Finite Group Databasebeta

← C2×#11722843 #13170590

Subgroups — C2×#11722843 #13170590

Order 1728.

1522 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1522 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×C4⋊#1113C211yesmaximal, characteristic
2C2×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
3C2×#88796C212yesmaximal, complemented
4C2×C3×#3717C212yesmaximal, complemented
5C2×C3×#3710C211yesmaximal, characteristic, complemented
6#11722843C218yesmaximal, direct product factor
7C2×#63940—31nomaximal
8C2×D3×#204—32nomaximal
9C22×#1113—21no—
10C2×#6308C2211yescharacteristic, complemented
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic, complemented
14C2×#6295—21no—
15C4⋊#1113—22no—
16C4⋊#1113C2214yes—
17C2×#6293—21no—
18C2×Dic3×C3⋊D3—21no—
19C3⋊D3×Dic6—22no—
20C3⋊D3×Dic6C2214yescomplemented
21C2×C32×C3⋊C8C2212yescomplemented
22#88796C2218yescomplemented
23C3×#3717C2218yescomplemented
24C3×#3710C2214yescomplemented
25C2×Q8×C3⋊D3—31no—
26C2×#3802—31no—
27C2×D3×D12—32no—
28C2×D3×Dic6—32no—
29C2×D3×Dic6—61no—
30C2×#3744—32no—
31C2×C3×#215—32no—
32C2×C3×#204—32no—
33C2×C3×D3×C8—32no—
34#63940—38no—
35C2×#3717D312yescomplemented
36C2×#3716—32no—
37C2×#3716—32no—
38C2×#3716—62no—
39C2×#3710D311yescharacteristic, complemented
40D3×#204—316no—
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3D411yescharacteristic
43C2×#1113—24no—
44C2×#1113—44no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47#6308—21no—
48#6308C2312yescomplemented
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#6293—24no—
59Dic3×C3⋊D3—24no—
60C32×C3⋊C8C2314yescomplemented
61C2×D3×SD16—92no—
62C22×#204—91no—
63C22×D32—62no—
64C22×D32—61no—
65C2×C32×Q8—31no—
66Q8×C3⋊D3—34no—
67Q8×C3⋊D3—62no—
68C2×C4⋊(C3⋊D3)C2×D312yes—
69C2×C4×C3⋊D3—61no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×#634—32no—
72C2×#634—31no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—62no—
76C2×#634—121no—
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×#627—62no—
85C2×#627—62no—
86C2×#626—62no—
87C2×#625—61no—
88C2×D3×Dic3—62no—
89C2×D3×Dic3—62no—
90#3802—34no—
91#3802—62no—
92D3×D12—38no—
93D3×D12—64no—
94D3×Dic6—38no—
95D3×Dic6—64no—
96D3×Dic6—64no—
97D3×Dic6—64no—
98C2×C32×C8—32no—
99#3744—38no—
100C3×#215—38no—
101C2×C3×C3⋊C8—22no—
102C2×C3×C3⋊C8C2×D312yescomplemented
103C2×C3×C3⋊C8C2×D312yescomplemented
104C3×#204—38no—
105C3×D3×C8—38no—
106#3717C2×D318yescomplemented
107#3716—38no—
108#3716—38no—
109#3716—68no—
110#3710C2×D314yescomplemented
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3—24no—
113C2×C3×C3⋊D3—44no—
114C2×C3×C3⋊D3SD1614yescomplemented
115C2×C3×C3⋊D3C2×D412yes—
116#1113—48no—
117C33×C4C2×D411yescharacteristic
118C33×C4C2411yescharacteristic
119C3⋊(C3⋊Dic3)—22no—
120C3×C3⋊Dic3C2×D412yes—
121C32×Dic3—22no—
122C2×D3×Q8—92no—
123C2×D3×Q8—181no—
124C2×D3×D4—92no—
125C22×D12—91no—
126C22×Dic6—91no—
127C2×C3×SD16—92no—
128C22×C3×C8—91no—
129C2×#215—32no—
130C2×#214—92no—
131D3×SD16—916no—
132C2×#204—32no—
133C2×#204—98no—
134C2×#204—92no—
135C2×#204—92no—
136C2×#204—182no—
137C2×D3×C8—32no—
138C22×C3⋊D3—22no—
139C22×C3⋊D3D1211yescharacteristic
140C22×C3×D3—32no—
141C22×C3×D3—61no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—122no—
145C2×D32—68no—
146C2×D32—64no—
147C2×D32—128no—
148C2×D32—124no—
149C32×Q8—32no—
150C32×Q8—61no—
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—41no—
154C2×C32×C4—61no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D312yes—
157C2×C3⋊Dic3—61no—
158C2×C3⋊Dic3—62no—
159C2×C3⋊Dic3—62no—
160C2×C3⋊Dic3—122no—
161C2×C3⋊Dic3—122no—
162C2×C3⋊Dic3D1211yescharacteristic
163C4⋊(C3⋊D3)—22no—
164C4⋊(C3⋊D3)C22×D314yes—
165C4×C3⋊D3—64no—
166C4×C3⋊D3D1212yes—
167C4×C3⋊D3C22×D312yes—
168#634—32no—
169#634—34no—
170#634—62no—
171#634—64no—
172#634—61no—
173#634—64no—
174#634—62no—
175#634—62no—
176#634—122no—
177#634—122no—
178#634—122no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—61no—
183C3×D12—34no—
184C3×D12—34no—
185C3×D12—64no—
186C3×D12—62no—
187C3×D12—62no—
188C3×D12—122no—
189C3×C4×D3—34no—
190C3×C4×D3—34no—
191C3×C4×D3—62no—
192C3×C4×D3—62no—
193C3×Dic6—22no—
194C3×Dic6—22no—
195C3×Dic6—22no—
196C3×Dic6C22×D314yescomplemented
197#627—68no—
198#627—68no—
199#626—68no—
200#625—64no—
201D3×Dic3—68no—
202D3×Dic3—68no—
203C32×C8—34no—
204C3×C3⋊C8—24no—
205C3×C3⋊C8C22×D314yescomplemented
206C3×C3⋊C8C22×D314yescomplemented
207C22×SD16—271noSylow
208C2×C33C2×SD1612yescomplemented
209C2×C33C22×D411yescharacteristic
210C3×C3⋊D3—44no—
211C3×C3⋊D3C2×SD1614yescomplemented
212C23×D3—182no—
213C23×D3—181no—
214C2×C3×Q8—32no—
215C2×C3×Q8—61no—
216C2×C3×D4—92no—
217C22×C3×C4—91no—
218C2×#67—182no—
219C22×Dic3—181no—
220D3×Q8—98no—
221D3×Q8—184no—
222D3×Q8—184no—
223D3×Q8—184no—
224D3×D4—98no—
225D3×D4—184no—
226C2×D12—32no—
227C2×D12—32no—
228C2×D12—62no—
229C2×D12—94no—
230C2×D12—182no—
231C2×D12—181no—
232C2×C4×D3—32no—
233C2×C4×D3—61no—
234C2×C4×D3—182no—
235C2×C4×D3—182no—
236C2×Dic6—91no—
237C2×Dic6—92no—
238C2×Dic6—94no—
239C2×Dic6—181no—
240C2×Dic6—182no—
241C2×Dic6—182no—
242C2×Dic6—181no—
243C2×Dic6—182no—
244C2×Dic6—182no—
245C2×Dic6—361no—
246C2×Dic6D3211yescharacteristic, complemented
247C3×SD16—98no—
248C2×C3×C8—32no—
249C2×C3×C8—32no—
250C2×C3×C8—62no—
251C2×C3×C8—94no—
252#215—38no—
253#214—98no—
254C2×C3⋊C8D3212yescomplemented
255#204—38no—
256#204—98no—
257#204—98no—
258#204—188no—
259D3×C8—38no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32C2×D1211yescharacteristic
264C22×C32D3×D412yes—
265C2×C3⋊D3—24no—
266C2×C3⋊D3—44no—
267C2×C3⋊D3#20414yescomplemented
268C2×C3⋊D3C2×D1212yes—
269C2×C3×D3—38no—
270C2×C3×D3—34no—
271C2×C3×D3—64no—
272C2×C3×D3—62no—
273C2×C3×D3—64no—
274C2×C3×D3—64no—
275C2×C3×D3—124no—
276C2×C3×D3—124no—
277C2×C3×D3—124no—
278C2×C3×D3—244no—
279D32—1216no—
280D32—128no—
281C32×C4—22no—
282C32×C4—21no—
283C32×C4—22no—
284C32×C4—21no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—62no—
288C32×C4C2×D1211yescharacteristic
289C32×C4D3×D412yes—
290C32×C4C23×D311yescharacteristic
291C32×C4C23×D312yes—
292C3⋊Dic3—62no—
293C3⋊Dic3—64no—
294C3⋊Dic3—64no—
295C3⋊Dic3—124no—
296C3⋊Dic3—124no—
297C3⋊Dic3C2×D1212yes—
298C3×Dic3—24no—
299C3×Dic3—24no—
300C3×Dic3—34no—
301C3×Dic3—62no—
302C22×Q8—271no—
303C22×D4—271no—
304C2×SD16—92no—
305C2×SD16—272no—
306C2×SD16—278no—
307C22×C8—91no—
308C33C22×SD1611yescharacteristic, complemented, Sylow
309C23×C3—91no—
310C23×C3—182no—
311C22×D3—32no—
312C22×D3—61no—
313C22×D3—62no—
314C22×D3—62no—
315C22×D3—122no—
316C22×D3—188no—
317C22×D3—184no—
318C22×D3—368no—
319C22×D3—364no—
320C3×Q8—34no—
321C3×Q8—62no—
322C3×Q8—62no—
323C3×Q8—62no—
324C3×D4—94no—
325C3×D4—182no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—91no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4C2×D3212yes—
335C2×C3×C4C2×D3211yescharacteristic
336#67—188no—
337C2×Dic3—21no—
338C2×Dic3—32no—
339C2×Dic3—61no—
340C2×Dic3—184no—
341C2×Dic3—182no—
342C2×Dic3—182no—
343C2×Dic3—181no—
344C2×Dic3—362no—
345C2×Dic3—362no—
346D12—34no—
347D12—34no—
348D12—64no—
349D12—62no—
350D12—62no—
351D12—122no—
352D12—184no—
353C4×D3—34no—
354C4×D3—34no—
355C4×D3—62no—
356C4×D3—62no—
357C4×D3—188no—
358C4×D3—188no—
359Dic6—21no—
360Dic6—92no—
361Dic6—94no—
362Dic6—182no—
363Dic6—184no—
364Dic6—184no—
365Dic6—188no—
366Dic6—188no—
367Dic6—182no—
368Dic6—182no—
369Dic6—181no—
370Dic6—362no—
371Dic6—362no—
372Dic6—362no—
373Dic6C2×D3212yescomplemented
374C3×C8—34no—
375C3×C8—34no—
376C3×C8—64no—
377C3⋊C8C2×D3214yescomplemented
378C2×C32—22no—
379C2×C32—24no—
380C2×C32—21no—
381C2×C32—22no—
382C2×C32—42no—
383C2×C32—41no—
384C2×C32C2×#20412yescomplemented
385C2×C32D3×SD1614yescomplemented
386C2×C32C22×D1211yescharacteristic
387C2×C32C2×D3×D412yes—
388C3⋊D3—44no—
389C3⋊D3C2×#20414yescomplemented
390C3×D3—38no—
391C3×D3—64no—
392C3×D3—124no—
393C3×D3—124no—
394C3×D3—244no—
395C24—541no—
396C2×Q8—31no—
397C2×Q8—274no—
398C2×Q8—271no—
399C2×Q8—541no—
400C2×Q8—542no—
401C2×D4—92no—
402C2×D4—274no—
403C2×D4—542no—
404C2×D4—541no—
405C22×C4—91no—
406C22×C4—541no—
407SD16—98no—
408SD16—278no—
409C2×C8—32no—
410C2×C8—94no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—41no—
414C22×C3—92no—
415C22×C3—94no—
416C22×C3—184no—
417C22×C3—364no—
418C22×C3D3×D1212yes—
419C22×C3#381111yescharacteristic
420C2×D3—34no—
421C2×D3—38no—
422C2×D3—62no—
423C2×D3—64no—
424C2×D3—64no—
425C2×D3—64no—
426C2×D3—124no—
427C2×D3—124no—
428C2×D3—124no—
429C2×D3—244no—
430C2×D3—3616no—
431C2×D3—368no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—64no—
439C3×C4—64no—
440C3×C4—92no—
441C3×C4D3×D1212yes—
442C3×C4#381111yescharacteristic
443C3×C4C22×D3212yes—
444C3×C4C22×D3211yescharacteristic
445Dic3—22no—
446Dic3—34no—
447Dic3—62no—
448Dic3—184no—
449Dic3—184no—
450Dic3—182no—
451Dic3—364no—
452Dic3—364no—
453C32—22no—
454C32—21no—
455C32—41no—
456C32C22×#20411yescharacteristic, complemented
457C32C2×D3×SD1612yescomplemented
458C23—91no—
459C23—182no—
460C23—544no—
461C23—1084no—
462Q8—32no—
463Q8—61no—
464Q8—272no—
465Q8—544no—
466Q8—541no—
467D4—94no—
468D4—182no—
469D4—544no—
470C2×C4—61no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—544no—
475C2×C4—541no—
476C2×C4D3311yescharacteristic
477C8—34no—
478C2×C3—22no—
479C2×C3—24no—
480C2×C3—21no—
481C2×C3—22no—
482C2×C3—42no—
483C2×C3—41no—
484C2×C3—94no—
485C2×C3—364no—
486C2×C3D3×#20414yescomplemented
487C2×C3#6394012yescomplemented
488C2×C3C2×D3×D1212yes—
489C2×C3C2×#381111yescharacteristic
490D3—38no—
491D3—64no—
492D3—124no—
493D3—124no—
494D3—244no—
495C22—94no—
496C22—92no—
497C22—184no—
498C22—364no—
499C22—1088no—
500C22#8895511yescharacteristic
501C4—62no—
502C4—92no—
503C4—542no—
504C4#8895511yescharacteristic
505C4C2×D3311yescharacteristic
506C3—21no—
507C3—22no—
508C3—41no—
509C3C2×D3×#20412yescomplemented
510C3C2×#6394011yescharacteristic, complemented
511C2—94no—
512C2—364no—
513C2#1172284312yesdirect product factor
514C2C2×#8895511yescharacteristic
subgroupG/H
centerC22#88955
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C4⋊#1113C2
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
2. C2×C3⋊D3×Dic6C2semidirect
3. C2×#88796C2semidirect
4. C2×C3×#3717C2semidirect
5. C2×C3×#3710C2semidirect
6. #11722843C2direct (N × H)
10. C2×#6308C22semidirect
13. C2×C32×Dic6C22semidirect
20. C3⋊D3×Dic6C22semidirect
21. C2×C32×C3⋊C8C22semidirect
22. #88796C22semidirect
23. C3×#3717C22semidirect
24. C3×#3710C22semidirect
35. C2×#3717D3semidirect
39. C2×#3710D3semidirect
48. #6308C23semidirect
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. #3710C2×D3semidirect
114. C2×C3×C3⋊D3SD16semidirect
196. C3×Dic6C22×D3semidirect
205. C3×C3⋊C8C22×D3semidirect
206. C3×C3⋊C8C22×D3semidirect
208. C2×C33C2×SD16semidirect
211. C3×C3⋊D3C2×SD16semidirect
246. C2×Dic6D32semidirect
254. C2×C3⋊C8D32semidirect
267. C2×C3⋊D3#204semidirect
308. C33C22×SD16semidirect
373. Dic6C2×D32semidirect
377. C3⋊C8C2×D32semidirect
384. C2×C32C2×#204semidirect
385. C2×C32D3×SD16semidirect
389. C3⋊D3C2×#204semidirect
456. C32C22×#204semidirect
457. C32C2×D3×SD16semidirect
486. C2×C3D3×#204semidirect
487. C2×C3#63940semidirect
509. C3C2×D3×#204semidirect
510. C3C2×#63940semidirect
513. C2#11722843direct (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×C3⋊D3×Dic62
C2×#887962
C2×C3×#37172
C2×C3×#37102
#117228432
C2×#639403
C2×D3×#2043