Finite Group Databasebeta

← C2×#11722505 #13164742

Subgroups — C2×#11722505 #13164742

Order 1728.

3351 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#272C211yesmaximal, characteristic, complemented
2C2×#3830⋊C3C212yesmaximal, complemented
3C2×#88677C212yesmaximal, complemented
4C22×#6199C212yesmaximal, complemented
5C2×#88671C214yesmaximal, complemented
6#11722505C2116yesmaximal, direct product factor
7C2×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic, complemented
8C2×#88652C212yesmaximal, complemented
9C2×#88651C211yesmaximal, characteristic, complemented
10C2×D3×#237—32nomaximal
11C23×#272C2212yes—
12C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×#6235C2212yescomplemented
15D4×#272C2218yescomplemented
16C2×C4⋊#272C2211yescharacteristic, complemented
17C2×C4×#272C2211yescharacteristic
18C2×#638⋊C3C2214yescomplemented
19C22×(C3⋊Dic3)⋊C3C2214yescomplemented
20#3830⋊C3C22116yescomplemented
21C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
22C2×#634⋊C3C2212yescomplemented
23#88677C22116yescomplemented
24C2×#6200C2214yescomplemented
25C2×#6199C2211yescharacteristic
26C2×#6199C2212yes—
27C2×#6199C22116yescomplemented
28C2×#6199C2214yescomplemented
29#88671C22132yescomplemented
30C2×((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
31C2×#6196C2212yescomplemented
32C4×((C3⋊D3)⋊C3)⋊C2C2218yescomplemented
33#88652C22116yescomplemented
34#88651C2218yescomplemented
35C2×#3830D312yescomplemented
36C2×C3×#237—32no—
37C2×C3×D3×D4—32no—
38C2×C3×D3×D4—61no—
39C2×D3×#67—34no—
40C2×#3805—34no—
41C2×#3804—34no—
42C22×D3×Dic3—34no—
43D3×#237—332no—
44C2×C4×D32—32no—
45C2×#3796—32no—
46C2×#3795—32no—
47C2×D3×Dic6—32no—
48C23×H(3,𝔽3)C2312yes—
49C22×#272—28no—
50C22×#272C2312yes—
51C22×#272C2318yes—
52C22×#272C2311yescharacteristic
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×((C3⋊D3)⋊C3)⋊C2—24no—
55C2×((C3⋊D3)⋊C3)⋊C2C2314yes—
56D4×H(3,𝔽3)C2314yescomplemented
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58#6235C2318yescomplemented
59C2×#1084C2311yescharacteristic
60C4⋊#272C2314yescomplemented
61C4×#272C2314yes—
62#638⋊C3C23116yescomplemented
63C2×(C3⋊Dic3)⋊C3C2314yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C23116yescomplemented
66C4×(C3⋊D3)⋊C3C2318yescomplemented
67#634⋊C3C2318yescomplemented
68#6200C23116yescomplemented
69#6199C2318yes—
70#6199C2314yes—
71#6199C23116yescomplemented
72((C3⋊D3)⋊C3)⋊C4C2318yescomplemented
73#6196C2318yescomplemented
74C2×D3×#41—92no—
75C22×#237—91no—
76C23×C3×D3—34no—
77C23×C3×D3—62no—
78C22×D32—32no—
79C2×C32×D4—21no—
80C2×C32×D4C2×D312yescomplemented
81C2×#638C2×D314yescomplemented
82C22×C3⋊Dic3C2×D314yescomplemented
83#3830C2×D3116yescomplemented
84C2×C4×C3⋊D3C2×D312yescomplemented
85C2×#634C2×D312yescomplemented
86C2×C3×#67—34no—
87C2×C3×#67—34no—
88C2×C3×#67—62no—
89C22×C3×Dic3—34no—
90C3×#237—316no—
91C3×D3×D4—316no—
92C3×D3×D4—68no—
93C2×C3×D12—32no—
94C2×C3×D12—61no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—61no—
98C2×C3×Dic6—32no—
99D3×#67—332no—
100C2×#627—34no—
101C2×#626—34no—
102C2×#625—34no—
103C2×#624—32no—
104#3805—332no—
105#3804—332no—
106C2×D3×Dic3—34no—
107C2×D3×Dic3—34no—
108C2×D3×Dic3—34no—
109C2×D3×Dic3—32no—
110C2×D3×Dic3—32no—
111C2×D3×Dic3—332no—
112C4×D32—316no—
113#3796—316no—
114#3795—316no—
115D3×Dic6—316no—
116C22×H(3,𝔽3)—24no—
117C22×H(3,𝔽3)C2414yes—
118C22×H(3,𝔽3)C2411yescharacteristic
119C2×#272—24no—
120C2×#272—216no—
121C2×#272#4114yescomplemented
122C2×#272C2414yes—
123C2×#272C2412yes—
124C2×(C3⋊D3)⋊C3—24no—
125C2×(C3⋊D3)⋊C3C2414yes—
126((C3⋊D3)⋊C3)⋊C2—28no—
127C4×H(3,𝔽3)C2412yes—
128#1084C2412yes—
129(C3⋊Dic3)⋊C3C2418yes—
130(C3⋊Dic3)⋊C3C2414yes—
131C2×C3×#41—92no—
132C22×C3×D4—91no—
133C22×#67—92no—
134C23×Dic3—92no—
135D3×#41—932no—
136C2×#239—92no—
137C2×D3×Q8—92no—
138C2×#237—32no—
139C2×#237—92no—
140C2×#237—92no—
141C2×#237—916no—
142C2×#237—94no—
143C2×#237—94no—
144C2×D3×D4—32no—
145C2×D3×D4—61no—
146C2×D3×D4—94no—
147C2×#235—92no—
148C2×#235—94no—
149C22×C4×D3—91no—
150C22×C4×D3—92no—
151C22×C4×D3—94no—
152C22×Dic6—91no—
153C23×C32—22no—
154C23×C32C22×D314yes—
155C22×C3⋊D3C22×D312yes—
156C22×C3×D3—32no—
157C22×C3×D3—32no—
158C22×C3×D3—34no—
159C22×C3×D3—316no—
160C22×C3×D3—61no—
161C22×C3×D3—62no—
162C22×C3×D3—68no—
163C22×C3×D3—616no—
164C22×C3×D3—128no—
165C2×D32—38no—
166C2×D32—68no—
167C32×D4—24no—
168C32×D4C22×D318yescomplemented
169C2×C32×C4—21no—
170C2×C32×C4C22×D312yes—
171#638C22×D3116yescomplemented
172C2×C3⋊Dic3C22×D312yes—
173C2×C3⋊Dic3C22×D314yes—
174C2×C3⋊Dic3C22×D3116yescomplemented
175C4×C3⋊D3C22×D318yescomplemented
176#634C22×D318yescomplemented
177C3×#67—316no—
178C3×#67—316no—
179C3×#67—68no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—34no—
182C2×C3×Dic3—316no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—61no—
185C3×D12—38no—
186C3×D12—64no—
187C3×C4×D3—38no—
188C3×C4×D3—38no—
189C3×C4×D3—64no—
190C3×Dic6—38no—
191#627—316no—
192#626—316no—
193#625—316no—
194#624—38no—
195D3×Dic3—38no—
196D3×Dic3—316no—
197D3×Dic3—38no—
198D3×Dic3—316no—
199D3×Dic3—316no—
200C22×#41—271noSylow
201C2×H(3,𝔽3)—24no—
202C2×H(3,𝔽3)C2×#4112yescomplemented
203C2×H(3,𝔽3)C2511yescharacteristic
204#272—24no—
205#272C2×#4114yescomplemented
206(C3⋊D3)⋊C3—24no—
207C24×C3—92no—
208C23×D3—34no—
209C23×D3—62no—
210C23×D3—91no—
211C23×D3—92no—
212C3×#41—916no—
213C2×C3×Q8—92no—
214C2×C3×D4—32no—
215C2×C3×D4—61no—
216C2×C3×D4—94no—
217C2×C3×D4—92no—
218C2×C3×D4—98no—
219C2×C3×D4—91no—
220C2×C3×D4D3211yescharacteristic
221C22×C3×C4—92no—
222C22×C3×C4—94no—
223C22×C3×C4—91no—
224C2×#67—34no—
225C2×#67—34no—
226C2×#67—62no—
227C2×#67—94no—
228C2×#67—94no—
229C2×#67—916no—
230C2×#67—94no—
231C22×Dic3—34no—
232C22×Dic3—94no—
233C22×Dic3—92no—
234C22×Dic3—91no—
235C22×Dic3—94no—
236C22×Dic3—916no—
237C22×Dic3—92no—
238#239—916no—
239D3×Q8—916no—
240#237—316no—
241#237—916no—
242#237—916no—
243#237—932no—
244#237—932no—
245D3×D4—316no—
246D3×D4—68no—
247D3×D4—932no—
248#235—916no—
249#235—932no—
250C2×D12—32no—
251C2×D12—61no—
252C2×D12—94no—
253C2×C4×D3—32no—
254C2×C4×D3—32no—
255C2×C4×D3—61no—
256C2×C4×D3—92no—
257C2×C4×D3—98no—
258C2×C4×D3—916no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—92no—
262C2×C4×D3—94no—
263C2×C4×D3—94no—
264C2×C4×D3—94no—
265C2×C4×D3—932no—
266C2×Dic6—32no—
267C2×Dic6—92no—
268C2×Dic6—98no—
269C2×Dic6—92no—
270C2×Dic6—94no—
271C22×C32—28no—
272C22×C32—21no—
273C22×C32—24no—
274C22×C32—44no—
275C22×C32C23×D318yes—
276C22×C32C23×D312yes—
277C2×C3⋊D3—24no—
278C2×C3⋊D3C23×D314yes—
279C2×C3×D3—38no—
280C2×C3×D3—34no—
281C2×C3×D3—34no—
282C2×C3×D3—38no—
283C2×C3×D3—64no—
284C2×C3×D3—62no—
285C2×C3×D3—64no—
286C2×C3×D3—632no—
287C2×C3×D3—64no—
288C2×C3×D3—68no—
289C2×C3×D3—1216no—
290C2×C3×D3—124no—
291D32—616no—
292C32×C4—22no—
293C32×C4C23×D314yes—
294C3⋊Dic3C23×D314yes—
295C3⋊Dic3C23×D318yes—
296C3×Dic3—34no—
297C3×Dic3—38no—
298C3×Dic3—34no—
299C3×Dic3—62no—
300C2×#41—92no—
301C2×#41—274no—
302C2×#41—272no—
303C2×#41—2716no—
304C22×Q8—271no—
305C22×D4—91no—
306C22×D4—272no—
307C23×C4—271no—
308C23×C4—272no—
309H(3,𝔽3)C22×#4111yescharacteristic, complemented, Sylow
310C23×C3—34no—
311C23×C3—62no—
312C23×C3—92no—
313C23×C3—91no—
314C23×C3—98no—
315C23×C3—92no—
316C23×C3—188no—
317C23×C3C2×D3212yes—
318C22×D3—32no—
319C22×D3—34no—
320C22×D3—316no—
321C22×D3—32no—
322C22×D3—61no—
323C22×D3—62no—
324C22×D3—68no—
325C22×D3—616no—
326C22×D3—92no—
327C22×D3—94no—
328C22×D3—98no—
329C22×D3—128no—
330C22×D3—188no—
331C22×D3—184no—
332C3×Q8—98no—
333C3×D4—38no—
334C3×D4—64no—
335C3×D4—916no—
336C3×D4—98no—
337C3×D4—94no—
338C3×D4C2×D3214yes—
339C2×C3×C4—32no—
340C2×C3×C4—61no—
341C2×C3×C4—98no—
342C2×C3×C4—94no—
343C2×C3×C4—92no—
344C2×C3×C4—916no—
345C2×C3×C4—91no—
346C2×C3×C4—94no—
347C2×C3×C4C2×D3211yescharacteristic
348#67—316no—
349#67—316no—
350#67—68no—
351#67—916no—
352#67—916no—
353#67—916no—
354C2×Dic3—32no—
355C2×Dic3—34no—
356C2×Dic3—32no—
357C2×Dic3—316no—
358C2×Dic3—61no—
359C2×Dic3—94no—
360C2×Dic3—92no—
361C2×Dic3—916no—
362C2×Dic3—98no—
363C2×Dic3—94no—
364C2×Dic3—916no—
365C2×Dic3—98no—
366D12—38no—
367D12—64no—
368D12—916no—
369C4×D3—38no—
370C4×D3—38no—
371C4×D3—64no—
372C4×D3—98no—
373C4×D3—98no—
374C4×D3—98no—
375C4×D3—98no—
376C4×D3—916no—
377C4×D3—916no—
378C4×D3—916no—
379Dic6—38no—
380Dic6—98no—
381Dic6—98no—
382Dic6—916no—
383C2×C32—28no—
384C2×C32—22no—
385C2×C32—21no—
386C2×C32—44no—
387C2×C32D3×#4114yescomplemented
388C2×C32C24×D312yes—
389C3⋊D3—24no—
390C3×D3—38no—
391C3×D3—64no—
392C3×D3—64no—
393C3×D3—68no—
394C3×D3—124no—
395C24—92no—
396C24—271no—
397#41—916no—
398#41—2732no—
399#41—2716no—
400C2×Q8—92no—
401C2×Q8—272no—
402C2×Q8—278no—
403C2×D4—92no—
404C2×D4—94no—
405C2×D4—98no—
406C2×D4—91no—
407C2×D4—274no—
408C2×D4—2716no—
409C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
410C22×C4—94no—
411C22×C4—91no—
412C22×C4—92no—
413C22×C4—278no—
414C22×C4—271no—
415C22×C4—272no—
416C22×C4—2716no—
417C22×C4—272no—
418C22×C4—274no—
419C22×C3—24no—
420C22×C3—38no—
421C22×C3—32no—
422C22×C3—68no—
423C22×C3—64no—
424C22×C3—61no—
425C22×C3—94no—
426C22×C3—94no—
427C22×C3—92no—
428C22×C3—94no—
429C22×C3—124no—
430C22×C3—1816no—
431C22×C3—184no—
432C22×C3—184no—
433C22×C3C22×D3214yes—
434C22×C3C22×D3211yescharacteristic
435C2×D3—34no—
436C2×D3—38no—
437C2×D3—38no—
438C2×D3—34no—
439C2×D3—62no—
440C2×D3—64no—
441C2×D3—64no—
442C2×D3—632no—
443C2×D3—68no—
444C2×D3—64no—
445C2×D3—94no—
446C2×D3—1216no—
447C2×D3—124no—
448C2×D3—184no—
449C2×D3—1816no—
450C2×D3—188no—
451C3×C4—34no—
452C3×C4—62no—
453C3×C4—98no—
454C3×C4—94no—
455C3×C4—92no—
456C3×C4C22×D3212yes—
457Dic3—34no—
458Dic3—38no—
459Dic3—34no—
460Dic3—62no—
461Dic3—98no—
462Dic3—94no—
463C32—21no—
464C32C2×D3×#4112yescomplemented
465C23—91no—
466C23—98no—
467C23—92no—
468C23—92no—
469C23—188no—
470C23—274no—
471C23—544no—
472C23C2×((C3⋊D3)⋊C3)⋊C212yes—
473Q8—98no—
474Q8—278no—
475D4—98no—
476D4—916no—
477D4—94no—
478D4—2716no—
479D4C2×((C3⋊D3)⋊C3)⋊C214yescomplemented
480C2×C4—916no—
481C2×C4—92no—
482C2×C4—91no—
483C2×C4—94no—
484C2×C4—94no—
485C2×C4—98no—
486C2×C4—274no—
487C2×C4—278no—
488C2×C4—278no—
489C2×C4—2716no—
490C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
491C2×C3—24no—
492C2×C3—34no—
493C2×C3—32no—
494C2×C3—68no—
495C2×C3—62no—
496C2×C3—61no—
497C2×C3—94no—
498C2×C3—124no—
499C2×C3—184no—
500C2×C3—184no—
501C2×C3#6431212yes—
502C2×C3C23×D3211yescharacteristic
503D3—38no—
504D3—64no—
505D3—68no—
506D3—64no—
507D3—124no—
508D3—184no—
509C22—24no—
510C22—94no—
511C22—92no—
512C22—94no—
513C22—94no—
514C22—1816no—
515C22—184no—
516C22—184no—
517C22—548no—
518C22C22×((C3⋊D3)⋊C3)⋊C214yes—
519C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
520C4—94no—
521C4—92no—
522C4—98no—
523C4C22×((C3⋊D3)⋊C3)⋊C212yes—
524C3—32no—
525C3—61no—
526C3C2×#6431211yescharacteristic
527C2—24no—
528C2—94no—
529C2—184no—
530C2—184no—
531C2#1172250512yesdirect product factor
532C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3C22×D32
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×#272C2semidirect
2. C2×#3830⋊C3C2semidirect
3. C2×#88677C2semidirect
4. C22×#6199C2semidirect
5. C2×#88671C2semidirect
6. #11722505C2direct (N × H)
7. C2×C4×((C3⋊D3)⋊C3)⋊C2C2semidirect
8. C2×#88652C2semidirect
9. C2×#88651C2semidirect
13. C2×D4×H(3,𝔽3)C22semidirect
14. C2×#6235C22semidirect
15. D4×#272C22semidirect
16. C2×C4⋊#272C22semidirect
18. C2×#638⋊C3C22semidirect
19. C22×(C3⋊Dic3)⋊C3C22semidirect
20. #3830⋊C3C22semidirect
21. C2×C4×(C3⋊D3)⋊C3C22semidirect
22. C2×#634⋊C3C22semidirect
23. #88677C22semidirect
24. C2×#6200C22semidirect
27. C2×#6199C22semidirect
28. C2×#6199C22semidirect
29. #88671C22semidirect
30. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
31. C2×#6196C22semidirect
32. C4×((C3⋊D3)⋊C3)⋊C2C22semidirect
33. #88652C22semidirect
34. #88651C22semidirect
35. C2×#3830D3semidirect
56. D4×H(3,𝔽3)C23semidirect
58. #6235C23semidirect
60. C4⋊#272C23semidirect
62. #638⋊C3C23semidirect
65. C2×(C3⋊Dic3)⋊C3C23semidirect
66. C4×(C3⋊D3)⋊C3C23semidirect
67. #634⋊C3C23semidirect
68. #6200C23semidirect
71. #6199C23semidirect
72. ((C3⋊D3)⋊C3)⋊C4C23semidirect
73. #6196C23semidirect
80. C2×C32×D4C2×D3semidirect
81. C2×#638C2×D3semidirect
82. C22×C3⋊Dic3C2×D3semidirect
83. #3830C2×D3semidirect
84. C2×C4×C3⋊D3C2×D3semidirect
85. C2×#634C2×D3semidirect
121. C2×#272#41semidirect
168. C32×D4C22×D3semidirect
171. #638C22×D3semidirect
174. C2×C3⋊Dic3C22×D3semidirect
175. C4×C3⋊D3C22×D3semidirect
176. #634C22×D3semidirect
202. C2×H(3,𝔽3)C2×#41semidirect
205. #272C2×#41semidirect
309. H(3,𝔽3)C22×#41semidirect
387. C2×C32D3×#41semidirect
409. C2×D4((C3⋊D3)⋊C3)⋊C2semidirect
464. C32C2×D3×#41semidirect
479. D4C2×((C3⋊D3)⋊C3)⋊C2semidirect
531. C2#11722505direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×#3830⋊C32
C2×#886772
C22×#61992
C2×#886712
#117225052
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×#886522
C2×#886512
C2×D3×#2373