Finite Group Databasebeta

← C2×#11721061 #13150595

Subgroups — C2×#11721061 #13150595

Order 1728.

1111 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×#88538C211yesmaximal, characteristic, complemented
4#11721061C218yesmaximal, direct product factor
5C2×#88432C212yesmaximal, complemented
6C2×#88431C212yesmaximal, complemented
7C2×#63830—31nomaximal
8C2×#63825—31nomaximal
9C23×(C3⋊D3)⋊C3—21no—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12#3828⋊C3—22no—
13#3828⋊C3C2214yescomplemented
14C2×C4⋊((C3⋊D3)⋊C3)—22no—
15C2×C4⋊((C3⋊D3)⋊C3)—21no—
16C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
17C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
19C2×C4×(C3⋊D3)⋊C3—21no—
20C2×#634⋊C3C2211yescharacteristic, complemented
21#88538C2214yescomplemented
22C2×#6180C2212yescomplemented
23#88432C2218yescomplemented
24#88431C2218yescomplemented
25C2×#3828D311yescharacteristic, complemented
26C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
27C2×C3×#235—31no—
28C22×C3×D12—31no—
29C2×C3×#208—31no—
30C2×C3×#208—31no—
31C2×C3×#208—61no—
32C2×#3717—32no—
33#63830—38no—
34C2×#3716—32no—
35#63825—38no—
36C2×#3714—32no—
37C2×#3714—32no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×(C3⋊D3)⋊C3—22no—
40C22×(C3⋊D3)⋊C3—24no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3—44no—
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)D412yes—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)C2312yes—
47#638⋊C3—24no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C4⋊((C3⋊D3)⋊C3)—22no—
50C4⋊((C3⋊D3)⋊C3)—24no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)C2314yes—
53C4⋊((C3⋊D3)⋊C3)C2312yescomplemented
54C4×(C3⋊D3)⋊C3—24no—
55#634⋊C3—21no—
56#634⋊C3C2312yescomplemented
57#6180C2314yescomplemented
58C2×#949—91no—
59C2×#932—91no—
60C2×#908—91no—
61C23×C3⋊D3—21no—
62C23×C3×D3—61no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C2×#638—21no—
67#3828—22no—
68#3828C2×D314yescomplemented
69C2×C4⋊(C3⋊D3)—22no—
70C2×C4⋊(C3⋊D3)—21no—
71C2×C4⋊(C3⋊D3)C2×D312yes—
72C2×C4⋊(C3⋊D3)C2×D314yescomplemented
73C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
74C2×C4×C3⋊D3—21no—
75C2×#634C2×D311yescharacteristic, complemented
76C2×C3×#67—61no—
77C3×#235—34no—
78C3×#235—62no—
79C2×C3×D12—31no—
80C2×C3×D12—32no—
81C2×C3×D12—34no—
82C2×C3×D12—61no—
83C2×C3×D12—62no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—31no—
86C3×#208—34no—
87C3×#208—34no—
88C3×#208—64no—
89C2×C3×C3⋊C8—32no—
90C2×C3×C3⋊C8—32no—
91C2×C3×C3⋊C8—62no—
92#3717—38no—
93#3716—38no—
94#3714—38no—
95#3714—38no—
96C22×H(3,𝔽3)—22no—
97C22×H(3,𝔽3)C2×D412yes—
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C2×(C3⋊D3)⋊C3—22no—
100C2×(C3⋊D3)⋊C3—24no—
101C2×(C3⋊D3)⋊C3—48no—
102C2×(C3⋊D3)⋊C3—42no—
103C2×(C3⋊D3)⋊C3—44no—
104C4×H(3,𝔽3)C2×D411yescharacteristic
105C4×H(3,𝔽3)C2×D412yes—
106C4×H(3,𝔽3)C2411yescharacteristic
107(C3⋊Dic3)⋊C3—22no—
108C2×C3×#41—91no—
109C22×C3×D4—91no—
110C2×#235—31no—
111C2×#235—91no—
112C22×D12—31no—
113C22×D12—91no—
114C2×C3×M16—91no—
115#949—98no—
116C2×#215—92no—
117C2×#214—92no—
118#932—98no—
119C2×#213—92no—
120C2×#213—92no—
121C2×#208—31no—
122C2×#208—31no—
123C2×#208—61no—
124#908—98no—
125C2×D24—92no—
126C2×#204—92no—
127C23×C32—21no—
128C23×C32#6711yescharacteristic
129C23×C32#6711yescharacteristic
130C22×C3⋊D3—22no—
131C22×C3⋊D3—24no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—44no—
134C22×C3×D3—64no—
135C22×C3×D3—62no—
136C22×C3×D3—61no—
137C22×C3×D3—124no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4#6711yescharacteristic
143C2×C32×C4#6711yescharacteristic
144C2×C32×C4#6712yes—
145C2×C32×C4#6712yes—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D312yes—
149C2×C32×C4C22×D312yes—
150#638—24no—
151C2×C3⋊Dic3—21no—
152C4⋊(C3⋊D3)—22no—
153C4⋊(C3⋊D3)—24no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)C22×D314yes—
156C4⋊(C3⋊D3)C22×D312yescomplemented
157C4×C3⋊D3—24no—
158#634—21no—
159#634C22×D312yescomplemented
160C3×#67—64no—
161C2×C3×Dic3—61no—
162C3×D12—34no—
163C3×D12—32no—
164C3×D12—62no—
165C3×D12—64no—
166C3×D12—61no—
167C3×C4×D3—64no—
168C3×Dic6—32no—
169C3×Dic6—61no—
170C3×C3⋊C8—34no—
171C3×C3⋊C8—34no—
172C3×C3⋊C8—64no—
173C2×AGL(1,ℤ/8)—271noSylow
174C2×H(3,𝔽3)—22no—
175C2×H(3,𝔽3)AGL(1,ℤ/8)12yescomplemented
176C2×H(3,𝔽3)C22×D411yescharacteristic
177(C3⋊D3)⋊C3—42no—
178(C3⋊D3)⋊C3—44no—
179C24×C3—181no—
180C23×D3—61no—
181C23×D3—181no—
182C3×#41—94no—
183C3×#41—182no—
184C2×C3×Q8—91no—
185C2×C3×D4—91no—
186C2×C3×D4—92no—
187C2×C3×D4—94no—
188C2×C3×D4—181no—
189C2×C3×D4—182no—
190C2×C3×D4—181no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—61no—
194C22×C3×C4—181no—
195C22×C3×C4D3211yescharacteristic
196C2×#67—61no—
197C2×#67—181no—
198#235—34no—
199#235—62no—
200#235—94no—
201#235—182no—
202C2×D12—32no—
203C2×D12—31no—
204C2×D12—34no—
205C2×D12—61no—
206C2×D12—62no—
207C2×D12—91no—
208C2×D12—92no—
209C2×D12—94no—
210C2×D12—181no—
211C2×D12—182no—
212C2×C4×D3—61no—
213C2×C4×D3—181no—
214C2×Dic6—31no—
215C2×Dic6—91no—
216C3×M16—94no—
217C2×C3×C8—92no—
218#215—98no—
219#214—98no—
220#213—98no—
221#213—98no—
222#208—34no—
223#208—34no—
224#208—64no—
225C2×C3⋊C8—32no—
226C2×C3⋊C8—32no—
227C2×C3⋊C8—62no—
228D24—98no—
229#204—98no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—21no—
233C22×C32—22no—
234C22×C32—42no—
235C22×C32C2×#6712yes—
236C22×C32C2×#6712yes—
237C22×C32C2×#6711yescharacteristic
238C22×C32C2×#6711yescharacteristic
239C2×C3⋊D3—22no—
240C2×C3⋊D3—24no—
241C2×C3⋊D3—48no—
242C2×C3⋊D3—44no—
243C2×C3⋊D3—42no—
244C2×C3×D3—62no—
245C2×C3×D3—64no—
246C2×C3×D3—128no—
247C2×C3×D3—122no—
248C2×C3×D3—124no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—22no—
252C32×C4C2×#6711yescharacteristic
253C32×C4C2×#6712yes—
254C32×C4C2×#6711yescharacteristic
255C32×C4C2×#6712yes—
256C32×C4C23×D311yescharacteristic
257C32×C4C23×D311yescharacteristic
258C3⋊Dic3—22no—
259C3×Dic3—62no—
260C2×#41—91no—
261C22×D4—91no—
262AGL(1,ℤ/8)—278no—
263C2×SD16—272no—
264C2×D8—272no—
265C2×M16—91no—
266H(3,𝔽3)C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
267C23×C3—31no—
268C23×C3—31no—
269C23×C3—61no—
270C23×C3—181no—
271C23×C3—184no—
272C23×C3—182no—
273C23×C3—364no—
274C23×C3#62511yescharacteristic
275C22×D3—61no—
276C22×D3—64no—
277C22×D3—62no—
278C22×D3—124no—
279C22×D3—181no—
280C22×D3—184no—
281C22×D3—182no—
282C22×D3—364no—
283C3×Q8—92no—
284C3×Q8—181no—
285C3×D4—92no—
286C3×D4—94no—
287C3×D4—182no—
288C3×D4—184no—
289C3×D4—181no—
290C3×D4—184no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—32no—
294C2×C3×C4—32no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—62no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—62no—
303C2×C3×C4—184no—
304C2×C3×C4—181no—
305C2×C3×C4#62511yescharacteristic
306C2×C3×C4#62512yes—
307C2×C3×C4C2×D3211yescharacteristic
308C2×C3×C4C2×D3212yes—
309#67—64no—
310#67—184no—
311C2×Dic3—61no—
312C2×Dic3—181no—
313D12—34no—
314D12—32no—
315D12—62no—
316D12—64no—
317D12—61no—
318D12—94no—
319D12—92no—
320D12—184no—
321D12—182no—
322D12—181no—
323C4×D3—64no—
324C4×D3—184no—
325Dic6—32no—
326Dic6—61no—
327Dic6—92no—
328Dic6—181no—
329C3×C8—94no—
330C3⋊C8—34no—
331C3⋊C8—34no—
332C3⋊C8—64no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—21no—
336C2×C32—22no—
337C2×C32—42no—
338C2×C32#93212yescomplemented
339C2×C32#94912yescomplemented
340C2×C32C22×#6711yescharacteristic
341C2×C32C22×#6711yescharacteristic
342C3⋊D3—44no—
343C3⋊D3—42no—
344C3×D3—122no—
345C3×D3—124no—
346C24—181no—
347#41—94no—
348#41—182no—
349C2×Q8—91no—
350C2×D4—91no—
351C2×D4—92no—
352C2×D4—94no—
353C2×D4—181no—
354C2×D4—182no—
355C2×D4—181no—
356C22×C4—181no—
357C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
358SD16—278no—
359D8—278no—
360M16—94no—
361C2×C8—92no—
362C22×C3—22no—
363C22×C3—32no—
364C22×C3—32no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—62no—
368C22×C3—62no—
369C22×C3—62no—
370C22×C3—61no—
371C22×C3—122no—
372C22×C3—182no—
373C22×C3—184no—
374C22×C3—364no—
375C22×C3—362no—
376C22×C3—368no—
377C22×C3C2×#62512yes—
378C22×C3C2×#62511yescharacteristic
379C2×D3—62no—
380C2×D3—64no—
381C2×D3—128no—
382C2×D3—124no—
383C2×D3—122no—
384C2×D3—182no—
385C2×D3—184no—
386C2×D3—364no—
387C2×D3—362no—
388C2×D3—368no—
389C3×C4—31no—
390C3×C4—31no—
391C3×C4—31no—
392C3×C4—32no—
393C3×C4—32no—
394C3×C4—31no—
395C3×C4—61no—
396C3×C4—62no—
397C3×C4—61no—
398C3×C4—182no—
399C3×C4C2×#62511yescharacteristic
400C3×C4C2×#62512yes—
401C3×C4C22×D3211yescharacteristic
402Dic3—62no—
403Dic3—182no—
404C32—21no—
405C32C2×#93211yescharacteristic, complemented
406C32C2×#94911yescharacteristic, complemented
407C23—184no—
408C23—181no—
409C23—182no—
410C23—364no—
411C23#619811yescharacteristic
412Q8—92no—
413Q8—181no—
414D4—92no—
415D4—94no—
416D4—184no—
417D4—184no—
418D4—182no—
419D4—181no—
420C2×C4—181no—
421C2×C4—184no—
422C2×C4#619812yes—
423C2×C4#619811yescharacteristic
424C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
425C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
426C8—94no—
427C2×C3—22no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—32no—
431C2×C3—32no—
432C2×C3—62no—
433C2×C3—62no—
434C2×C3—61no—
435C2×C3—62no—
436C2×C3—122no—
437C2×C3—364no—
438C2×C3—362no—
439C2×C3#6382312yes—
440C2×C3C22×#62511yescharacteristic
441D3—124no—
442D3—122no—
443D3—364no—
444D3—362no—
445C22—22no—
446C22—182no—
447C22—184no—
448C22—368no—
449C22—364no—
450C22—362no—
451C22C2×#619812yes—
452C22C2×#619811yescharacteristic
453C4—182no—
454C4C2×#619812yes—
455C4C2×#619811yescharacteristic
456C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
457C3—31no—
458C3—31no—
459C3—61no—
460C3C2×#6382311yescharacteristic
461C2—22no—
462C2—364no—
463C2—362no—
464C2#1172106112yesdirect product factor
465C2C22×#619811yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×AGL(1,ℤ/8)
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3828⋊C3C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C2×#88538C2semidirect
4. #11721061C2direct (N × H)
5. C2×#88432C2semidirect
6. C2×#88431C2semidirect
13. #3828⋊C3C22semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
18. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
20. C2×#634⋊C3C22semidirect
21. #88538C22semidirect
22. C2×#6180C22semidirect
23. #88432C22semidirect
24. #88431C22semidirect
25. C2×#3828D3semidirect
26. C22×C4⋊(C3⋊D3)D3semidirect
53. C4⋊((C3⋊D3)⋊C3)C23semidirect
56. #634⋊C3C23semidirect
57. #6180C23semidirect
68. #3828C2×D3semidirect
72. C2×C4⋊(C3⋊D3)C2×D3semidirect
73. C2×C4⋊(C3⋊D3)C2×D3semidirect
75. C2×#634C2×D3semidirect
156. C4⋊(C3⋊D3)C22×D3semidirect
159. #634C22×D3semidirect
175. C2×H(3,𝔽3)AGL(1,ℤ/8)semidirect
266. H(3,𝔽3)C2×AGL(1,ℤ/8)semidirect
338. C2×C32#932semidirect
339. C2×C32#949semidirect
405. C32C2×#932semidirect
406. C32C2×#949semidirect
464. C2#11721061direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C22×C4⋊((C3⋊D3)⋊C3)2
C2×#885382
#117210612
C2×#884322
C2×#884312
C2×#638303
C2×#638253