Finite Group Databasebeta

← C2×#11722872 #13170654

Subgroups — C2×#11722872 #13170654

Order 1728.

1530 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1530 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×#89014C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C2×#88809C211yesmaximal, characteristic, complemented
4C2×#88794C212yesmaximal, complemented
5C2×C3×#3713C212yesmaximal, complemented
6#11722872C218yesmaximal, direct product factor
7C2×#63932—32nomaximal
8C2×#63927—31nomaximal
9C22×C3×D32—21no—
10C2×#6308C2211yescharacteristic, complemented
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2212yes—
13C2×C32×D12C2211yescharacteristic, complemented
14C2×#6290—21no—
15C2×D3×C3⋊Dic3—21no—
16#89014—22no—
17#89014C2214yescomplemented
18C2×C3×#625—21no—
19C3×#3802—22no—
20C3×#3802C2214yescomplemented
21C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
22#88809C2214yescomplemented
23#88794C2218yescomplemented
24C3×#3713C2218yescomplemented
25C2×#3830—31no—
26C2×C3×D3×D4—32no—
27C2×#3802D311yescharacteristic, complemented
28C2×#3795—32no—
29C2×#3795—61no—
30C2×#3754—32no—
31C2×C3×#213—32no—
32C2×C3×#213—32no—
33#63932—316no—
34#63927—38no—
35C2×#3716—32no—
36C2×#3716—32no—
37C2×#3716—62no—
38C2×#3713D312yescomplemented
39C2×#3712—31no—
40C2×#3711—32no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C22×C32×D3—22no—
44C2×C3×D32—24no—
45C2×C3×D32—44no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48#6308—21no—
49#6308C2312yescomplemented
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3D412yes—
52C3×C4×C3⋊D3C2312yes—
53C32×D12—21no—
54C32×D12—22no—
55C32×D12C2314yes—
56C32×D12C2312yescomplemented
57#6290—24no—
58D3×C3⋊Dic3—24no—
59C3×#625—24no—
60C3×C3⋊(C3⋊C8)C2314yescomplemented
61C2×#932—91no—
62C2×#911—92no—
63C23×C3×D3—62no—
64C22×D32—21no—
65C2×C32×D4—31no—
66C2×C32×D4—32no—
67C2×#638—61no—
68C22×C3⋊Dic3—61no—
69#3830—34no—
70#3830—62no—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×#634—32no—
73C2×#634—31no—
74C2×#634—61no—
75C2×#634—62no—
76C2×#634—121no—
77C2×C3×#67—62no—
78C3×D3×D4—38no—
79C3×D3×D4—64no—
80C2×C3×D12—21no—
81C2×C3×D12—22no—
82C2×C3×D12C2×D312yes—
83C2×C3×D12C2×D312yes—
84C2×C3×D12C2×D312yescomplemented
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—61no—
87C2×#625—21no—
88C2×#625—62no—
89C2×#625—62no—
90C2×D3×Dic3—62no—
91C2×D3×Dic3—62no—
92#3802—22no—
93#3802C2×D314yescomplemented
94#3795—38no—
95#3795—64no—
96#3795—64no—
97#3795—64no—
98#3754—38no—
99C2×C3⋊(C3⋊C8)C2×D312yescomplemented
100C3×#213—38no—
101C3×#213—38no—
102C2×C3×C3⋊C8—32no—
103C2×C3×C3⋊C8—32no—
104C2×C3×C3⋊C8—62no—
105#3716—38no—
106#3716—38no—
107#3716—68no—
108#3713C2×D318yescomplemented
109#3712—34no—
110#3711—38no—
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3C2×D412yes—
114C2×C32×D3—22no—
115C2×C32×D3—24no—
116C2×C32×D3—42no—
117C2×C32×D3—44no—
118C3×D32—48no—
119C33×C4C2×D411yescharacteristic
120C33×C4C2411yescharacteristic
121C3⋊(C3⋊Dic3)—22no—
122C3×C3⋊Dic3C2×D412yes—
123C22×C3×D4—91no—
124C2×#237—92no—
125C2×#237—181no—
126C2×D3×D4—32no—
127C2×#235—91no—
128C2×C3×D8—92no—
129C2×#214—92no—
130C2×#214—92no—
131C2×#214—182no—
132#932—98no—
133C2×#213—32no—
134C2×#213—32no—
135C2×#208—91no—
136#911—916no—
137C2×#204—92no—
138C2×ΓL(1,𝔽25)—92no—
139C23×C32—61no—
140C23×C32—62no—
141C22×C3⋊D3#6711yescharacteristic
142C22×C3×D3—22no—
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3—22no—
146C22×C3×D3—32no—
147C22×C3×D3—42no—
148C22×C3×D3—61no—
149C22×C3×D3—68no—
150C22×C3×D3—128no—
151C2×D32—24no—
152C2×D32—44no—
153C32×D4—32no—
154C32×D4—34no—
155C32×D4—61no—
156C32×D4—62no—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—41no—
160C2×C32×C4C22×D312yes—
161C2×C32×C4C22×D311yescharacteristic
162#638—64no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—64no—
165C2×C3⋊Dic3—62no—
166C2×C3⋊Dic3—62no—
167C2×C3⋊Dic3—122no—
168C2×C3⋊Dic3—122no—
169C2×C3⋊Dic3#6711yescharacteristic
170C4×C3⋊D3#6712yes—
171C4×C3⋊D3C22×D312yes—
172#634—32no—
173#634—34no—
174#634—62no—
175#634—64no—
176#634—61no—
177#634—62no—
178#634—62no—
179#634—122no—
180#634—122no—
181#634—122no—
182C3×#67—68no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—61no—
185C3×D12—22no—
186C3×D12—22no—
187C3×D12—22no—
188C3×D12—22no—
189C3×D12—24no—
190C3×D12—22no—
191C3×D12—42no—
192C3×D12C22×D314yes—
193C3×D12C22×D314yes—
194C3×D12C22×D314yescomplemented
195C3×C4×D3—34no—
196C3×C4×D3—34no—
197C3×C4×D3—62no—
198C3×C4×D3—62no—
199#625—24no—
200#625—68no—
201#625—68no—
202D3×Dic3—68no—
203D3×Dic3—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—22no—
212C32×D3—42no—
213C32×D3—44no—
214C24×C3—181no—
215C23×D3—62no—
216C2×C3×D4—32no—
217C2×C3×D4—32no—
218C2×C3×D4—32no—
219C2×C3×D4—61no—
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—
227C2×#67—182no—
228C2×#67—182no—
229C22×Dic3—182no—
230C22×Dic3—182no—
231#237—98no—
232#237—184no—
233#237—184no—
234#237—184no—
235D3×D4—38no—
236D3×D4—64no—
237#235—94no—
238#235—182no—
239C2×D12D3212yes—
240C2×D12D3211yescharacteristic, complemented
241C2×C4×D3—32no—
242C2×C4×D3—61no—
243C2×C4×D3—181no—
244C2×Dic6—92no—
245C2×Dic6—91no—
246C2×Dic6—181no—
247C2×Dic6—182no—
248C2×Dic6—361no—
249C3×D8—98no—
250C2×C3×C8—92no—
251#214—98no—
252#214—98no—
253#214—188no—
254#213—38no—
255#213—38no—
256#208—94no—
257C2×C3⋊C8—32no—
258C2×C3⋊C8—32no—
259C2×C3⋊C8—62no—
260#204—98no—
261ΓL(1,𝔽25)—98no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—62no—
266C22×C32—64no—
267C22×C32—122no—
268C22×C32—124no—
269C22×C32D3×D412yes—
270C22×C32C2×#6711yescharacteristic
271C2×C3⋊D3—22no—
272C2×C3⋊D3C2×#6712yes—
273C2×C3×D3—24no—
274C2×C3×D3—24no—
275C2×C3×D3—24no—
276C2×C3×D3—24no—
277C2×C3×D3—34no—
278C2×C3×D3—44no—
279C2×C3×D3—44no—
280C2×C3×D3—44no—
281C2×C3×D3—44no—
282C2×C3×D3—44no—
283C2×C3×D3—62no—
284C2×C3×D3—64no—
285C2×C3×D3—84no—
286C2×C3×D3—122no—
287C2×C3×D3—1216no—
288D32—48no—
289C32×C4—22no—
290C32×C4—21no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—41no—
294C32×C4—41no—
295C32×C4D3×D412yes—
296C32×C4C2×#6711yescharacteristic
297C32×C4C23×D312yes—
298C32×C4C23×D311yescharacteristic
299C3⋊Dic3—62no—
300C3⋊Dic3—64no—
301C3⋊Dic3—64no—
302C3⋊Dic3—124no—
303C3⋊Dic3—124no—
304C3⋊Dic3C2×#6712yes—
305C3×Dic3—34no—
306C3×Dic3—62no—
307C2×#41—271no—
308C22×D4—91no—
309AGL(1,ℤ/8)—278no—
310C2×SD16—272no—
311C2×D8—92no—
312C2×M16—271no—
313C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
314C23×C3—62no—
315C23×C3—62no—
316C23×C3—62no—
317C23×C3—62no—
318C23×C3—91no—
319C23×C3—122no—
320C23×C3—184no—
321C23×C3—364no—
322C22×D3—22no—
323C22×D3—21no—
324C22×D3—32no—
325C22×D3—61no—
326C22×D3—68no—
327C22×D3—128no—
328C3×D4—34no—
329C3×D4—34no—
330C3×D4—34no—
331C3×D4—62no—
332C3×D4—64no—
333C3×D4—62no—
334C3×D4—62no—
335C3×D4—62no—
336C3×D4—62no—
337C3×D4—122no—
338C3×D4—184no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—41no—
342C2×C3×C4—91no—
343C2×C3×C4—92no—
344C2×C3×C4—92no—
345C2×C3×C4C2×D3212yes—
346C2×C3×C4C2×D3211yescharacteristic
347#67—68no—
348#67—184no—
349#67—188no—
350#67—188no—
351C2×Dic3—32no—
352C2×Dic3—61no—
353C2×Dic3—188no—
354C2×Dic3—188no—
355C2×Dic3—182no—
356C2×Dic3—182no—
357C2×Dic3—181no—
358C2×Dic3—362no—
359C2×Dic3—362no—
360D12—22no—
361D12—21no—
362D12C2×D3214yes—
363D12C2×D3212yescomplemented
364C4×D3—34no—
365C4×D3—34no—
366C4×D3—62no—
367C4×D3—62no—
368C4×D3—184no—
369Dic6—94no—
370Dic6—92no—
371Dic6—182no—
372Dic6—184no—
373Dic6—182no—
374Dic6—182no—
375Dic6—181no—
376Dic6—362no—
377Dic6—362no—
378Dic6—362no—
379C3×C8—94no—
380C3⋊C8—34no—
381C3⋊C8—34no—
382C3⋊C8—64no—
383C2×C32—22no—
384C2×C32—21no—
385C2×C32—22no—
386C2×C32—24no—
387C2×C32—42no—
388C2×C32—41no—
389C2×C32—122no—
390C2×C32—124no—
391C2×C32#91114yescomplemented
392C2×C32#93212yescomplemented
393C2×C32C2×D3×D412yes—
394C2×C32C22×#6711yescharacteristic
395C3⋊D3—22no—
396C3×D3—44no—
397C3×D3—44no—
398C3×D3—44no—
399C3×D3—44no—
400C3×D3—64no—
401C3×D3—84no—
402C3×D3—122no—
403C24—181no—
404#41—274no—
405#41—542no—
406C2×Q8—271no—
407C2×D4—31no—
408C2×D4—32no—
409C2×D4—94no—
410C2×D4—182no—
411C2×D4—181no—
412C2×D4—541no—
413C22×C4—91no—
414C22×C4—541no—
415SD16—278no—
416D8—98no—
417M16—274no—
418C2×C8—92no—
419C22×C3—21no—
420C22×C3—22no—
421C22×C3—41no—
422C22×C3—64no—
423C22×C3—64no—
424C22×C3—64no—
425C22×C3—64no—
426C22×C3—92no—
427C22×C3—124no—
428C22×C3—124no—
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—22no—
439C2×D3—34no—
440C2×D3—42no—
441C2×D3—44no—
442C2×D3—62no—
443C2×D3—64no—
444C2×D3—122no—
445C2×D3—1216no—
446C3×C4—22no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—22no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4—92no—
453C3×C4D3×#6712yes—
454C3×C4#381111yescharacteristic
455C3×C4C22×D3212yes—
456C3×C4C22×D3211yescharacteristic
457Dic3—34no—
458Dic3—62no—
459Dic3—184no—
460Dic3—184no—
461Dic3—182no—
462Dic3—364no—
463Dic3—364no—
464C32—21no—
465C32—22no—
466C32—41no—
467C32C2×#91112yescomplemented
468C32C2×#93211yescharacteristic, complemented
469C23—61no—
470C23—62no—
471C23—91no—
472C23—184no—
473C23—364no—
474Q8—272no—
475Q8—541no—
476D4—32no—
477D4—34no—
478D4—62no—
479D4—61no—
480D4—184no—
481D4—544no—
482C2×C4—91no—
483C2×C4—92no—
484C2×C4—92no—
485C2×C4—544no—
486C2×C4—541no—
487C2×C4D3311yescharacteristic
488C8—94no—
489C2×C3—22no—
490C2×C3—24no—
491C2×C3—21no—
492C2×C3—22no—
493C2×C3—42no—
494C2×C3—41no—
495C2×C3—124no—
496C2×C3—124no—
497C2×C3—124no—
498C2×C3—124no—
499C2×C3—182no—
500C2×C3—244no—
501C2×C3#6392712yescomplemented
502C2×C3#6393214yescomplemented
503C2×C3C2×D3×#6712yes—
504C2×C3C2×#381111yescharacteristic
505D3—42no—
506D3—44no—
507D3—64no—
508D3—122no—
509C22—62no—
510C22—64no—
511C22—92no—
512C22—124no—
513C22—122no—
514C22—182no—
515C22—368no—
516C22#8895211yescharacteristic
517C4—92no—
518C4—542no—
519C4#8895211yescharacteristic
520C4C2×D3311yescharacteristic
521C3—21no—
522C3—22no—
523C3—41no—
524C3C2×#6392711yescharacteristic, complemented
525C3C2×#6393212yescomplemented
526C2—124no—
527C2—122no—
528C2—182no—
529C2#1172287212yesdirect product factor
530C2C2×#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×#89014C2semidirect
2. C2×C3×#3802C2semidirect
3. C2×#88809C2semidirect
4. C2×#88794C2semidirect
5. C2×C3×#3713C2semidirect
6. #11722872C2direct (N × H)
10. C2×#6308C22semidirect
13. C2×C32×D12C22semidirect
17. #89014C22semidirect
20. C3×#3802C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. #88809C22semidirect
23. #88794C22semidirect
24. C3×#3713C22semidirect
27. C2×#3802D3semidirect
38. C2×#3713D3semidirect
49. #6308C23semidirect
56. C32×D12C23semidirect
60. C3×C3⋊(C3⋊C8)C23semidirect
84. C2×C3×D12C2×D3semidirect
93. #3802C2×D3semidirect
99. C2×C3⋊(C3⋊C8)C2×D3semidirect
108. #3713C2×D3semidirect
194. C3×D12C22×D3semidirect
204. C3⋊(C3⋊C8)C22×D3semidirect
209. C2×C33AGL(1,ℤ/8)semidirect
240. C2×D12D32semidirect
313. C33C2×AGL(1,ℤ/8)semidirect
363. D12C2×D32semidirect
391. C2×C32#911semidirect
392. C2×C32#932semidirect
467. C32C2×#911semidirect
468. C32C2×#932semidirect
501. C2×C3#63927semidirect
502. C2×C3#63932semidirect
524. C3C2×#63927semidirect
525. C3C2×#63932semidirect
529. C2#11722872direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890142
C2×C3×#38022
C2×#888092
C2×#887942
C2×C3×#37132
#117228722
C2×#639323
C2×#639273