Finite Group Databasebeta

← C2×#11722842 #13170584

Subgroups — C2×#11722842 #13170584

Order 1728.

1618 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1618 conjugacy classes of subgroups — too many to draw. Only its 212 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×#89040C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×D12C211yesmaximal, characteristic
3C2×#88797C212yesmaximal, complemented
4C2×C3×#3716C212yesmaximal, complemented
5C2×C3×#3710C211yesmaximal, characteristic, complemented
6#11722842C218yesmaximal, direct product factor
7C2×#63930—31nomaximal
8C2×D3×#204—32nomaximal
9C22×D3×C3⋊D3—21no—
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2212yescomplemented
13C2×C32×D12C2211yescharacteristic
14C2×#6296—21no—
15C2×#6294—21no—
16#89040—22no—
17#89040C2214yescomplemented
18C2×#6291—21no—
19C3⋊D3×D12—22no—
20C3⋊D3×D12C2214yes—
21C2×C32×C3⋊C8C2212yescomplemented
22#88797C2218yescomplemented
23C3×#3716C2218yescomplemented
24C3×#3710C2214yescomplemented
25C2×D4×C3⋊D3—31no—
26C2×D3×D12—32no—
27C2×D3×D12—61no—
28C2×#3797—31no—
29C2×D3×Dic6—32no—
30C2×#3744—32no—
31C2×C3×#214—32no—
32C2×C3×#204—32no—
33C2×C3×D3×C8—32no—
34#63930—38no—
35C2×#3717—32no—
36C2×#3717—32no—
37C2×#3717—62no—
38C2×#3716D312yescomplemented
39C2×#3710D311yescharacteristic, complemented
40D3×#204—316no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C22×C32×D3—21no—
44C2×D3×C3⋊D3—24no—
45C2×D3×C3⋊D3—44no—
46C2×C33×C4C2311yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))C2312yes—
49C2×C3×C3⋊Dic3—22no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3D412yes—
52C3×C4×C3⋊D3C2312yes—
53C3×#634—22no—
54C3×#634C2314yescomplemented
55C32×D12—21no—
56C32×D12C2312yes—
57#6296—24no—
58#6294—24no—
59#6291—24no—
60C32×C3⋊C8C2314yescomplemented
61C2×D3×SD16—92no—
62C22×#204—91no—
63C23×C3⋊D3—61no—
64C22×D32—62no—
65C22×D32—62no—
66C2×C32×D4—31no—
67C2×#638—61no—
68D4×C3⋊D3—34no—
69D4×C3⋊D3—62no—
70C2×C4⋊(C3⋊D3)—32no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—62no—
74C2×C4⋊(C3⋊D3)—121no—
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×#634C2×D312yescomplemented
77C2×C3×D12—21no—
78C2×C3×D12C2×D312yes—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—32no—
83C2×C3×Dic6—62no—
84C2×#627—62no—
85C2×#627—61no—
86C2×#626—62no—
87C2×#626—62no—
88C2×#624—61no—
89C2×D3×Dic3—62no—
90D3×D12—38no—
91D3×D12—64no—
92D3×D12—64no—
93D3×D12—64no—
94#3797—34no—
95#3797—62no—
96D3×Dic6—38no—
97D3×Dic6—64no—
98C2×C32×C8—32no—
99#3744—38no—
100C3×#214—38no—
101C2×C3×C3⋊C8—22no—
102C2×C3×C3⋊C8C2×D312yescomplemented
103C2×C3×C3⋊C8C2×D312yescomplemented
104C3×#204—38no—
105C3×D3×C8—38no—
106#3717—38no—
107#3717—38no—
108#3717—68no—
109#3716C2×D318yescomplemented
110#3710C2×D314yescomplemented
111C22×C33C2×D411yescharacteristic
112C2×C3⋊(C3⋊D3)—22no—
113C2×C3⋊(C3⋊D3)—42no—
114C2×C3×C3⋊D3SD1614yescomplemented
115C2×C3×C3⋊D3C2×D412yes—
116C2×C32×D3—22no—
117C2×C32×D3—42no—
118D3×C3⋊D3—48no—
119C33×C4C2×D411yescharacteristic
120C33×C4C2411yescharacteristic
121C3×C3⋊Dic3—24no—
122C3×C3⋊Dic3C2×D412yes—
123C2×D3×Q8—92no—
124C2×D3×D4—92no—
125C2×D3×D4—181no—
126C22×D12—91no—
127C22×Dic6—91no—
128C2×C3×SD16—92no—
129C22×C3×C8—91no—
130C2×#215—92no—
131C2×#214—32no—
132D3×SD16—916no—
133C2×#204—32no—
134C2×#204—98no—
135C2×#204—92no—
136C2×#204—92no—
137C2×#204—182no—
138C2×D3×C8—32no—
139C23×C32—61no—
140C22×C3⋊D3—61no—
141C22×C3⋊D3—64no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—62no—
144C22×C3⋊D3—122no—
145C22×C3⋊D3—122no—
146C22×C3⋊D3—124no—
147C22×C3⋊D3D1211yescharacteristic
148C22×C3×D3—22no—
149C22×C3×D3—22no—
150C22×C3×D3—32no—
151C22×C3×D3—61no—
152C2×D32—68no—
153C2×D32—68no—
154C2×D32—128no—
155C2×D32—128no—
156C32×D4—32no—
157C32×D4—61no—
158C2×C32×C4—22no—
159C2×C32×C4—21no—
160C2×C32×C4—41no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D312yes—
163#638—64no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3D1211yescharacteristic
166C4⋊(C3⋊D3)—32no—
167C4⋊(C3⋊D3)—34no—
168C4⋊(C3⋊D3)—62no—
169C4⋊(C3⋊D3)—64no—
170C4⋊(C3⋊D3)—61no—
171C4⋊(C3⋊D3)—62no—
172C4⋊(C3⋊D3)—62no—
173C4⋊(C3⋊D3)—122no—
174C4⋊(C3⋊D3)—122no—
175C4⋊(C3⋊D3)—122no—
176C4×C3⋊D3D1212yes—
177C4×C3⋊D3C22×D312yes—
178#634—22no—
179#634C22×D314yescomplemented
180C2×C3×Dic3—32no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—122no—
185C3×D12—22no—
186C3×D12—22no—
187C3×D12—22no—
188C3×D12C22×D314yes—
189C3×C4×D3—34no—
190C3×C4×D3—34no—
191C3×C4×D3—62no—
192C3×C4×D3—62no—
193C3×Dic6—34no—
194C3×Dic6—34no—
195C3×Dic6—64no—
196C3×Dic6—62no—
197C3×Dic6—62no—
198C3×Dic6—122no—
199#627—68no—
200#627—64no—
201#626—68no—
202#626—68no—
203#624—64no—
204D3×Dic3—68no—
205C32×C8—34no—
206C3×C3⋊C8—24no—
207C3×C3⋊C8C22×D314yescomplemented
208C3×C3⋊C8C22×D314yescomplemented
209C22×SD16—271noSylow
210C2×C33C2×SD1612yescomplemented
211C2×C33C22×D411yescharacteristic
212C3⋊(C3⋊D3)—42no—
213C3×C3⋊D3C2×SD1614yescomplemented
214C32×D3—42no—
215C23×D3—181no—
216C23×D3—182no—
217C23×D3—182no—
218C2×C3×Q8—92no—
219C2×C3×D4—32no—
220C2×C3×D4—61no—
221C22×C3×C4—91no—
222C2×#67—182no—
223C2×#67—182no—
224C22×Dic3—181no—
225D3×Q8—98no—
226D3×Q8—184no—
227D3×D4—98no—
228D3×D4—184no—
229D3×D4—184no—
230D3×D4—184no—
231C2×D12—91no—
232C2×D12—92no—
233C2×D12—94no—
234C2×D12—181no—
235C2×D12—182no—
236C2×D12—182no—
237C2×D12—181no—
238C2×D12—361no—
239C2×D12D3211yescharacteristic
240C2×C4×D3—32no—
241C2×C4×D3—61no—
242C2×C4×D3—182no—
243C2×Dic6—32no—
244C2×Dic6—32no—
245C2×Dic6—62no—
246C2×Dic6—94no—
247C2×Dic6—182no—
248C2×Dic6—182no—
249C2×Dic6—181no—
250C3×SD16—98no—
251C2×C3×C8—32no—
252C2×C3×C8—32no—
253C2×C3×C8—62no—
254C2×C3×C8—94no—
255#215—98no—
256#214—38no—
257C2×C3⋊C8D3212yescomplemented
258#204—38no—
259#204—98no—
260#204—98no—
261#204—188no—
262D3×C8—38no—
263C22×C32—22no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32—62no—
267C22×C32—122no—
268C22×C32C2×D1211yescharacteristic
269C22×C32D3×D412yes—
270C2×C3⋊D3—62no—
271C2×C3⋊D3—64no—
272C2×C3⋊D3—64no—
273C2×C3⋊D3—124no—
274C2×C3⋊D3—124no—
275C2×C3⋊D3—122no—
276C2×C3⋊D3—128no—
277C2×C3⋊D3—124no—
278C2×C3⋊D3—124no—
279C2×C3⋊D3—244no—
280C2×C3⋊D3—244no—
281C2×C3⋊D3#20414yescomplemented
282C2×C3⋊D3C2×D1212yes—
283C2×C3×D3—24no—
284C2×C3×D3—24no—
285C2×C3×D3—38no—
286C2×C3×D3—34no—
287C2×C3×D3—44no—
288C2×C3×D3—44no—
289C2×C3×D3—64no—
290C2×C3×D3—62no—
291D32—1216no—
292D32—1216no—
293C32×C4—22no—
294C32×C4—21no—
295C32×C4—22no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4C2×D1211yescharacteristic
300C32×C4D3×D412yes—
301C32×C4C23×D311yescharacteristic
302C32×C4C23×D312yes—
303C3⋊Dic3—24no—
304C3⋊Dic3C2×D1212yes—
305C3×Dic3—34no—
306C3×Dic3—62no—
307C3×Dic3—64no—
308C3×Dic3—64no—
309C3×Dic3—124no—
310C22×Q8—271no—
311C22×D4—271no—
312C2×SD16—92no—
313C2×SD16—272no—
314C2×SD16—278no—
315C22×C8—91no—
316C33C22×SD1611yescharacteristic, complemented, Sylow
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—91no—
320C22×D3—21no—
321C22×D3—32no—
322C22×D3—61no—
323C22×D3—184no—
324C22×D3—188no—
325C22×D3—188no—
326C22×D3—182no—
327C22×D3—182no—
328C22×D3—181no—
329C22×D3—362no—
330C22×D3—362no—
331C22×D3—368no—
332C22×D3—364no—
333C22×D3—368no—
334C3×Q8—94no—
335C3×Q8—182no—
336C3×D4—34no—
337C3×D4—62no—
338C3×D4—62no—
339C3×D4—62no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—41no—
343C2×C3×C4—91no—
344C2×C3×C4—92no—
345C2×C3×C4—92no—
346C2×C3×C4—182no—
347C2×C3×C4C2×D3212yes—
348C2×C3×C4C2×D3211yescharacteristic
349#67—188no—
350#67—188no—
351C2×Dic3—32no—
352C2×Dic3—61no—
353C2×Dic3—62no—
354C2×Dic3—62no—
355C2×Dic3—122no—
356C2×Dic3—184no—
357D12—21no—
358D12—92no—
359D12—94no—
360D12—182no—
361D12—184no—
362D12—182no—
363D12—184no—
364D12—182no—
365D12—181no—
366D12—362no—
367D12—362no—
368D12—362no—
369D12C2×D3212yes—
370C4×D3—34no—
371C4×D3—34no—
372C4×D3—62no—
373C4×D3—62no—
374C4×D3—188no—
375Dic6—34no—
376Dic6—34no—
377Dic6—64no—
378Dic6—62no—
379Dic6—62no—
380Dic6—122no—
381Dic6—188no—
382Dic6—184no—
383C3×C8—34no—
384C3×C8—34no—
385C3×C8—64no—
386C3⋊C8C2×D3214yescomplemented
387C2×C32—22no—
388C2×C32—24no—
389C2×C32—21no—
390C2×C32—22no—
391C2×C32—42no—
392C2×C32—41no—
393C2×C32—122no—
394C2×C32C2×#20412yescomplemented
395C2×C32D3×SD1614yescomplemented
396C2×C32C22×D1211yescharacteristic
397C2×C32C2×D3×D412yes—
398C3⋊D3—122no—
399C3⋊D3—124no—
400C3⋊D3—124no—
401C3⋊D3—244no—
402C3⋊D3—244no—
403C3⋊D3C2×#20414yescomplemented
404C3×D3—38no—
405C3×D3—44no—
406C3×D3—44no—
407C3×D3—64no—
408C24—541no—
409C2×Q8—92no—
410C2×Q8—274no—
411C2×Q8—541no—
412C2×Q8—542no—
413C2×D4—31no—
414C2×D4—274no—
415C2×D4—271no—
416C2×D4—542no—
417C2×D4—541no—
418C22×C4—91no—
419C22×C4—541no—
420SD16—98no—
421SD16—278no—
422C2×C8—32no—
423C2×C8—94no—
424C22×C3—21no—
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—64no—
428C22×C3—64no—
429C22×C3—92no—
430C22×C3—94no—
431C22×C3—124no—
432C22×C3—124no—
433C22×C3D3×D1212yes—
434C22×C3#381111yescharacteristic
435C2×D3—22no—
436C2×D3—34no—
437C2×D3—38no—
438C2×D3—42no—
439C2×D3—62no—
440C2×D3—64no—
441C2×D3—184no—
442C2×D3—184no—
443C2×D3—182no—
444C2×D3—364no—
445C2×D3—364no—
446C2×D3—3616no—
447C2×D3—368no—
448C2×D3—3616no—
449C2×D3—364no—
450C2×D3—364no—
451C2×D3—362no—
452C2×D3—724no—
453C2×D3—724no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—22no—
457C3×C4—21no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—92no—
461C3×C4—184no—
462C3×C4D3×D1212yes—
463C3×C4#381111yescharacteristic
464C3×C4C22×D3212yes—
465C3×C4C22×D3211yescharacteristic
466Dic3—34no—
467Dic3—62no—
468Dic3—64no—
469Dic3—64no—
470Dic3—124no—
471C32—21no—
472C32—22no—
473C32—41no—
474C32C22×#20411yescharacteristic, complemented
475C32C2×D3×SD1612yescomplemented
476C23—61no—
477C23—91no—
478C23—544no—
479C23—541no—
480C23—1084no—
481Q8—94no—
482Q8—182no—
483Q8—544no—
484D4—32no—
485D4—61no—
486D4—272no—
487D4—544no—
488D4—541no—
489C2×C4—91no—
490C2×C4—92no—
491C2×C4—92no—
492C2×C4—182no—
493C2×C4—544no—
494C2×C4D3311yescharacteristic
495C8—34no—
496C2×C3—22no—
497C2×C3—24no—
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—42no—
501C2×C3—41no—
502C2×C3—94no—
503C2×C3—124no—
504C2×C3—124no—
505C2×C3D3×#20414yescomplemented
506C2×C3#6393012yescomplemented
507C2×C3C2×D3×D1212yes—
508C2×C3C2×#381111yescharacteristic
509D3—38no—
510D3—42no—
511D3—64no—
512D3—364no—
513D3—362no—
514D3—364no—
515D3—724no—
516D3—724no—
517C22—62no—
518C22—94no—
519C22—92no—
520C22—122no—
521C22—542no—
522C22—1088no—
523C22—1082no—
524C22#8895511yescharacteristic
525C4—92no—
526C4—184no—
527C4#8895511yescharacteristic
528C4C2×D3311yescharacteristic
529C3—21no—
530C3—22no—
531C3—41no—
532C3C2×D3×#20412yescomplemented
533C3C2×#6393011yescharacteristic, complemented
534C2—94no—
535C2—122no—
536C2—1082no—
537C2#1172284212yesdirect product factor
538C2C2×#8895511yescharacteristic
subgroupG/H
centerC22#88955
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3⋊D3×D12C2
hypercenterC2×C4D33
hypocenterC33C22×SD16
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×#89040C2semidirect
3. C2×#88797C2semidirect
4. C2×C3×#3716C2semidirect
5. C2×C3×#3710C2semidirect
6. #11722842C2direct (N × H)
12. C2×C3×#634C22semidirect
17. #89040C22semidirect
21. C2×C32×C3⋊C8C22semidirect
22. #88797C22semidirect
23. C3×#3716C22semidirect
24. C3×#3710C22semidirect
38. C2×#3716D3semidirect
39. C2×#3710D3semidirect
54. C3×#634C23semidirect
60. C32×C3⋊C8C23semidirect
76. C2×#634C2×D3semidirect
102. C2×C3×C3⋊C8C2×D3semidirect
103. C2×C3×C3⋊C8C2×D3semidirect
109. #3716C2×D3semidirect
110. #3710C2×D3semidirect
114. C2×C3×C3⋊D3SD16semidirect
179. #634C22×D3semidirect
207. C3×C3⋊C8C22×D3semidirect
208. C3×C3⋊C8C22×D3semidirect
210. C2×C33C2×SD16semidirect
213. C3×C3⋊D3C2×SD16semidirect
257. C2×C3⋊C8D32semidirect
281. C2×C3⋊D3#204semidirect
316. C33C22×SD16semidirect
386. C3⋊C8C2×D32semidirect
394. C2×C32C2×#204semidirect
395. C2×C32D3×SD16semidirect
403. C3⋊D3C2×#204semidirect
474. C32C22×#204semidirect
475. C32C2×D3×SD16semidirect
505. C2×C3D3×#204semidirect
506. C2×C3#63930semidirect
532. C3C2×D3×#204semidirect
533. C3C2×#63930semidirect
537. C2#11722842direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3⋊D3×D122
C2×#887972
C2×C3×#37162
C2×C3×#37102
#117228422
C2×#639303
C2×D3×#2043