Finite Group Databasebeta

← C2×#11722880 #13170676

Subgroups — C2×#11722880 #13170676

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×C4⋊#1113C211yesmaximal, characteristic, complemented
2C2×#89014C211yesmaximal, characteristic, complemented
3C2×#88796C212yesmaximal, complemented
4C2×C3×#3714C212yesmaximal, complemented
5C2×C3×#3712C211yesmaximal, characteristic, complemented
6#11722880C218yesmaximal, direct product factor
7C2×#63927—31nomaximal
8C2×#63795—32nomaximal
9C22×#1113—21no—
10C2×#6308C2211yescharacteristic, complemented
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic, complemented
14C2×#6295—21no—
15C4⋊#1113—22no—
16C4⋊#1113C2214yescomplemented
17C2×#6290—21no—
18C2×D3×C3⋊Dic3—21no—
19#89014—22no—
20#89014C2214yescomplemented
21C2×C32×C3⋊C8C2212yescomplemented
22#88796C2218yescomplemented
23C3×#3714C2218yescomplemented
24C3×#3712C2214yescomplemented
25C2×#3830—31no—
26C2×#3802—31no—
27C2×D3×D12—32no—
28C2×#3795—32no—
29C2×#3795—61no—
30C2×#3744—32no—
31C2×C3×#213—32no—
32C2×C3×D24—32no—
33C2×C3×ΓL(1,𝔽25)—32no—
34#63927—38no—
35C2×#3716—32no—
36C2×#3716—32no—
37C2×#3716—62no—
38C2×#3714D312yescomplemented
39C2×#3712D311yescharacteristic, complemented
40#63795—316no—
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3D411yescharacteristic
43C22×C32×D3—21no—
44C2×#1113—24no—
45C2×#1113—44no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48#6308—21no—
49#6308C2312yescomplemented
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—22no—
52C3×C4⋊(C3⋊D3)C2314yes—
53C3×C4×C3⋊D3D412yes—
54C3×C4×C3⋊D3C2312yes—
55C32×D12—21no—
56C32×D12C2312yescomplemented
57#6295—24no—
58#6290—24no—
59D3×C3⋊Dic3—24no—
60C32×C3⋊C8C2314yescomplemented
61C2×#911—92no—
62C2×#908—91no—
63C22×D32—62no—
64C22×D32—61no—
65C2×C32×D4—31no—
66C2×#638—61no—
67C22×C3⋊Dic3—61no—
68#3830—34no—
69#3830—62no—
70C2×C4⋊(C3⋊D3)C2×D312yes—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×#634—32no—
73C2×#634—31no—
74C2×#634—61no—
75C2×#634—62no—
76C2×#634—121no—
77C2×C3×D12—21no—
78C2×C3×D12—32no—
79C2×C3×D12—32no—
80C2×C3×D12—62no—
81C2×C3×D12C2×D312yescomplemented
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—61no—
84C2×#626—62no—
85C2×#625—62no—
86C2×#625—62no—
87C2×#625—61no—
88C2×D3×Dic3—62no—
89C2×D3×Dic3—62no—
90#3802—34no—
91#3802—62no—
92D3×D12—38no—
93D3×D12—64no—
94#3795—38no—
95#3795—64no—
96#3795—64no—
97#3795—64no—
98C2×C32×C8—32no—
99#3744—38no—
100C3×#213—38no—
101C2×C3×C3⋊C8—22no—
102C2×C3×C3⋊C8C2×D312yescomplemented
103C2×C3×C3⋊C8C2×D312yescomplemented
104C3×D24—38no—
105C3×ΓL(1,𝔽25)—38no—
106#3716—38no—
107#3716—38no—
108#3716—68no—
109#3714C2×D318yescomplemented
110#3712C2×D314yescomplemented
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—24no—
114C2×C3×C3⋊D3—44no—
115C2×C3×C3⋊D3C2×D412yes—
116C2×C32×D3—22no—
117C2×C32×D3—42no—
118#1113—48no—
119C33×C4C2×D411yescharacteristic
120C33×C4C2411yescharacteristic
121C3⋊(C3⋊Dic3)—22no—
122C3×C3⋊Dic3C2×D412yes—
123C2×#237—92no—
124C2×#237—181no—
125C2×D3×D4—92no—
126C2×#235—91no—
127C22×D12—91no—
128C2×C3×D8—92no—
129C2×C3×M16—91no—
130C2×#214—92no—
131C2×#213—32no—
132#911—916no—
133#908—98no—
134C2×D24—32no—
135C2×#204—92no—
136C2×#204—92no—
137C2×#204—182no—
138C2×ΓL(1,𝔽25)—32no—
139C23×C32—61no—
140C22×C3⋊D3—22no—
141C22×C3⋊D3D1211yescharacteristic
142C22×C3×D3—22no—
143C22×C3×D3—22no—
144C22×C3×D3—32no—
145C22×C3×D3—61no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3—122no—
149C2×D32—68no—
150C2×D32—64no—
151C2×D32—128no—
152C2×D32—124no—
153C32×D4—32no—
154C32×D4—61no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D312yes—
160#638—64no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—64no—
163C2×C3⋊Dic3—62no—
164C2×C3⋊Dic3—62no—
165C2×C3⋊Dic3—122no—
166C2×C3⋊Dic3—122no—
167C2×C3⋊Dic3D1211yescharacteristic
168C4⋊(C3⋊D3)—22no—
169C4⋊(C3⋊D3)C22×D314yes—
170C4×C3⋊D3D1212yes—
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—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—61no—
184C3×D12—22no—
185C3×D12—22no—
186C3×D12—22no—
187C3×D12—34no—
188C3×D12—34no—
189C3×D12—64no—
190C3×D12—62no—
191C3×D12—62no—
192C3×D12—122no—
193C3×D12C22×D314yescomplemented
194C3×C4×D3—34no—
195C3×C4×D3—34no—
196C3×C4×D3—62no—
197C3×C4×D3—62no—
198#626—68no—
199#625—68no—
200#625—68no—
201#625—64no—
202D3×Dic3—68no—
203D3×Dic3—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—22no—
212C3×C3⋊D3—44no—
213C32×D3—42no—
214C23×D3—182no—
215C23×D3—181no—
216C2×C3×D4—32no—
217C2×C3×D4—61no—
218C2×C3×D4—92no—
219C22×C3×C4—91no—
220C2×#67—182no—
221C2×#67—181no—
222C2×#67—182no—
223C2×#67—182no—
224C22×Dic3—182no—
225C22×Dic3—182no—
226#237—98no—
227#237—184no—
228#237—184no—
229#237—184no—
230D3×D4—98no—
231D3×D4—184no—
232#235—94no—
233#235—182no—
234C2×D12—32no—
235C2×D12—32no—
236C2×D12—62no—
237C2×D12—94no—
238C2×D12—182no—
239C2×D12—181no—
240C2×D12D3211yescharacteristic, complemented
241C2×C4×D3—32no—
242C2×C4×D3—61no—
243C2×C4×D3—181no—
244C2×Dic6—91no—
245C2×Dic6—92no—
246C2×Dic6—181no—
247C2×Dic6—182no—
248C2×Dic6—361no—
249C3×D8—98no—
250C3×M16—94no—
251C2×C3×C8—32no—
252C2×C3×C8—32no—
253C2×C3×C8—62no—
254#214—98no—
255#213—38no—
256C2×C3⋊C8D3212yescomplemented
257D24—38no—
258#204—98no—
259#204—98no—
260#204—188no—
261ΓL(1,𝔽25)—38no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32—62no—
266C22×C32—122no—
267C22×C32C2×D1211yescharacteristic
268C22×C32D3×D412yes—
269C2×C3⋊D3—22no—
270C2×C3⋊D3—24no—
271C2×C3⋊D3—44no—
272C2×C3⋊D3C2×D1212yes—
273C2×C3×D3—24no—
274C2×C3×D3—24no—
275C2×C3×D3—34no—
276C2×C3×D3—44no—
277C2×C3×D3—44no—
278C2×C3×D3—62no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C2×C3×D3—64no—
282C2×C3×D3—122no—
283C2×C3×D3—124no—
284C2×C3×D3—124no—
285C2×C3×D3—124no—
286C2×C3×D3—244no—
287D32—1216no—
288D32—128no—
289C32×C4—22no—
290C32×C4—21no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—41no—
294C32×C4—41no—
295C32×C4C2×D1211yescharacteristic
296C32×C4D3×D412yes—
297C32×C4C23×D311yescharacteristic
298C32×C4C23×D312yes—
299C3⋊Dic3—62no—
300C3⋊Dic3—64no—
301C3⋊Dic3—64no—
302C3⋊Dic3—124no—
303C3⋊Dic3—124no—
304C3⋊Dic3C2×D1212yes—
305C3×Dic3—34no—
306C3×Dic3—62no—
307C2×#41—271no—
308C22×D4—271no—
309AGL(1,ℤ/8)—278no—
310C2×SD16—272no—
311C2×D8—92no—
312C2×M16—91no—
313C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
314C23×C3—62no—
315C23×C3—62no—
316C23×C3—91no—
317C23×C3—182no—
318C22×D3—21no—
319C22×D3—32no—
320C22×D3—61no—
321C22×D3—62no—
322C22×D3—62no—
323C22×D3—122no—
324C22×D3—188no—
325C22×D3—184no—
326C22×D3—368no—
327C22×D3—364no—
328C3×D4—34no—
329C3×D4—62no—
330C3×D4—62no—
331C3×D4—62no—
332C3×D4—94no—
333C3×D4—182no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—41no—
337C2×C3×C4—91no—
338C2×C3×C4—92no—
339C2×C3×C4—92no—
340C2×C3×C4C2×D3212yes—
341C2×C3×C4C2×D3211yescharacteristic
342#67—188no—
343#67—184no—
344#67—188no—
345#67—188no—
346C2×Dic3—32no—
347C2×Dic3—61no—
348C2×Dic3—188no—
349C2×Dic3—188no—
350C2×Dic3—182no—
351C2×Dic3—182no—
352C2×Dic3—181no—
353C2×Dic3—362no—
354C2×Dic3—362no—
355D12—21no—
356D12—34no—
357D12—34no—
358D12—64no—
359D12—62no—
360D12—62no—
361D12—122no—
362D12—184no—
363D12C2×D3212yescomplemented
364C4×D3—34no—
365C4×D3—34no—
366C4×D3—62no—
367C4×D3—62no—
368C4×D3—184no—
369Dic6—92no—
370Dic6—94no—
371Dic6—182no—
372Dic6—184no—
373Dic6—182no—
374Dic6—182no—
375Dic6—181no—
376Dic6—362no—
377Dic6—362no—
378Dic6—362no—
379C3×C8—34no—
380C3×C8—34no—
381C3×C8—64no—
382C3⋊C8C2×D3214yescomplemented
383C2×C32—22no—
384C2×C32—24no—
385C2×C32—21no—
386C2×C32—22no—
387C2×C32—42no—
388C2×C32—41no—
389C2×C32—122no—
390C2×C32#90812yescomplemented
391C2×C32#91114yescomplemented
392C2×C32C22×D1211yescharacteristic
393C2×C32C2×D3×D412yes—
394C3⋊D3—22no—
395C3⋊D3—44no—
396C3×D3—44no—
397C3×D3—44no—
398C3×D3—64no—
399C3×D3—122no—
400C3×D3—124no—
401C3×D3—124no—
402C3×D3—244no—
403C24—541no—
404#41—274no—
405#41—542no—
406C2×Q8—271no—
407C2×D4—31no—
408C2×D4—92no—
409C2×D4—274no—
410C2×D4—542no—
411C2×D4—541no—
412C2×D4—541no—
413C22×C4—91no—
414C22×C4—541no—
415SD16—278no—
416D8—98no—
417M16—94no—
418C2×C8—32no—
419C22×C3—21no—
420C22×C3—22no—
421C22×C3—41no—
422C22×C3—64no—
423C22×C3—64no—
424C22×C3—92no—
425C22×C3—124no—
426C22×C3—124no—
427C22×C3—182no—
428C22×C3—184no—
429C22×C3—364no—
430C22×C3D3×D1212yes—
431C22×C3#381111yescharacteristic
432C2×D3—22no—
433C2×D3—34no—
434C2×D3—42no—
435C2×D3—62no—
436C2×D3—64no—
437C2×D3—64no—
438C2×D3—64no—
439C2×D3—122no—
440C2×D3—124no—
441C2×D3—124no—
442C2×D3—124no—
443C2×D3—244no—
444C2×D3—3616no—
445C2×D3—368no—
446C3×C4—22no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—22no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4—92no—
453C3×C4D3×D1212yes—
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×#90811yescharacteristic, complemented
468C32C2×#91112yescomplemented
469C23—61no—
470C23—91no—
471C23—182no—
472C23—544no—
473C23—1084no—
474Q8—272no—
475Q8—541no—
476D4—32no—
477D4—61no—
478D4—94no—
479D4—182no—
480D4—544no—
481D4—544no—
482C2×C4—91no—
483C2×C4—92no—
484C2×C4—92no—
485C2×C4—544no—
486C2×C4—541no—
487C2×C4D3311yescharacteristic
488C8—34no—
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—182no—
498C2×C3—364no—
499C2×C3#6379514yescomplemented
500C2×C3#6392712yescomplemented
501C2×C3C2×D3×D1212yes—
502C2×C3C2×#381111yescharacteristic
503D3—42no—
504D3—64no—
505D3—122no—
506D3—124no—
507D3—124no—
508D3—244no—
509C22—62no—
510C22—92no—
511C22—122no—
512C22—182no—
513C22—184no—
514C22—364no—
515C22—1088no—
516C22#8895511yescharacteristic
517C4—92no—
518C4—542no—
519C4#8895511yescharacteristic
520C4C2×D3311yescharacteristic
521C3—21no—
522C3—22no—
523C3—41no—
524C3C2×#6379512yescomplemented
525C3C2×#6392711yescharacteristic, complemented
526C2—122no—
527C2—182no—
528C2—364no—
529C2#1172288012yesdirect product factor
530C2C2×#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×#89014C2semidirect
3. C2×#88796C2semidirect
4. C2×C3×#3714C2semidirect
5. C2×C3×#3712C2semidirect
6. #11722880C2direct (N × H)
10. C2×#6308C22semidirect
13. C2×C32×D12C22semidirect
16. C4⋊#1113C22semidirect
20. #89014C22semidirect
21. C2×C32×C3⋊C8C22semidirect
22. #88796C22semidirect
23. C3×#3714C22semidirect
24. C3×#3712C22semidirect
38. C2×#3714D3semidirect
39. C2×#3712D3semidirect
49. #6308C23semidirect
56. C32×D12C23semidirect
60. C32×C3⋊C8C23semidirect
81. C2×C3×D12C2×D3semidirect
102. C2×C3×C3⋊C8C2×D3semidirect
103. C2×C3×C3⋊C8C2×D3semidirect
109. #3714C2×D3semidirect
110. #3712C2×D3semidirect
193. C3×D12C22×D3semidirect
206. C3×C3⋊C8C22×D3semidirect
207. C3×C3⋊C8C22×D3semidirect
209. C2×C33AGL(1,ℤ/8)semidirect
240. C2×D12D32semidirect
256. C2×C3⋊C8D32semidirect
313. C33C2×AGL(1,ℤ/8)semidirect
363. D12C2×D32semidirect
382. C3⋊C8C2×D32semidirect
390. C2×C32#908semidirect
391. C2×C32#911semidirect
467. C32C2×#908semidirect
468. C32C2×#911semidirect
499. C2×C3#63795semidirect
500. C2×C3#63927semidirect
524. C3C2×#63795semidirect
525. C3C2×#63927semidirect
529. C2#11722880direct (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×#890142
C2×#887962
C2×C3×#37142
C2×C3×#37122
#117228802
C2×#639273
C2×#637953