Finite Group Databasebeta

← C2×#11722881 #13170677

Subgroups — C2×#11722881 #13170677

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×C4⋊#1113C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3C2×#88791C212yesmaximal, complemented
4C2×C3×#3717C212yesmaximal, complemented
5C2×C3×#3712C211yesmaximal, characteristic, complemented
6#11722881C218yesmaximal, direct product factor
7C2×#63941—31nomaximal
8C2×#63794—32nomaximal
9C22×#1113—21no—
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic, complemented
14C2×#6295—21no—
15C4⋊#1113—22no—
16C4⋊#1113C2214yescomplemented
17C2×#6292—21no—
18C2×#6289—21no—
19#89018—22no—
20#89018C2214yescomplemented
21C2×C32×C3⋊C8C2212yescomplemented
22#88791C2218yescomplemented
23C3×#3717C2218yescomplemented
24C3×#3712C2214yescomplemented
25C2×#3832—31no—
26C2×#3802—31no—
27C2×D3×D12—32no—
28C2×#3799—32no—
29C2×#3799—61no—
30C2×#3745—32no—
31C2×C3×#215—32no—
32C2×C3×#204—32no—
33C2×C3×ΓL(1,𝔽25)—32no—
34#63941—38no—
35C2×#3717D312yescomplemented
36C2×#3714—32no—
37C2×#3714—32no—
38C2×#3714—62no—
39C2×#3712D311yescharacteristic, complemented
40#63794—316no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3D411yescharacteristic
44C2×#1113—24no—
45C2×#1113—44no—
46C2×C33×C4C2311yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))C2312yescomplemented
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4⋊(C3⋊D3)—22no—
51C3×C4⋊(C3⋊D3)C2314yes—
52C3×C4×C3⋊D3D412yes—
53C3×C4×C3⋊D3C2312yes—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6C2312yescomplemented
57#6295—24no—
58#6292—24no—
59#6289—24no—
60C32×C3⋊C8C2314yescomplemented
61C2×#914—92no—
62C2×#908—91no—
63C22×D32—62no—
64C22×D32—61no—
65C2×C32×Q8—31no—
66#3832—34no—
67#3832—62no—
68C2×C4⋊(C3⋊D3)—32no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—62no—
73C2×C4⋊(C3⋊D3)—121no—
74C2×C4⋊(C3⋊D3)C2×D312yes—
75C2×C4×C3⋊D3—61no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C2×C3×D12—32no—
78C2×C3×D12—32no—
79C2×C3×D12—62no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—61no—
82C2×C3×Dic6—21no—
83C2×C3×Dic6C2×D312yescomplemented
84C2×#626—62no—
85C2×#626—62no—
86C2×#626—62no—
87C2×#625—61no—
88C2×#624—62no—
89C2×#624—62no—
90#3802—34no—
91#3802—62no—
92D3×D12—38no—
93D3×D12—64no—
94#3799—38no—
95#3799—64no—
96#3799—64no—
97#3799—64no—
98C2×C32×C8—32no—
99#3745—38no—
100C3×#215—38no—
101C2×C3×C3⋊C8—22no—
102C2×C3×C3⋊C8C2×D312yescomplemented
103C2×C3×C3⋊C8C2×D312yescomplemented
104C3×#204—38no—
105C3×ΓL(1,𝔽25)—38no—
106#3717C2×D318yescomplemented
107#3714—38no—
108#3714—38no—
109#3714—68no—
110#3712C2×D314yescomplemented
111C22×C33C2×D411yescharacteristic
112C2×C3⋊(C3⋊D3)—22no—
113C2×C3⋊(C3⋊D3)—42no—
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—24no—
116C2×C3×C3⋊D3—44no—
117C2×C3×C3⋊D3C2×D412yes—
118#1113—48no—
119C33×C4C2×D411yescharacteristic
120C33×C4C2411yescharacteristic
121C3×C3⋊Dic3C2×D412yes—
122C32×Dic3—22no—
123C2×#239—92no—
124C2×#239—181no—
125C2×D3×D4—92no—
126C2×#235—91no—
127C22×D12—91no—
128C2×C3×SD16—92no—
129C2×C3×M16—91no—
130C2×#215—32no—
131C2×#213—92no—
132#914—916no—
133#908—98no—
134C2×D24—92no—
135C2×D24—92no—
136C2×D24—182no—
137C2×#204—32no—
138C2×ΓL(1,𝔽25)—32no—
139C22×C3⋊D3—22no—
140C22×C3⋊D3—61no—
141C22×C3⋊D3—62no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—122no—
144C22×C3⋊D3—122no—
145C22×C3⋊D3D1211yescharacteristic
146C22×C3×D3—32no—
147C22×C3×D3—61no—
148C22×C3×D3—62no—
149C22×C3×D3—62no—
150C22×C3×D3—122no—
151C2×D32—68no—
152C2×D32—64no—
153C2×D32—128no—
154C2×D32—124no—
155C32×Q8—32no—
156C32×Q8—61no—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—41no—
160C2×C32×C4—61no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D312yes—
163C2×C3⋊Dic3D1211yescharacteristic
164C4⋊(C3⋊D3)—22no—
165C4⋊(C3⋊D3)—32no—
166C4⋊(C3⋊D3)—34no—
167C4⋊(C3⋊D3)—62no—
168C4⋊(C3⋊D3)—64no—
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⋊D3)C22×D314yes—
177C4×C3⋊D3—64no—
178C4×C3⋊D3D1212yes—
179C4×C3⋊D3C22×D312yes—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—61no—
184C3×D12—34no—
185C3×D12—34no—
186C3×D12—64no—
187C3×D12—62no—
188C3×D12—62no—
189C3×D12—122no—
190C3×C4×D3—34no—
191C3×C4×D3—34no—
192C3×C4×D3—62no—
193C3×C4×D3—62no—
194C3×Dic6—22no—
195C3×Dic6—22no—
196C3×Dic6—22no—
197C3×Dic6C22×D314yescomplemented
198#626—68no—
199#626—68no—
200#626—68no—
201#625—64no—
202#624—68no—
203#624—68no—
204C32×C8—34no—
205C3×C3⋊C8—24no—
206C3×C3⋊C8C22×D314yescomplemented
207C3×C3⋊C8C22×D314yescomplemented
208C2×AGL(1,ℤ/8)—271noSylow
209C2×C33AGL(1,ℤ/8)12yescomplemented
210C2×C33C22×D411yescharacteristic
211C3⋊(C3⋊D3)—42no—
212C3×C3⋊D3—22no—
213C3×C3⋊D3—44no—
214C23×D3—182no—
215C23×D3—181no—
216C2×C3×Q8—32no—
217C2×C3×Q8—61no—
218C2×C3×D4—92no—
219C22×C3×C4—91no—
220C2×#67—181no—
221C2×#67—182no—
222#239—98no—
223#239—184no—
224#239—184no—
225#239—184no—
226D3×D4—98no—
227D3×D4—184no—
228#235—94no—
229#235—182no—
230C2×D12—32no—
231C2×D12—32no—
232C2×D12—62no—
233C2×D12—94no—
234C2×D12—91no—
235C2×D12—92no—
236C2×D12—181no—
237C2×D12—182no—
238C2×D12—182no—
239C2×D12—182no—
240C2×D12—182no—
241C2×D12—181no—
242C2×D12—361no—
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—
250C3×M16—94no—
251C2×C3×C8—32no—
252C2×C3×C8—32no—
253C2×C3×C8—62no—
254#215—38no—
255#213—98no—
256C2×C3⋊C8D3212yescomplemented
257D24—98no—
258D24—98no—
259D24—188no—
260#204—38no—
261ΓL(1,𝔽25)—38no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32C2×D1211yescharacteristic
266C22×C32D3×D412yes—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—24no—
269C2×C3⋊D3—44no—
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×D1212yes—
281C2×C3×D3—34no—
282C2×C3×D3—62no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C2×C3×D3—64no—
286C2×C3×D3—122no—
287C2×C3×D3—124no—
288C2×C3×D3—124no—
289C2×C3×D3—124no—
290C2×C3×D3—244no—
291D32—1216no—
292D32—128no—
293C32×C4—22no—
294C32×C4—21no—
295C32×C4—22no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—62no—
300C32×C4C2×D1211yescharacteristic
301C32×C4D3×D412yes—
302C32×C4C23×D311yescharacteristic
303C32×C4C23×D312yes—
304C3⋊Dic3C2×D1212yes—
305C3×Dic3—24no—
306C3×Dic3—24no—
307C3×Dic3—34no—
308C3×Dic3—62no—
309C2×#41—271no—
310C22×D4—271no—
311AGL(1,ℤ/8)—278no—
312C2×SD16—92no—
313C2×D8—272no—
314C2×M16—91no—
315C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
316C23×C3—91no—
317C23×C3—182no—
318C22×D3—32no—
319C22×D3—61no—
320C22×D3—62no—
321C22×D3—62no—
322C22×D3—122no—
323C22×D3—188no—
324C22×D3—184no—
325C22×D3—182no—
326C22×D3—182no—
327C22×D3—181no—
328C22×D3—362no—
329C22×D3—362no—
330C22×D3—368no—
331C22×D3—364no—
332C3×Q8—34no—
333C3×Q8—62no—
334C3×Q8—62no—
335C3×Q8—62no—
336C3×D4—94no—
337C3×D4—182no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—41no—
341C2×C3×C4—62no—
342C2×C3×C4—62no—
343C2×C3×C4—91no—
344C2×C3×C4—92no—
345C2×C3×C4—92no—
346C2×C3×C4C2×D3212yes—
347C2×C3×C4C2×D3211yescharacteristic
348#67—184no—
349#67—188no—
350C2×Dic3—21no—
351C2×Dic3—32no—
352C2×Dic3—61no—
353D12—34no—
354D12—34no—
355D12—64no—
356D12—62no—
357D12—62no—
358D12—92no—
359D12—94no—
360D12—122no—
361D12—182no—
362D12—184no—
363D12—182no—
364D12—188no—
365D12—188no—
366D12—184no—
367D12—182no—
368D12—181no—
369D12—362no—
370D12—362no—
371D12—362no—
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—34no—
382C3×C8—34no—
383C3×C8—64no—
384C3⋊C8C2×D3214yescomplemented
385C2×C32—22no—
386C2×C32—24no—
387C2×C32—21no—
388C2×C32—22no—
389C2×C32—42no—
390C2×C32—41no—
391C2×C32#90812yescomplemented
392C2×C32#91414yescomplemented
393C2×C32C22×D1211yescharacteristic
394C2×C32C2×D3×D412yes—
395C3⋊D3—22no—
396C3⋊D3—44no—
397C3⋊D3—122no—
398C3⋊D3—124no—
399C3⋊D3—124no—
400C3⋊D3—244no—
401C3⋊D3—244no—
402C3×D3—64no—
403C3×D3—122no—
404C3×D3—124no—
405C3×D3—124no—
406C3×D3—244no—
407C24—541no—
408#41—274no—
409#41—542no—
410C2×Q8—31no—
411C2×D4—92no—
412C2×D4—274no—
413C2×D4—271no—
414C2×D4—541no—
415C2×D4—542no—
416C2×D4—541no—
417C22×C4—91no—
418C22×C4—541no—
419SD16—98no—
420D8—278no—
421M16—94no—
422C2×C8—32no—
423C22×C3—21no—
424C22×C3—22no—
425C22×C3—41no—
426C22×C3—92no—
427C22×C3—182no—
428C22×C3—184no—
429C22×C3—364no—
430C22×C3D3×D1212yes—
431C22×C3#381111yescharacteristic
432C2×D3—34no—
433C2×D3—62no—
434C2×D3—64no—
435C2×D3—64no—
436C2×D3—64no—
437C2×D3—122no—
438C2×D3—124no—
439C2×D3—124no—
440C2×D3—124no—
441C2×D3—184no—
442C2×D3—184no—
443C2×D3—182no—
444C2×D3—244no—
445C2×D3—364no—
446C2×D3—364no—
447C2×D3—3616no—
448C2×D3—368no—
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×D1212yes—
464C3×C4#381111yescharacteristic
465C3×C4C22×D3212yes—
466C3×C4C22×D3211yescharacteristic
467Dic3—22no—
468Dic3—34no—
469Dic3—62no—
470C32—22no—
471C32—21no—
472C32—41no—
473C32C2×#90811yescharacteristic, complemented
474C32C2×#91412yescomplemented
475C23—91no—
476C23—182no—
477C23—544no—
478C23—541no—
479C23—1084no—
480Q8—32no—
481Q8—61no—
482D4—94no—
483D4—182no—
484D4—272no—
485D4—544no—
486D4—544no—
487D4—541no—
488C2×C4—61no—
489C2×C4—91no—
490C2×C4—92no—
491C2×C4—92no—
492C2×C4—544no—
493C2×C4D3311yescharacteristic
494C8—34no—
495C2×C3—22no—
496C2×C3—24no—
497C2×C3—21no—
498C2×C3—22no—
499C2×C3—42no—
500C2×C3—41no—
501C2×C3—182no—
502C2×C3—364no—
503C2×C3#6379414yescomplemented
504C2×C3#6394112yescomplemented
505C2×C3C2×D3×D1212yes—
506C2×C3C2×#381111yescharacteristic
507D3—64no—
508D3—122no—
509D3—124no—
510D3—124no—
511D3—244no—
512D3—364no—
513D3—364no—
514D3—362no—
515D3—724no—
516D3—724no—
517C22—92no—
518C22—182no—
519C22—184no—
520C22—364no—
521C22—542no—
522C22—1088no—
523C22—1082no—
524C22#8895511yescharacteristic
525C4—62no—
526C4—92no—
527C4#8895511yescharacteristic
528C4C2×D3311yescharacteristic
529C3—21no—
530C3—22no—
531C3—41no—
532C3C2×#6379412yescomplemented
533C3C2×#6394111yescharacteristic, complemented
534C2—182no—
535C2—364no—
536C2—1082no—
537C2#1172288112yesdirect product factor
538C2C2×#8895511yescharacteristic
subgroupG/H
centerC22#88955
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×C4⋊#1113C2semidirect
2. C2×#89018C2semidirect
3. C2×#88791C2semidirect
4. C2×C3×#3717C2semidirect
5. C2×C3×#3712C2semidirect
6. #11722881C2direct (N × H)
10. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
13. C2×C32×Dic6C22semidirect
16. C4⋊#1113C22semidirect
20. #89018C22semidirect
21. C2×C32×C3⋊C8C22semidirect
22. #88791C22semidirect
23. C3×#3717C22semidirect
24. C3×#3712C22semidirect
35. C2×#3717D3semidirect
39. C2×#3712D3semidirect
48. C4⋊(C3⋊(C3⋊D3))C23semidirect
56. C32×Dic6C23semidirect
60. C32×C3⋊C8C23semidirect
83. C2×C3×Dic6C2×D3semidirect
102. C2×C3×C3⋊C8C2×D3semidirect
103. C2×C3×C3⋊C8C2×D3semidirect
106. #3717C2×D3semidirect
110. #3712C2×D3semidirect
197. C3×Dic6C22×D3semidirect
206. C3×C3⋊C8C22×D3semidirect
207. C3×C3⋊C8C22×D3semidirect
209. C2×C33AGL(1,ℤ/8)semidirect
248. C2×Dic6D32semidirect
256. C2×C3⋊C8D32semidirect
315. C33C2×AGL(1,ℤ/8)semidirect
380. Dic6C2×D32semidirect
384. C3⋊C8C2×D32semidirect
391. C2×C32#908semidirect
392. C2×C32#914semidirect
473. C32C2×#908semidirect
474. C32C2×#914semidirect
503. C2×C3#63794semidirect
504. C2×C3#63941semidirect
532. C3C2×#63794semidirect
533. C3C2×#63941semidirect
537. C2#11722881direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×#890182
C2×#887912
C2×C3×#37172
C2×C3×#37122
#117228812
C2×#639413
C2×#637943