Finite Group Databasebeta

← #10596237⋊C3 #13153466

Subgroups — #10596237⋊C3 #13153466

Order 1728.

928 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722627C212yesmaximal, complemented
2C2×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#64151⋊C3C212yesmaximal, complemented
4#64128⋊C3C212yesmaximal, complemented
5#64127⋊C3C214yesmaximal, complemented
6Q16×(C3⋊D3)⋊C3C212yesmaximal, complemented
7#64113⋊C3C211yesmaximal, characteristic, complemented
8#64110⋊C3C211yesmaximal, characteristic, complemented
9#10596237C311yesmaximal, characteristic, complemented
10C3×#5766—31nomaximal
11C2×Q8×H(3,𝔽3)C2212yes—
12#88722—22no—
13#88722C2214yes—
14Q8×(C3⋊D3)⋊C3—22no—
15Q8×(C3⋊D3)⋊C3C2214yes—
16#3828⋊C3—24no—
17#3828⋊C3C2211yescharacteristic
18#3828⋊C3C2212yes—
19Q16×H(3,𝔽3)C2212yes—
20Q16×H(3,𝔽3)C2212yes—
21C2×C8×H(3,𝔽3)C2211yescharacteristic
22#3755⋊C3C2214yes—
23#3756⋊C3C2214yes—
24#3748⋊C3C2211yescharacteristic
25#3745⋊C3C2211yescharacteristic
26#3744⋊C3C2212yes—
27#3743⋊C3C2212yes—
28C8×(C3⋊D3)⋊C3C2212yes—
29#3746⋊C3C2211yescharacteristic
30#64364C2×C312yescomplemented
31C3×#1007—32no—
32C2×C32×Q16—21no—
33C2×C32×Q16D311yescharacteristic, complemented
34C2×C32×Q16C2×C311yescharacteristic, complemented
35#64151C2×C312yescomplemented
36#64128C2×C312yescomplemented
37#64127C2×C314yescomplemented
38Q16×C3⋊D3C2×C312yescomplemented
39#64113C2×C311yescharacteristic, complemented
40#64110C2×C311yescharacteristic, complemented
41C3×#942—32no—
42C3×#919—32no—
43C3×#918—34no—
44C3×D3×Q16—32no—
45C3×#904—31no—
46C3×#901—31no—
47Q8×H(3,𝔽3)—22no—
48Q8×H(3,𝔽3)C2314yes—
49C2×C4×H(3,𝔽3)—22no—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51#638⋊C3—22no—
52#638⋊C3D412yes—
53C4⋊((C3⋊D3)⋊C3)—24no—
54C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
55C4⋊((C3⋊D3)⋊C3)C2312yes—
56C4×(C3⋊D3)⋊C3—24no—
57C4×(C3⋊D3)⋊C3—22no—
58C4×(C3⋊D3)⋊C3—24no—
59C4×(C3⋊D3)⋊C3C2312yes—
60#634⋊C3—24no—
61#634⋊C3D411yescharacteristic
62#634⋊C3C2312yes—
63C8×H(3,𝔽3)C2311yescharacteristic
64C8×H(3,𝔽3)C2311yescharacteristic
65(C3⋊(C3⋊C8))⋊C3C2312yes—
66C3×#578—91no—
67#5766—31no—
68#5766—91no—
69C2×C32×Q8—22no—
70C2×C32×Q8C2×D312yes—
71C2×C32×Q8C22×C312yes—
72#3832—22no—
73#3832C22×C314yes—
74Q8×C3⋊D3—22no—
75Q8×C3⋊D3C22×C314yes—
76#3828—24no—
77#3828C22×C311yescharacteristic
78#3828C22×C312yes—
79C3×#239—34no—
80C3×#239—62no—
81C3×D3×Q8—34no—
82C3×D3×Q8—62no—
83C3×#235—31no—
84C3×#235—32no—
85C3×#235—64no—
86C32×Q16—22no—
87C32×Q16—22no—
88C32×Q16C2×D312yes—
89C32×Q16C2×D312yes—
90C32×Q16C22×C312yes—
91C32×Q16C22×C312yes—
92C2×C32×C8—21no—
93C2×C32×C8C2×D311yescharacteristic
94C2×C32×C8C22×C311yescharacteristic
95#3756C22×C314yes—
96#3755C22×C314yes—
97#3748C22×C311yescharacteristic
98#3746C22×C311yescharacteristic
99#3745C22×C311yescharacteristic
100#3744C22×C312yes—
101#3743C22×C312yes—
102C8×C3⋊D3C22×C312yes—
103C3×#216—34no—
104C3×#215—34no—
105C3×#208—31no—
106C3×Dic12—31no—
107C3×D24—31no—
108C3×#204—32no—
109C3×ΓL(1,𝔽25)—32no—
110C3×D3×C8—32no—
111C22×H(3,𝔽3)C2×D411yescharacteristic
112C2×(C3⋊D3)⋊C3—22no—
113C2×(C3⋊D3)⋊C3C2×D412yes—
114C4×H(3,𝔽3)—24no—
115C4×H(3,𝔽3)C2×D411yescharacteristic
116C4×H(3,𝔽3)C2411yescharacteristic
117(C3⋊Dic3)⋊C3—22no—
118(C3⋊Dic3)⋊C3C2×D412yes—
119C3×2−1+4—92no—
120#1007—32no—
121#1007—92no—
122C3×#137—94no—
123C3×#137—92no—
124C3×#135—91no—
125C3×#135—92no—
126C2×C3×Q16—31no—
127C2×C3×Q16—31no—
128C2×C3×Q16—61no—
129C2×C3×Q16—92no—
130C2×C3×Q16C3×D311yescharacteristic
131C3×#131—91no—
132#942—32no—
133#942—92no—
134#919—32no—
135#919—92no—
136#918—34no—
137#918—94no—
138D3×Q16—32no—
139D3×Q16—92no—
140#904—31no—
141#904—91no—
142#901—31no—
143#901—91no—
144C32×Q8—22no—
145C32×Q8—22no—
146C32×Q8—24no—
147C32×Q8—42no—
148C32×Q8C22×D314yes—
149C32×Q8C23×C314yes—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—42no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C23×C311yescharacteristic
156#638—22no—
157#638C3×D412yes—
158C4⋊(C3⋊D3)—24no—
159C4⋊(C3⋊D3)C3×D411yescharacteristic
160C4⋊(C3⋊D3)C23×C312yes—
161C4×C3⋊D3—24no—
162C4×C3⋊D3—22no—
163C4×C3⋊D3—24no—
164C4×C3⋊D3C23×C312yes—
165#634—24no—
166#634C3×D411yescharacteristic
167#634C23×C312yes—
168C3×#67—32no—
169C3×#67—62no—
170C3×D12—31no—
171C3×D12—32no—
172C3×D12—64no—
173C3×C4×D3—32no—
174C3×C4×D3—64no—
175C3×C4×D3—64no—
176C3×C4×D3—62no—
177C3×Dic6—31no—
178C3×Dic6—32no—
179C3×Dic6—64no—
180C32×C8—21no—
181C32×C8—21no—
182C32×C8C22×D311yescharacteristic
183C32×C8C22×D311yescharacteristic
184C32×C8C23×C311yescharacteristic
185C32×C8C23×C311yescharacteristic
186C3⋊(C3⋊C8)C23×C312yes—
187C3×C3⋊C8—32no—
188#578—91noSylow
189C2×H(3,𝔽3)—21no—
190C2×H(3,𝔽3)C22×D411yescharacteristic
191(C3⋊D3)⋊C3—22no—
192(C3⋊D3)⋊C3—42no—
193C3×#41—91no—
194C3×#41—92no—
195C3×#41—94no—
196C3×#41—184no—
197C3×#41—182no—
198C2×C3×Q8—32no—
199C2×C3×Q8—32no—
200C2×C3×Q8—62no—
201C2×C3×Q8—94no—
202C2×C3×Q8—182no—
203C2×C3×Q8C2×C3×D312yes—
204#239—34no—
205#239—62no—
206#239—94no—
207#239—182no—
208D3×Q8—34no—
209D3×Q8—62no—
210D3×Q8—94no—
211D3×Q8—182no—
212#235—31no—
213#235—32no—
214#235—64no—
215#235—91no—
216#235—92no—
217#235—184no—
218C3×Q16—32no—
219C3×Q16—32no—
220C3×Q16—32no—
221C3×Q16—32no—
222C3×Q16—62no—
223C3×Q16—62no—
224C3×Q16—91no—
225C3×Q16—94no—
226C3×Q16C2×C3×D312yes—
227C3×Q16C2×C3×D312yes—
228C3×SD16—92no—
229C3×SD16—94no—
230C3×D8—91no—
231C3×M16—92no—
232C3×M16—91no—
233C2×C3×C8—31no—
234C2×C3×C8—31no—
235C2×C3×C8—61no—
236C2×C3×C8—92no—
237C2×C3×C8C2×C3×D311yescharacteristic
238#216—34no—
239#216—94no—
240#215—34no—
241#215—94no—
242#208—31no—
243#208—91no—
244Dic12—31no—
245Dic12—91no—
246D24—31no—
247D24—91no—
248#204—32no—
249#204—92no—
250ΓL(1,𝔽25)—32no—
251ΓL(1,𝔽25)—92no—
252D3×C8—32no—
253D3×C8—92no—
254C22×C32—21no—
255C22×C32D3×D411yescharacteristic
256C22×C32C2×C3×D411yescharacteristic
257C2×C3⋊D3—22no—
258C2×C3⋊D3C2×C3×D412yes—
259C2×C3×D3—32no—
260C2×C3×D3—62no—
261C32×C4—24no—
262C32×C4—24no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—44no—
266C32×C4D3×D411yescharacteristic
267C32×C4C2×C3×D411yescharacteristic
268C32×C4C23×D311yescharacteristic
269C32×C4C24×C311yescharacteristic
270C3⋊Dic3—22no—
271C3⋊Dic3C2×C3×D412yes—
272C3×Dic3—32no—
273C3×Dic3—62no—
2742−1+4—92no—
275#137—92no—
276#137—94no—
277#135—92no—
278#135—91no—
279C2×Q16—92no—
280C2×Q16(C3⋊D3)⋊C311yescharacteristic, complemented
281#131—91no—
282H(3,𝔽3)#57811yescharacteristic, complemented, Sylow
283C3×Q8—22no—
284C3×Q8—34no—
285C3×Q8—34no—
286C3×Q8—62no—
287C3×Q8—62no—
288C3×Q8—64no—
289C3×Q8—91no—
290C3×Q8—92no—
291C3×Q8—122no—
292C3×Q8—184no—
293C3×Q8C22×C3×D314yes—
294C3×D4—91no—
295C3×D4—92no—
296C3×D4—92no—
297C3×D4—184no—
298C3×D4—182no—
299C2×C3×C4—22no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—61no—
305C2×C3×C4—92no—
306C2×C3×C4—122no—
307C2×C3×C4—182no—
308C2×C3×C4—184no—
309C2×C3×C4—184no—
310C2×C3×C4C22×C3×D311yescharacteristic
311#67—32no—
312#67—62no—
313#67—92no—
314#67—182no—
315D12—32no—
316D12—31no—
317D12—64no—
318D12—91no—
319D12—92no—
320D12—184no—
321C4×D3—32no—
322C4×D3—64no—
323C4×D3—64no—
324C4×D3—62no—
325C4×D3—92no—
326C4×D3—184no—
327C4×D3—182no—
328C4×D3—184no—
329Dic6—31no—
330Dic6—32no—
331Dic6—64no—
332Dic6—91no—
333Dic6—92no—
334Dic6—184no—
335C3×C8—31no—
336C3×C8—31no—
337C3×C8—31no—
338C3×C8—31no—
339C3×C8—61no—
340C3×C8—61no—
341C3×C8—92no—
342C3×C8C22×C3×D311yescharacteristic
343C3×C8C22×C3×D311yescharacteristic
344C3⋊C8—32no—
345C3⋊C8—92no—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32C2×D3×D411yescharacteristic
351C2×C32C22×C3×D411yescharacteristic
352C3⋊D3—22no—
353C3⋊D3—42no—
354C3×D3—62no—
355C3×D3—122no—
356#41—91no—
357#41—94no—
358#41—92no—
359#41—184no—
360#41—182no—
361C2×Q8—94no—
362C2×Q8—182no—
363C2×Q8C2×(C3⋊D3)⋊C312yes—
364Q16—94no—
365Q16—91no—
366Q16C2×(C3⋊D3)⋊C312yes—
367Q16C2×(C3⋊D3)⋊C312yes—
368SD16—94no—
369SD16—92no—
370D8—91no—
371M16—91no—
372M16—92no—
373C2×C8—92no—
374C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—61no—
378C22×C3—92no—
379C22×C3—182no—
380C22×C3C3×D3×D411yescharacteristic
381C2×D3—32no—
382C2×D3—62no—
383C2×D3—92no—
384C2×D3—182no—
385C3×C4—24no—
386C3×C4—31no—
387C3×C4—31no—
388C3×C4—31no—
389C3×C4—31no—
390C3×C4—64no—
391C3×C4—64no—
392C3×C4—61no—
393C3×C4—61no—
394C3×C4—92no—
395C3×C4—124no—
396C3×C4—182no—
397C3×C4C3×D3×D411yescharacteristic
398C3×C4C23×C3×D311yescharacteristic
399Dic3—32no—
400Dic3—62no—
401Dic3—92no—
402Dic3—182no—
403C32—21no—
404C32#576611yescharacteristic, complemented
405C32C3×#57811yescharacteristic, complemented
406Q8—22no—
407Q8—91no—
408Q8—92no—
409Q8—184no—
410Q8C22×(C3⋊D3)⋊C314yes—
411D4—92no—
412D4—92no—
413D4—91no—
414D4—184no—
415D4—182no—
416C2×C4—22no—
417C2×C4—92no—
418C2×C4—184no—
419C2×C4—184no—
420C2×C4—182no—
421C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
422C8—92no—
423C8C22×(C3⋊D3)⋊C311yescharacteristic
424C8C22×(C3⋊D3)⋊C311yescharacteristic
425C2×C3—21no—
426C2×C3—31no—
427C2×C3—31no—
428C2×C3—61no—
429C2×C3—61no—
430C2×C3—62no—
431C2×C3—61no—
432C2×C3—182no—
433C2×C3—362no—
434C2×C3C2×C3×D3×D411yescharacteristic
435D3—62no—
436D3—122no—
437D3—182no—
438D3—362no—
439C22—92no—
440C22—182no—
441C22D4×(C3⋊D3)⋊C311yescharacteristic
442C4—24no—
443C4—92no—
444C4—182no—
445C4D4×(C3⋊D3)⋊C311yescharacteristic
446C4C23×(C3⋊D3)⋊C311yescharacteristic
447C3—31no—
448C3—31no—
449C3—61no—
450C3C3×#576611yescharacteristic
451C2—21no—
452C2—182no—
453C2—362no—
454C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2C2×D4×(C3⋊D3)⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingC2×Q16×H(3,𝔽3)C2
socleC2×C3C2×C3×D3×D4
hypercenterC2×Q16(C3⋊D3)⋊C3
hypocenterC32C3×#578
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. #11722627C2semidirect
2. C2×Q16×H(3,𝔽3)C2semidirect
3. #64151⋊C3C2semidirect
4. #64128⋊C3C2semidirect
5. #64127⋊C3C2semidirect
6. Q16×(C3⋊D3)⋊C3C2semidirect
7. #64113⋊C3C2semidirect
8. #64110⋊C3C2semidirect
9. #10596237C3semidirect
30. #64364C2×C3semidirect
33. C2×C32×Q16D3semidirect
34. C2×C32×Q16C2×C3semidirect
35. #64151C2×C3semidirect
36. #64128C2×C3semidirect
37. #64127C2×C3semidirect
38. Q16×C3⋊D3C2×C3semidirect
39. #64113C2×C3semidirect
40. #64110C2×C3semidirect
280. C2×Q16(C3⋊D3)⋊C3semidirect
282. H(3,𝔽3)#578semidirect
404. C32#5766semidirect
405. C32C3×#578semidirect

Maximal subgroups

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

subgroupindex
#117226272
C2×Q16×H(3,𝔽3)2
#64151⋊C32
#64128⋊C32
#64127⋊C32
Q16×(C3⋊D3)⋊C32
#64113⋊C32
#64110⋊C32
#105962373
C3×#57663