Finite Group Databasebeta

← C2×#11722838 #13170570

Subgroups — C2×#11722838 #13170570

Order 1728.

1618 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1618 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×D12C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C2×#88808C211yesmaximal, characteristic, complemented
4C2×#88795C212yesmaximal, complemented
5C2×C3×#3715C212yesmaximal, complemented
6#11722838C218yesmaximal, direct product factor
7C2×D3×#215—32nomaximal
8C2×#63930—31nomaximal
9C22×D3×C3⋊D3—21no—
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2212yescomplemented
14C2×#6291—21no—
15C3⋊D3×D12—22no—
16C3⋊D3×D12C2214yes—
17C2×C3×#627—21no—
18C2×C3×#624—21no—
19C3×#3797—22no—
20C3×#3797C2214yescomplemented
21C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
22#88808C2214yescomplemented
23#88795C2218yescomplemented
24C3×#3715C2218yescomplemented
25C2×D4×C3⋊D3—31no—
26C2×C3×D3×Q8—32no—
27C2×D3×D12—32no—
28C2×D3×D12—61no—
29C2×#3797D311yescharacteristic, complemented
30C2×#3755—32no—
31C2×C3×#215—32no—
32C2×C3×#214—32no—
33D3×#215—316no—
34#63930—38no—
35C2×#3717—32no—
36C2×#3717—32no—
37C2×#3717—62no—
38C2×#3715D312yescomplemented
39C2×#3710—31no—
40C2×D3×C3⋊C8—32no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C22×C32×D3—21no—
44C2×D3×C3⋊D3—24no—
45C2×D3×C3⋊D3—44no—
46C2×C33×C4C2311yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))C2312yes—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3D412yes—
51C3×C4×C3⋊D3C2312yes—
52C2×C32×Dic3—22no—
53C32×D12—21no—
54C32×D12C2312yes—
55C32×Dic6—22no—
56C32×Dic6C2314yescomplemented
57#6291—24no—
58C3×#627—24no—
59C3×#624—24no—
60C3×C3⋊(C3⋊C8)C2314yescomplemented
61C22×#215—91no—
62C2×D3×SD16—92no—
63C23×C3⋊D3—61no—
64C22×D32—62no—
65C22×D32—62no—
66C2×C32×Q8—32no—
67C2×C32×D4—31no—
68C2×#638—61no—
69D4×C3⋊D3—34no—
70D4×C3⋊D3—62no—
71C2×C4⋊(C3⋊D3)—32no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—62no—
75C2×C4⋊(C3⋊D3)—121no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C3×D3×Q8—38no—
78C3×D3×Q8—64no—
79C2×C3×D12—21no—
80C2×C3×D12C2×D312yes—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—62no—
83C2×C3×C4×D3—61no—
84C2×C3×Dic6—22no—
85C2×C3×Dic6—62no—
86C2×C3×Dic6C2×D312yescomplemented
87C2×C3×Dic6C2×D312yescomplemented
88C2×#627—21no—
89C2×#626—62no—
90C2×#626—62no—
91C2×#624—21no—
92D3×D12—38no—
93D3×D12—64no—
94D3×D12—64no—
95D3×D12—64no—
96#3797—22no—
97#3797C2×D314yescomplemented
98#3755—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#3717—38no—
106#3717—38no—
107#3717—68no—
108#3715C2×D318yescomplemented
109#3710—34no—
110D3×C3⋊C8—38no—
111C22×C33C2×D411yescharacteristic
112C2×C3⋊(C3⋊D3)—22no—
113C2×C3⋊(C3⋊D3)—42no—
114C2×C3×C3⋊D3SD1614yescomplemented
115C2×C3×C3⋊D3C2×D412yes—
116C2×C32×D3—22no—
117C2×C32×D3—42no—
118D3×C3⋊D3—48no—
119C33×C4C2×D411yescharacteristic
120C33×C4C2411yescharacteristic
121C3×C3⋊Dic3C2×D412yes—
122C32×Dic3—24no—
123C22×C3×Q8—91no—
124C2×D3×Q8—32no—
125C2×D3×D4—92no—
126C2×D3×D4—181no—
127C22×D12—91no—
128C2×C3×SD16—92no—
129C2×#215—32no—
130C2×#215—92no—
131C2×#215—92no—
132C2×#215—98no—
133C2×#215—182no—
134C2×#214—32no—
135C22×C3⋊C8—91no—
136D3×SD16—916no—
137C2×#204—92no—
138C2×D3×C8—92no—
139C23×C32—61no—
140C22×C3⋊D3—61no—
141C22×C3⋊D3—64no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—62no—
144C22×C3⋊D3—122no—
145C22×C3⋊D3—122no—
146C22×C3⋊D3—124no—
147C22×C3⋊D3#6711yescharacteristic
148C22×C3×D3—22no—
149C22×C3×D3—22no—
150C22×C3×D3—32no—
151C22×C3×D3—61no—
152C2×D32—68no—
153C2×D32—68no—
154C2×D32—128no—
155C2×D32—128no—
156C32×Q8—34no—
157C32×Q8—62no—
158C32×D4—32no—
159C32×D4—61no—
160C2×C32×C4—22no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—62no—
164C2×C32×C4C22×D312yes—
165C2×C32×C4C22×D311yescharacteristic
166#638—64no—
167C2×C3⋊Dic3#6711yescharacteristic
168C4⋊(C3⋊D3)—32no—
169C4⋊(C3⋊D3)—34no—
170C4⋊(C3⋊D3)—62no—
171C4⋊(C3⋊D3)—64no—
172C4⋊(C3⋊D3)—61no—
173C4⋊(C3⋊D3)—62no—
174C4⋊(C3⋊D3)—62no—
175C4⋊(C3⋊D3)—122no—
176C4⋊(C3⋊D3)—122no—
177C4⋊(C3⋊D3)—122no—
178C4×C3⋊D3#6712yes—
179C4×C3⋊D3C22×D312yes—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—42no—
184C2×C3×Dic3—61no—
185C3×D12—22no—
186C3×D12—22no—
187C3×D12—22no—
188C3×D12C22×D314yes—
189C3×C4×D3—34no—
190C3×C4×D3—34no—
191C3×C4×D3—62no—
192C3×C4×D3—62no—
193C3×C4×D3—68no—
194C3×Dic6—22no—
195C3×Dic6—22no—
196C3×Dic6—24no—
197C3×Dic6—42no—
198C3×Dic6—68no—
199C3×Dic6C22×D314yescomplemented
200C3×Dic6C22×D314yescomplemented
201#627—24no—
202#626—68no—
203#626—68no—
204#624—24no—
205C3⋊(C3⋊C8)C22×D314yescomplemented
206C3×C3⋊C8—34no—
207C3×C3⋊C8—34no—
208C3×C3⋊C8—64no—
209C22×SD16—271noSylow
210C2×C33C2×SD1612yescomplemented
211C2×C33C22×D411yescharacteristic
212C3⋊(C3⋊D3)—42no—
213C3×C3⋊D3C2×SD1614yescomplemented
214C32×D3—42no—
215C23×D3—181no—
216C23×D3—182no—
217C23×D3—182no—
218C2×C3×Q8—32no—
219C2×C3×Q8—32no—
220C2×C3×Q8—62no—
221C2×C3×Q8—94no—
222C2×C3×Q8—181no—
223C2×C3×Q8—182no—
224C2×C3×D4—32no—
225C2×C3×D4—61no—
226C22×C3×C4—91no—
227C22×C3×C4—181no—
228C2×#67—182no—
229C2×#67—182no—
230D3×Q8—38no—
231D3×Q8—64no—
232D3×D4—98no—
233D3×D4—184no—
234D3×D4—184no—
235D3×D4—184no—
236C2×D12—92no—
237C2×D12—91no—
238C2×D12—94no—
239C2×D12—181no—
240C2×D12—182no—
241C2×D12—182no—
242C2×D12—181no—
243C2×D12—361no—
244C2×D12D3211yescharacteristic
245C2×C4×D3—32no—
246C2×C4×D3—61no—
247C2×C4×D3—62no—
248C2×Dic6—62no—
249C2×Dic6D3212yescomplemented
250C3×SD16—98no—
251C2×C3×C8—92no—
252#215—38no—
253#215—98no—
254#215—98no—
255#215—188no—
256#214—38no—
257C2×C3⋊C8—32no—
258C2×C3⋊C8—32no—
259C2×C3⋊C8—62no—
260C2×C3⋊C8—94no—
261#204—98no—
262D3×C8—98no—
263C22×C32—22no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32—62no—
267C22×C32—122no—
268C22×C32D3×D412yes—
269C22×C32C2×#6711yescharacteristic
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—128no—
277C2×C3⋊D3—124no—
278C2×C3⋊D3—124no—
279C2×C3⋊D3—244no—
280C2×C3⋊D3—244no—
281C2×C3⋊D3#21514yescomplemented
282C2×C3⋊D3C2×#6712yes—
283C2×C3×D3—24no—
284C2×C3×D3—24no—
285C2×C3×D3—38no—
286C2×C3×D3—34no—
287C2×C3×D3—44no—
288C2×C3×D3—44no—
289C2×C3×D3—64no—
290C2×C3×D3—62no—
291D32—1216no—
292D32—1216no—
293C32×C4—22no—
294C32×C4—21no—
295C32×C4—22no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—64no—
300C32×C4D3×D412yes—
301C32×C4C2×#6711yescharacteristic
302C32×C4C23×D312yes—
303C32×C4C23×D311yescharacteristic
304C3⋊Dic3C2×#6712yes—
305C3×Dic3—24no—
306C3×Dic3—24no—
307C3×Dic3—34no—
308C3×Dic3—44no—
309C3×Dic3—62no—
310C22×Q8—91no—
311C22×D4—271no—
312C2×SD16—92no—
313C2×SD16—272no—
314C2×SD16—278no—
315C22×C8—271no—
316C33C22×SD1611yescharacteristic, complemented, Sylow
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—91no—
320C22×D3—21no—
321C22×D3—32no—
322C22×D3—61no—
323C22×D3—184no—
324C22×D3—188no—
325C22×D3—188no—
326C22×D3—182no—
327C22×D3—182no—
328C22×D3—181no—
329C22×D3—362no—
330C22×D3—362no—
331C22×D3—368no—
332C22×D3—364no—
333C22×D3—368no—
334C3×Q8—34no—
335C3×Q8—34no—
336C3×Q8—64no—
337C3×Q8—62no—
338C3×Q8—62no—
339C3×Q8—122no—
340C3×Q8—184no—
341C3×D4—34no—
342C3×D4—62no—
343C3×D4—62no—
344C3×D4—62no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—41no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—91no—
351C2×C3×C4—92no—
352C2×C3×C4—92no—
353C2×C3×C4—122no—
354C2×C3×C4—184no—
355C2×C3×C4C2×D3212yes—
356C2×C3×C4C2×D3211yescharacteristic
357#67—188no—
358#67—188no—
359C2×Dic3—22no—
360C2×Dic3—32no—
361C2×Dic3—61no—
362D12—21no—
363D12—94no—
364D12—92no—
365D12—182no—
366D12—184no—
367D12—182no—
368D12—184no—
369D12—182no—
370D12—181no—
371D12—362no—
372D12—362no—
373D12—362no—
374D12C2×D3212yes—
375C4×D3—34no—
376C4×D3—34no—
377C4×D3—62no—
378C4×D3—62no—
379C4×D3—68no—
380Dic6—22no—
381Dic6—68no—
382Dic6C2×D3214yescomplemented
383C3×C8—94no—
384C3⋊C8—34no—
385C3⋊C8—34no—
386C3⋊C8—64no—
387C2×C32—22no—
388C2×C32—21no—
389C2×C32—22no—
390C2×C32—24no—
391C2×C32—42no—
392C2×C32—41no—
393C2×C32—122no—
394C2×C32D3×SD1614yescomplemented
395C2×C32C2×#21512yescomplemented
396C2×C32C2×D3×D412yes—
397C2×C32C22×#6711yescharacteristic
398C3⋊D3—122no—
399C3⋊D3—124no—
400C3⋊D3—124no—
401C3⋊D3—244no—
402C3⋊D3—244no—
403C3⋊D3C2×#21514yescomplemented
404C3×D3—38no—
405C3×D3—44no—
406C3×D3—44no—
407C3×D3—64no—
408C24—541no—
409C2×Q8—32no—
410C2×Q8—94no—
411C2×Q8—181no—
412C2×Q8—182no—
413C2×D4—31no—
414C2×D4—274no—
415C2×D4—271no—
416C2×D4—542no—
417C2×D4—541no—
418C22×C4—91no—
419C22×C4—181no—
420SD16—98no—
421SD16—278no—
422C2×C8—92no—
423C2×C8—274no—
424C22×C3—21no—
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—64no—
428C22×C3—64no—
429C22×C3—92no—
430C22×C3—94no—
431C22×C3—124no—
432C22×C3—124no—
433C22×C3D3×#6712yes—
434C22×C3#381111yescharacteristic
435C2×D3—22no—
436C2×D3—34no—
437C2×D3—38no—
438C2×D3—42no—
439C2×D3—62no—
440C2×D3—64no—
441C2×D3—184no—
442C2×D3—184no—
443C2×D3—182no—
444C2×D3—364no—
445C2×D3—364no—
446C2×D3—3616no—
447C2×D3—368no—
448C2×D3—3616no—
449C2×D3—364no—
450C2×D3—362no—
451C2×D3—364no—
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×C4—124no—
464C3×C4D3×#6712yes—
465C3×C4#381111yescharacteristic
466C3×C4C22×D3212yes—
467C3×C4C22×D3211yescharacteristic
468Dic3—24no—
469Dic3—34no—
470Dic3—62no—
471C32—21no—
472C32—22no—
473C32—41no—
474C32C2×D3×SD1612yescomplemented
475C32C22×#21511yescharacteristic, complemented
476C23—61no—
477C23—91no—
478C23—544no—
479C23—541no—
480C23—1084no—
481Q8—34no—
482Q8—62no—
483Q8—184no—
484D4—32no—
485D4—61no—
486D4—272no—
487D4—544no—
488D4—541no—
489C2×C4—62no—
490C2×C4—91no—
491C2×C4—92no—
492C2×C4—92no—
493C2×C4—184no—
494C2×C4D3311yescharacteristic
495C8—94no—
496C2×C3—22no—
497C2×C3—24no—
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—42no—
501C2×C3—41no—
502C2×C3—94no—
503C2×C3—124no—
504C2×C3—124no—
505C2×C3#6393012yescomplemented
506C2×C3D3×#21514yescomplemented
507C2×C3C2×D3×#6712yes—
508C2×C3C2×#381111yescharacteristic
509D3—38no—
510D3—42no—
511D3—64no—
512D3—364no—
513D3—362no—
514D3—364no—
515D3—724no—
516D3—724no—
517C22—62no—
518C22—94no—
519C22—92no—
520C22—122no—
521C22—542no—
522C22—1088no—
523C22—1082no—
524C22#8895211yescharacteristic
525C4—64no—
526C4—92no—
527C4#8895211yescharacteristic
528C4C2×D3311yescharacteristic
529C3—21no—
530C3—22no—
531C3—41no—
532C3C2×#6393011yescharacteristic, complemented
533C3C2×D3×#21512yescomplemented
534C2—94no—
535C2—122no—
536C2—1082no—
537C2#1172283812yesdirect product factor
538C2C2×#8895211yescharacteristic
subgroupG/H
centerC22#88952
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3⋊D3×D12C2
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×#3797C2semidirect
3. C2×#88808C2semidirect
4. C2×#88795C2semidirect
5. C2×C3×#3715C2semidirect
6. #11722838C2direct (N × H)
13. C2×C32×Dic6C22semidirect
20. C3×#3797C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. #88808C22semidirect
23. #88795C22semidirect
24. C3×#3715C22semidirect
29. C2×#3797D3semidirect
38. C2×#3715D3semidirect
56. C32×Dic6C23semidirect
60. C3×C3⋊(C3⋊C8)C23semidirect
86. C2×C3×Dic6C2×D3semidirect
87. C2×C3×Dic6C2×D3semidirect
97. #3797C2×D3semidirect
99. C2×C3⋊(C3⋊C8)C2×D3semidirect
108. #3715C2×D3semidirect
114. C2×C3×C3⋊D3SD16semidirect
199. C3×Dic6C22×D3semidirect
200. C3×Dic6C22×D3semidirect
205. C3⋊(C3⋊C8)C22×D3semidirect
210. C2×C33C2×SD16semidirect
213. C3×C3⋊D3C2×SD16semidirect
249. C2×Dic6D32semidirect
281. C2×C3⋊D3#215semidirect
316. C33C22×SD16semidirect
382. Dic6C2×D32semidirect
394. C2×C32D3×SD16semidirect
395. C2×C32C2×#215semidirect
403. C3⋊D3C2×#215semidirect
474. C32C2×D3×SD16semidirect
475. C32C22×#215semidirect
505. C2×C3#63930semidirect
506. C2×C3D3×#215semidirect
532. C3C2×#63930semidirect
533. C3C2×D3×#215semidirect
537. C2#11722838direct (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×D122
C2×C3×#37972
C2×#888082
C2×#887952
C2×C3×#37152
#117228382
C2×D3×#2153
C2×#639303