Finite Group Databasebeta

← C2×#11722831 #13170540

Subgroups — C2×#11722831 #13170540

Order 1728.

1602 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1602 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, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C2×#88789C212yesmaximal, complemented
4C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×#3714C212yesmaximal, complemented
6#11722831C218yesmaximal, direct product factor
7C2×D3×#213—32nomaximal
8C2×#63797—31nomaximal
9C22×#1113—21no—
10C22×C3×D32—21no—
11C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2212yescomplemented
14C2×#6295—21no—
15C4⋊#1113—22no—
16C4⋊#1113C2214yescomplemented
17C2×C3×#625—21no—
18C3×#3802—22no—
19C3×#3802C2214yescomplemented
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C2×C32×C3⋊C8C2211yescharacteristic, complemented
22#88789C2218yescomplemented
23C3⋊D3×C3⋊C8C2214yescomplemented
24C3×#3714C2218yescomplemented
25C2×C3×D3×D4—32no—
26C2×#3802—31no—
27C2×#3802D311yescharacteristic, complemented
28C2×D3×D12—32no—
29C2×#3753—32no—
30C2×C8×C3⋊D3—31no—
31C2×C3×#213—32no—
32C2×C3×D24—32no—
33D3×#213—316no—
34C2×#3714D312yescomplemented
35C2×#3713—32no—
36C2×#3713—32no—
37C2×#3713—62no—
38C2×D3×C3⋊C8—32no—
39C2×D3×C3⋊C8—61no—
40#63797—38no—
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3D411yescharacteristic
43C22×C32×D3—22no—
44C2×#1113—24no—
45C2×#1113—44no—
46C2×C3×D32—24no—
47C2×C3×D32—44no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4⋊(C3⋊D3)—22no—
51C3×C4⋊(C3⋊D3)C2314yescomplemented
52C3×C4×C3⋊D3D412yes—
53C3×C4×C3⋊D3C2312yes—
54C32×D12—22no—
55C32×D12C2314yescomplemented
56#6295—24no—
57C3×#625—24no—
58C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
59C32×C3⋊C8C2312yescomplemented
60C22×#213—91no—
61C2×D3×D8—92no—
62C23×C3×D3—62no—
63C22×D32—21no—
64C22×D32—62no—
65C22×D32—61no—
66C2×C32×D4—32no—
67C2×C4⋊(C3⋊D3)C2×D312yescomplemented
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C3×#67—62no—
70C3×D3×D4—38no—
71C3×D3×D4—64no—
72C2×C3×D12—22no—
73C2×C3×D12—32no—
74C2×C3×D12—32no—
75C2×C3×D12—62no—
76C2×C3×D12C2×D312yescomplemented
77C2×C3×D12C2×D312yescomplemented
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—61no—
80C2×#626—62no—
81C2×#625—21no—
82C2×#625—61no—
83#3802—22no—
84#3802—34no—
85#3802—62no—
86#3802C2×D314yescomplemented
87D3×D12—38no—
88D3×D12—64no—
89C2×C32×C8—31no—
90#3753—38no—
91C2×C3⋊(C3⋊C8)—32no—
92C2×C3⋊(C3⋊C8)—31no—
93C2×C3⋊(C3⋊C8)—61no—
94C2×C3⋊(C3⋊C8)—62no—
95C2×C3⋊(C3⋊C8)—121no—
96C8×C3⋊D3—34no—
97C3×#213—38no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2×D312yescomplemented
100C3×D24—38no—
101#3714C2×D318yescomplemented
102#3713—38no—
103#3713—38no—
104#3713—68no—
105D3×C3⋊C8—38no—
106D3×C3⋊C8—64no—
107C22×C33C2×D411yescharacteristic
108C2×C3×C3⋊D3—24no—
109C2×C3×C3⋊D3—44no—
110C2×C3×C3⋊D3D814yescomplemented
111C2×C3×C3⋊D3C2×D412yes—
112C2×C32×D3—24no—
113C2×C32×D3—44no—
114#1113—48no—
115C3×D32—48no—
116C33×C4C2×D411yescharacteristic
117C33×C4C2411yescharacteristic
118C3×C3⋊Dic3C2×D412yes—
119C22×C3×D4—91no—
120C2×D3×D4—32no—
121C2×D3×D4—92no—
122C22×D12—91no—
123C2×C3×D8—92no—
124C2×#213—32no—
125C2×#213—98no—
126C2×#213—92no—
127C2×#213—92no—
128C2×#213—92no—
129C2×#213—182no—
130C22×C3⋊C8—91no—
131D3×D8—916no—
132C2×D24—32no—
133C2×D3×C8—92no—
134C2×D3×C8—181no—
135C23×C32—62no—
136C22×C3⋊D3—22no—
137C22×C3⋊D3#6711yescharacteristic
138C22×C3×D3—22no—
139C22×C3×D3—22no—
140C22×C3×D3—32no—
141C22×C3×D3—42no—
142C22×C3×D3—61no—
143C22×C3×D3—68no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3—122no—
147C22×C3×D3—128no—
148C2×D32—24no—
149C2×D32—44no—
150C2×D32—68no—
151C2×D32—64no—
152C2×D32—128no—
153C2×D32—124no—
154C32×D4—34no—
155C32×D4—62no—
156C2×C32×C4—21no—
157C2×C32×C4—22no—
158C2×C32×C4—41no—
159C2×C32×C4C22×D312yes—
160C2×C32×C4C22×D311yescharacteristic
161C2×C3⋊Dic3#6711yescharacteristic
162C4⋊(C3⋊D3)—22no—
163C4⋊(C3⋊D3)C22×D314yescomplemented
164C4×C3⋊D3#6712yes—
165C4×C3⋊D3C22×D312yes—
166C3×#67—68no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—61no—
169C3×D12—22no—
170C3×D12—22no—
171C3×D12—24no—
172C3×D12—34no—
173C3×D12—34no—
174C3×D12—42no—
175C3×D12—64no—
176C3×D12—62no—
177C3×D12—62no—
178C3×D12—122no—
179C3×D12C22×D314yescomplemented
180C3×D12C22×D314yescomplemented
181C3×C4×D3—34no—
182C3×C4×D3—34no—
183C3×C4×D3—62no—
184C3×C4×D3—62no—
185#626—68no—
186#625—24no—
187#625—64no—
188C32×C8—32no—
189C3⋊(C3⋊C8)—34no—
190C3⋊(C3⋊C8)—32no—
191C3⋊(C3⋊C8)—62no—
192C3⋊(C3⋊C8)—64no—
193C3⋊(C3⋊C8)—122no—
194C3×C3⋊C8—22no—
195C3×C3⋊C8C22×D314yescomplemented
196C22×D8—271noSylow
197C2×C33C2×D812yescomplemented
198C2×C33C22×D411yescharacteristic
199C3×C3⋊D3—44no—
200C3×C3⋊D3C2×D814yescomplemented
201C32×D3—44no—
202C24×C3—181no—
203C23×D3—62no—
204C23×D3—182no—
205C23×D3—181no—
206C2×C3×D4—32no—
207C2×C3×D4—32no—
208C2×C3×D4—62no—
209C2×C3×D4—94no—
210C2×C3×D4—92no—
211C2×C3×D4—182no—
212C2×C3×D4—181no—
213C22×C3×C4—91no—
214C2×#67—62no—
215C2×#67—182no—
216D3×D4—38no—
217D3×D4—64no—
218D3×D4—98no—
219D3×D4—184no—
220C2×D12—32no—
221C2×D12—32no—
222C2×D12—62no—
223C2×D12—94no—
224C2×D12—182no—
225C2×D12—181no—
226C2×D12D3212yescomplemented
227C2×C4×D3—32no—
228C2×C4×D3—61no—
229C3×D8—98no—
230C2×C3×C8—32no—
231C2×C3×C8—61no—
232#213—38no—
233#213—98no—
234#213—98no—
235#213—98no—
236#213—188no—
237C2×C3⋊C8—92no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—94no—
240C2×C3⋊C8—181no—
241C2×C3⋊C8—182no—
242C2×C3⋊C8—361no—
243C2×C3⋊C8D3211yescharacteristic, complemented
244D24—38no—
245D3×C8—98no—
246D3×C8—184no—
247C22×C32—21no—
248C22×C32—22no—
249C22×C32—41no—
250C22×C32—64no—
251C22×C32—124no—
252C22×C32D3×D412yes—
253C22×C32C2×#6711yescharacteristic
254C2×C3⋊D3—24no—
255C2×C3⋊D3—44no—
256C2×C3⋊D3#21314yescomplemented
257C2×C3⋊D3C2×#6712yes—
258C2×C3×D3—24no—
259C2×C3×D3—24no—
260C2×C3×D3—38no—
261C2×C3×D3—34no—
262C2×C3×D3—44no—
263C2×C3×D3—44no—
264C2×C3×D3—44no—
265C2×C3×D3—64no—
266C2×C3×D3—62no—
267C2×C3×D3—64no—
268C2×C3×D3—64no—
269C2×C3×D3—84no—
270C2×C3×D3—124no—
271C2×C3×D3—1216no—
272C2×C3×D3—124no—
273C2×C3×D3—124no—
274C2×C3×D3—244no—
275D32—48no—
276D32—1216no—
277D32—128no—
278C32×C4—21no—
279C32×C4—22no—
280C32×C4—21no—
281C32×C4—22no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4D3×D412yes—
285C32×C4C2×#6711yescharacteristic
286C32×C4C23×D312yes—
287C32×C4C23×D311yescharacteristic
288C3⋊Dic3C2×#6712yes—
289C3×Dic3—34no—
290C3×Dic3—62no—
291C22×D4—91no—
292C22×D4—271no—
293C2×D8—92no—
294C2×D8—278no—
295C2×D8—272no—
296C22×C8—271no—
297C33C22×D811yescharacteristic, complemented, Sylow
298C23×C3—62no—
299C23×C3—62no—
300C23×C3—91no—
301C23×C3—122no—
302C23×C3—184no—
303C23×C3—182no—
304C23×C3—364no—
305C22×D3—22no—
306C22×D3—32no—
307C22×D3—61no—
308C22×D3—68no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—122no—
312C22×D3—128no—
313C22×D3—188no—
314C22×D3—184no—
315C22×D3—368no—
316C22×D3—364no—
317C3×D4—34no—
318C3×D4—34no—
319C3×D4—64no—
320C3×D4—62no—
321C3×D4—62no—
322C3×D4—94no—
323C3×D4—122no—
324C3×D4—184no—
325C3×D4—182no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—41no—
329C2×C3×C4—91no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C2×D3212yes—
334#67—68no—
335#67—188no—
336C2×Dic3—32no—
337C2×Dic3—61no—
338D12—22no—
339D12—34no—
340D12—34no—
341D12—64no—
342D12—62no—
343D12—62no—
344D12—122no—
345D12—184no—
346D12C2×D3214yescomplemented
347C4×D3—34no—
348C4×D3—34no—
349C4×D3—62no—
350C4×D3—62no—
351C3×C8—34no—
352C3×C8—62no—
353C3⋊C8—94no—
354C3⋊C8—92no—
355C3⋊C8—182no—
356C3⋊C8—184no—
357C3⋊C8—362no—
358C3⋊C8C2×D3212yescomplemented
359C2×C32—21no—
360C2×C32—22no—
361C2×C32—22no—
362C2×C32—24no—
363C2×C32—42no—
364C2×C32—41no—
365C2×C32—124no—
366C2×C32D3×D814yescomplemented
367C2×C32C2×#21312yescomplemented
368C2×C32C2×D3×D412yes—
369C2×C32C22×#6711yescharacteristic
370C3⋊D3—44no—
371C3⋊D3C2×#21314yescomplemented
372C3×D3—38no—
373C3×D3—44no—
374C3×D3—44no—
375C3×D3—64no—
376C3×D3—84no—
377C3×D3—124no—
378C3×D3—124no—
379C3×D3—244no—
380C24—181no—
381C24—541no—
382C2×D4—32no—
383C2×D4—94no—
384C2×D4—92no—
385C2×D4—182no—
386C2×D4—181no—
387C2×D4—274no—
388C2×D4—542no—
389C2×D4—541no—
390C22×C4—91no—
391D8—98no—
392D8—278no—
393C2×C8—31no—
394C2×C8—274no—
395C2×C8—271no—
396C22×C3—21no—
397C22×C3—22no—
398C22×C3—41no—
399C22×C3—64no—
400C22×C3—64no—
401C22×C3—92no—
402C22×C3—94no—
403C22×C3—124no—
404C22×C3—124no—
405C22×C3—124no—
406C22×C3—184no—
407C22×C3—244no—
408C22×C3—368no—
409C22×C3—364no—
410C22×C3#380211yescharacteristic
411C22×C3D3×#6712yes—
412C2×D3—24no—
413C2×D3—34no—
414C2×D3—38no—
415C2×D3—44no—
416C2×D3—62no—
417C2×D3—64no—
418C2×D3—64no—
419C2×D3—64no—
420C2×D3—124no—
421C2×D3—1216no—
422C2×D3—124no—
423C2×D3—124no—
424C2×D3—244no—
425C2×D3—3616no—
426C2×D3—368no—
427C3×C4—21no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—22no—
431C3×C4—41no—
432C3×C4—41no—
433C3×C4—92no—
434C3×C4#380211yescharacteristic
435C3×C4D3×#6712yes—
436C3×C4C22×D3211yescharacteristic
437C3×C4C22×D3212yes—
438Dic3—34no—
439Dic3—62no—
440C32—22no—
441C32—21no—
442C32—41no—
443C32C2×D3×D812yescomplemented
444C32C22×#21311yescharacteristic, complemented
445C23—62no—
446C23—91no—
447C23—184no—
448C23—182no—
449C23—364no—
450C23—544no—
451C23—1084no—
452D4—34no—
453D4—62no—
454D4—94no—
455D4—184no—
456D4—182no—
457D4—544no—
458C2×C4—91no—
459C2×C4—92no—
460C2×C4—92no—
461C2×C4D3311yescharacteristic
462C8—32no—
463C8—272no—
464C2×C3—22no—
465C2×C3—24no—
466C2×C3—21no—
467C2×C3—22no—
468C2×C3—42no—
469C2×C3—41no—
470C2×C3—94no—
471C2×C3—124no—
472C2×C3—124no—
473C2×C3—244no—
474C2×C3—364no—
475C2×C3#6379712yescomplemented
476C2×C3D3×#21314yescomplemented
477C2×C3C2×#380211yescharacteristic
478C2×C3C2×D3×#6712yes—
479D3—38no—
480D3—44no—
481D3—64no—
482D3—124no—
483D3—124no—
484D3—244no—
485C22—64no—
486C22—94no—
487C22—92no—
488C22—124no—
489C22—184no—
490C22—368no—
491C22—364no—
492C22—1088no—
493C22#8895311yescharacteristic
494C4—92no—
495C4#8895311yescharacteristic
496C4C2×D3311yescharacteristic
497C3—21no—
498C3—22no—
499C3—41no—
500C3C2×#6379711yescharacteristic, complemented
501C3C2×D3×#21312yescomplemented
502C2—94no—
503C2—124no—
504C2—364no—
505C2#1172283112yesdirect product factor
506C2C2×#8895311yescharacteristic
subgroupG/H
centerC22#88953
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
hypercenterC2×C4D33
hypocenterC33C22×D8
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×C3×#3802C2semidirect
3. C2×#88789C2semidirect
4. C2×C3⋊D3×C3⋊C8C2semidirect
5. C2×C3×#3714C2semidirect
6. #11722831C2direct (N × H)
11. C2×C3×C4⋊(C3⋊D3)C22semidirect
13. C2×C32×D12C22semidirect
16. C4⋊#1113C22semidirect
19. C3×#3802C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. C2×C32×C3⋊C8C22semidirect
22. #88789C22semidirect
23. C3⋊D3×C3⋊C8C22semidirect
24. C3×#3714C22semidirect
27. C2×#3802D3semidirect
34. C2×#3714D3semidirect
51. C3×C4⋊(C3⋊D3)C23semidirect
55. C32×D12C23semidirect
58. C3⋊(C3⋊(C3⋊C8))C23semidirect
59. C32×C3⋊C8C23semidirect
67. C2×C4⋊(C3⋊D3)C2×D3semidirect
76. C2×C3×D12C2×D3semidirect
77. C2×C3×D12C2×D3semidirect
86. #3802C2×D3semidirect
99. C2×C3×C3⋊C8C2×D3semidirect
101. #3714C2×D3semidirect
110. C2×C3×C3⋊D3D8semidirect
163. C4⋊(C3⋊D3)C22×D3semidirect
179. C3×D12C22×D3semidirect
180. C3×D12C22×D3semidirect
195. C3×C3⋊C8C22×D3semidirect
197. C2×C33C2×D8semidirect
200. C3×C3⋊D3C2×D8semidirect
226. C2×D12D32semidirect
243. C2×C3⋊C8D32semidirect
256. C2×C3⋊D3#213semidirect
297. C33C22×D8semidirect
346. D12C2×D32semidirect
358. C3⋊C8C2×D32semidirect
366. C2×C32D3×D8semidirect
367. C2×C32C2×#213semidirect
371. C3⋊D3C2×#213semidirect
443. C32C2×D3×D8semidirect
444. C32C22×#213semidirect
475. C2×C3#63797semidirect
476. C2×C3D3×#213semidirect
500. C3C2×#63797semidirect
501. C3C2×D3×#213semidirect
505. C2#11722831direct (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×#38022
C2×#887892
C2×C3⋊D3×C3⋊C82
C2×C3×#37142
#117228312
C2×D3×#2133
C2×#637973