect
27. #64116C2×C3semidirect
32. C32×#104D3semidirect
33. C32×#104C2×C3semidirect
34. #63649C2×C3semidirect
35. #63632C2×C3semidirect
36. #63630C2×C3semidirect
70. #3756C3×C4semidirect
71. #3755C3×C4semidirect
72. #3754C3×C4semidirect
73. #3753C3×C4semidirect
241. #104(C3⋊D3)⋊C3semidirect
243. H(3,𝔽3)#444semidirect
356. C32#4823semidirect
357. C32C3×#444semidirect

Maximal subgroups

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

subgroupindex
#64159⋊C32
#117216892
#64116⋊C32
H(3,𝔽3)×#1042
#63649⋊C32
#63632⋊C32
#63630⋊C32
#105925223
C3×#48233