Finite Group Databasebeta

← C2×#11721073 #13150645

Subgroups — C2×#11721073 #13150645

Order 1728.

1095 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88740C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4C2×#88439C211yesmaximal, characteristic, complemented
5#11721073C218yesmaximal, direct product factor
6C2×#88438C211yesmaximal, characteristic, complemented
7C2×#88437C211yesmaximal, characteristic, complemented
8C2×#88436C211yesmaximal, characteristic, complemented
9C2×#63829—31nomaximal
10C2×#63822—31nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235—21no—
13#88740—22no—
14#88740C2214yescomplemented
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272—21no—
17C2×#6229C2211yescharacteristic
18C2×#638⋊C3—21no—
19#3828⋊C3—22no—
20#3828⋊C3C2214yescomplemented
21C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
22C2×C4×(C3⋊D3)⋊C3—21no—
23C2×#634⋊C3C2211yescharacteristic
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
25C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
26C2×(C3⋊(C3⋊C8))⋊C3C2214yescomplemented
27#88439C2214yescomplemented
28#88438C2214yescomplemented
29#88437C2214yescomplemented
30#88436C2214yescomplemented
31C2×#3828D311yescharacteristic, complemented
32C2×C3×#235—31no—
33C2×C3×#235—31no—
34C2×C3×#235—31no—
35C2×C3×#235—61no—
36C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
37C22×C3×C3⋊C8—31no—
38C2×#3719—31no—
39C2×#3718—31no—
40C2×#3717—31no—
41#63829—38no—
42C2×#3716—31no—
43C2×#3715—31no—
44C2×#3715—31no—
45C2×#3714—31no—
46#63822—38no—
47C2×#3713—31no—
48C23×H(3,𝔽3)D411yescharacteristic
49C22×#272—21no—
50C22×(C3⋊D3)⋊C3—21no—
51C2×C4×H(3,𝔽3)D411yescharacteristic
52C2×C4×H(3,𝔽3)D412yes—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2312yes—
55#6235—24no—
56C2×#1084—21no—
57C4⋊#272—21no—
58C4⋊#272C2312yes—
59C4×#272—24no—
60#6229—21no—
61#6229C2312yes—
62#638⋊C3—24no—
63C2×(C3⋊Dic3)⋊C3—21no—
64C4⋊((C3⋊D3)⋊C3)—21no—
65C4⋊((C3⋊D3)⋊C3)C2312yes—
66C4×(C3⋊D3)⋊C3—24no—
67#634⋊C3—21no—
68#634⋊C3C2312yes—
69(C3⋊(C3⋊C8))⋊C3C2312yes—
70(C3⋊(C3⋊C8))⋊C3C2312yes—
71C2×#950—91no—
72C2×#950—91no—
73C2×#904—91no—
74C22×C32×C4—21no—
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C2×#638—21no—
78#3828—22no—
79#3828C2×D314yescomplemented
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
81C2×C4×C3⋊D3—21no—
82C2×#634C2×D311yescharacteristic
83C2×C3×#67—62no—
84C2×C3×#67—61no—
85C2×C3×#67—61no—
86C2×C3×#67—61no—
87C3×#235—34no—
88C3×#235—34no—
89C3×#235—34no—
90C3×#235—64no—
91C3×#235—64no—
92C3×#235—62no—
93C3×#235—62no—
94C3×#235—62no—
95C2×C3×D12—31no—
96C2×C3×D12—31no—
97C2×C3×D12—31no—
98C2×C3×D12—61no—
99C2×C3×C4×D3—62no—
100C2×C3×C4×D3—61no—
101C2×C3×C4×D3—61no—
102C2×C3×C4×D3—61no—
103C2×C3×Dic6—31no—
104C2×C3×Dic6—31no—
105C2×C3×Dic6—31no—
106C2×C3×Dic6—61no—
107C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
108C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
109C2×C3⋊(C3⋊C8)C2×D314yescomplemented
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—34no—
113#3719—34no—
114#3718—34no—
115#3717—34no—
116#3716—34no—
117#3715—34no—
118#3715—34no—
119#3714—34no—
120#3713—34no—
121C22×H(3,𝔽3)—22no—
122C22×H(3,𝔽3)C2×D412yes—
123C22×H(3,𝔽3)C2×D411yescharacteristic
124C2×#272—22no—
125C2×#272—42no—
126C2×(C3⋊D3)⋊C3—22no—
127C2×(C3⋊D3)⋊C3—42no—
128C4×H(3,𝔽3)C2×D411yescharacteristic
129C4×H(3,𝔽3)C2×D412yes—
130C4×H(3,𝔽3)C2411yescharacteristic
131#1084—22no—
132(C3⋊Dic3)⋊C3—22no—
133C2×C3×#41—91no—
134C2×C3×#41—91no—
135C2×#235—31no—
136C2×#235—31no—
137C2×#235—31no—
138C2×#235—61no—
139C2×#235—91no—
140C22×C3×C8—91no—
141#950—98no—
142#950—98no—
143C2×#216—91no—
144C2×#216—91no—
145C2×#215—91no—
146C2×#215—91no—
147C2×#214—91no—
148C2×#214—91no—
149C2×#213—91no—
150C2×#213—91no—
151C22×C3⋊C8—31no—
152C22×C3⋊C8—91no—
153C2×Dic12—91no—
154#904—98no—
155C2×D24—91no—
156C2×#204—91no—
157C2×#204—91no—
158C23×C32—21no—
159C23×C32D1211yescharacteristic
160C23×C32#6711yescharacteristic
161C22×C3⋊D3—21no—
162C22×C3×D3—62no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4D1211yescharacteristic
171C2×C32×C4D1212yes—
172C2×C32×C4#6711yescharacteristic
173C2×C32×C4#6712yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D312yes—
178#638—24no—
179C2×C3⋊Dic3—21no—
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)C22×D312yes—
182C4×C3⋊D3—24no—
183#634—21no—
184#634C22×D312yes—
185C3×#67—68no—
186C3×#67—64no—
187C3×#67—64no—
188C3×#67—64no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C3×D12—32no—
194C3×D12—32no—
195C3×D12—32no—
196C3×D12—62no—
197C3×D12—62no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—61no—
201C3×C4×D3—68no—
202C3×C4×D3—64no—
203C3×C4×D3—64no—
204C3×C4×D3—64no—
205C3×Dic6—32no—
206C3×Dic6—32no—
207C3×Dic6—32no—
208C3×Dic6—62no—
209C3×Dic6—62no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3⋊(C3⋊C8)C22×D312yes—
214C3⋊(C3⋊C8)C22×D312yes—
215C3×C3⋊C8—32no—
216C3×C3⋊C8—32no—
217C2×#135—271noSylow
218C2×H(3,𝔽3)—22no—
219C2×H(3,𝔽3)#13512yescomplemented
220C2×H(3,𝔽3)C22×D411yescharacteristic
221#272—42no—
222(C3⋊D3)⋊C3—42no—
223C3×#41—94no—
224C3×#41—94no—
225C3×#41—182no—
226C3×#41—182no—
227C2×C3×Q8—91no—
228C2×C3×Q8—91no—
229C2×C3×D4—91no—
230C2×C3×D4—91no—
231C2×C3×D4—181no—
232C2×C3×D4—181no—
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—61no—
236C22×C3×C4—181no—
237C22×C3×C4—181no—
238C22×C3×C4D3211yescharacteristic
239C2×#67—61no—
240C2×#67—61no—
241C2×#67—62no—
242C2×#67—61no—
243C2×#67—181no—
244#235—34no—
245#235—34no—
246#235—34no—
247#235—64no—
248#235—62no—
249#235—62no—
250#235—64no—
251#235—62no—
252#235—94no—
253#235—182no—
254C2×D12—31no—
255C2×D12—31no—
256C2×D12—31no—
257C2×D12—61no—
258C2×D12—91no—
259C2×C4×D3—61no—
260C2×C4×D3—61no—
261C2×C4×D3—62no—
262C2×C4×D3—61no—
263C2×C4×D3—181no—
264C2×Dic6—31no—
265C2×Dic6—31no—
266C2×Dic6—31no—
267C2×Dic6—61no—
268C2×Dic6—91no—
269C2×C3×C8—91no—
270C2×C3×C8—91no—
271C2×C3×C8—94no—
272#216—94no—
273#216—94no—
274#215—94no—
275#215—94no—
276#214—94no—
277#214—94no—
278#213—94no—
279#213—94no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—34no—
283C2×C3⋊C8—91no—
284C2×C3⋊C8—91no—
285C2×C3⋊C8—94no—
286Dic12—94no—
287D24—94no—
288#204—94no—
289#204—94no—
290C22×C32—22no—
291C22×C32—22no—
292C22×C32—21no—
293C22×C32—22no—
294C22×C32—42no—
295C22×C32C2×D1212yes—
296C22×C32C2×D1211yescharacteristic
297C22×C32C2×#6712yes—
298C22×C32C2×#6711yescharacteristic
299C2×C3⋊D3—22no—
300C2×C3⋊D3—42no—
301C2×C3×D3—64no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C2×C3×D3—62no—
305C2×C3×D3—124no—
306C2×C3×D3—122no—
307C2×C3×D3—122no—
308C2×C3×D3—122no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—22no—
312C32×C4C2×D1211yescharacteristic
313C32×C4C2×D1212yes—
314C32×C4C2×#6711yescharacteristic
315C32×C4C2×#6712yes—
316C32×C4C23×D311yescharacteristic
317C32×C4C23×D311yescharacteristic
318C3⋊Dic3—22no—
319C3×Dic3—64no—
320C3×Dic3—62no—
321C3×Dic3—62no—
322C3×Dic3—62no—
323C2×#41—91no—
324C2×#41—91no—
325#135—278no—
326C2×Q16—271no—
327C2×SD16—271no—
328C2×SD16—271no—
329C2×D8—271no—
330C22×C8—91no—
331H(3,𝔽3)C2×#13511yescharacteristic, complemented, Sylow
332C23×C3—31no—
333C23×C3—31no—
334C23×C3—61no—
335C23×C3—181no—
336C23×C3—181no—
337C23×C3#62611yescharacteristic
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—62no—
341C22×D3—61no—
342C22×D3—181no—
343C3×Q8—92no—
344C3×Q8—92no—
345C3×Q8—181no—
346C3×Q8—181no—
347C3×D4—92no—
348C3×D4—92no—
349C3×D4—181no—
350C3×D4—184no—
351C3×D4—184no—
352C3×D4—181no—
353C2×C3×C4—32no—
354C2×C3×C4—32no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—32no—
359C2×C3×C4—31no—
360C2×C3×C4—32no—
361C2×C3×C4—62no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—62no—
365C2×C3×C4—184no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4—184no—
369C2×C3×C4#62612yes—
370C2×C3×C4#62611yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3212yes—
373#67—64no—
374#67—64no—
375#67—68no—
376#67—64no—
377#67—184no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—62no—
381C2×Dic3—61no—
382C2×Dic3—181no—
383D12—32no—
384D12—32no—
385D12—32no—
386D12—62no—
387D12—61no—
388D12—61no—
389D12—62no—
390D12—61no—
391D12—92no—
392D12—181no—
393C4×D3—64no—
394C4×D3—64no—
395C4×D3—68no—
396C4×D3—64no—
397C4×D3—184no—
398Dic6—32no—
399Dic6—32no—
400Dic6—32no—
401Dic6—62no—
402Dic6—61no—
403Dic6—61no—
404Dic6—62no—
405Dic6—61no—
406Dic6—92no—
407Dic6—181no—
408C3×C8—92no—
409C3×C8—92no—
410C3⋊C8—32no—
411C3⋊C8—32no—
412C3⋊C8—92no—
413C3⋊C8—92no—
414C2×C32—22no—
415C2×C32—22no—
416C2×C32—21no—
417C2×C32—22no—
418C2×C32—42no—
419C2×C32#90412yescomplemented
420C2×C32#95012yescomplemented
421C2×C32C22×D1211yescharacteristic
422C2×C32C22×#6711yescharacteristic
423C3⋊D3—42no—
424C3×D3—124no—
425C3×D3—122no—
426C3×D3—122no—
427C3×D3—122no—
428#41—94no—
429#41—94no—
430#41—182no—
431#41—182no—
432C2×Q8—91no—
433C2×Q8—91no—
434C2×D4—91no—
435C2×D4—91no—
436C2×D4—181no—
437C2×D4—181no—
438C22×C4—181no—
439C22×C4—181no—
440C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
441Q16—274no—
442SD16—274no—
443SD16—274no—
444D8—274no—
445C2×C8—91no—
446C2×C8—91no—
447C2×C8—94no—
448C22×C3—22no—
449C22×C3—31no—
450C22×C3—31no—
451C22×C3—32no—
452C22×C3—32no—
453C22×C3—62no—
454C22×C3—62no—
455C22×C3—61no—
456C22×C3—62no—
457C22×C3—122no—
458C22×C3—182no—
459C22×C3—182no—
460C22×C3—362no—
461C22×C3—362no—
462C22×C3C2×#62612yes—
463C22×C3C2×#62611yescharacteristic
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—64no—
467C2×D3—62no—
468C2×D3—122no—
469C2×D3—122no—
470C2×D3—124no—
471C2×D3—122no—
472C2×D3—182no—
473C2×D3—362no—
474C3×C4—31no—
475C3×C4—31no—
476C3×C4—31no—
477C3×C4—31no—
478C3×C4—32no—
479C3×C4—32no—
480C3×C4—61no—
481C3×C4—61no—
482C3×C4—62no—
483C3×C4—182no—
484C3×C4—182no—
485C3×C4C2×#62611yescharacteristic
486C3×C4C2×#62612yes—
487C3×C4C22×D3211yescharacteristic
488Dic3—62no—
489Dic3—62no—
490Dic3—64no—
491Dic3—62no—
492Dic3—182no—
493C32—21no—
494C32C2×#90411yescharacteristic, complemented
495C32C2×#95011yescharacteristic, complemented
496C23—181no—
497C23—181no—
498C23#620011yescharacteristic
499Q8—92no—
500Q8—92no—
501Q8—181no—
502Q8—181no—
503D4—92no—
504D4—92no—
505D4—184no—
506D4—184no—
507D4—181no—
508D4—181no—
509C2×C4—181no—
510C2×C4—181no—
511C2×C4—184no—
512C2×C4—184no—
513C2×C4#620012yes—
514C2×C4#620011yescharacteristic
515C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
516C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
517C8—92no—
518C8—92no—
519C2×C3—22no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—32no—
523C2×C3—32no—
524C2×C3—62no—
525C2×C3—62no—
526C2×C3—61no—
527C2×C3—62no—
528C2×C3—122no—
529C2×C3—362no—
530C2×C3—362no—
531C2×C3#6382912yes—
532C2×C3C22×#62611yescharacteristic
533D3—122no—
534D3—122no—
535D3—124no—
536D3—122no—
537D3—362no—
538C22—22no—
539C22—182no—
540C22—182no—
541C22—362no—
542C22—362no—
543C22C2×#620012yes—
544C22C2×#620011yescharacteristic
545C4—182no—
546C4—182no—
547C4C2×#620012yes—
548C4C2×#620011yescharacteristic
549C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
550C3—31no—
551C3—31no—
552C3—61no—
553C3C2×#6382911yescharacteristic
554C2—22no—
555C2—362no—
556C2—362no—
557C2#1172107312yesdirect product factor
558C2C22×#620011yescharacteristic
subgroupG/H
centerC2×C4#6200
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#135
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88740C2semidirect
2. C2×#3828⋊C3C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. C2×#88439C2semidirect
5. #11721073C2direct (N × H)
6. C2×#88438C2semidirect
7. C2×#88437C2semidirect
8. C2×#88436C2semidirect
14. #88740C22semidirect
20. #3828⋊C3C22semidirect
26. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
27. #88439C22semidirect
28. #88438C22semidirect
29. #88437C22semidirect
30. #88436C22semidirect
31. C2×#3828D3semidirect
36. C22×C3⋊(C3⋊C8)D3semidirect
79. #3828C2×D3semidirect
109. C2×C3⋊(C3⋊C8)C2×D3semidirect
219. C2×H(3,𝔽3)#135semidirect
331. H(3,𝔽3)C2×#135semidirect
419. C2×C32#904semidirect
420. C2×C32#950semidirect
494. C32C2×#904semidirect
495. C32C2×#950semidirect
557. C2#11721073direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887402
C2×#3828⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
C2×#884392
#117210732
C2×#884382
C2×#884372
C2×#884362
C2×#638293
C2×#638223