Finite Group Databasebeta

← C2×#11722832 #13170542

Subgroups — C2×#11722832 #13170542

Order 1728.

1954 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1954 conjugacy classes of subgroups — too many to draw. Only its 196 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89026C212yesmaximal, complemented
2C2×#88810C211yesmaximal, characteristic, complemented
3C2×#88785C211yesmaximal, characteristic, complemented
4C2×C3×#3714C212yesmaximal, complemented
5C2×C3×#3713C211yesmaximal, characteristic, complemented
6#11722832C218yesmaximal, direct product factor
7C2×#63926—32nomaximal
8C2×#63797—31nomaximal
9C22×D3×C3⋊D3—22no—
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
12C2×C32×D12C2212yescomplemented
13C2×#6290—22no—
14#89026—24no—
15#89026C2218yescomplemented
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
17C2×C32×C3⋊C8C2211yescharacteristic, complemented
18#88810C2214yescomplemented
19#88785C2214yescomplemented
20C3×#3714C2218yescomplemented
21C3×#3713C2214yescomplemented
22C2×D4×C3⋊D3—32no—
23C2×#3802—32no—
24C2×#3802—32no—
25C2×#3802—62no—
26C2×C8×C3⋊D3—31no—
27C2×C3×#213—32no—
28C2×C3×#213—32no—
29C2×C3×D24—32no—
30#63926—316no—
31C2×#3714—32no—
32C2×#3714D312yescomplemented
33C2×#3713—31no—
34C2×#3713D311yescharacteristic, complemented
35C2×#3710—32no—
36C2×#3710—61no—
37#63797—38no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C22×C3×C3⋊D3—22no—
40C22×C32×D3—22no—
41C2×D3×C3⋊D3—28no—
42C2×D3×C3⋊D3—48no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4×C3⋊(C3⋊D3)D412yes—
46C4×C3⋊(C3⋊D3)C2312yes—
47C3×C4⋊(C3⋊D3)—22no—
48C3×C4⋊(C3⋊D3)C2314yescomplemented
49C32×D12—22no—
50C32×D12C2314yescomplemented
51#6290—28no—
52C3×C3⋊(C3⋊C8)C2312yescomplemented
53C32×C3⋊C8C2312yescomplemented
54C2×D3×D8—92no—
55C2×D3×D8—91no—
56C23×C3⋊D3—62no—
57C22×D32—62no—
58C22×D32—62no—
59C22×D32—64no—
60C2×C32×D4—32no—
61C2×#638—62no—
62D4×C3⋊D3—38no—
63D4×C3⋊D3—64no—
64C2×C4⋊(C3⋊D3)C2×D312yescomplemented
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—32no—
67C2×C4×C3⋊D3—62no—
68C2×C4×C3⋊D3—61no—
69C2×C4×C3⋊D3—121no—
70C2×C3×D12—22no—
71C2×C3×D12—32no—
72C2×C3×D12—32no—
73C2×C3×D12—62no—
74C2×C3×D12C2×D312yescomplemented
75C2×C3×D12C2×D312yescomplemented
76C2×#625—62no—
77C2×#625—62no—
78C2×#625—64no—
79#3802—38no—
80#3802—38no—
81#3802—64no—
82#3802—64no—
83#3802—68no—
84#3802—68no—
85C2×C32×C8—31no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
87C8×C3⋊D3—34no—
88C3×#213—38no—
89C3×#213—38no—
90C2×C3×C3⋊C8—21no—
91C2×C3×C3⋊C8—32no—
92C2×C3×C3⋊C8—61no—
93C2×C3×C3⋊C8C2×D312yescomplemented
94C3×D24—38no—
95#3714—38no—
96#3714C2×D318yescomplemented
97#3713—34no—
98#3713C2×D314yescomplemented
99#3710—38no—
100#3710—64no—
101C22×C33C2×D411yescharacteristic
102C2×C3⋊(C3⋊D3)D814yescomplemented
103C2×C3⋊(C3⋊D3)C2×D412yes—
104C2×C3×C3⋊D3—24no—
105C2×C3×C3⋊D3—44no—
106C2×C32×D3—24no—
107C2×C32×D3—44no—
108D3×C3⋊D3—416no—
109C33×C4C2×D411yescharacteristic
110C33×C4C2411yescharacteristic
111C3⋊(C3⋊Dic3)C2×D412yes—
112C2×D3×D4—92no—
113C2×D3×D4—92no—
114C2×D3×D4—92no—
115C2×D3×D4—182no—
116C2×C3×D8—92no—
117C2×C3×D8—91no—
118C2×#213—32no—
119C2×#213—32no—
120C2×#213—92no—
121D3×D8—916no—
122D3×D8—98no—
123C2×D24—32no—
124C2×D24—91no—
125C2×D3×C8—92no—
126C2×D3×C8—91no—
127C2×D3×C8—181no—
128C23×C32—62no—
129C22×C3⋊D3—22no—
130C22×C3⋊D3—31no—
131C22×C3⋊D3—32no—
132C22×C3⋊D3—62no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—68no—
135C22×C3⋊D3—121no—
136C22×C3⋊D3—128no—
137C22×C3×D3—24no—
138C22×C3×D3—22no—
139C22×C3×D3—22no—
140C22×C3×D3—64no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C2×D32—68no—
144C2×D32—68no—
145C2×D32—616no—
146C2×D32—128no—
147C2×D32—128no—
148C2×D32—1216no—
149C32×D4—34no—
150C32×D4—62no—
151C2×C32×C4—21no—
152C2×C32×C4—22no—
153C2×C32×C4—41no—
154C2×C32×C4C22×D312yes—
155C2×C32×C4C22×D311yescharacteristic
156#638—68no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—32no—
159C2×C3⋊Dic3—62no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—121no—
162C4⋊(C3⋊D3)—22no—
163C4⋊(C3⋊D3)C22×D314yescomplemented
164C4×C3⋊D3—32no—
165C4×C3⋊D3—34no—
166C4×C3⋊D3—34no—
167C4×C3⋊D3—32no—
168C4×C3⋊D3—62no—
169C4×C3⋊D3—64no—
170C4×C3⋊D3—62no—
171C4×C3⋊D3—64no—
172C4×C3⋊D3—122no—
173C4×C3⋊D3—122no—
174C3×D12—24no—
175C3×D12—22no—
176C3×D12—22no—
177C3×D12—24no—
178C3×D12—34no—
179C3×D12—34no—
180C3×D12—64no—
181C3×D12—64no—
182C3×D12—62no—
183C3×D12—62no—
184C3×D12C22×D314yescomplemented
185C3×D12C22×D314yescomplemented
186#625—68no—
187#625—68no—
188#625—616no—
189C32×C8—32no—
190C3⋊(C3⋊C8)C22×D312yescomplemented
191C3×C3⋊C8—22no—
192C3×C3⋊C8—34no—
193C3×C3⋊C8—62no—
194C3×C3⋊C8C22×D314yescomplemented
195C22×D8—271noSylow
196C2×C33C2×D812yescomplemented
197C2×C33C22×D411yescharacteristic
198C3⋊(C3⋊D3)C2×D814yescomplemented
199C3×C3⋊D3—44no—
200C32×D3—44no—
201C23×D3—182no—
202C23×D3—182no—
203C23×D3—182no—
204C23×D3—184no—
205C2×C3×D4—32no—
206C2×C3×D4—32no—
207C2×C3×D4—62no—
208C2×C3×D4—92no—
209C2×#67—182no—
210C2×#67—182no—
211C2×#67—182no—
212C2×#67—184no—
213D3×D4—98no—
214D3×D4—98no—
215D3×D4—98no—
216D3×D4—188no—
217D3×D4—184no—
218D3×D4—184no—
219D3×D4—184no—
220D3×D4—188no—
221C2×D12—32no—
222C2×D12—32no—
223C2×D12—62no—
224C2×D12D3212yescomplemented
225C2×C4×D3—91no—
226C2×C4×D3—92no—
227C2×C4×D3—181no—
228C2×C4×D3—182no—
229C2×C4×D3—361no—
230C3×D8—98no—
231C3×D8—94no—
232C2×C3×C8—32no—
233C2×C3×C8—61no—
234C2×C3×C8—91no—
235#213—38no—
236#213—38no—
237#213—98no—
238C2×C3⋊C8—32no—
239C2×C3⋊C8—61no—
240C2×C3⋊C8D3211yescharacteristic, complemented
241D24—38no—
242D24—94no—
243D3×C8—98no—
244D3×C8—94no—
245D3×C8—184no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—41no—
249C22×C32—64no—
250C22×C32—124no—
251C22×C32D3×D412yes—
252C22×C32D3×D411yescharacteristic
253C2×C3⋊D3—24no—
254C2×C3⋊D3—34no—
255C2×C3⋊D3—38no—
256C2×C3⋊D3—32no—
257C2×C3⋊D3—34no—
258C2×C3⋊D3—44no—
259C2×C3⋊D3—68no—
260C2×C3⋊D3—64no—
261C2×C3⋊D3—64no—
262C2×C3⋊D3—62no—
263C2×C3⋊D3—124no—
264C2×C3⋊D3—122no—
265C2×C3⋊D3—1216no—
266C2×C3×D3—28no—
267C2×C3×D3—24no—
268C2×C3×D3—24no—
269C2×C3×D3—48no—
270C2×C3×D3—44no—
271C2×C3×D3—44no—
272C2×C3×D3—68no—
273C2×C3×D3—64no—
274C2×C3×D3—64no—
275C2×C3×D3—128no—
276C2×C3×D3—124no—
277C2×C3×D3—124no—
278D32—1216no—
279D32—1216no—
280D32—1232no—
281C32×C4—21no—
282C32×C4—22no—
283C32×C4—21no—
284C32×C4—22no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4D3×D412yes—
288C32×C4D3×D411yescharacteristic
289C32×C4C23×D312yes—
290C32×C4C23×D311yescharacteristic
291C3⋊Dic3—32no—
292C3⋊Dic3—34no—
293C3⋊Dic3—64no—
294C3⋊Dic3—62no—
295C3⋊Dic3—122no—
296C22×D4—272no—
297C2×D8—92no—
298C2×D8—91no—
299C2×D8—271no—
300C2×D8—278no—
301C22×C8—271no—
302C33C22×D811yescharacteristic, complemented, Sylow
303C23×C3—64no—
304C23×C3—62no—
305C23×C3—62no—
306C23×C3—182no—
307C22×D3—22no—
308C22×D3—64no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—91no—
312C22×D3—92no—
313C22×D3—181no—
314C22×D3—182no—
315C22×D3—188no—
316C22×D3—188no—
317C22×D3—188no—
318C22×D3—1816no—
319C22×D3—361no—
320C22×D3—368no—
321C22×D3—368no—
322C22×D3—368no—
323C22×D3—3616no—
324C3×D4—34no—
325C3×D4—34no—
326C3×D4—64no—
327C3×D4—64no—
328C3×D4—62no—
329C3×D4—62no—
330C3×D4—94no—
331C3×D4—182no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—41no—
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C2×D3212yes—
337#67—188no—
338#67—188no—
339#67—188no—
340#67—1816no—
341C2×Dic3—91no—
342C2×Dic3—92no—
343C2×Dic3—181no—
344C2×Dic3—182no—
345C2×Dic3—361no—
346D12—22no—
347D12—34no—
348D12—34no—
349D12—64no—
350D12—64no—
351D12—62no—
352D12—62no—
353D12C2×D3214yescomplemented
354C4×D3—92no—
355C4×D3—94no—
356C4×D3—92no—
357C4×D3—94no—
358C4×D3—182no—
359C4×D3—182no—
360C4×D3—184no—
361C4×D3—184no—
362C4×D3—362no—
363C4×D3—362no—
364C3×C8—34no—
365C3×C8—62no—
366C3×C8—92no—
367C3⋊C8—34no—
368C3⋊C8—62no—
369C3⋊C8C2×D3212yescomplemented
370C2×C32—21no—
371C2×C32—22no—
372C2×C32—22no—
373C2×C32—24no—
374C2×C32—42no—
375C2×C32—41no—
376C2×C32—124no—
377C2×C32D3×D814yescomplemented
378C2×C32D3×D812yescomplemented
379C2×C32C2×D3×D412yes—
380C2×C32C2×D3×D411yescharacteristic
381C3⋊D3—34no—
382C3⋊D3—38no—
383C3⋊D3—44no—
384C3⋊D3—68no—
385C3⋊D3—64no—
386C3⋊D3—124no—
387C3×D3—48no—
388C3×D3—44no—
389C3×D3—44no—
390C3×D3—128no—
391C3×D3—124no—
392C3×D3—124no—
393C24—542no—
394C2×D4—32no—
395C2×D4—92no—
396C2×D4—278no—
397C2×D4—544no—
398C2×D4—542no—
399C22×C4—271no—
400D8—98no—
401D8—94no—
402D8—274no—
403C2×C8—31no—
404C2×C8—91no—
405C2×C8—274no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—68no—
410C22×C3—64no—
411C22×C3—64no—
412C22×C3—128no—
413C22×C3—124no—
414C22×C3—124no—
415C22×C3—184no—
416C22×C3—364no—
417C22×C3#380211yescharacteristic
418C22×C3#381112yes—
419C2×D3—24no—
420C2×D3—44no—
421C2×D3—68no—
422C2×D3—64no—
423C2×D3—64no—
424C2×D3—92no—
425C2×D3—94no—
426C2×D3—94no—
427C2×D3—98no—
428C2×D3—128no—
429C2×D3—124no—
430C2×D3—124no—
431C2×D3—184no—
432C2×D3—182no—
433C2×D3—188no—
434C2×D3—184no—
435C2×D3—362no—
436C2×D3—364no—
437C2×D3—3616no—
438C2×D3—3616no—
439C2×D3—3616no—
440C2×D3—3632no—
441C3×C4—21no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—21no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4#380211yescharacteristic
448C3×C4#381112yes—
449C3×C4C22×D3211yescharacteristic
450C3×C4C22×D3212yes—
451Dic3—92no—
452Dic3—94no—
453Dic3—184no—
454Dic3—182no—
455Dic3—362no—
456C32—22no—
457C32—21no—
458C32—41no—
459C32C2×D3×D811yescharacteristic, complemented
460C32C2×D3×D812yescomplemented
461C23—62no—
462C23—182no—
463C23—271no—
464C23—548no—
465C23—1088no—
466D4—34no—
467D4—62no—
468D4—94no—
469D4—182no—
470D4—548no—
471C2×C4—271no—
472C2×C4—272no—
473C2×C4—272no—
474C2×C4D3311yescharacteristic
475C8—32no—
476C8—92no—
477C2×C3—24no—
478C2×C3—22no—
479C2×C3—22no—
480C2×C3—21no—
481C2×C3—42no—
482C2×C3—41no—
483C2×C3—128no—
484C2×C3—124no—
485C2×C3—124no—
486C2×C3—364no—
487C2×C3#6379712yescomplemented
488C2×C3#6392614yescomplemented
489C2×C3C2×#380211yescharacteristic
490C2×C3C2×#381112yes—
491D3—44no—
492D3—98no—
493D3—94no—
494D3—128no—
495D3—124no—
496D3—124no—
497D3—188no—
498D3—184no—
499D3—364no—
500C22—64no—
501C22—124no—
502C22—184no—
503C22—274no—
504C22—272no—
505C22—364no—
506C22—10816no—
507C22#8895411yescharacteristic
508C4—272no—
509C4#8895411yescharacteristic
510C4C2×D3311yescharacteristic
511C3—22no—
512C3—21no—
513C3—41no—
514C3C2×#6379711yescharacteristic, complemented
515C3C2×#6392612yescomplemented
516C2—124no—
517C2—274no—
518C2—364no—
519C2#1172283212yesdirect product factor
520C2C2×#8895411yescharacteristic
subgroupG/H
centerC22#88954
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
hypercenterC2×C4D33
hypocenterC33C22×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89026C2semidirect
2. C2×#88810C2semidirect
3. C2×#88785C2semidirect
4. C2×C3×#3714C2semidirect
5. C2×C3×#3713C2semidirect
6. #11722832C2direct (N × H)
11. C2×C3×C4⋊(C3⋊D3)C22semidirect
12. C2×C32×D12C22semidirect
15. #89026C22semidirect
16. C2×C3×C3⋊(C3⋊C8)C22semidirect
17. C2×C32×C3⋊C8C22semidirect
18. #88810C22semidirect
19. #88785C22semidirect
20. C3×#3714C22semidirect
21. C3×#3713C22semidirect
32. C2×#3714D3semidirect
34. C2×#3713D3semidirect
48. C3×C4⋊(C3⋊D3)C23semidirect
50. C32×D12C23semidirect
52. C3×C3⋊(C3⋊C8)C23semidirect
53. C32×C3⋊C8C23semidirect
64. C2×C4⋊(C3⋊D3)C2×D3semidirect
74. C2×C3×D12C2×D3semidirect
75. C2×C3×D12C2×D3semidirect
86. C2×C3⋊(C3⋊C8)C2×D3semidirect
93. C2×C3×C3⋊C8C2×D3semidirect
96. #3714C2×D3semidirect
98. #3713C2×D3semidirect
102. C2×C3⋊(C3⋊D3)D8semidirect
163. C4⋊(C3⋊D3)C22×D3semidirect
184. C3×D12C22×D3semidirect
185. C3×D12C22×D3semidirect
190. C3⋊(C3⋊C8)C22×D3semidirect
194. C3×C3⋊C8C22×D3semidirect
196. C2×C33C2×D8semidirect
198. C3⋊(C3⋊D3)C2×D8semidirect
224. C2×D12D32semidirect
240. C2×C3⋊C8D32semidirect
302. C33C22×D8semidirect
353. D12C2×D32semidirect
369. C3⋊C8C2×D32semidirect
377. C2×C32D3×D8semidirect
378. C2×C32D3×D8semidirect
459. C32C2×D3×D8semidirect
460. C32C2×D3×D8semidirect
487. C2×C3#63797semidirect
488. C2×C3#63926semidirect
514. C3C2×#63797semidirect
515. C3C2×#63926semidirect
519. C2#11722832direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890262
C2×#888102
C2×#887852
C2×C3×#37142
C2×C3×#37132
#117228322
C2×#639263
C2×#637973