Finite Group Databasebeta

← C2×#13049721 #72220271

Subgroups — C2×#13049721 #72220271

Order 2592.

4546 conjugacy classes of subgroups; only the 379 normal subgroups are shown.

normal subgroups only

This group has 4546 conjugacy classes of subgroups — too many to draw. Only its 379 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
1C23×C32×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×C3⋊#1113C211yesmaximal, characteristic, complemented
3C2×C3×#6312C212yesmaximal, complemented
4#13049721C218yesmaximal, direct product factor
5C2×#10620157C212yesmaximal, complemented
6C2×#10620155C211yesmaximal, characteristic, complemented
7C2×#89049—34nomaximal
8C2×D3×#638—32nomaximal
9C23×C34C2211yescharacteristic, complemented
10C22×C3×C3⋊(C3⋊D3)C2212yescomplemented
11C22×C32×C3⋊D3—24no—
12C22×C32×C3⋊D3C2211yescharacteristic
13C22×C32×C3⋊D3C2214yescomplemented
14C22×C32×C3⋊D3C2211yescharacteristic, complemented
15C2×C3⋊#1113—24no—
16C2×C3⋊#1113C2214yescomplemented
17C3×#6312C2218yescomplemented
18C2×C3×C3⋊(C3⋊Dic3)C2212yescomplemented
19#10620157C2218yescomplemented
20#10620155C2214yescomplemented
21C23×C3×C3⋊D3D314yescomplemented
22C23×C32×D3—32no—
23C23×C32×D3—61no—
24C22×#1113—34no—
25C22×D3×C3⋊D3—32no—
26C2×#6312D312yescomplemented
27C2×C3×#638—32no—
28C2×C3×#638—38no—
29C2×C3×#638—68no—
30#89049—332no—
31C2×#6295—38no—
32C2×#6294—34no—
33D3×#638—316no—
34C2×#6291—32no—
35C2×#6290—32no—
36C2×D3×C3⋊Dic3—32no—
37C22×C34—22no—
38C22×C34C2311yescharacteristic
39C22×C34C2312yescomplemented
40C2×C3×C3⋊(C3⋊D3)—24no—
41C2×C3×C3⋊(C3⋊D3)C2314yescomplemented
42C2×C32×C3⋊D3—22no—
43C2×C32×C3⋊D3—28no—
44C2×C32×C3⋊D3D414yescomplemented
45C2×C32×C3⋊D3C2312yes—
46C2×C32×C3⋊D3C2312yescomplemented
47C3⋊#1113—28no—
48C3×C3⋊(C3⋊Dic3)C2314yescomplemented
49C22×#638—91no—
50C2×#3811—91no—
51C2×D3×#67—98no—
52C23×C33—21no—
53C23×C33—28no—
54C23×C33—44no—
55C23×C33C2×D312yescomplemented
56C23×C33C2×D314yescomplemented
57C22×C3⋊(C3⋊D3)C2×D312yescomplemented
58C22×C3×C3⋊D3—216no—
59C22×C3×C3⋊D3—32no—
60C22×C3×C3⋊D3—38no—
61C22×C3×C3⋊D3—68no—
62C22×C3×C3⋊D3C2×D314yes—
63C22×C3×C3⋊D3C2×D3116yescomplemented
64C22×C3×C3⋊D3C2×D314yescomplemented
65C22×C32×D3—32no—
66C22×C32×D3—32no—
67C22×C32×D3—38no—
68C22×C32×D3—68no—
69C22×C32×D3—61no—
70C22×C32×D3—64no—
71C22×C32×D3—61no—
72C22×C32×D3—68no—
73C2×#1113—316no—
74C2×#1113—616no—
75C2×D3×C3⋊D3—38no—
76C2×D3×C3⋊D3—68no—
77#6312C2×D318yescomplemented
78C2×C3⋊(C3⋊Dic3)C2×D312yescomplemented
79C3×#638—38no—
80C3×#638—332no—
81C3×#638—632no—
82C2×C3×C3⋊Dic3—32no—
83C2×C3×C3⋊Dic3—38no—
84C2×C3×C3⋊Dic3—68no—
85#6295—332no—
86#6294—316no—
87#6291—38no—
88#6290—38no—
89D3×C3⋊Dic3—38no—
90C2×C34—22no—
91C2×C34C2×D412yescomplemented
92C2×C34C2411yescharacteristic
93C3×C3⋊(C3⋊D3)—24no—
94C32×C3⋊D3—22no—
95C32×C3⋊D3C2×D414yescomplemented
96C24×C32—91no—
97C23×C3⋊D3—91no—
98C23×C3⋊D3C3⋊D311yescharacteristic, complemented
99C23×C3×D3—38no—
100C23×C3×D3—64no—
101C22×D32—98no—
102C22×D32—91no—
103C2×#638—32no—
104C2×#638—38no—
105C2×#638—68no—
106C2×#638—92no—
107C2×#638—98no—
108C22×C3⋊Dic3—91no—
109C2×C3×#67—98no—
110C2×C3×#67—92no—
111C2×C3×#67—188no—
112#3811—98no—
113D3×#67—964no—
114C2×#626—98no—
115C2×#626—92no—
116C2×#625—98no—
117C2×#624—91no—
118C2×D3×Dic3—98no—
119C22×C33—24no—
120C22×C33—28no—
121C22×C33—21no—
122C22×C33—22no—
123C22×C33—24no—
124C22×C33—28no—
125C22×C33—216no—
126C22×C33—416no—
127C22×C33—44no—
128C22×C33—48no—
129C22×C33—416no—
130C22×C33C22×D312yes—
131C22×C33C22×D314yes—
132C22×C33C22×D314yescomplemented
133C22×C33C22×D318yescomplemented
134C2×C3⋊(C3⋊D3)—24no—
135C2×C3⋊(C3⋊D3)C22×D314yescomplemented
136C2×C3×C3⋊D3—28no—
137C2×C3×C3⋊D3—232no—
138C2×C3×C3⋊D3—34no—
139C2×C3×C3⋊D3—316no—
140C2×C3×C3⋊D3—64no—
141C2×C3×C3⋊D3—616no—
142C2×C3×C3⋊D3—616no—
143C2×C3×C3⋊D3—1216no—
144C2×C3×C3⋊D3#67116yescomplemented
145C2×C3×C3⋊D3C22×D318yes—
146C2×C3×C3⋊D3C22×D318yescomplemented
147C2×C32×D3—38no—
148C2×C32×D3—34no—
149C2×C32×D3—34no—
150C2×C32×D3—616no—
151C2×C32×D3—64no—
152C2×C32×D3—62no—
153C2×C32×D3—62no—
154C2×C32×D3—616no—
155C2×C32×D3—64no—
156C2×C32×D3—64no—
157#1113—632no—
158D3×C3⋊D3—616no—
159C3⋊(C3⋊Dic3)C22×D314yescomplemented
160C3×C3⋊Dic3—34no—
161C3×C3⋊Dic3—316no—
162C3×C3⋊Dic3—616no—
163C22×#67—274no—
164C2×D3×D4—272no—
165C34C22×D411yescharacteristic, complemented, Sylow
166C23×C32—28no—
167C23×C32—28no—
168C23×C32—24no—
169C23×C32—44no—
170C23×C32—412no—
171C23×C32—44no—
172C23×C32—91no—
173C23×C32—91no—
174C23×C32—94no—
175C23×C32—184no—
176C23×C32D3218yescomplemented
177C23×C32D3211yescharacteristic, complemented
178C23×C32C2×C3⋊D311yescharacteristic, complemented
179C22×C3⋊D3—24no—
180C22×C3⋊D3—32no—
181C22×C3⋊D3—38no—
182C22×C3⋊D3—68no—
183C22×C3⋊D3—94no—
184C22×C3⋊D3—184no—
185C22×C3⋊D3C2×C3⋊D311yescharacteristic
186C22×C3⋊D3C2×C3⋊D314yescomplemented
187C22×C3⋊D3C2×C3⋊D311yescharacteristic, complemented
188C22×C3×D3—38no—
189C22×C3×D3—38no—
190C22×C3×D3—332no—
191C22×C3×D3—64no—
192C22×C3×D3—64no—
193C22×C3×D3—616no—
194C22×C3×D3—632no—
195C22×C3×D3—632no—
196C22×C3×D3—98no—
197C22×C3×D3—92no—
198C22×C3×D3—188no—
199C2×D32—932no—
200C2×D32—94no—
201C2×D32—1832no—
202C2×D32—184no—
203#638—38no—
204#638—332no—
205#638—632no—
206#638—98no—
207C2×C3⋊Dic3—32no—
208C2×C3⋊Dic3—38no—
209C2×C3⋊Dic3—68no—
210C2×C3⋊Dic3—94no—
211C3×#67—932no—
212C3×#67—98no—
213C3×#67—1832no—
214C2×C3×Dic3—98no—
215C2×C3×Dic3—92no—
216C2×C3×Dic3—188no—
217#626—932no—
218#626—98no—
219#625—932no—
220#624—94no—
221D3×Dic3—932no—
222C2×C33—24no—
223C2×C33—24no—
224C2×C33—28no—
225C2×C33—28no—
226C2×C33—21no—
227C2×C33—216no—
228C2×C33—22no—
229C2×C33—48no—
230C2×C33—44no—
231C2×C33—416no—
232C2×C33—416no—
233C2×C33D3×D414yescomplemented
234C2×C33C2×#6718yescomplemented
235C2×C33C23×D312yes—
236C2×C33C23×D314yes—
237C3⋊(C3⋊D3)—24no—
238C3×C3⋊D3—28no—
239C3×C3⋊D3—64no—
240C3×C3⋊D3—616no—
241C3×C3⋊D3—1216no—
242C3×C3⋊D3C2×#67116yescomplemented
243C32×D3—38no—
244C32×D3—64no—
245C32×D3—64no—
246C32×D3—64no—
247C24×C3—94no—
248C23×D3—32no—
249C23×D3—61no—
250C23×D3—272no—
251C23×D3—274no—
252C2×C3×D4—272no—
253C2×#67—92no—
254C2×#67—98no—
255C2×#67—188no—
256C2×#67—272no—
257C2×#67—278no—
258C2×#67—2732no—
259C22×Dic3—274no—
260D3×D4—2716no—
261C2×D12—272no—
262C2×C4×D3—272no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—28no—
266C22×C32—216no—
267C22×C32—216no—
268C22×C32—216no—
269C22×C32—28no—
270C22×C32—216no—
271C22×C32—24no—
272C22×C32—28no—
273C22×C32—44no—
274C22×C32—48no—
275C22×C32—412no—
276C22×C32—424no—
277C22×C32—416no—
278C22×C32—416no—
279C22×C32—416no—
280C22×C32—416no—
281C22×C32—448no—
282C22×C32—44no—
283C22×C32—48no—
284C22×C32—92no—
285C22×C32—94no—
286C22×C32—92no—
287C22×C32—188no—
288C22×C32—182no—
289C22×C32C2×D3218yes—
290C22×C32C2×D3211yescharacteristic
291C22×C32C2×D32116yescomplemented
292C22×C32C2×D3212yescomplemented
293C22×C32C22×C3⋊D311yescharacteristic
294C22×C32C22×C3⋊D312yescomplemented
295C2×C3⋊D3—22no—
296C2×C3⋊D3—28no—
297C2×C3⋊D3—34no—
298C2×C3⋊D3—316no—
299C2×C3⋊D3—616no—
300C2×C3⋊D3—64no—
301C2×C3⋊D3—616no—
302C2×C3⋊D3—1216no—
303C2×C3⋊D3—188no—
304C2×C3⋊D3#63814yescomplemented
305C2×C3⋊D3C22×C3⋊D312yes—
306C2×C3⋊D3C22×C3⋊D312yescomplemented
307C2×C3×D3—316no—
308C2×C3×D3—332no—
309C2×C3×D3—316no—
310C2×C3×D3—616no—
311C2×C3×D3—68no—
312C2×C3×D3—68no—
313C2×C3×D3—664no—
314C2×C3×D3—664no—
315C2×C3×D3—616no—
316C2×C3×D3—616no—
317C2×C3×D3—916no—
318C2×C3×D3—94no—
319C2×C3×D3—1816no—
320C2×C3×D3—1816no—
321C2×C3×D3—184no—
322C2×C3×D3—3616no—
323D32—1864no—
324D32—188no—
325C3⋊Dic3—34no—
326C3⋊Dic3—316no—
327C3⋊Dic3—616no—
328C3×Dic3—916no—
329C3×Dic3—94no—
330C3×Dic3—1816no—
331C22×D4—811noSylow
332C33—28no—
333C33—21no—
334C33—44no—
335C33C2×D3×D412yescomplemented
336C33C22×#6714yescomplemented
337C23×C3—21no—
338C23×C3—28no—
339C23×C3—44no—
340C23×C3—94no—
341C23×C3—94no—
342C23×C3—916no—
343C23×C3—1816no—
344C23×C3—272no—
345C23×C3D3×C3⋊D312yescomplemented
346C23×C3#111314yescomplemented
347C22×D3—32no—
348C22×D3—32no—
349C22×D3—38no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—64no—
353C22×D3—68no—
354C22×D3—68no—
355C22×D3—92no—
356C22×D3—98no—
357C22×D3—188no—
358C22×D3—278no—
359C22×D3—2716no—
360C22×D3—548no—
361C22×D3—5416no—
362C3×D4—278no—
363C2×C3×C4—272no—
364#67—98no—
365#67—932no—
366#67—1832no—
367#67—278no—
368#67—2732no—
369C2×Dic3—92no—
370C2×Dic3—98no—
371C2×Dic3—188no—
372C2×Dic3—2716no—
373D12—278no—
374C4×D3—278no—
375C2×C32—22no—
376C2×C32—22no—
377C2×C32—28no—
378C2×C32—216no—
379C2×C32—216no—
380C2×C32—216no—
381C2×C32—28no—
382C2×C32—24no—
383C2×C32—216no—
384C2×C32—28no—
385C2×C32—48no—
386C2×C32—44no—
387C2×C32—412no—
388C2×C32—424no—
389C2×C32—416no—
390C2×C32—416no—
391C2×C32—416no—
392C2×C32—416no—
393C2×C32—448no—
394C2×C32—44no—
395C2×C32—48no—
396C2×C32—94no—
397C2×C32—182no—
398C2×C32D3×#67116yescomplemented
399C2×C32#381112yescomplemented
400C2×C32C2×#63812yescomplemented
401C2×C32C22×D3218yes—
402C2×C32C22×D3211yescharacteristic
403C2×C32C23×C3⋊D311yescharacteristic
404C3⋊D3—22no—
405C3⋊D3—64no—
406C3⋊D3—616no—
407C3⋊D3—1216no—
408C3⋊D3C2×#63814yescomplemented
409C3×D3—332no—
410C3×D3—616no—
411C3×D3—616no—
412C3×D3—616no—
413C3×D3—1816no—
414C3×D3—184no—
415C3×D3—3616no—
416C24—91no—
417C24—811no—
418C2×D4—272no—
419C2×D4—812no—
420C2×D4—818no—
421C22×C4—811no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—28no—
425C22×C3—28no—
426C22×C3—21no—
427C22×C3—22no—
428C22×C3—216no—
429C22×C3—416no—
430C22×C3—416no—
431C22×C3—44no—
432C22×C3—48no—
433C22×C3—98no—
434C22×C3—916no—
435C22×C3—98no—
436C22×C3—1832no—
437C22×C3—188no—
438C22×C3—274no—
439C22×C3—544no—
440C22×C3C2×D3×C3⋊D312yes—
441C22×C3C2×D3×C3⋊D314yescomplemented
442C22×C3C2×#111314yes—
443C22×C3C2×#111318yescomplemented
444C2×D3—34no—
445C2×D3—38no—
446C2×D3—34no—
447C2×D3—64no—
448C2×D3—62no—
449C2×D3—62no—
450C2×D3—616no—
451C2×D3—616no—
452C2×D3—64no—
453C2×D3—64no—
454C2×D3—94no—
455C2×D3—916no—
456C2×D3—1816no—
457C2×D3—184no—
458C2×D3—1816no—
459C2×D3—3616no—
460C2×D3—5416no—
461C2×D3—5432no—
462C3×C4—274no—
463Dic3—94no—
464Dic3—916no—
465Dic3—1816no—
466C32—28no—
467C32—28no—
468C32—24no—
469C32—44no—
470C32—412no—
471C32—44no—
472C32C2×D3×#6718yescomplemented
473C32C2×#381111yescharacteristic, complemented
474C32C22×#63811yescharacteristic, complemented
475C23—91no—
476C23—91no—
477C23—94no—
478C23—184no—
479C23—272no—
480C23—814no—
481C23—1624no—
482C23C3⋊#111311yescharacteristic, complemented
483D4—278no—
484D4—818no—
485C2×C4—272no—
486C2×C4—814no—
487C2×C3—24no—
488C2×C3—24no—
489C2×C3—28no—
490C2×C3—28no—
491C2×C3—21no—
492C2×C3—22no—
493C2×C3—216no—
494C2×C3—416no—
495C2×C3—416no—
496C2×C3—44no—
497C2×C3—48no—
498C2×C3—916no—
499C2×C3—188no—
500C2×C3—544no—
501C2×C3D3×#63814yescomplemented
502C2×C3#8904918yescomplemented
503C2×C3C22×D3×C3⋊D312yes—
504C2×C3C22×#111314yes—
505D3—38no—
506D3—64no—
507D3—64no—
508D3—64no—
509D3—184no—
510D3—1816no—
511D3—3616no—
512C22—22no—
513C22—92no—
514C22—94no—
515C22—92no—
516C22—188no—
517C22—182no—
518C22—274no—
519C22—544no—
520C22—1628no—
521C22C2×C3⋊#111311yescharacteristic
522C22C2×C3⋊#111312yescomplemented
523C4—274no—
524C3—21no—
525C3—28no—
526C3—44no—
527C3C2×D3×#63812yescomplemented
528C3C2×#8904914yescomplemented
529C2—22no—
530C2—94no—
531C2—182no—
532C2—544no—
533C2#1304972112yesdirect product factor
534C2C22×C3⋊#111311yescharacteristic
subgroupG/H
centerC22C2×C3⋊#1113
commutator, cosocleC2×C34C24
FrattiniC2C22×C3⋊#1113
FittingC23×C34C22
socleC22×C34C23
hypercenterC23C3⋊#1113
hypocenterC34C22×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C22 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×C32×C3⋊D3C2semidirect
2. C22×C3⋊#1113C2semidirect
3. C2×C3×#6312C2semidirect
4. #13049721C2direct (N × H)
5. C2×#10620157C2semidirect
6. C2×#10620155C2semidirect
9. C23×C34C22semidirect
10. C22×C3×C3⋊(C3⋊D3)C22semidirect
13. C22×C32×C3⋊D3C22semidirect
14. C22×C32×C3⋊D3C22semidirect
16. C2×C3⋊#1113C22semidirect
17. C3×#6312C22semidirect
18. C2×C3×C3⋊(C3⋊Dic3)C22semidirect
19. #10620157C22semidirect
20. #10620155C22semidirect
21. C23×C3×C3⋊D3D3semidirect
26. C2×#6312D3semidirect
39. C22×C34C23semidirect
41. C2×C3×C3⋊(C3⋊D3)C23semidirect
44. C2×C32×C3⋊D3D4semidirect
46. C2×C32×C3⋊D3C23semidirect
48. C3×C3⋊(C3⋊Dic3)C23semidirect
55. C23×C33C2×D3semidirect
56. C23×C33C2×D3semidirect
57. C22×C3⋊(C3⋊D3)C2×D3semidirect
63. C22×C3×C3⋊D3C2×D3semidirect
64. C22×C3×C3⋊D3C2×D3semidirect
77. #6312C2×D3semidirect
78. C2×C3⋊(C3⋊Dic3)C2×D3semidirect
91. C2×C34C2×D4semidirect
95. C32×C3⋊D3C2×D4semidirect
98. C23×C3⋊D3C3⋊D3semidirect
132. C22×C33C22×D3semidirect
133. C22×C33C22×D3semidirect
135. C2×C3⋊(C3⋊D3)C22×D3semidirect
144. C2×C3×C3⋊D3#67semidirect
146. C2×C3×C3⋊D3C22×D3semidirect
159. C3⋊(C3⋊Dic3)C22×D3semidirect
165. C34C22×D4semidirect
176. C23×C32D32semidirect
177. C23×C32D32semidirect
178. C23×C32C2×C3⋊D3semidirect
186. C22×C3⋊D3C2×C3⋊D3semidirect
187. C22×C3⋊D3C2×C3⋊D3semidirect
233. C2×C33D3×D4semidirect
234. C2×C33C2×#67semidirect
242. C3×C3⋊D3C2×#67semidirect
291. C22×C32C2×D32semidirect
292. C22×C32C2×D32semidirect
294. C22×C32C22×C3⋊D3semidirect
304. C2×C3⋊D3#638semidirect
306. C2×C3⋊D3C22×C3⋊D3semidirect
335. C33C2×D3×D4semidirect
336. C33C22×#67semidirect
345. C23×C3D3×C3⋊D3semidirect
346. C23×C3#1113semidirect
398. C2×C32D3×#67semidirect
399. C2×C32#3811semidirect
400. C2×C32C2×#638semidirect
408. C3⋊D3C2×#638semidirect
441. C22×C3C2×D3×C3⋊D3semidirect
443. C22×C3C2×#1113semidirect
472. C32C2×D3×#67semidirect
473. C32C2×#3811semidirect
474. C32C22×#638semidirect
482. C23C3⋊#1113semidirect
501. C2×C3D3×#638semidirect
502. C2×C3#89049semidirect
522. C22C2×C3⋊#1113semidirect
527. C3C2×D3×#638semidirect
528. C3C2×#89049semidirect
533. C2#13049721direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×C3⋊D32
C22×C3⋊#11132
C2×C3×#63122
#130497212
C2×#106201572
C2×#106201552
C2×#890493
C2×D3×#6383