Finite Group Databasebeta

← C2×#11722955 #13172322

Subgroups — C2×#11722955 #13172322

Order 1728.

2258 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6295C211yesmaximal, characteristic, complemented
2C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C22×C3×#626C212yesmaximal, complemented
4C2×#88803C212yesmaximal, complemented
5C2×C3×#3724C211yesmaximal, characteristic, complemented
6#11722955C218yesmaximal, direct product factor
7C2×#63980—31nomaximal
8C2×#63886—32nomaximal
9C23×C3×C3⋊D3C2212yes—
10C23×C32×D3C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C2×#6295—24no—
15C2×#6295C2214yescomplemented
16C2×D3×C3⋊Dic3—24no—
17C2×D3×C3⋊Dic3C2214yescomplemented
18C2×C3×#626—28no—
19C2×C3×#626C2218yescomplemented
20#88803C2218yescomplemented
21C3×#3724C2214yescomplemented
22C23×C3⋊Dic3—31no—
23C22×C3×#67—32no—
24C22×C3×D12—32no—
25C22×#626—32no—
26C22×#626D312yescomplemented
27C22×#625—31no—
28C22×D3×Dic3—32no—
29C22×D3×Dic3—61no—
30C2×#3752—32no—
31C2×C3×Dic3⋊C4—32no—
32#63980—38no—
33C2×#3724D311yescharacteristic, complemented
34C2×(C2×D3)⋊Dic3—32no—
35C2×(C2×D3)⋊Dic3—32no—
36C2×(C2×D3)⋊Dic3—62no—
37#63886—316no—
38C23×C33C2311yescharacteristic
39C22×C3×C3⋊D3—24no—
40C22×C3×C3⋊D3—24no—
41C22×C3×C3⋊D3—24no—
42C22×C3×C3⋊D3C2314yes—
43C22×C32×D3—22no—
44C22×C32×D3—22no—
45C22×C32×D3D414yescomplemented
46C22×C32×D3C2312yes—
47C2×C3⋊(C3⋊Dic3)—22no—
48C2×C3⋊(C3⋊Dic3)C2312yes—
49C2×C3×C3⋊Dic3D414yescomplemented
50C2×C3×C3⋊Dic3C2312yes—
51C2×C32×Dic3—24no—
52C2×C32×Dic3C2314yes—
53#6295—28no—
54D3×C3⋊Dic3—28no—
55C3×#626—216no—
56C2×#939—92no—
57C2×#895—91no—
58C24×C32—31no—
59C23×C3⋊D3C2×D312yes—
60C23×C3×D3—21no—
61C23×C3×D3—32no—
62C23×C3×D3—32no—
63C23×C3×D3—62no—
64C23×C3×D3C2×D312yes—
65C22×C32×C4—32no—
66C22×C3⋊Dic3—32no—
67C22×C3⋊Dic3—31no—
68C22×C3⋊Dic3—34no—
69C22×C3⋊Dic3—61no—
70C22×C3⋊Dic3—64no—
71C22×C3⋊Dic3—62no—
72C22×C3⋊Dic3—121no—
73C22×C3⋊Dic3C2×D311yescharacteristic
74C2×C3×#67—38no—
75C2×C3×#67—68no—
76C22×C3×Dic3—22no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—61no—
79C22×C3×Dic3C2×D312yes—
80C22×C3×Dic3C2×D312yes—
81C2×C3×D12—38no—
82C2×C3×D12—68no—
83C2×#626—28no—
84C2×#626—38no—
85C2×#626—68no—
86C2×#626C2×D318yescomplemented
87C2×#625—34no—
88C2×#625—64no—
89C2×D3×Dic3—38no—
90C2×D3×Dic3—68no—
91C2×D3×Dic3—64no—
92C2×D3×Dic3—68no—
93#3752—38no—
94C3×Dic3⋊C4—38no—
95#3724C2×D314yescomplemented
96(C2×D3)⋊Dic3—38no—
97(C2×D3)⋊Dic3—38no—
98(C2×D3)⋊Dic3—68no—
99C22×C33C2×D412yes—
100C22×C33C2×D412yes—
101C22×C33#4112yes—
102C22×C33C2411yescharacteristic
103C2×C3×C3⋊D3—24no—
104C2×C3×C3⋊D3—24no—
105C2×C3×C3⋊D3—24no—
106C2×C3×C3⋊D3—48no—
107C2×C32×D3—28no—
108C2×C32×D3—22no—
109C2×C32×D3—22no—
110C2×C32×D3C2×D414yescomplemented
111C3⋊(C3⋊Dic3)—22no—
112C3×C3⋊Dic3C2×D414yescomplemented
113C32×Dic3—24no—
114C22×C3×D4—92no—
115C22×#67—32no—
116C22×#67—92no—
117C23×Dic3—92no—
118C23×Dic3—181no—
119C22×D12—32no—
120C22×D12—91no—
121C22×C4×D3—91no—
122C2×C3×C4⋊C4—91no—
123C2×#217—92no—
124#939—916no—
125C2×(C2×D3)⋊C4—92no—
126C2×(C2×D3)⋊C4—92no—
127C2×(C2×D3)⋊C4—182no—
128C2×Dic3⋊C4—32no—
129#895—98no—
130C23×C32—21no—
131C23×C32—22no—
132C23×C32—34no—
133C23×C32—32no—
134C23×C32—41no—
135C23×C32—62no—
136C23×C32—62no—
137C23×C32C22×D311yescharacteristic
138C23×C32C22×D312yes—
139C22×C3⋊D3—24no—
140C22×C3⋊D3—24no—
141C22×C3⋊D3—24no—
142C22×C3⋊D3C22×D314yes—
143C22×C3×D3—24no—
144C22×C3×D3—24no—
145C22×C3×D3—24no—
146C22×C3×D3—24no—
147C22×C3×D3—24no—
148C22×C3×D3—22no—
149C22×C3×D3—34no—
150C22×C3×D3—34no—
151C22×C3×D3—64no—
152C22×C3×D3—68no—
153C22×C3×D3—64no—
154C22×C3×D3—64no—
155C22×C3×D3—64no—
156C22×C3×D3—64no—
157C22×C3×D3—64no—
158C22×C3×D3—64no—
159C22×C3×D3—124no—
160C22×C3×D3—124no—
161C22×C3×D3#6718yescomplemented
162C22×C3×D3C22×D314yes—
163C2×C32×C4—34no—
164C2×C32×C4—64no—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—34no—
167C2×C3⋊Dic3—62no—
168C2×C3⋊Dic3—64no—
169C2×C3⋊Dic3—62no—
170C2×C3⋊Dic3—68no—
171C2×C3⋊Dic3—64no—
172C2×C3⋊Dic3—64no—
173C2×C3⋊Dic3—122no—
174C2×C3⋊Dic3—124no—
175C2×C3⋊Dic3—124no—
176C2×C3⋊Dic3D1214yescomplemented
177C2×C3⋊Dic3C22×D312yes—
178C3×#67—616no—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—38no—
183C2×C3×Dic3—34no—
184C2×C3×Dic3—44no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3C22×D314yes—
188C2×C3×Dic3C22×D314yes—
189C3×D12—616no—
190#626—216no—
191#626—616no—
192#625—68no—
193D3×Dic3—616no—
194D3×Dic3—616no—
195C2×#121—271noSylow
196C2×C33#12114yescomplemented
197C2×C33C22×D411yescharacteristic
198C2×C33C22×D411yescharacteristic
199C2×C33C2×#4111yescharacteristic
200C3×C3⋊D3—44no—
201C32×D3—24no—
202C24×C3—32no—
203C24×C3—61no—
204C24×C3—92no—
205C23×D3—32no—
206C23×D3—32no—
207C23×D3—62no—
208C23×D3D3211yescharacteristic
209C2×C3×D4—98no—
210C2×C3×D4—188no—
211C22×C3×C4—32no—
212C22×C3×C4—32no—
213C22×C3×C4—62no—
214C22×C3×C4—91no—
215C2×#67—38no—
216C2×#67—68no—
217C2×#67—98no—
218C2×#67—188no—
219C22×Dic3—32no—
220C22×Dic3—61no—
221C22×Dic3—98no—
222C22×Dic3—91no—
223C22×Dic3—92no—
224C22×Dic3—184no—
225C22×Dic3—181no—
226C22×Dic3—182no—
227C22×Dic3—188no—
228C22×Dic3—188no—
229C22×Dic3—361no—
230C22×Dic3D3212yes—
231C2×D12—38no—
232C2×D12—68no—
233C2×D12—94no—
234C2×D12—184no—
235C2×C4×D3—94no—
236C2×C4×D3—184no—
237C3×C4⋊C4—94no—
238#217—98no—
239(C2×D3)⋊C4—98no—
240(C2×D3)⋊C4—98no—
241(C2×D3)⋊C4—188no—
242Dic3⋊C4—38no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—24no—
246C22×C32—24no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—24no—
250C22×C32—22no—
251C22×C32—34no—
252C22×C32—42no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—41no—
256C22×C32—68no—
257C22×C32—62no—
258C22×C32—62no—
259C22×C32C2×D1212yes—
260C22×C32D3×D414yes—
261C22×C32D3×D412yes—
262C22×C32#23714yes—
263C22×C32#23912yes—
264C22×C32C2×#6714yes—
265C22×C32C23×D311yescharacteristic
266C22×C32C23×D312yes—
267C2×C3⋊D3—24no—
268C2×C3⋊D3—24no—
269C2×C3⋊D3—24no—
270C2×C3⋊D3—48no—
271C2×C3×D3—24no—
272C2×C3×D3—24no—
273C2×C3×D3—216no—
274C2×C3×D3—216no—
275C2×C3×D3—24no—
276C2×C3×D3—24no—
277C2×C3×D3—24no—
278C2×C3×D3—68no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C2×C3×D3—64no—
282C2×C3×D3—64no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C2×C3×D3—124no—
286C2×C3×D3—124no—
287C2×C3×D3—1216no—
288C2×C3×D3—128no—
289C2×C3×D3—128no—
290C2×C3×D3C2×#6718yescomplemented
291C32×C4—64no—
292C3⋊Dic3—62no—
293C3⋊Dic3—64no—
294C3⋊Dic3—64no—
295C3⋊Dic3—124no—
296C3⋊Dic3—124no—
297C3⋊Dic3C2×D1214yescomplemented
298C3×Dic3—24no—
299C3×Dic3—24no—
300C3×Dic3—38no—
301C3×Dic3—44no—
302C3×Dic3—64no—
303C22×D4—92no—
304C22×D4—271no—
305C23×C4—271no—
306#121—278no—
307C2×C4⋊C4—91no—
308C2×C22⋊C4—272no—
309C33C2×#12111yescharacteristic, complemented, Sylow
310C23×C3—21no—
311C23×C3—22no—
312C23×C3—34no—
313C23×C3—38no—
314C23×C3—41no—
315C23×C3—62no—
316C23×C3—64no—
317C23×C3—64no—
318C23×C3—64no—
319C23×C3—64no—
320C23×C3—64no—
321C23×C3—94no—
322C23×C3—184no—
323C23×C3—184no—
324C23×C3—184no—
325C23×C3C2×D3212yes—
326C23×C3C2×D3211yescharacteristic
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—34no—
330C22×D3—34no—
331C22×D3—64no—
332C22×D3—68no—
333C22×D3—64no—
334C22×D3—64no—
335C22×D3—64no—
336C22×D3—64no—
337C22×D3—64no—
338C22×D3—64no—
339C22×D3—124no—
340C22×D3—124no—
341C22×D3#62514yescomplemented
342C22×D3C2×D3212yes—
343C3×D4—1816no—
344C2×C3×C4—34no—
345C2×C3×C4—34no—
346C2×C3×C4—64no—
347C2×C3×C4—64no—
348C2×C3×C4—64no—
349C2×C3×C4—92no—
350C2×C3×C4—94no—
351C2×C3×C4—124no—
352#67—616no—
353#67—1816no—
354C2×Dic3—24no—
355C2×Dic3—34no—
356C2×Dic3—38no—
357C2×Dic3—64no—
358C2×Dic3—62no—
359C2×Dic3—92no—
360C2×Dic3—94no—
361C2×Dic3—182no—
362C2×Dic3—184no—
363C2×Dic3—1816no—
364C2×Dic3—1816no—
365C2×Dic3—184no—
366C2×Dic3—184no—
367C2×Dic3—182no—
368C2×Dic3—362no—
369C2×Dic3—364no—
370C2×Dic3—364no—
371C2×Dic3C2×D3214yes—
372D12—616no—
373D12—188no—
374C4×D3—188no—
375C2×C32—22no—
376C2×C32—22no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—22no—
380C2×C32—21no—
381C2×C32—28no—
382C2×C32—24no—
383C2×C32—44no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—64no—
388C2×C32#89514yescomplemented
389C2×C32#93918yescomplemented
390C2×C32C22×D1211yescharacteristic
391C2×C32C2×D3×D412yes—
392C2×C32C2×D3×D411yescharacteristic
393C2×C32C2×#23712yes—
394C2×C32C2×#23911yescharacteristic
395C2×C32C22×#6712yes—
396C3⋊D3—44no—
397C3×D3—28no—
398C3×D3—28no—
399C3×D3—128no—
400C3×D3—124no—
401C3×D3—124no—
402C24—31no—
403C24—92no—
404C2×D4—98no—
405C2×D4—188no—
406C2×D4—274no—
407C2×D4—544no—
408C22×C4—32no—
409C22×C4—91no—
410C22×C4—274no—
411C22×C4—271no—
412C22×C4—544no—
413C4⋊C4—94no—
414C22⋊C4—278no—
415C22×C3—21no—
416C22×C3—22no—
417C22×C3—24no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—22no—
423C22×C3—38no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—64no—
429C22×C3—64no—
430C22×C3—64no—
431C22×C3—616no—
432C22×C3—64no—
433C22×C3—64no—
434C22×C3—616no—
435C22×C3—184no—
436C22×C3—184no—
437C22×C3—184no—
438C22×C3—368no—
439C22×C3#379614yes—
440C22×C3D3×D1214yes—
441C22×C3#380512yes—
442C22×C3C2×#62512yes—
443C22×C3D3×#6714yes—
444C22×C3#381112yes—
445C22×C3C22×D3212yes—
446C22×C3C22×D3211yescharacteristic
447C2×D3—22no—
448C2×D3—22no—
449C2×D3—28no—
450C2×D3—68no—
451C2×D3—64no—
452C2×D3—64no—
453C2×D3—64no—
454C2×D3—64no—
455C2×D3—64no—
456C2×D3—64no—
457C2×D3—124no—
458C2×D3—124no—
459C2×D3—1216no—
460C2×D3—128no—
461C2×D3—128no—
462C2×D3C2×#62514yescomplemented
463C3×C4—64no—
464C3×C4—64no—
465C3×C4—94no—
466C3×C4—124no—
467Dic3—24no—
468Dic3—38no—
469Dic3—64no—
470Dic3—184no—
471Dic3—184no—
472Dic3—182no—
473Dic3—364no—
474Dic3—364no—
475C32—22no—
476C32—21no—
477C32—41no—
478C32C2×#89511yescharacteristic, complemented
479C32C2×#93912yescomplemented
480C23—32no—
481C23—34no—
482C23—62no—
483C23—62no—
484C23—94no—
485C23—184no—
486C23—184no—
487C23—184no—
488C23D3311yescharacteristic
489D4—1816no—
490D4—548no—
491C2×C4—34no—
492C2×C4—64no—
493C2×C4—92no—
494C2×C4—94no—
495C2×C4—272no—
496C2×C4—548no—
497C2×C4—542no—
498C2×C3—22no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—22no—
502C2×C3—22no—
503C2×C3—21no—
504C2×C3—28no—
505C2×C3—24no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—44no—
510C2×C3—68no—
511C2×C3—68no—
512C2×C3—364no—
513C2×C3#6388618yescomplemented
514C2×C3#6398014yescomplemented
515C2×C3C2×#379612yes—
516C2×C3C2×D3×D1212yes—
517C2×C3C2×#380511yescharacteristic
518C2×C3C22×#62511yescharacteristic
519C2×C3C2×D3×#6712yes—
520C2×C3C2×#381111yescharacteristic
521D3—24no—
522D3—128no—
523D3—124no—
524D3—124no—
525C22—34no—
526C22—62no—
527C22—62no—
528C22—68no—
529C22—184no—
530C22—184no—
531C22—184no—
532C22—368no—
533C22D3×#62512yes—
534C22#8895512yes—
535C22#8896412yes—
536C22C2×D3311yescharacteristic
537C4—64no—
538C4—94no—
539C4—542no—
540C3—22no—
541C3—21no—
542C3—41no—
543C3C2×#6388612yescomplemented
544C3C2×#6398011yescharacteristic, complemented
545C2—64no—
546C2—364no—
547C2#1172295514yesdirect product factor
548C2C2×D3×#62511yescharacteristic
549C2C2×#8895511yescharacteristic
550C2C2×#8896411yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutator, cosocleC22×C33C24
FrattiniC22C2×D33
Fitting, socleC23×C33C23
hypocenterC33C2×#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6295C2semidirect
2. C22×D3×C3⋊Dic3C2semidirect
3. C22×C3×#626C2semidirect
4. C2×#88803C2semidirect
5. C2×C3×#3724C2semidirect
6. #11722955C2direct (N × H)
15. C2×#6295C22semidirect
17. C2×D3×C3⋊Dic3C22semidirect
19. C2×C3×#626C22semidirect
20. #88803C22semidirect
21. C3×#3724C22semidirect
26. C22×#626D3semidirect
33. C2×#3724D3semidirect
45. C22×C32×D3D4semidirect
49. C2×C3×C3⋊Dic3D4semidirect
86. C2×#626C2×D3semidirect
95. #3724C2×D3semidirect
110. C2×C32×D3C2×D4semidirect
112. C3×C3⋊Dic3C2×D4semidirect
161. C22×C3×D3#67semidirect
176. C2×C3⋊Dic3D12semidirect
196. C2×C33#121semidirect
290. C2×C3×D3C2×#67semidirect
297. C3⋊Dic3C2×D12semidirect
309. C33C2×#121semidirect
341. C22×D3#625semidirect
388. C2×C32#895semidirect
389. C2×C32#939semidirect
462. C2×D3C2×#625semidirect
478. C32C2×#895semidirect
479. C32C2×#939semidirect
513. C2×C3#63886semidirect
514. C2×C3#63980semidirect
543. C3C2×#63886semidirect
544. C3C2×#63980semidirect
547. C2#11722955direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62952
C22×D3×C3⋊Dic32
C22×C3×#6262
C2×#888032
C2×C3×#37242
#117229552
C2×#639803
C2×#638863