Finite Group Databasebeta

← C2×#10593902 #11902726

Subgroups — C2×#10593902 #11902726

Order 1152.

3721 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3799C212yesmaximal, complemented
2C2×#64083C211yesmaximal, characteristic, complemented
3C2×C3×C4×D12C211yesmaximal, characteristic, complemented
4C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
5C2×#63911C212yesmaximal, complemented
6C2×#63910C214yesmaximal, complemented
7C2×#63909C212yesmaximal, complemented
8C2×#63894C212yesmaximal, complemented
9#10593902C2116yesmaximal, direct product factor
10C2×#5577—31nomaximal
11C2×#5553—31nomaximal
12C22×C4⋊(C3⋊D3)C2212yes—
13C22×C4⋊(C3⋊D3)C2212yes—
14C22×C4⋊(C3⋊D3)C2212yes—
15C22×C3×D12C2211yescharacteristic
16C22×C3×C4×D3C2212yes—
17C22×C3×Dic6C2211yescharacteristic
18C22×#626C2214yes—
19C22×#626C2214yes—
20C22×#624C2214yes—
21C2×#3799—216no—
22C2×#3799C22116yescomplemented
23C2×C32×C42C2211yescharacteristic
24#64083C2218yescomplemented
25C2×C3×(C2×D3)⋊C4C2212yes—
26C2×C3×C4⋊Dic3C2211yescharacteristic
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×C4×Dic3C2212yes—
30C3×C4×D12C2218yescomplemented
31C3×C4×Dic6C2218yescomplemented
32C2×#3722C2214yes—
33#63911C22116yescomplemented
34#63910C22132yescomplemented
35#63909C22116yescomplemented
36#63894C22116yescomplemented
37C22×#239—32no—
38C22×#235—32no—
39C2×C3×C4×Q8—31no—
40C2×C3×C4×D4—31no—
41C2×#940—32no—
42C2×#930—32no—
43#5577—316no—
44#5553—316no—
45C2×#895—32no—
46C2×#895—34no—
47C2×#893—32no—
48C2×#884—34no—
49C2×#874—32no—
50C2×#874—31no—
51C2×#874—31no—
52C2×#874—61no—
53C2×C4×D12—32no—
54C2×C4×D12D311yescharacteristic, complemented
55C2×C4×Dic6D311yescharacteristic, complemented
56C23×C3⋊D3C2314yes—
57C23×C3×D3C2312yes—
58C22×C32×C4C2311yescharacteristic
59C22×C32×C4C2311yescharacteristic
60C22×C32×C4C2311yescharacteristic
61C2×C4⋊(C3⋊D3)—28no—
62C2×C4⋊(C3⋊D3)—28no—
63C2×C4⋊(C3⋊D3)—44no—
64C2×C4⋊(C3⋊D3)C2318yes—
65C2×C4⋊(C3⋊D3)C2318yes—
66C2×C4⋊(C3⋊D3)C2318yes—
67C22×C3×Dic3C2312yes—
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2312yes—
70C2×C3×D12—24no—
71C2×C3×D12C2314yes—
72C2×C3×C4×D3—28no—
73C2×C3×C4×D3C2318yes—
74C2×C3×Dic6D418yescomplemented
75C2×C3×Dic6C2314yes—
76C2×#626—216no—
77C2×#626—48no—
78C2×#626C23116yes—
79C2×#626C23116yes—
80C2×#624—216no—
81C2×#624C23116yes—
82#3799—264no—
83C32×C42C2314yes—
84C3×(C2×D3)⋊C4C2318yes—
85C3×C4⋊Dic3C2314yes—
86C3×C4⋊Dic3C2314yes—
87C3×Dic3⋊C4C2318yes—
88C3×C4×Dic3C2318yes—
89#3722C23116yes—
90C2×#550—91noSylow
91C22×C3×Q8—31no—
92C22×C3×D4—31no—
93C23×C3×C4—32no—
94C22×#67—34no—
95C22×#67—34no—
96C2×#239—316no—
97C2×#239—616no—
98C2×#235—316no—
99C2×#235—616no—
100C22×D12—34no—
101C22×D12—32no—
102C22×D12—32no—
103C22×D12—32no—
104C22×D12—32no—
105C22×D12—34no—
106C22×D12—32no—
107C22×D12—32no—
108C22×D12—62no—
109C22×D12—62no—
110C22×D12—62no—
111C22×D12C2×D311yescharacteristic
112C22×C4×D3—34no—
113C22×C4×D3—34no—
114C22×C4×D3C2×D312yes—
115C22×Dic6C2×D311yescharacteristic
116C3×C4×Q8—38no—
117C3×C4×D4—38no—
118C2×C3×C4⋊C4—32no—
119C2×C3×C4⋊C4—31no—
120C2×C3×C4⋊C4—31no—
121C2×C3×C22⋊C4—32no—
122C2×C3×C42—21no—
123C2×C3×C42—32no—
124C2×C3×C42C2×D311yescharacteristic
125C2×C3×C42C2×D311yescharacteristic
126#940—316no—
127#930—316no—
128C2×(C2×D3)⋊C4—34no—
129C2×(C2×D3)⋊C4—34no—
130C2×(C2×D3)⋊C4C2×D312yes—
131C2×C4⋊Dic3C2×D311yescharacteristic
132C2×C4⋊Dic3C2×D311yescharacteristic
133C2×Dic3⋊C4C2×D312yes—
134C2×C4×Dic3C2×D312yes—
135#895—316no—
136#895—332no—
137#893—316no—
138#884—332no—
139#874—316no—
140#874—38no—
141#874—38no—
142#874—68no—
143C4×D12—316no—
144C4×D12C2×D318yescomplemented
145C4×Dic6C2×D318yescomplemented
146C23×C32C2411yescharacteristic
147C22×C3⋊D3—28no—
148C22×C3⋊D3—28no—
149C22×C3⋊D3—28no—
150C22×C3⋊D3C2418yes—
151C22×C3×D3—24no—
152C22×C3×D3—24no—
153C22×C3×D3—24no—
154C22×C3×D3C2414yes—
155C2×C32×C4—22no—
156C2×C32×C4C2×D414yes—
157C2×C32×C4#4114yescomplemented
158C2×C32×C4C2412yes—
159C2×C32×C4C2412yes—
160C2×C32×C4C2412yes—
161C4⋊(C3⋊D3)—216no—
162C4⋊(C3⋊D3)—216no—
163C4⋊(C3⋊D3)—48no—
164C2×C3×Dic3—24no—
165C2×C3×Dic3—24no—
166C2×C3×Dic3C2×D418yes—
167C2×C3×Dic3C2414yes—
168C2×C3×Dic3C2414yes—
169C2×C3×Dic3C2414yes—
170C3×D12—28no—
171C3×C4×D3—216no—
172C3×Dic6C2×D4116yescomplemented
173#626—232no—
174#626—416no—
175#624—232no—
176C22×#41—92no—
177#550—916no—
178C2×#127—92no—
179C2×#127—91no—
180C2×#121—94no—
181C2×#121—92no—
182C2×C4×Q8—31no—
183C2×C4×D4—31no—
184C2×C4×D4—92no—
185C24×C3—32no—
186C23×D3—34no—
187C23×D3—34no—
188C23×D3—64no—
189C23×D3C22×D312yes—
190C2×C3×Q8—38no—
191C2×C3×Q8—34no—
192C2×C3×D4—34no—
193C2×C3×D4—64no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—32no—
198C22×C3×C4—32no—
199C22×C3×C4—32no—
200C22×C3×C4—38no—
201C22×C3×C4—68no—
202C22×C3×C4C22×D311yescharacteristic
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C22×D311yescharacteristic
205C22×C3×C4C22×D311yescharacteristic
206C22×C3×C4C22×D311yescharacteristic
207C22×C3×C4C22×D311yescharacteristic
208C2×#67—316no—
209C2×#67—316no—
210C2×#67—616no—
211C2×#67—128no—
212C22×Dic3C22×D312yes—
213C22×Dic3C22×D312yes—
214C22×Dic3C22×D312yes—
215#239—664no—
216#235—664no—
217C2×D12—24no—
218C2×D12—316no—
219C2×D12—38no—
220C2×D12—38no—
221C2×D12—38no—
222C2×D12—38no—
223C2×D12—316no—
224C2×D12—38no—
225C2×D12—38no—
226C2×D12—68no—
227C2×D12—68no—
228C2×D12—68no—
229C2×D12—616no—
230C2×D12—68no—
231C2×D12—68no—
232C2×D12—68no—
233C2×D12—68no—
234C2×D12—128no—
235C2×D12—128no—
236C2×D12—124no—
237C2×D12—124no—
238C2×D12—128no—
239C2×D12—128no—
240C2×D12C22×D314yes—
241C2×C4×D3—28no—
242C2×C4×D3—316no—
243C2×C4×D3—316no—
244C2×C4×D3—616no—
245C2×C4×D3—616no—
246C2×C4×D3C22×D318yes—
247C2×Dic6D1218yescomplemented
248C2×Dic6C22×D314yes—
249C3×C4⋊C4—34no—
250C3×C4⋊C4—38no—
251C3×C4⋊C4—34no—
252C3×C22⋊C4—38no—
253C3×C42—24no—
254C3×C42—38no—
255C3×C42C22×D314yes—
256C3×C42C22×D314yes—
257(C2×D3)⋊C4—316no—
258(C2×D3)⋊C4—316no—
259(C2×D3)⋊C4C22×D318yes—
260C4⋊Dic3C22×D314yes—
261C4⋊Dic3C22×D314yes—
262Dic3⋊C4C22×D318yes—
263C4×Dic3C22×D318yes—
264C22×C32C22×D412yes—
265C22×C32C2×#4112yes—
266C22×C32H(5,𝔽2)12yes—
267C22×C32C2511yescharacteristic
268C2×C3⋊D3—28no—
269C2×C3⋊D3—28no—
270C2×C3⋊D3—28no—
271C2×C3⋊D3—416no—
272C2×C3×D3—24no—
273C2×C3×D3—24no—
274C2×C3×D3—24no—
275C2×C3×D3—48no—
276C32×C4—22no—
277C32×C4C22×D414yes—
278C32×C4C2×#4114yescomplemented
279C3×Dic3—24no—
280C3×Dic3—24no—
281C3×Dic3C22×D418yes—
282C2×#41—916no—
283C2×#41—1816no—
284C22×Q8—31no—
285C22×D4—31no—
286C22×D4—94no—
287C22×D4—94no—
288C22×D4—92no—
289C22×D4—92no—
290C22×D4—92no—
291C23×C4—32no—
292C23×C4—94no—
293#127—916no—
294#127—98no—
295#121—932no—
296#121—916no—
297C4×Q8—38no—
298C4×D4—38no—
299C4×D4—916no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—31no—
303C2×C22⋊C4—32no—
304C2×C22⋊C4—94no—
305C2×C42—32no—
306C2×C42D3211yescharacteristic
307C23×C3—21no—
308C23×C3—34no—
309C23×C3—64no—
310C23×C3—64no—
311C23×C3—64no—
312C23×C3C23×D311yescharacteristic
313C23×C3C23×D311yescharacteristic
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—38no—
318C22×D3—38no—
319C22×D3—68no—
320C22×D3—68no—
321C22×D3—68no—
322C22×D3—68no—
323C22×D3—68no—
324C22×D3—68no—
325C22×D3—68no—
326C22×D3—128no—
327C22×D3—128no—
328C22×D3—128no—
329C22×D3C23×D314yes—
330C3×Q8—316no—
331C3×D4—68no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—38no—
341C2×C3×C4—34no—
342C2×C3×C4—34no—
343C2×C3×C4—34no—
344C2×C3×C4—64no—
345C2×C3×C4—64no—
346C2×C3×C4—616no—
347C2×C3×C4C2×D1214yes—
348C2×C3×C4#23514yescomplemented
349C2×C3×C4D3×D414yes—
350C2×C3×C4#23914yescomplemented
351C2×C3×C4C23×D312yes—
352C2×C3×C4C23×D312yes—
353C2×C3×C4C23×D312yes—
354C2×C3×C4C23×D312yes—
355C2×C3×C4C23×D312yes—
356C2×C3×C4C23×D312yes—
357#67—632no—
358#67—1216no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3C2×D1218yes—
362C2×Dic3C23×D314yes—
363C2×Dic3C23×D314yes—
364C2×Dic3C23×D314yes—
365D12—28no—
366D12—632no—
367D12—616no—
368D12—616no—
369D12—616no—
370D12—616no—
371D12—1216no—
372D12—1216no—
373D12—128no—
374D12—128no—
375D12—1216no—
376D12—1216no—
377C4×D3—216no—
378C4×D3—632no—
379C4×D3—632no—
380Dic6C2×D12116yescomplemented
381C2×C32#55014yescomplemented
382C2×C32C23×D411yescharacteristic
383C2×C32C22×#4111yescharacteristic
384C2×C32C2×H(5,𝔽2)11yescharacteristic
385C3⋊D3—48no—
386C3×D3—44no—
387C24—32no—
388C24—94no—
389#41—1864no—
390C2×Q8—34no—
391C2×Q8—38no—
392C2×D4—34no—
393C2×D4—64no—
394C2×D4—916no—
395C2×D4—916no—
396C2×D4—98no—
397C2×D4—98no—
398C2×D4—98no—
399C2×D4—1816no—
400C2×D4—188no—
401C2×D4—188no—
402C2×D4—364no—
403C2×D4—368no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—38no—
408C22×C4—68no—
409C22×C4—916no—
410C22×C4—1816no—
411C22×C4C2×D3211yescharacteristic
412C22×C4C2×D3211yescharacteristic
413C22×C4C2×D3211yescharacteristic
414C4⋊C4—34no—
415C4⋊C4—38no—
416C4⋊C4—34no—
417C22⋊C4—38no—
418C22⋊C4—916no—
419C42—38no—
420C42C2×D3214yes—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—21no—
425C22×C3—64no—
426C22×C3—64no—
427C22×C3—64no—
428C22×C3—128no—
429C22×C3C22×D1212yes—
430C22×C3C2×#23512yes—
431C22×C3C2×D3×D412yes—
432C22×C3C2×#23912yes—
433C22×C3#100912yes—
434C22×C3#100912yes—
435C22×C3C24×D311yescharacteristic
436C22×C3C24×D311yescharacteristic
437C2×D3—24no—
438C2×D3—24no—
439C2×D3—24no—
440C2×D3—48no—
441C2×D3—68no—
442C2×D3—68no—
443C2×D3—68no—
444C2×D3—68no—
445C2×D3—68no—
446C2×D3—68no—
447C2×D3—128no—
448C2×D3—128no—
449C2×D3—128no—
450C2×D3—1216no—
451C2×D3—1216no—
452C2×D3—2416no—
453C3×C4—24no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—24no—
457C3×C4—24no—
458C3×C4—38no—
459C3×C4—64no—
460C3×C4—64no—
461C3×C4C22×D1214yes—
462C3×C4C2×#23514yescomplemented
463C3×C4C2×D3×D414yes—
464C3×C4C2×#23914yescomplemented
465Dic3—24no—
466Dic3—24no—
467Dic3C22×D1218yes—
468C32C2×#55011yescharacteristic, complemented, Sylow
469C23—34no—
470C23—64no—
471C23—64no—
472C23—64no—
473C23—98no—
474C23—188no—
475C23—188no—
476C23—188no—
477C23C22×D3211yescharacteristic
478Q8—316no—
479D4—68no—
480D4—1832no—
481D4—1816no—
482D4—1816no—
483D4—368no—
484D4—3616no—
485C2×C4—22no—
486C2×C4—38no—
487C2×C4—34no—
488C2×C4—34no—
489C2×C4—34no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—616no—
493C2×C4—1832no—
494C2×C4#379914yescomplemented
495C2×C4D3×D1214yes—
496C2×C4C22×D3212yes—
497C2×C4C22×D3212yes—
498C2×C4C22×D3212yes—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—24no—
503C2×C3—124no—
504C2×C3#555314yescomplemented
505C2×C3#557714yescomplemented
506C2×C3C23×D1211yescharacteristic
507C2×C3C22×#23511yescharacteristic
508C2×C3C22×D3×D411yescharacteristic
509C2×C3C22×#23911yescharacteristic
510C2×C3C2×#100911yescharacteristic
511C2×C3C2×#100911yescharacteristic
512D3—44no—
513D3—128no—
514D3—128no—
515D3—248no—
516C22—64no—
517C22—64no—
518C22—64no—
519C22—128no—
520C22—188no—
521C22—188no—
522C22—188no—
523C22—3616no—
524C22C2×#379912yes—
525C22C2×D3×D1212yes—
526C22#6430812yes—
527C22C23×D3211yescharacteristic
528C4—22no—
529C4—38no—
530C4—64no—
531C4—64no—
532C4C2×#379914yescomplemented
533C4C2×D3×D1214yes—
534C3—21no—
535C3C2×#555311yescharacteristic, complemented
536C3C2×#557711yescharacteristic, complemented
537C2—124no—
538C2—368no—
539C2#1059390214yesdirect product factor
540C2C22×#379911yescharacteristic
541C2C22×D3×D1211yescharacteristic
542C2C2×#6430811yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC2×C32×C42C22
socleC23×C32C24
hypercenterC2×C42D32
hypocenterC32C2×#550
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C22×#3799C2semidirect
2. C2×#64083C2semidirect
3. C2×C3×C4×D12C2semidirect
4. C2×C3×C4×Dic6C2semidirect
5. C2×#63911C2semidirect
6. C2×#63910C2semidirect
7. C2×#63909C2semidirect
8. C2×#63894C2semidirect
9. #10593902C2direct (N × H)
22. C2×#3799C22semidirect
24. #64083C22semidirect
30. C3×C4×D12C22semidirect
31. C3×C4×Dic6C22semidirect
33. #63911C22semidirect
34. #63910C22semidirect
35. #63909C22semidirect
36. #63894C22semidirect
54. C2×C4×D12D3semidirect
55. C2×C4×Dic6D3semidirect
74. C2×C3×Dic6D4semidirect
144. C4×D12C2×D3semidirect
145. C4×Dic6C2×D3semidirect
157. C2×C32×C4#41semidirect
172. C3×Dic6C2×D4semidirect
247. C2×Dic6D12semidirect
278. C32×C4C2×#41semidirect
348. C2×C3×C4#235semidirect
350. C2×C3×C4#239semidirect
380. Dic6C2×D12semidirect
381. C2×C32#550semidirect
462. C3×C4C2×#235semidirect
464. C3×C4C2×#239semidirect
468. C32C2×#550semidirect
494. C2×C4#3799semidirect
504. C2×C3#5553semidirect
505. C2×C3#5577semidirect
532. C4C2×#3799semidirect
535. C3C2×#5553semidirect
536. C3C2×#5577semidirect
539. C2#10593902direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#37992
C2×#640832
C2×C3×C4×D122
C2×C3×C4×Dic62
C2×#639112
C2×#639102
C2×#639092
C2×#638942
#105939022
C2×#55773
C2×#55533