Finite Group Databasebeta

← C2×#13049504 #72220267

Subgroups — C2×#13049504 #72220267

Order 2592.

4546 conjugacy classes of subgroups; only the 364 normal subgroups are shown.

normal subgroups only

This group has 4546 conjugacy classes of subgroups — too many to draw. Only its 364 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
1C22×C3⋊#1113C211yesmaximal, characteristic, complemented
2C22×C3⋊D32C211yesmaximal, characteristic, complemented
3C2×#10620147C212yesmaximal, complemented
4C2×C3×#6292C212yesmaximal, complemented
5C2×C32×#624C211yesmaximal, characteristic, complemented
6#13049504C218yesmaximal, direct product factor
7C2×D3×C4⋊(C3⋊D3)—32nomaximal
8C2×#88955—34nomaximal
9C22×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
10C22×C3×C3⋊(C3⋊D3)C2212yescomplemented
11C22×C32×C3⋊D3C2211yescharacteristic
12C22×C32×C3⋊D3C2211yescharacteristic, complemented
13C2×C3⋊#1113—24no—
14C2×C3⋊#1113C2214yescomplemented
15C2×C3⋊D32—24no—
16C2×C3⋊D32C2214yescomplemented
17C2×C33×Dic3C2212yescomplemented
18#10620147C2218yescomplemented
19C3×#6292C2218yescomplemented
20C32×#624C2214yescomplemented
21C22×#1113—34no—
22C22×D3×C3⋊D3—32no—
23C22×D3×C3⋊D3—32no—
24C22×D3×C3⋊D3—34no—
25C22×D3×C3⋊D3—61no—
26C2×C4⋊(C3⋊(C3⋊D3))—32no—
27C2×C3×C4⋊(C3⋊D3)—32no—
28C2×C32×C4×D3—32no—
29C2×#6292—32no—
30C2×#6292—38no—
31C2×#6292—68no—
32C2×#6292D312yescomplemented
33D3×C4⋊(C3⋊D3)—316no—
34C2×C3×#626—38no—
35C2×C3×#624D314yescomplemented
36#88955—332no—
37C22×C34C2311yescharacteristic
38C2×C3⋊(C3⋊(C3⋊D3))—22no—
39C2×C3⋊(C3⋊(C3⋊D3))C2312yescomplemented
40C2×C3×C3⋊(C3⋊D3)—24no—
41C2×C3×C3⋊(C3⋊D3)C2314yescomplemented
42C2×C32×C3⋊D3—22no—
43C2×C32×C3⋊D3D414yescomplemented
44C2×C32×C3⋊D3C2312yes—
45C2×C32×C3⋊D3C2312yescomplemented
46C3⋊#1113—28no—
47C3⋊D32—28no—
48C33×Dic3C2314yescomplemented
49C22×C4⋊(C3⋊D3)—91no—
50C2×#3811—91no—
51C2×D3×D12—98no—
52C22×C3⋊(C3⋊D3)—32no—
53C22×C3⋊(C3⋊D3)—34no—
54C22×C3⋊(C3⋊D3)—61no—
55C22×C3⋊(C3⋊D3)—68no—
56C22×C3⋊(C3⋊D3)—124no—
57C22×C3⋊(C3⋊D3)C2×D312yescomplemented
58C22×C3×C3⋊D3—21no—
59C22×C3×C3⋊D3—32no—
60C22×C3×C3⋊D3—38no—
61C22×C3×C3⋊D3—68no—
62C22×C3×C3⋊D3C2×D314yes—
63C22×C3×C3⋊D3C2×D312yescomplemented
64C22×C32×D3—32no—
65C22×C32×D3—34no—
66C22×C32×D3—61no—
67C2×#1113—316no—
68C2×#1113—616no—
69C2×D3×C3⋊D3—38no—
70C2×D3×C3⋊D3—316no—
71C2×D3×C3⋊D3—38no—
72C2×D3×C3⋊D3—68no—
73C2×D3×C3⋊D3—68no—
74C2×D3×C3⋊D3—64no—
75C2×D3×C3⋊D3—68no—
76C2×D3×C3⋊D3—616no—
77C2×C33×C4—32no—
78C4⋊(C3⋊(C3⋊D3))—38no—
79C3×C4⋊(C3⋊D3)—38no—
80C2×C32×Dic3—28no—
81C2×C32×Dic3C2×D318yescomplemented
82C2×C32×Dic3C2×D312yescomplemented
83C32×C4×D3—38no—
84#6292—38no—
85#6292—332no—
86#6292—632no—
87#6292C2×D318yescomplemented
88C3×#626—332no—
89C3×#624C2×D3116yescomplemented
90C2×C34C2×D412yescomplemented
91C2×C34C2411yescharacteristic
92C3⋊(C3⋊(C3⋊D3))—22no—
93C3×C3⋊(C3⋊D3)—24no—
94C32×C3⋊D3—22no—
95C32×C3⋊D3C2×D414yescomplemented
96C23×C3⋊D3—91no—
97C23×C3⋊D3—91no—
98C23×C3⋊D3—91no—
99C22×D32—98no—
100C22×D32—98no—
101C22×D32—91no—
102C22×D32—184no—
103C22×C32×C4—91no—
104C2×C4⋊(C3⋊D3)—32no—
105C2×C4⋊(C3⋊D3)—98no—
106C2×C4⋊(C3⋊D3)—92no—
107C2×C4⋊(C3⋊D3)—98no—
108C2×C4⋊(C3⋊D3)—188no—
109C2×C3×#67—92no—
110C2×C3×D12—98no—
111C2×C3×C4×D3—38no—
112#3811—98no—
113C2×#626—38no—
114C2×#626—98no—
115C2×#626—92no—
116C2×#626—188no—
117C2×#624C3⋊D311yescharacteristic, complemented
118D3×D12—964no—
119C22×C33—28no—
120C22×C33—21no—
121C22×C33—44no—
122C22×C33C22×D314yes—
123C22×C33C22×D312yes—
124C2×C3⋊(C3⋊D3)—24no—
125C2×C3⋊(C3⋊D3)—34no—
126C2×C3⋊(C3⋊D3)—38no—
127C2×C3⋊(C3⋊D3)—62no—
128C2×C3⋊(C3⋊D3)—64no—
129C2×C3⋊(C3⋊D3)—68no—
130C2×C3⋊(C3⋊D3)—64no—
131C2×C3⋊(C3⋊D3)—616no—
132C2×C3⋊(C3⋊D3)—1216no—
133C2×C3⋊(C3⋊D3)—128no—
134C2×C3⋊(C3⋊D3)—1216no—
135C2×C3⋊(C3⋊D3)C22×D314yescomplemented
136C2×C3×C3⋊D3—24no—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—24no—
139C2×C3×C3⋊D3—34no—
140C2×C3×C3⋊D3—316no—
141C2×C3×C3⋊D3—616no—
142C2×C3×C3⋊D3—64no—
143C2×C3×C3⋊D3—616no—
144C2×C3×C3⋊D3—1216no—
145C2×C3×C3⋊D3D12116yescomplemented
146C2×C3×C3⋊D3C22×D318yes—
147C2×C3×C3⋊D3C22×D314yescomplemented
148C2×C32×D3—38no—
149C2×C32×D3—34no—
150C2×C32×D3—38no—
151C2×C32×D3—64no—
152C2×C32×D3—62no—
153C2×C32×D3—68no—
154#1113—632no—
155D3×C3⋊D3—616no—
156D3×C3⋊D3—616no—
157D3×C3⋊D3—616no—
158D3×C3⋊D3—632no—
159C33×C4—34no—
160C32×Dic3—216no—
161C32×Dic3C22×D3116yescomplemented
162C32×Dic3C22×D314yescomplemented
163C2×D3×D4—272no—
164C22×D12—274no—
165C34C22×D411yescharacteristic, complemented, Sylow
166C23×C32—91no—
167C23×C32—91no—
168C22×C3⋊D3—32no—
169C22×C3⋊D3—38no—
170C22×C3⋊D3—68no—
171C22×C3⋊D3—94no—
172C22×C3⋊D3—91no—
173C22×C3⋊D3—94no—
174C22×C3⋊D3—98no—
175C22×C3⋊D3—91no—
176C22×C3⋊D3—94no—
177C22×C3⋊D3—188no—
178C22×C3⋊D3—184no—
179C22×C3⋊D3—188no—
180C22×C3⋊D3—184no—
181C22×C3⋊D3—184no—
182C22×C3⋊D3—184no—
183C22×C3⋊D3—364no—
184C22×C3⋊D3—364no—
185C22×C3⋊D3—3612no—
186C22×C3⋊D3D3211yescharacteristic, complemented
187C22×C3⋊D3C2×C3⋊D311yescharacteristic
188C22×C3×D3—38no—
189C22×C3×D3—38no—
190C22×C3×D3—64no—
191C22×C3×D3—64no—
192C22×C3×D3—98no—
193C22×C3×D3—92no—
194C22×C3×D3—188no—
195C2×D32—932no—
196C2×D32—932no—
197C2×D32—94no—
198C2×D32—1816no—
199C2×D32—1832no—
200C2×D32—1832no—
201C2×D32—1832no—
202C2×D32—184no—
203C2×C32×C4—32no—
204C2×C32×C4—38no—
205C2×C32×C4—68no—
206C2×C32×C4—94no—
207C4⋊(C3⋊D3)—38no—
208C4⋊(C3⋊D3)—98no—
209C4⋊(C3⋊D3)—932no—
210C4⋊(C3⋊D3)—1832no—
211C3×#67—98no—
212C2×C3×Dic3—28no—
213C2×C3×Dic3D3218yescomplemented
214C2×C3×Dic3C2×C3⋊D312yescomplemented
215C3×D12—932no—
216C3×C4×D3—332no—
217#626—332no—
218#626—932no—
219#626—98no—
220#626—1832no—
221#624C2×C3⋊D314yescomplemented
222C2×C33—28no—
223C2×C33—21no—
224C2×C33—216no—
225C2×C33—22no—
226C2×C33—48no—
227C2×C33—44no—
228C2×C33C2×D1218yescomplemented
229C2×C33D3×D414yescomplemented
230C2×C33C23×D312yes—
231C2×C33C23×D314yes—
232C3⋊(C3⋊D3)—24no—
233C3⋊(C3⋊D3)—64no—
234C3⋊(C3⋊D3)—68no—
235C3⋊(C3⋊D3)—64no—
236C3⋊(C3⋊D3)—1216no—
237C3⋊(C3⋊D3)—1216no—
238C3×C3⋊D3—24no—
239C3×C3⋊D3—24no—
240C3×C3⋊D3—64no—
241C3×C3⋊D3—616no—
242C3×C3⋊D3—1216no—
243C3×C3⋊D3C2×D12116yescomplemented
244C32×D3—38no—
245C32×D3—64no—
246C32×D3—68no—
247C23×D3—272no—
248C23×D3—272no—
249C23×D3—274no—
250C23×D3—274no—
251C23×D3—541no—
252C2×C3×D4—272no—
253C22×C3×C4—94no—
254C2×#67—92no—
255C2×#67—272no—
256D3×D4—2716no—
257C2×D12—98no—
258C2×D12—2732no—
259C2×D12—278no—
260C2×D12—272no—
261C2×D12—548no—
262C2×C4×D3—32no—
263C22×C32—28no—
264C22×C32—24no—
265C22×C32—28no—
266C22×C32—44no—
267C22×C32—412no—
268C22×C32—44no—
269C22×C32—92no—
270C22×C32—94no—
271C22×C32—92no—
272C22×C32—182no—
273C22×C32C2×D3218yes—
274C22×C32C2×D3211yescharacteristic
275C22×C32C22×C3⋊D311yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—34no—
278C2×C3⋊D3—316no—
279C2×C3⋊D3—616no—
280C2×C3⋊D3—64no—
281C2×C3⋊D3—616no—
282C2×C3⋊D3—92no—
283C2×C3⋊D3—916no—
284C2×C3⋊D3—92no—
285C2×C3⋊D3—1216no—
286C2×C3⋊D3—1816no—
287C2×C3⋊D3—188no—
288C2×C3⋊D3—1816no—
289C2×C3⋊D3—188no—
290C2×C3⋊D3—188no—
291C2×C3⋊D3—182no—
292C2×C3⋊D3—182no—
293C2×C3⋊D3—188no—
294C2×C3⋊D3—1816no—
295C2×C3⋊D3—1816no—
296C2×C3⋊D3—368no—
297C2×C3⋊D3—368no—
298C2×C3⋊D3—3624no—
299C2×C3⋊D3—3616no—
300C2×C3⋊D3—3616no—
301C2×C3⋊D3—3616no—
302C2×C3⋊D3—3616no—
303C2×C3⋊D3—3648no—
304C2×C3⋊D3C4⋊(C3⋊D3)14yescomplemented
305C2×C3⋊D3C2×D3212yescomplemented
306C2×C3⋊D3C22×C3⋊D312yes—
307C2×C3×D3—316no—
308C2×C3×D3—332no—
309C2×C3×D3—316no—
310C2×C3×D3—68no—
311C2×C3×D3—616no—
312C2×C3×D3—68no—
313C2×C3×D3—616no—
314C2×C3×D3—616no—
315C2×C3×D3—916no—
316C2×C3×D3—94no—
317C2×C3×D3—1816no—
318C2×C3×D3—1816no—
319C2×C3×D3—184no—
320C2×C3×D3—3616no—
321D32—1864no—
322D32—1864no—
323D32—1864no—
324D32—188no—
325C32×C4—34no—
326C32×C4—316no—
327C32×C4—616no—
328C3×Dic3—216no—
329C3×Dic3C2×D32116yescomplemented
330C3×Dic3C22×C3⋊D314yescomplemented
331C22×D4—811noSylow
332C33—28no—
333C33—21no—
334C33—44no—
335C33C22×D1214yescomplemented
336C33C2×D3×D412yescomplemented
337C23×C3—94no—
338C23×C3—92no—
339C23×C3—181no—
340C23×C3—272no—
341C22×D3—32no—
342C22×D3—34no—
343C22×D3—61no—
344C22×D3—98no—
345C22×D3—92no—
346C22×D3—188no—
347C22×D3—278no—
348C22×D3—278no—
349C22×D3—2716no—
350C22×D3—2716no—
351C22×D3—274no—
352C22×D3—272no—
353C22×D3—544no—
354C22×D3—541no—
355C22×D3—548no—
356C22×D3—548no—
357C22×D3—548no—
358C22×D3—5416no—
359C22×D3—548no—
360C22×D3—5416no—
361C22×D3—1084no—
362C3×D4—278no—
363C2×C3×C4—38no—
364C2×C3×C4—32no—
365C2×C3×C4—68no—
366C2×C3×C4—916no—
367#67—98no—
368#67—278no—
369C2×Dic3D3×C3⋊D312yescomplemented
370D12—932no—
371D12—278no—
372D12—2732no—
373D12—5432no—
374C4×D3—38no—
375C2×C32—28no—
376C2×C32—216no—
377C2×C32—24no—
378C2×C32—28no—
379C2×C32—28no—
380C2×C32—216no—
381C2×C32—48no—
382C2×C32—44no—
383C2×C32—412no—
384C2×C32—424no—
385C2×C32—44no—
386C2×C32—48no—
387C2×C32—94no—
388C2×C32—182no—
389C2×C32D3×D12116yescomplemented
390C2×C32#381112yescomplemented
391C2×C32C2×C4⋊(C3⋊D3)12yescomplemented
392C2×C32C22×D3218yes—
393C2×C32C22×D3211yescharacteristic
394C2×C32C23×C3⋊D311yescharacteristic
395C3⋊D3—22no—
396C3⋊D3—64no—
397C3⋊D3—616no—
398C3⋊D3—1216no—
399C3⋊D3—182no—
400C3⋊D3—182no—
401C3⋊D3—1816no—
402C3⋊D3—1816no—
403C3⋊D3—3616no—
404C3⋊D3—3616no—
405C3⋊D3—3616no—
406C3⋊D3—3616no—
407C3⋊D3—3648no—
408C3⋊D3C2×C4⋊(C3⋊D3)14yescomplemented
409C3×D3—332no—
410C3×D3—616no—
411C3×D3—616no—
412C3×D3—616no—
413C3×D3—1816no—
414C3×D3—184no—
415C3×D3—3616no—
416C24—811no—
417C24—811no—
418C2×D4—272no—
419C2×D4—812no—
420C2×D4—818no—
421C22×C4—91no—
422C22×C3—28no—
423C22×C3—21no—
424C22×C3—44no—
425C22×C3—98no—
426C22×C3—916no—
427C22×C3—94no—
428C22×C3—182no—
429C22×C3—184no—
430C22×C3—184no—
431C22×C3—274no—
432C22×C3—544no—
433C22×C3D3314yes—
434C22×C3C2×D3×C3⋊D312yes—
435C2×D3—34no—
436C2×D3—38no—
437C2×D3—38no—
438C2×D3—64no—
439C2×D3—62no—
440C2×D3—68no—
441C2×D3—916no—
442C2×D3—94no—
443C2×D3—1816no—
444C2×D3—1816no—
445C2×D3—184no—
446C2×D3—278no—
447C2×D3—274no—
448C2×D3—3616no—
449C2×D3—542no—
450C2×D3—5416no—
451C2×D3—5416no—
452C2×D3—5416no—
453C2×D3—5432no—
454C2×D3—5416no—
455C2×D3—5432no—
456C2×D3—544no—
457C2×D3—548no—
458C2×D3—544no—
459C2×D3—1088no—
460C2×D3—10816no—
461C2×D3—10816no—
462C3×C4—316no—
463C3×C4—34no—
464C3×C4—616no—
465Dic3C2×D3×C3⋊D314yescomplemented
466C32—28no—
467C32—28no—
468C32—24no—
469C32—44no—
470C32—412no—
471C32—44no—
472C32C2×D3×D1218yescomplemented
473C32C2×#381111yescharacteristic, complemented
474C32C22×C4⋊(C3⋊D3)11yescharacteristic, complemented
475C23—91no—
476C23—91no—
477C23—272no—
478C23—814no—
479C23—814no—
480C23—811no—
481C23—1624no—
482C23—1624no—
483D4—278no—
484D4—818no—
485C2×C4—32no—
486C2×C4—94no—
487C2×C3—28no—
488C2×C3—21no—
489C2×C3—22no—
490C2×C3—216no—
491C2×C3—44no—
492C2×C3—48no—
493C2×C3—916no—
494C2×C3—184no—
495C2×C3—184no—
496C2×C3—544no—
497C2×C3#8895518yescomplemented
498C2×C3D3×C4⋊(C3⋊D3)14yescomplemented
499C2×C3C2×D3314yes—
500C2×C3C22×D3×C3⋊D312yes—
501D3—38no—
502D3—64no—
503D3—68no—
504D3—184no—
505D3—1816no—
506D3—3616no—
507D3—544no—
508D3—544no—
509D3—548no—
510D3—10816no—
511D3—10816no—
512C22—92no—
513C22—94no—
514C22—92no—
515C22—182no—
516C22—274no—
517C22—544no—
518C22—812no—
519C22—1628no—
520C22—1628no—
521C22—1622no—
522C22D32×C3⋊D311yescharacteristic
523C4—34no—
524C3—21no—
525C3—28no—
526C3—44no—
527C3C2×#8895514yescomplemented
528C3C2×D3×C4⋊(C3⋊D3)12yescomplemented
529C2—94no—
530C2—182no—
531C2—544no—
532C2—1622no—
533C2#1304950412yesdirect product factor
534C2C2×D32×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC22D32×C3⋊D3
commutator, cosocleC2×C34C24
FrattiniC2C2×D32×C3⋊D3
Fitting, socleC22×C34C23
hypocenterC34C22×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊#1113C2semidirect
2. C22×C3⋊D32C2semidirect
3. C2×#10620147C2semidirect
4. C2×C3×#6292C2semidirect
5. C2×C32×#624C2semidirect
6. #13049504C2direct (N × H)
9. C22×C3⋊(C3⋊(C3⋊D3))C22semidirect
10. C22×C3×C3⋊(C3⋊D3)C22semidirect
12. C22×C32×C3⋊D3C22semidirect
14. C2×C3⋊#1113C22semidirect
16. C2×C3⋊D32C22semidirect
17. C2×C33×Dic3C22semidirect
18. #10620147C22semidirect
19. C3×#6292C22semidirect
20. C32×#624C22semidirect
32. C2×#6292D3semidirect
35. C2×C3×#624D3semidirect
39. C2×C3⋊(C3⋊(C3⋊D3))C23semidirect
41. C2×C3×C3⋊(C3⋊D3)C23semidirect
43. C2×C32×C3⋊D3D4semidirect
45. C2×C32×C3⋊D3C23semidirect
48. C33×Dic3C23semidirect
57. C22×C3⋊(C3⋊D3)C2×D3semidirect
63. C22×C3×C3⋊D3C2×D3semidirect
81. C2×C32×Dic3C2×D3semidirect
82. C2×C32×Dic3C2×D3semidirect
87. #6292C2×D3semidirect
89. C3×#624C2×D3semidirect
90. C2×C34C2×D4semidirect
95. C32×C3⋊D3C2×D4semidirect
117. C2×#624C3⋊D3semidirect
135. C2×C3⋊(C3⋊D3)C22×D3semidirect
145. C2×C3×C3⋊D3D12semidirect
147. C2×C3×C3⋊D3C22×D3semidirect
161. C32×Dic3C22×D3semidirect
162. C32×Dic3C22×D3semidirect
165. C34C22×D4semidirect
186. C22×C3⋊D3D32semidirect
213. C2×C3×Dic3D32semidirect
214. C2×C3×Dic3C2×C3⋊D3semidirect
221. #624C2×C3⋊D3semidirect
228. C2×C33C2×D12semidirect
229. C2×C33D3×D4semidirect
243. C3×C3⋊D3C2×D12semidirect
304. C2×C3⋊D3C4⋊(C3⋊D3)semidirect
305. C2×C3⋊D3C2×D32semidirect
329. C3×Dic3C2×D32semidirect
330. C3×Dic3C22×C3⋊D3semidirect
335. C33C22×D12semidirect
336. C33C2×D3×D4semidirect
369. C2×Dic3D3×C3⋊D3semidirect
389. C2×C32D3×D12semidirect
390. C2×C32#3811semidirect
391. C2×C32C2×C4⋊(C3⋊D3)semidirect
408. C3⋊D3C2×C4⋊(C3⋊D3)semidirect
465. Dic3C2×D3×C3⋊D3semidirect
472. C32C2×D3×D12semidirect
473. C32C2×#3811semidirect
474. C32C22×C4⋊(C3⋊D3)semidirect
497. C2×C3#88955semidirect
498. C2×C3D3×C4⋊(C3⋊D3)semidirect
527. C3C2×#88955semidirect
528. C3C2×D3×C4⋊(C3⋊D3)semidirect
533. C2#13049504direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊#11132
C22×C3⋊D322
C2×#106201472
C2×C3×#62922
C2×C32×#6242
#130495042
C2×D3×C4⋊(C3⋊D3)3
C2×#889553