Finite Group Databasebeta

← C2×#11722873 #13170657

Subgroups — C2×#11722873 #13170657

Order 1728.

1562 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1562 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×#89018C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C2×#88809C211yesmaximal, characteristic, complemented
4C2×#88790C212yesmaximal, complemented
5C2×C3×#3715C212yesmaximal, complemented
6#11722873C218yesmaximal, direct product factor
7C2×#63941—31nomaximal
8C2×#63934—32nomaximal
9C22×C3×D32—21no—
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2212yes—
13C2×C32×Dic6C2211yescharacteristic, complemented
14C2×#6292—21no—
15C2×#6289—21no—
16#89018—22no—
17#89018C2214yescomplemented
18C2×C3×#625—21no—
19C3×#3802—22no—
20C3×#3802C2214yescomplemented
21C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
22#88809C2214yescomplemented
23#88790C2218yescomplemented
24C3×#3715C2218yescomplemented
25C2×#3832—31no—
26C2×C3×D3×D4—32no—
27C2×#3802D311yescharacteristic, complemented
28C2×#3799—32no—
29C2×#3799—61no—
30C2×#3753—32no—
31C2×C3×#215—32no—
32C2×C3×#214—32no—
33#63941—38no—
34#63934—316no—
35C2×#3715D312yescomplemented
36C2×#3714—32no—
37C2×#3714—32no—
38C2×#3714—62no—
39C2×#3712—31no—
40C2×#3711—32no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C22×C32×D3—22no—
44C2×C3×D32—24no—
45C2×C3×D32—44no—
46C2×C33×C4C2311yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))C2312yescomplemented
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3D412yes—
51C3×C4×C3⋊D3C2312yes—
52C2×C32×Dic3—21no—
53C32×D12—22no—
54C32×D12C2314yes—
55C32×Dic6—21no—
56C32×Dic6C2312yescomplemented
57#6292—24no—
58#6289—24no—
59C3×#625—24no—
60C3×C3⋊(C3⋊C8)C2314yescomplemented
61C2×#932—91no—
62C2×#914—92no—
63C23×C3×D3—62no—
64C22×D32—21no—
65C2×C32×Q8—31no—
66C2×C32×D4—32no—
67#3832—34no—
68#3832—62no—
69C2×C4⋊(C3⋊D3)—32no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—62no—
74C2×C4⋊(C3⋊D3)—121no—
75C2×C4×C3⋊D3—61no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C2×C3×#67—62no—
78C3×D3×D4—38no—
79C3×D3×D4—64no—
80C2×C3×D12—22no—
81C2×C3×D12C2×D312yes—
82C2×C3×D12C2×D312yes—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—21no—
86C2×C3×Dic6C2×D312yescomplemented
87C2×#626—62no—
88C2×#626—62no—
89C2×#625—21no—
90C2×#624—62no—
91C2×#624—62no—
92#3802—22no—
93#3802C2×D314yescomplemented
94#3799—38no—
95#3799—64no—
96#3799—64no—
97#3799—64no—
98#3753—38no—
99C2×C3⋊(C3⋊C8)C2×D312yescomplemented
100C3×#215—38no—
101C3×#214—38no—
102C2×C3×C3⋊C8—32no—
103C2×C3×C3⋊C8—32no—
104C2×C3×C3⋊C8—62no—
105#3715C2×D318yescomplemented
106#3714—38no—
107#3714—38no—
108#3714—68no—
109#3712—34no—
110#3711—38no—
111C22×C33C2×D411yescharacteristic
112C2×C3⋊(C3⋊D3)—22no—
113C2×C3⋊(C3⋊D3)—42no—
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3C2×D412yes—
116C2×C32×D3—24no—
117C2×C32×D3—44no—
118C3×D32—48no—
119C33×C4C2×D411yescharacteristic
120C33×C4C2411yescharacteristic
121C3×C3⋊Dic3C2×D412yes—
122C32×Dic3—22no—
123C22×C3×D4—91no—
124C2×#239—92no—
125C2×#239—181no—
126C2×D3×D4—32no—
127C2×#235—91no—
128C2×C3×SD16—92no—
129C2×#215—32no—
130C2×#214—32no—
131#932—98no—
132C2×#213—92no—
133C2×#213—92no—
134C2×#213—182no—
135C2×#208—91no—
136#914—916no—
137C2×D24—92no—
138C2×ΓL(1,𝔽25)—92no—
139C23×C32—62no—
140C22×C3⋊D3—61no—
141C22×C3⋊D3—62no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—122no—
144C22×C3⋊D3—122no—
145C22×C3⋊D3#6711yescharacteristic
146C22×C3×D3—22no—
147C22×C3×D3—22no—
148C22×C3×D3—32no—
149C22×C3×D3—42no—
150C22×C3×D3—61no—
151C22×C3×D3—68no—
152C22×C3×D3—128no—
153C2×D32—24no—
154C2×D32—44no—
155C32×Q8—32no—
156C32×Q8—61no—
157C32×D4—34no—
158C32×D4—62no—
159C2×C32×C4—22no—
160C2×C32×C4—21no—
161C2×C32×C4—41no—
162C2×C32×C4—61no—
163C2×C32×C4C22×D312yes—
164C2×C32×C4C22×D311yescharacteristic
165C2×C3⋊Dic3#6711yescharacteristic
166C4⋊(C3⋊D3)—32no—
167C4⋊(C3⋊D3)—34no—
168C4⋊(C3⋊D3)—62no—
169C4⋊(C3⋊D3)—64no—
170C4⋊(C3⋊D3)—64no—
171C4⋊(C3⋊D3)—61no—
172C4⋊(C3⋊D3)—62no—
173C4⋊(C3⋊D3)—62no—
174C4⋊(C3⋊D3)—122no—
175C4⋊(C3⋊D3)—122no—
176C4⋊(C3⋊D3)—122no—
177C4×C3⋊D3—64no—
178C4×C3⋊D3#6712yes—
179C4×C3⋊D3C22×D312yes—
180C3×#67—68no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—61no—
185C3×D12—22no—
186C3×D12—22no—
187C3×D12—24no—
188C3×D12—42no—
189C3×D12C22×D314yes—
190C3×D12C22×D314yes—
191C3×C4×D3—34no—
192C3×C4×D3—34no—
193C3×C4×D3—62no—
194C3×C4×D3—62no—
195C3×Dic6—22no—
196C3×Dic6—22no—
197C3×Dic6—22no—
198C3×Dic6C22×D314yescomplemented
199#626—68no—
200#626—68no—
201#625—24no—
202#624—68no—
203#624—68no—
204C3⋊(C3⋊C8)C22×D314yescomplemented
205C3×C3⋊C8—34no—
206C3×C3⋊C8—34no—
207C3×C3⋊C8—64no—
208C2×AGL(1,ℤ/8)—271noSylow
209C2×C33AGL(1,ℤ/8)12yescomplemented
210C2×C33C22×D411yescharacteristic
211C3⋊(C3⋊D3)—42no—
212C3×C3⋊D3—22no—
213C32×D3—44no—
214C24×C3—181no—
215C23×D3—62no—
216C2×C3×Q8—32no—
217C2×C3×Q8—61no—
218C2×C3×D4—32no—
219C2×C3×D4—32no—
220C2×C3×D4—62no—
221C2×C3×D4—94no—
222C2×C3×D4—182no—
223C2×C3×D4—181no—
224C22×C3×C4—91no—
225C2×#67—62no—
226C2×#67—181no—
227#239—98no—
228#239—184no—
229#239—184no—
230#239—184no—
231D3×D4—38no—
232D3×D4—64no—
233#235—94no—
234#235—182no—
235C2×D12—92no—
236C2×D12—91no—
237C2×D12—181no—
238C2×D12—182no—
239C2×D12—182no—
240C2×D12—182no—
241C2×D12—361no—
242C2×D12D3212yes—
243C2×C4×D3—32no—
244C2×C4×D3—61no—
245C2×C4×D3—181no—
246C2×C4×D3—182no—
247C2×C4×D3—182no—
248C2×Dic6D3211yescharacteristic, complemented
249C3×SD16—98no—
250C2×C3×C8—92no—
251#215—38no—
252#214—38no—
253#213—98no—
254#213—98no—
255#213—188no—
256#208—94no—
257C2×C3⋊C8—32no—
258C2×C3⋊C8—32no—
259C2×C3⋊C8—62no—
260D24—98no—
261ΓL(1,𝔽25)—98no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—64no—
266C22×C32—124no—
267C22×C32D3×D412yes—
268C22×C32C2×#6711yescharacteristic
269C2×C3⋊D3—22no—
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—124no—
277C2×C3⋊D3—124no—
278C2×C3⋊D3—244no—
279C2×C3⋊D3—244no—
280C2×C3⋊D3C2×#6712yes—
281C2×C3×D3—24no—
282C2×C3×D3—24no—
283C2×C3×D3—34no—
284C2×C3×D3—44no—
285C2×C3×D3—44no—
286C2×C3×D3—44no—
287C2×C3×D3—62no—
288C2×C3×D3—64no—
289C2×C3×D3—84no—
290C2×C3×D3—122no—
291C2×C3×D3—1216no—
292D32—48no—
293C32×C4—22no—
294C32×C4—21no—
295C32×C4—22no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—62no—
300C32×C4D3×D412yes—
301C32×C4C2×#6711yescharacteristic
302C32×C4C23×D312yes—
303C32×C4C23×D311yescharacteristic
304C3⋊Dic3C2×#6712yes—
305C3×Dic3—24no—
306C3×Dic3—24no—
307C3×Dic3—34no—
308C3×Dic3—62no—
309C2×#41—271no—
310C22×D4—91no—
311AGL(1,ℤ/8)—278no—
312C2×SD16—92no—
313C2×D8—272no—
314C2×M16—271no—
315C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
316C23×C3—62no—
317C23×C3—62no—
318C23×C3—91no—
319C23×C3—122no—
320C23×C3—184no—
321C23×C3—364no—
322C22×D3—22no—
323C22×D3—32no—
324C22×D3—61no—
325C22×D3—68no—
326C22×D3—128no—
327C22×D3—182no—
328C22×D3—182no—
329C22×D3—181no—
330C22×D3—362no—
331C22×D3—362no—
332C3×Q8—34no—
333C3×Q8—62no—
334C3×Q8—62no—
335C3×Q8—62no—
336C3×D4—34no—
337C3×D4—34no—
338C3×D4—64no—
339C3×D4—62no—
340C3×D4—62no—
341C3×D4—122no—
342C3×D4—184no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—91no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4C2×D3212yes—
352C2×C3×C4C2×D3211yescharacteristic
353#67—68no—
354#67—184no—
355C2×Dic3—21no—
356C2×Dic3—32no—
357C2×Dic3—61no—
358D12—22no—
359D12—94no—
360D12—92no—
361D12—182no—
362D12—184no—
363D12—182no—
364D12—188no—
365D12—188no—
366D12—182no—
367D12—181no—
368D12—362no—
369D12—362no—
370D12—362no—
371D12C2×D3214yes—
372C4×D3—34no—
373C4×D3—34no—
374C4×D3—62no—
375C4×D3—62no—
376C4×D3—184no—
377C4×D3—188no—
378C4×D3—188no—
379Dic6—21no—
380Dic6C2×D3212yescomplemented
381C3×C8—94no—
382C3⋊C8—34no—
383C3⋊C8—34no—
384C3⋊C8—64no—
385C2×C32—22no—
386C2×C32—21no—
387C2×C32—24no—
388C2×C32—22no—
389C2×C32—42no—
390C2×C32—41no—
391C2×C32—124no—
392C2×C32#91414yescomplemented
393C2×C32#93212yescomplemented
394C2×C32C2×D3×D412yes—
395C2×C32C22×#6711yescharacteristic
396C3⋊D3—22no—
397C3⋊D3—122no—
398C3⋊D3—124no—
399C3⋊D3—124no—
400C3⋊D3—244no—
401C3⋊D3—244no—
402C3×D3—44no—
403C3×D3—44no—
404C3×D3—64no—
405C3×D3—84no—
406C3×D3—122no—
407C24—181no—
408#41—274no—
409#41—542no—
410C2×Q8—31no—
411C2×D4—32no—
412C2×D4—94no—
413C2×D4—182no—
414C2×D4—181no—
415C2×D4—271no—
416C2×D4—541no—
417C22×C4—91no—
418C22×C4—541no—
419SD16—98no—
420D8—278no—
421M16—274no—
422C2×C8—92no—
423C22×C3—21no—
424C22×C3—22no—
425C22×C3—41no—
426C22×C3—64no—
427C22×C3—64no—
428C22×C3—92no—
429C22×C3—124no—
430C22×C3—124no—
431C22×C3—124no—
432C22×C3—182no—
433C22×C3—244no—
434C22×C3—368no—
435C22×C3D3×#6712yes—
436C22×C3#381111yescharacteristic
437C2×D3—24no—
438C2×D3—34no—
439C2×D3—44no—
440C2×D3—62no—
441C2×D3—64no—
442C2×D3—122no—
443C2×D3—1216no—
444C2×D3—184no—
445C2×D3—184no—
446C2×D3—182no—
447C2×D3—364no—
448C2×D3—364no—
449C2×D3—364no—
450C2×D3—364no—
451C2×D3—362no—
452C2×D3—724no—
453C2×D3—724no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—22no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—64no—
461C3×C4—64no—
462C3×C4—92no—
463C3×C4D3×#6712yes—
464C3×C4#381111yescharacteristic
465C3×C4C22×D3212yes—
466C3×C4C22×D3211yescharacteristic
467Dic3—22no—
468Dic3—34no—
469Dic3—62no—
470C32—22no—
471C32—21no—
472C32—41no—
473C32C2×#91412yescomplemented
474C32C2×#93211yescharacteristic, complemented
475C23—62no—
476C23—91no—
477C23—184no—
478C23—364no—
479C23—541no—
480Q8—32no—
481Q8—61no—
482D4—34no—
483D4—62no—
484D4—184no—
485D4—272no—
486D4—544no—
487D4—541no—
488C2×C4—61no—
489C2×C4—91no—
490C2×C4—92no—
491C2×C4—92no—
492C2×C4—544no—
493C2×C4D3311yescharacteristic
494C8—94no—
495C2×C3—22no—
496C2×C3—24no—
497C2×C3—21no—
498C2×C3—22no—
499C2×C3—42no—
500C2×C3—41no—
501C2×C3—124no—
502C2×C3—124no—
503C2×C3—182no—
504C2×C3—244no—
505C2×C3#6393414yescomplemented
506C2×C3#6394112yescomplemented
507C2×C3C2×D3×#6712yes—
508C2×C3C2×#381111yescharacteristic
509D3—44no—
510D3—64no—
511D3—122no—
512D3—364no—
513D3—364no—
514D3—362no—
515D3—724no—
516D3—724no—
517C22—64no—
518C22—92no—
519C22—124no—
520C22—182no—
521C22—368no—
522C22—542no—
523C22—1082no—
524C22#8895211yescharacteristic
525C4—62no—
526C4—92no—
527C4#8895211yescharacteristic
528C4C2×D3311yescharacteristic
529C3—21no—
530C3—22no—
531C3—41no—
532C3C2×#6393412yescomplemented
533C3C2×#6394111yescharacteristic, complemented
534C2—124no—
535C2—182no—
536C2—1082no—
537C2#1172287312yesdirect product factor
538C2C2×#8895211yescharacteristic
subgroupG/H
centerC22#88952
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
hypercenterC2×C4D33
hypocenterC33C2×AGL(1,ℤ/8)
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×#89018C2semidirect
2. C2×C3×#3802C2semidirect
3. C2×#88809C2semidirect
4. C2×#88790C2semidirect
5. C2×C3×#3715C2semidirect
6. #11722873C2direct (N × H)
10. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
13. C2×C32×Dic6C22semidirect
17. #89018C22semidirect
20. C3×#3802C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. #88809C22semidirect
23. #88790C22semidirect
24. C3×#3715C22semidirect
27. C2×#3802D3semidirect
35. C2×#3715D3semidirect
48. C4⋊(C3⋊(C3⋊D3))C23semidirect
56. C32×Dic6C23semidirect
60. C3×C3⋊(C3⋊C8)C23semidirect
86. C2×C3×Dic6C2×D3semidirect
93. #3802C2×D3semidirect
99. C2×C3⋊(C3⋊C8)C2×D3semidirect
105. #3715C2×D3semidirect
198. C3×Dic6C22×D3semidirect
204. C3⋊(C3⋊C8)C22×D3semidirect
209. C2×C33AGL(1,ℤ/8)semidirect
248. C2×Dic6D32semidirect
315. C33C2×AGL(1,ℤ/8)semidirect
380. Dic6C2×D32semidirect
392. C2×C32#914semidirect
393. C2×C32#932semidirect
473. C32C2×#914semidirect
474. C32C2×#932semidirect
505. C2×C3#63934semidirect
506. C2×C3#63941semidirect
532. C3C2×#63934semidirect
533. C3C2×#63941semidirect
537. C2#11722873direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890182
C2×C3×#38022
C2×#888092
C2×#887902
C2×C3×#37152
#117228732
C2×#639413
C2×#639343