Finite Group Databasebeta

← C2×#11722837 #13170566

Subgroups — C2×#11722837 #13170566

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×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic
3C2×#88808C211yesmaximal, characteristic, complemented
4C2×#88794C212yesmaximal, complemented
5C2×C3×#3715C212yesmaximal, complemented
6#11722837C218yesmaximal, direct product factor
7C2×#63940—31nomaximal
8C2×D3×#214—32nomaximal
9C22×C3×D32—21no—
10C2×#6308C2211yescharacteristic, complemented
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2212yes—
13C2×C32×Dic6C2211yescharacteristic, complemented
14C2×#6293—21no—
15C2×Dic3×C3⋊D3—21no—
16C3⋊D3×Dic6—22no—
17C3⋊D3×Dic6C2214yescomplemented
18C2×C3×#625—21no—
19C3×#3802—22no—
20C3×#3802C2214yes—
21C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
22#88808C2214yescomplemented
23#88794C2218yescomplemented
24C3×#3715C2218yescomplemented
25C2×Q8×C3⋊D3—31no—
26C2×C3×D3×D4—32no—
27C2×#3802D311yescharacteristic
28C2×D3×Dic6—32no—
29C2×D3×Dic6—61no—
30C2×#3754—32no—
31C2×C3×#215—32no—
32C2×C3×#214—32no—
33#63940—38no—
34D3×#214—316no—
35C2×#3716—32no—
36C2×#3716—32no—
37C2×#3716—62no—
38C2×#3715D312yescomplemented
39C2×#3710—31no—
40C2×D3×C3⋊C8—32no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—22no—
43C2×C3×D32—24no—
44C2×C3×D32—44no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47#6308—21no—
48#6308C2312yescomplemented
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#6293—24no—
58Dic3×C3⋊D3—24no—
59C3×#625—24no—
60C3×C3⋊(C3⋊C8)C2314yescomplemented
61C22×#214—91no—
62C2×D3×SD16—92no—
63C23×C3×D3—62no—
64C22×D32—21no—
65C2×C32×Q8—31no—
66C2×C32×D4—32no—
67Q8×C3⋊D3—34no—
68Q8×C3⋊D3—62no—
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×#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×#627—62no—
88C2×#627—62no—
89C2×#625—21no—
90C2×D3×Dic3—62no—
91C2×D3×Dic3—62no—
92#3802—22no—
93#3802C2×D314yes—
94D3×Dic6—38no—
95D3×Dic6—64no—
96D3×Dic6—64no—
97D3×Dic6—64no—
98#3754—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#3716—38no—
106#3716—38no—
107#3716—68no—
108#3715C2×D318yescomplemented
109#3710—34no—
110D3×C3⋊C8—38no—
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3SD1614yescomplemented
113C2×C3×C3⋊D3C2×D412yes—
114C2×C32×D3—24no—
115C2×C32×D3—44no—
116C3×D32—48no—
117C33×C4C2×D411yescharacteristic
118C33×C4C2411yescharacteristic
119C3⋊(C3⋊Dic3)—22no—
120C3×C3⋊Dic3C2×D412yes—
121C32×Dic3—22no—
122C22×C3×D4—91no—
123C2×D3×Q8—92no—
124C2×D3×Q8—181no—
125C2×D3×D4—32no—
126C22×Dic6—91no—
127C2×C3×SD16—92no—
128C2×#215—32no—
129C2×#214—32no—
130C2×#214—92no—
131C2×#214—92no—
132C2×#214—98no—
133C2×#214—182no—
134C22×C3⋊C8—91no—
135D3×SD16—916no—
136C2×#204—92no—
137C2×D3×C8—92no—
138C23×C32—62no—
139C22×C3⋊D3#6711yescharacteristic
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—32no—
143C22×C3×D3—42no—
144C22×C3×D3—61no—
145C22×C3×D3—68no—
146C22×C3×D3—128no—
147C2×D32—24no—
148C2×D32—44no—
149C32×Q8—32no—
150C32×Q8—61no—
151C32×D4—34no—
152C32×D4—62no—
153C2×C32×C4—22no—
154C2×C32×C4—21no—
155C2×C32×C4—41no—
156C2×C32×C4—61no—
157C2×C32×C4C22×D312yes—
158C2×C32×C4C22×D311yescharacteristic
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—62no—
161C2×C3⋊Dic3—62no—
162C2×C3⋊Dic3—122no—
163C2×C3⋊Dic3—122no—
164C2×C3⋊Dic3#6711yescharacteristic
165C4×C3⋊D3—64no—
166C4×C3⋊D3#6712yes—
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—
179C3×#67—68no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—61no—
184C3×D12—22no—
185C3×D12—22no—
186C3×D12—24no—
187C3×D12—42no—
188C3×D12C22×D314yes—
189C3×D12C22×D314yes—
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#627—68no—
199#627—68no—
200#625—24no—
201D3×Dic3—68no—
202D3×Dic3—68no—
203C3⋊(C3⋊C8)C22×D314yescomplemented
204C3×C3⋊C8—34no—
205C3×C3⋊C8—34no—
206C3×C3⋊C8—64no—
207C22×SD16—271noSylow
208C2×C33C2×SD1612yescomplemented
209C2×C33C22×D411yescharacteristic
210C3×C3⋊D3C2×SD1614yescomplemented
211C32×D3—44no—
212C24×C3—181no—
213C23×D3—62no—
214C2×C3×Q8—32no—
215C2×C3×Q8—61no—
216C2×C3×D4—32no—
217C2×C3×D4—32no—
218C2×C3×D4—62no—
219C2×C3×D4—94no—
220C2×C3×D4—182no—
221C2×C3×D4—181no—
222C22×C3×C4—91no—
223C2×#67—62no—
224C22×Dic3—181no—
225D3×Q8—98no—
226D3×Q8—184no—
227D3×Q8—184no—
228D3×Q8—184no—
229D3×D4—38no—
230D3×D4—64no—
231C2×D12D3212yes—
232C2×C4×D3—32no—
233C2×C4×D3—61no—
234C2×C4×D3—182no—
235C2×C4×D3—182no—
236C2×Dic6—92no—
237C2×Dic6—91no—
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—92no—
249#215—38no—
250#214—38no—
251#214—98no—
252#214—98no—
253#214—188no—
254C2×C3⋊C8—32no—
255C2×C3⋊C8—32no—
256C2×C3⋊C8—62no—
257C2×C3⋊C8—94no—
258#204—98no—
259D3×C8—98no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32—64no—
264C22×C32—124no—
265C22×C32D3×D412yes—
266C22×C32C2×#6711yescharacteristic
267C2×C3⋊D3#21414yescomplemented
268C2×C3⋊D3C2×#6712yes—
269C2×C3×D3—24no—
270C2×C3×D3—24no—
271C2×C3×D3—38no—
272C2×C3×D3—34no—
273C2×C3×D3—44no—
274C2×C3×D3—44no—
275C2×C3×D3—44no—
276C2×C3×D3—64no—
277C2×C3×D3—62no—
278C2×C3×D3—84no—
279C2×C3×D3—1216no—
280D32—48no—
281C32×C4—22no—
282C32×C4—21no—
283C32×C4—22no—
284C32×C4—21no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—62no—
288C32×C4D3×D412yes—
289C32×C4C2×#6711yescharacteristic
290C32×C4C23×D312yes—
291C32×C4C23×D311yescharacteristic
292C3⋊Dic3—62no—
293C3⋊Dic3—64no—
294C3⋊Dic3—64no—
295C3⋊Dic3—124no—
296C3⋊Dic3—124no—
297C3⋊Dic3C2×#6712yes—
298C3×Dic3—24no—
299C3×Dic3—24no—
300C3×Dic3—34no—
301C3×Dic3—62no—
302C22×Q8—271no—
303C22×D4—91no—
304C2×SD16—92no—
305C2×SD16—272no—
306C2×SD16—278no—
307C22×C8—271no—
308C33C22×SD1611yescharacteristic, complemented, Sylow
309C23×C3—62no—
310C23×C3—62no—
311C23×C3—91no—
312C23×C3—122no—
313C23×C3—184no—
314C23×C3—364no—
315C22×D3—22no—
316C22×D3—32no—
317C22×D3—61no—
318C22×D3—68no—
319C22×D3—128no—
320C3×Q8—34no—
321C3×Q8—62no—
322C3×Q8—62no—
323C3×Q8—62no—
324C3×D4—34no—
325C3×D4—34no—
326C3×D4—64no—
327C3×D4—62no—
328C3×D4—62no—
329C3×D4—122no—
330C3×D4—184no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—41no—
334C2×C3×C4—62no—
335C2×C3×C4—62no—
336C2×C3×C4—91no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4C2×D3212yes—
340C2×C3×C4C2×D3211yescharacteristic
341#67—68no—
342C2×Dic3—21no—
343C2×Dic3—32no—
344C2×Dic3—61no—
345C2×Dic3—184no—
346C2×Dic3—182no—
347C2×Dic3—182no—
348C2×Dic3—181no—
349C2×Dic3—362no—
350C2×Dic3—362no—
351D12—22no—
352D12C2×D3214yes—
353C4×D3—34no—
354C4×D3—34no—
355C4×D3—62no—
356C4×D3—62no—
357C4×D3—188no—
358C4×D3—188no—
359Dic6—21no—
360Dic6—94no—
361Dic6—92no—
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—94no—
375C3⋊C8—34no—
376C3⋊C8—34no—
377C3⋊C8—64no—
378C2×C32—22no—
379C2×C32—21no—
380C2×C32—24no—
381C2×C32—22no—
382C2×C32—42no—
383C2×C32—41no—
384C2×C32—124no—
385C2×C32D3×SD1614yescomplemented
386C2×C32C2×#21412yescomplemented
387C2×C32C2×D3×D412yes—
388C2×C32C22×#6711yescharacteristic
389C3⋊D3C2×#21414yescomplemented
390C3×D3—38no—
391C3×D3—44no—
392C3×D3—44no—
393C3×D3—64no—
394C3×D3—84no—
395C24—181no—
396C2×Q8—31no—
397C2×Q8—274no—
398C2×Q8—271no—
399C2×Q8—541no—
400C2×Q8—542no—
401C2×D4—32no—
402C2×D4—94no—
403C2×D4—182no—
404C2×D4—181no—
405C22×C4—91no—
406C22×C4—541no—
407SD16—98no—
408SD16—278no—
409C2×C8—92no—
410C2×C8—274no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—41no—
414C22×C3—64no—
415C22×C3—64no—
416C22×C3—92no—
417C22×C3—94no—
418C22×C3—124no—
419C22×C3—124no—
420C22×C3—124no—
421C22×C3—244no—
422C22×C3—368no—
423C22×C3D3×#6712yes—
424C22×C3#381111yescharacteristic
425C2×D3—24no—
426C2×D3—34no—
427C2×D3—38no—
428C2×D3—44no—
429C2×D3—62no—
430C2×D3—64no—
431C2×D3—1216no—
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×#6712yes—
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—
456C32C2×D3×SD1612yescomplemented
457C32C22×#21411yescharacteristic, complemented
458C23—62no—
459C23—91no—
460C23—184no—
461C23—364no—
462Q8—32no—
463Q8—61no—
464Q8—272no—
465Q8—544no—
466Q8—541no—
467D4—34no—
468D4—62no—
469D4—184no—
470C2×C4—61no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—544no—
475C2×C4—541no—
476C2×C4D3311yescharacteristic
477C8—94no—
478C2×C3—22no—
479C2×C3—24no—
480C2×C3—21no—
481C2×C3—22no—
482C2×C3—42no—
483C2×C3—41no—
484C2×C3—94no—
485C2×C3—124no—
486C2×C3—124no—
487C2×C3—244no—
488C2×C3D3×#21414yescomplemented
489C2×C3#6394012yescomplemented
490C2×C3C2×D3×#6712yes—
491C2×C3C2×#381111yescharacteristic
492D3—38no—
493D3—44no—
494D3—64no—
495C22—64no—
496C22—94no—
497C22—92no—
498C22—124no—
499C22—368no—
500C22#8895211yescharacteristic
501C4—62no—
502C4—92no—
503C4—542no—
504C4#8895211yescharacteristic
505C4C2×D3311yescharacteristic
506C3—21no—
507C3—22no—
508C3—41no—
509C3C2×D3×#21412yescomplemented
510C3C2×#6394011yescharacteristic, complemented
511C2—94no—
512C2—124no—
513C2#1172283712yesdirect product factor
514C2C2×#8895211yescharacteristic
subgroupG/H
centerC22#88952
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
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
1. C2×C3⋊D3×Dic6C2semidirect
3. C2×#88808C2semidirect
4. C2×#88794C2semidirect
5. C2×C3×#3715C2semidirect
6. #11722837C2direct (N × H)
10. C2×#6308C22semidirect
13. C2×C32×Dic6C22semidirect
17. C3⋊D3×Dic6C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. #88808C22semidirect
23. #88794C22semidirect
24. C3×#3715C22semidirect
38. C2×#3715D3semidirect
48. #6308C23semidirect
56. C32×Dic6C23semidirect
60. C3×C3⋊(C3⋊C8)C23semidirect
86. C2×C3×Dic6C2×D3semidirect
99. C2×C3⋊(C3⋊C8)C2×D3semidirect
108. #3715C2×D3semidirect
112. C2×C3×C3⋊D3SD16semidirect
197. C3×Dic6C22×D3semidirect
203. C3⋊(C3⋊C8)C22×D3semidirect
208. C2×C33C2×SD16semidirect
210. C3×C3⋊D3C2×SD16semidirect
246. C2×Dic6D32semidirect
267. C2×C3⋊D3#214semidirect
308. C33C22×SD16semidirect
373. Dic6C2×D32semidirect
385. C2×C32D3×SD16semidirect
386. C2×C32C2×#214semidirect
389. C3⋊D3C2×#214semidirect
456. C32C2×D3×SD16semidirect
457. C32C22×#214semidirect
488. C2×C3D3×#214semidirect
489. C2×C3#63940semidirect
509. C3C2×D3×#214semidirect
510. C3C2×#63940semidirect
513. C2#11722837direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×C3×#38022
C2×#888082
C2×#887942
C2×C3×#37152
#117228372
C2×#639403
C2×D3×#2143