Finite Group Databasebeta

← #13177966

Subgroups — #13177966

Order 1728.

2252 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#64157C211yesmaximal, characteristic, complemented
3C32×#948C211yesmaximal, characteristic, complemented
4#11723358C213yesmaximal, complemented
5#11723357C213yesmaximal, complemented
6#11723356C213yesmaximal, complemented
7C2×#88785C213yesmaximal, complemented
8C3⋊D3×#131—31nomaximal
9#10595442—34nomaximal
10C33×#41C2211yescharacteristic
11#89080C411yescharacteristic
12Q8×C3⋊(C3⋊D3)C411yescharacteristic
13#89078C413yes—
14D4×C3⋊(C3⋊D3)C413yes—
15#89076C2213yes—
16C2×C4×C3⋊(C3⋊D3)C2213yes—
17C3×#3748C2213yescomplemented
18C2×C3×C3⋊(C3⋊C8)C2213yescomplemented
19C32×#208C2213yescomplemented
20C2×C32×C3⋊C8C2213yescomplemented
21#88788C2216yescomplemented
22#88785C2211yescharacteristic
23#88785C2213yes—
24#88785C2213yescomplemented
25#88785C2213yescomplemented
26#41×C3⋊D3—31no—
27#41×C3⋊D3—34no—
28#41×C3⋊D3—64no—
29C32×#131—31no—
30#64157D311yescharacteristic, complemented
31#64116—33no—
32M16×C3⋊D3—33no—
33#64110—33no—
34C2×C8×C3⋊D3—33no—
35C3×#948—34no—
36C3×#948D314yescomplemented
37#63818—312no—
38#63817—312no—
39#63817—312no—
40C2×#3710—312no—
41C22×C3⋊(C3⋊D3)C2×C413yes—
42C33×Q8C2×C411yescharacteristic
43C33×D4C2×C413yes—
44C2×C33×C4C2313yes—
45#6312C2×C416yes—
46C2×C3⋊(C3⋊Dic3)C2×C413yes—
47C4⋊(C3⋊(C3⋊D3))C2×C413yes—
48C4×C3⋊(C3⋊D3)C2×C413yes—
49C4×C3⋊(C3⋊D3)C2×C413yes—
50C4×C3⋊(C3⋊D3)C2311yescharacteristic
51C4×C3⋊(C3⋊D3)C2313yes—
52#6308C2×C413yes—
53C3×C3⋊(C3⋊C8)C2311yescharacteristic
54C3×C3⋊(C3⋊C8)C2313yes—
55C32×C3⋊C8C2311yescharacteristic
56C32×C3⋊C8C2313yes—
57D3×#131—94no—
58D3×#131—91no—
59C32×#41—24no—
60C32×#41C2×D314yes—
61C32×#41C2×D311yescharacteristic
62#3832—31no—
63#3832—34no—
64#3832—64no—
65Q8×C3⋊D3—31no—
66Q8×C3⋊D3—34no—
67Q8×C3⋊D3—64no—
68#3830—33no—
69#3830—312no—
70#3830—612no—
71D4×C3⋊D3—33no—
72D4×C3⋊D3—312no—
73D4×C3⋊D3—612no—
74#3828—33no—
75#3828—312no—
76#3828—612no—
77C2×C4×C3⋊D3—33no—
78C2×C4×C3⋊D3—312no—
79C2×C4×C3⋊D3—612no—
80C32×M16—33no—
81C2×C32×C8—33no—
82#3748C2×D313yescomplemented
83C2×C3⋊(C3⋊C8)C2×D313yescomplemented
84#3743—36no—
85C8×C3⋊D3—31no—
86C8×C3⋊D3—33no—
87C8×C3⋊D3—33no—
88C8×C3⋊D3—33no—
89C3×#208—312no—
90C3×#208C2×D3112yescomplemented
91C2×C3×C3⋊C8—312no—
92C2×C3×C3⋊C8C2×D3112yescomplemented
93#3712—324no—
94#3710—34no—
95#3710—312no—
96#3710—312no—
97#3710—312no—
98C22×C33C22×C413yes—
99C2×C3⋊(C3⋊D3)—26no—
100C2×C3⋊(C3⋊D3)C22×C413yes—
101C2×C3⋊(C3⋊D3)C22×C411yescharacteristic
102C33×C4C22×C413yes—
103C33×C4C2411yescharacteristic
104C3⋊(C3⋊Dic3)C22×C413yes—
105C3⋊(C3⋊Dic3)C22×C411yescharacteristic
106D3×#41—94no—
107D3×#41—91no—
108D3×#41—184no—
109C3×#131—34no—
110C3×#131—91no—
111#948—34no—
112#948C3⋊D311yescharacteristic, complemented
113#907—912no—
114#907—93no—
115D3×M16—912no—
116D3×M16—93no—
117#901—912no—
118#901—93no—
119C2×D3×C8—912no—
120C2×D3×C8—93no—
121C22×C3⋊D3—33no—
122C22×C3⋊D3—312no—
123C22×C3⋊D3—612no—
124C32×Q8—24no—
125C32×Q8C4×D314yes—
126C32×Q8C4×D311yescharacteristic
127C32×D4—212no—
128C32×D4C4×D3112yes—
129C32×D4C4×D313yes—
130C2×C32×C4—212no—
131C2×C32×C4C22×D3112yes—
132C2×C32×C4C22×D313yes—
133#638—36no—
134#638—324no—
135#638—624no—
136C2×C3⋊Dic3—33no—
137C2×C3⋊Dic3—312no—
138C2×C3⋊Dic3—612no—
139C4⋊(C3⋊D3)—33no—
140C4⋊(C3⋊D3)—312no—
141C4⋊(C3⋊D3)—612no—
142C4×C3⋊D3—33no—
143C4×C3⋊D3—33no—
144C4×C3⋊D3—31no—
145C4×C3⋊D3—33no—
146C4×C3⋊D3—312no—
147C4×C3⋊D3—312no—
148C4×C3⋊D3—34no—
149C4×C3⋊D3—312no—
150C4×C3⋊D3—612no—
151C4×C3⋊D3—612no—
152C4×C3⋊D3—64no—
153C4×C3⋊D3—612no—
154#634—33no—
155#634—312no—
156#634—612no—
157C32×C8—31no—
158C32×C8—33no—
159C3⋊(C3⋊C8)C22×D311yescharacteristic
160C3⋊(C3⋊C8)C22×D313yes—
161C3×C3⋊C8—34no—
162C3×C3⋊C8—312no—
163C3×C3⋊C8C22×D314yes—
164C3×C3⋊C8C22×D3112yes—
165C2×#131—271noSylow
166C2×C33—23no—
167C2×C33C23×C411yescharacteristic
168C3⋊(C3⋊D3)—23no—
169C3⋊(C3⋊D3)#13112yescomplemented
170C3×#41—24no—
171C3×#41D3214yes—
172C3×#41C2×C3⋊D311yescharacteristic
173#239—94no—
174#239—91no—
175#239—184no—
176D3×Q8—94no—
177D3×Q8—91no—
178D3×Q8—184no—
179#237—912no—
180#237—93no—
181#237—1812no—
182D3×D4—912no—
183D3×D4—93no—
184D3×D4—1812no—
185#235—912no—
186#235—93no—
187#235—1812no—
188C2×C4×D3—912no—
189C2×C4×D3—93no—
190C2×C4×D3—1812no—
191C3×M16—312no—
192C3×M16—93no—
193C2×C3×C8—312no—
194C2×C3×C8—93no—
195#208—312no—
196#208C2×C3⋊D313yescomplemented
197C2×C3⋊C8—312no—
198C2×C3⋊C8C2×C3⋊D313yescomplemented
199ΓL(1,𝔽25)—924no—
200ΓL(1,𝔽25)—96no—
201D3×C8—94no—
202D3×C8—912no—
203D3×C8—912no—
204D3×C8—912no—
205D3×C8—91no—
206D3×C8—93no—
207D3×C8—93no—
208D3×C8—93no—
209C22×C32—212no—
210C22×C32C2×C4×D3112yes—
211C22×C32C2×C4×D313yes—
212C2×C3⋊D3—31no—
213C2×C3⋊D3—33no—
214C2×C3⋊D3—34no—
215C2×C3⋊D3—312no—
216C2×C3⋊D3—64no—
217C2×C3⋊D3—612no—
218C2×C3⋊D3—66no—
219C2×C3⋊D3—624no—
220C2×C3⋊D3—1224no—
221C32×C4—24no—
222C32×C4—212no—
223C32×C4C2×C4×D3112yes—
224C32×C4C2×C4×D313yes—
225C32×C4C23×D314yes—
226C32×C4C23×D311yescharacteristic
227C3⋊Dic3—33no—
228C3⋊Dic3—31no—
229C3⋊Dic3—312no—
230C3⋊Dic3—34no—
231C3⋊Dic3—612no—
232C3⋊Dic3—64no—
233C2×#41—271no—
234#131—31no—
235#131—91no—
236#131—273no—
237#131—273no—
238C2×M16—273no—
239C22×C8—273no—
240C33C2×#13111yescharacteristic, complemented, Sylow
241C22×D3—912no—
242C22×D3—93no—
243C22×D3—1812no—
244C3×Q8—24no—
245C3×Q8#62414yes—
246C3×Q8C4×C3⋊D311yescharacteristic
247C3×D4—212no—
248C3×D4#624112yes—
249C3×D4C4×C3⋊D313yes—
250C2×C3×C4—212no—
251C2×C3×C4C2×D32112yes—
252C2×C3×C4C22×C3⋊D313yes—
253#67—924no—
254#67—96no—
255#67—1824no—
256C2×Dic3—912no—
257C2×Dic3—93no—
258C2×Dic3—1812no—
259D12—912no—
260D12—93no—
261D12—1812no—
262C4×D3—94no—
263C4×D3—912no—
264C4×D3—91no—
265C4×D3—93no—
266C4×D3—912no—
267C4×D3—912no—
268C4×D3—93no—
269C4×D3—93no—
270C4×D3—184no—
271C4×D3—1812no—
272C4×D3—1812no—
273C4×D3—1812no—
274Dic6—912no—
275Dic6—93no—
276Dic6—1812no—
277C3×C8—34no—
278C3×C8—312no—
279C3×C8—91no—
280C3×C8—93no—
281C3⋊C8—34no—
282C3⋊C8—312no—
283C3⋊C8C22×C3⋊D311yescharacteristic
284C3⋊C8C22×C3⋊D313yes—
285C2×C32—23no—
286C2×C32—24no—
287C2×C32—212no—
288C2×C32—412no—
289C2×C32C22×C4×D314yes—
290C2×C32C22×C4×D311yescharacteristic
291C3⋊D3—32no—
292C3⋊D3—38no—
293C3⋊D3—68no—
294C3⋊D3—63no—
295C3⋊D3—612no—
296C3⋊D3—1212no—
297#41—271no—
298#41—273no—
299#41—273no—
300#41D3×C3⋊D311yescharacteristic
301C2×Q8—271no—
302C2×D4—273no—
303C22×C4—273no—
304M16—33no—
305M16—93no—
306M16—276no—
307C2×C8—33no—
308C2×C8—93no—
309C2×C8—271no—
310C2×C8—273no—
311C2×C8—273no—
312C2×C8—273no—
313C22×C3—212no—
314C22×C3C2×#624112yes—
315C22×C3C2×C4×C3⋊D313yes—
316C2×D3—94no—
317C2×D3—91no—
318C2×D3—912no—
319C2×D3—93no—
320C2×D3—184no—
321C2×D3—1812no—
322C2×D3—186no—
323C2×D3—1824no—
324C2×D3—3624no—
325C3×C4—24no—
326C3×C4—212no—
327C3×C4C2×#624112yes—
328C3×C4C2×C4×C3⋊D313yes—
329C3×C4C22×D3214yes—
330C3×C4C23×C3⋊D311yescharacteristic
331Dic3—912no—
332Dic3—94no—
333Dic3—93no—
334Dic3—91no—
335Dic3—1812no—
336Dic3—184no—
337C32—24no—
338C32D3×#13114yescomplemented
339C32D3×#13111yescharacteristic, complemented
340C23—273no—
341Q8—273no—
342Q8#628911yescharacteristic
343D4—276no—
344D4—273no—
345D4#628913yes—
346C2×C4—273no—
347C2×C4—271no—
348C2×C4—273no—
349C2×C4—273no—
350C2×C4—273no—
351C2×C4C2×D3×C3⋊D313yes—
352C8—31no—
353C8—33no—
354C8—91no—
355C8—93no—
356C2×C3—212no—
357C2×C3—23no—
358C2×C3—24no—
359C2×C3—412no—
360C2×C3C22×#62414yes—
361C2×C3C22×C4×C3⋊D311yescharacteristic
362D3—98no—
363D3—92no—
364D3—188no—
365D3—1812no—
366D3—183no—
367D3—3612no—
368C22—271no—
369C22—273no—
370C22—546no—
371C22C2×#628913yes—
372C4—271no—
373C4—273no—
374C4C2×#628913yes—
375C4C22×D3×C3⋊D311yescharacteristic
376C3—24no—
377C3#1059544214yescomplemented
378C3C3⋊D3×#13111yescharacteristic, complemented
379C2—23no—
380C2—272no—
381C2—543no—
382C2C22×#628911yescharacteristic
subgroupG/H
center, FrattiniC4C22×D3×C3⋊D3
commutator, socleC2×C33C23×C4
FittingC33×#41C22
prime radical#41×C3⋊(C3⋊D3)C2
cosocleC33×C4C24
hypercenter#41D3×C3⋊D3
hypocenterC33C2×#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C3×#64157C2semidirect
3. C32×#948C2semidirect
4. #11723358C2semidirect
5. #11723357C2semidirect
6. #11723356C2semidirect
7. C2×#88785C2semidirect
17. C3×#3748C22semidirect
18. C2×C3×C3⋊(C3⋊C8)C22semidirect
19. C32×#208C22semidirect
20. C2×C32×C3⋊C8C22semidirect
21. #88788C22semidirect
24. #88785C22semidirect
25. #88785C22semidirect
30. #64157D3semidirect
36. C3×#948D3semidirect
82. #3748C2×D3semidirect
83. C2×C3⋊(C3⋊C8)C2×D3semidirect
90. C3×#208C2×D3semidirect
92. C2×C3×C3⋊C8C2×D3semidirect
112. #948C3⋊D3semidirect
169. C3⋊(C3⋊D3)#131semidirect
196. #208C2×C3⋊D3semidirect
198. C2×C3⋊C8C2×C3⋊D3semidirect
240. C33C2×#131semidirect
338. C32D3×#131semidirect
339. C32D3×#131semidirect
377. C3#10595442semidirect
378. C3C3⋊D3×#131semidirect

Maximal subgroups

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

subgroupindex
#41×C3⋊(C3⋊D3)2
C3×#641572
C32×#9482
#117233582
#117233572
#117233562
C2×#887852
C3⋊D3×#1313
#105954423