Finite Group Databasebeta

← C2×#10594250 #11903400

Subgroups — C2×#10594250 #11903400

Order 1152.

2437 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64090C211yesmaximal, characteristic
2C2×C3×#879C212yesmaximal, complemented
3C2×#63982C211yesmaximal, characteristic, complemented
4C2×#63957C211yesmaximal, characteristic, complemented
5C2×#63953C212yesmaximal
6C2×#63919C211yesmaximal, characteristic, complemented
7C2×#63874C211yesmaximal, characteristic, complemented
8C2×#63865C211yesmaximal, characteristic, complemented
9C2×#63848C212yesmaximal, complemented
10C2×#63838C211yesmaximal, characteristic, complemented
11C2×#63837C212yesmaximal, complemented
12#10594250C2116yesmaximal, direct product factor
13C2×#5702—32nomaximal
14C22×#638C2211yescharacteristic
15C23×C3⋊Dic3C2211yescharacteristic
16C22×C4×C3⋊D3C2211yescharacteristic
17C22×#627C2211yescharacteristic
18C22×#627C2212yes—
19C2×C32×C22⋊C4C2211yescharacteristic
20C2×#3752C2211yescharacteristic
21C2×#3750C2211yescharacteristic
22C2×C4×C3⋊Dic3C2211yescharacteristic
23#64090C2218yes—
24C2×C3×#217C2212yes—
25C2×C3×#217C2212yes—
26C2×C3×C4⋊Dic3C2212yes—
27C2×C3×Dic3⋊C4C2212yes—
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×C4×Dic3C2212yes—
30C3×#879C22116yescomplemented
31#63982C2218yescomplemented
32C2×#3724C2211yescharacteristic
33C2×#3724C2211yescharacteristic
34C2×#3724C2211yescharacteristic
35C2×Dic3⋊Dic3C2212yes—
36C2×#3722C2212yes—
37C2×#3722C2212yes—
38#63957C2218yescomplemented
39C2×Dic32C2212yes—
40C2×Dic32C2212yes—
41#63953C22116yes—
42#63919C2218yescomplemented
43#63874C2218yescomplemented
44#63865C2218yescomplemented
45#63848C22116yescomplemented
46#63838C2218yescomplemented
47#63837C22116yescomplemented
48C2×C3×#126—32no—
49#5702—332no—
50C2×#898—32no—
51C2×#898—32no—
52C2×#896—32no—
53C2×#888—32no—
54C2×#887—32no—
55C2×#887—32no—
56C2×#885—32no—
57C2×#885—32no—
58C2×#67⋊C4—32no—
59C2×#67⋊C4—61no—
60C2×#879D312yescomplemented
61C2×#878—32no—
62C2×#877—32no—
63C2×#872—32no—
64C2×C4×Dic6—32no—
65C24×C32C2311yescharacteristic
66C23×C3⋊D3C2311yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C32×C4C2311yescharacteristic
69C2×#638—24no—
70C2×#638C2314yes—
71C22×C3⋊Dic3—24no—
72C22×C3⋊Dic3C2314yes—
73C22×C3⋊Dic3C2311yescharacteristic
74C22×C3⋊Dic3C2311yescharacteristic
75C22×C3⋊Dic3C2311yescharacteristic
76C2×C4×C3⋊D3—24no—
77C2×C4×C3⋊D3C2314yes—
78C22×C3×Dic3C2312yes—
79C22×C3×Dic3C2312yes—
80C22×C3×Dic3C2312yes—
81C22×C3×Dic3C2312yes—
82C2×#627—28no—
83C2×#627—24no—
84C2×#627C2318yes—
85C2×#627C2314yes—
86C32×C22⋊C4C2314yes—
87#3752C2314yes—
88#3750C2314yes—
89C4×C3⋊Dic3C2314yes—
90C3×#217C2318yes—
91C3×#217C2318yes—
92C3×C4⋊Dic3C2318yes—
93C3×Dic3⋊C4C2318yes—
94C3×Dic3⋊C4C2318yes—
95C3×C4×Dic3C2318yes—
96#3724C2314yes—
97#3724C2314yes—
98#3724C2314yes—
99Dic3⋊Dic3C2318yes—
100#3722C2318yes—
101#3722C2318yes—
102Dic32C2318yes—
103Dic32C2318yes—
104C2×#556—91noSylow
105C22×#67—32no—
106C22×#67—61no—
107C23×Dic3—32no—
108C23×Dic3—61no—
109C22×C4×D3—32no—
110C22×C4×D3—61no—
111C22×Dic6—32no—
112C22×Dic6—32no—
113C22×Dic6—32no—
114C3×#126—316no—
115C2×C3×C4⋊C4—32no—
116C2×C3×C4⋊C4—32no—
117C2×C3×C4⋊C4—32no—
118C2×C3×C22⋊C4—21no—
119C2×C3×C22⋊C4—32no—
120C2×C3×C22⋊C4—32no—
121C2×C3×C22⋊C4C2×D312yes—
122C2×C3×C42—32no—
123C2×#217C2×D312yes—
124C2×#217C2×D312yes—
125C2×(C2×D3)⋊C4—32no—
126C2×(C2×D3)⋊C4—32no—
127C2×(C2×D3)⋊C4—32no—
128C2×(C2×D3)⋊C4—32no—
129C2×(C2×D3)⋊C4—32no—
130C2×(C2×D3)⋊C4—61no—
131C2×C4⋊Dic3—32no—
132C2×C4⋊Dic3C2×D312yes—
133C2×Dic3⋊C4—32no—
134C2×Dic3⋊C4—32no—
135C2×Dic3⋊C4—32no—
136C2×Dic3⋊C4—32no—
137C2×Dic3⋊C4—32no—
138C2×Dic3⋊C4—61no—
139C2×Dic3⋊C4C2×D312yes—
140C2×Dic3⋊C4C2×D312yes—
141C2×C4×Dic3—32no—
142C2×C4×Dic3—32no—
143C2×C4×Dic3—32no—
144C2×C4×Dic3—32no—
145C2×C4×Dic3—32no—
146C2×C4×Dic3—61no—
147C2×C4×Dic3C2×D312yes—
148#898—316no—
149#898—316no—
150#896—316no—
151#888—316no—
152#887—316no—
153#887—316no—
154#885—316no—
155#885—316no—
156#67⋊C4—316no—
157#67⋊C4—68no—
158#879C2×D3116yescomplemented
159#878—316no—
160#877—316no—
161#872—316no—
162C4×Dic6—316no—
163C23×C32—22no—
164C23×C32—22no—
165C23×C32—22no—
166C23×C32C2412yes—
167C23×C32C2411yescharacteristic
168C22×C3⋊D3—22no—
169C22×C3⋊D3—22no—
170C22×C3⋊D3—22no—
171C22×C3⋊D3C2412yes—
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4C2412yes—
175C2×C32×C4C2412yes—
176#638—28no—
177C2×C3⋊Dic3—22no—
178C2×C3⋊Dic3—28no—
179C2×C3⋊Dic3#4114yes—
180C2×C3⋊Dic3#4114yes—
181C2×C3⋊Dic3C2412yes—
182C2×C3⋊Dic3C2412yes—
183C2×C3⋊Dic3C2412yes—
184C4×C3⋊D3—28no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—24no—
188C2×C3×Dic3—24no—
189C2×C3×Dic3C2414yes—
190C2×C3×Dic3C2414yes—
191C2×C3×Dic3C2414yes—
192C2×C3×Dic3C2414yes—
193#627—216no—
194#627—28no—
195#556—916no—
196C2×C4⋊Q8—91no—
197C2×#126—32no—
198C2×#126—92no—
199C2×#124—92no—
200C2×C22⋊Q8—91no—
201C2×C22⋊Q8—91no—
202C2×C4×Q8—92no—
203C2×C4×Q8—91no—
204C2×C4×D4—91no—
205C2×#117—91no—
206C2×#117—91no—
207C24×C3—21no—
208C24×C3C22×D312yes—
209C23×D3—32no—
210C23×D3—61no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—32no—
214C22×C3×C4—32no—
215C22×C3×C4—32no—
216C22×C3×C4—32no—
217C22×C3×C4C22×D312yes—
218C22×C3×C4C22×D312yes—
219C2×#67—38no—
220C2×#67—64no—
221C2×#67—68no—
222C2×#67—68no—
223C22×Dic3—32no—
224C22×Dic3—32no—
225C22×Dic3—38no—
226C22×Dic3—32no—
227C22×Dic3—61no—
228C22×Dic3—61no—
229C22×Dic3—64no—
230C22×Dic3—61no—
231C22×Dic3—68no—
232C22×Dic3—124no—
233C22×Dic3C22×D312yes—
234C22×Dic3C22×D312yes—
235C22×Dic3C22×D312yes—
236C22×Dic3C22×D312yes—
237C2×C4×D3—38no—
238C2×C4×D3—64no—
239C2×C4×D3—68no—
240C2×C4×D3—124no—
241C2×Dic6—38no—
242C2×Dic6—38no—
243C2×Dic6—38no—
244C2×Dic6—68no—
245C2×Dic6—68no—
246C2×Dic6—68no—
247C3×C4⋊C4—38no—
248C3×C4⋊C4—38no—
249C3×C4⋊C4—38no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—38no—
252C3×C22⋊C4—38no—
253C3×C22⋊C4C22×D318yes—
254C3×C42—38no—
255#217C22×D318yes—
256#217C22×D318yes—
257(C2×D3)⋊C4—38no—
258(C2×D3)⋊C4—38no—
259(C2×D3)⋊C4—38no—
260(C2×D3)⋊C4—38no—
261(C2×D3)⋊C4—38no—
262(C2×D3)⋊C4—64no—
263C4⋊Dic3—38no—
264C4⋊Dic3C22×D318yes—
265Dic3⋊C4—38no—
266Dic3⋊C4—38no—
267Dic3⋊C4—38no—
268Dic3⋊C4—38no—
269Dic3⋊C4—38no—
270Dic3⋊C4—64no—
271Dic3⋊C4C22×D318yes—
272Dic3⋊C4C22×D318yes—
273C4×Dic3—38no—
274C4×Dic3—38no—
275C4×Dic3—38no—
276C4×Dic3—38no—
277C4×Dic3—38no—
278C4×Dic3—64no—
279C4×Dic3C22×D318yes—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—44no—
284C22×C32C2×#4112yes—
285C22×C32C2×#4112yes—
286C22×C322−1+412yes—
287C22×C32C2511yescharacteristic
288C2×C3⋊D3—22no—
289C2×C3⋊D3—22no—
290C2×C3⋊D3—22no—
291C2×C3⋊D3—44no—
292C32×C4—22no—
293C32×C4—22no—
294C3⋊Dic3—22no—
295C3⋊Dic3C2×#4114yes—
296C3⋊Dic3C2×#4114yes—
297C3×Dic3—24no—
298C3×Dic3—24no—
299C3×Dic3—24no—
300C3×Dic3—24no—
301C22×Q8—91no—
302C22×Q8—92no—
303C22×D4—91no—
304C23×C4—91no—
305C23×C4—91no—
306C4⋊Q8—98no—
307#126—316no—
308#126—916no—
309#124—916no—
310C22⋊Q8—98no—
311C22⋊Q8—98no—
312C4×Q8—916no—
313C4×Q8—98no—
314C4×D4—98no—
315#117—98no—
316#117—98no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—91no—
321C2×C4⋊C4—91no—
322C2×C4⋊C4—91no—
323C2×C4⋊C4—92no—
324C2×C4⋊C4—91no—
325C2×C22⋊C4—32no—
326C2×C22⋊C4—32no—
327C2×C22⋊C4—92no—
328C2×C22⋊C4—92no—
329C2×C22⋊C4—91no—
330C2×C22⋊C4D3211yescharacteristic
331C2×C42—32no—
332C2×C42—92no—
333C2×C42—92no—
334C2×C42—91no—
335C23×C3—24no—
336C23×C3—24no—
337C23×C3—24no—
338C23×C3—24no—
339C23×C3—22no—
340C23×C3—21no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3C23×D314yes—
344C23×C3C23×D312yes—
345C22×D3—34no—
346C22×D3—62no—
347C22×D3—64no—
348C22×D3—64no—
349C22×D3—64no—
350C22×D3—64no—
351C22×D3—122no—
352C22×D3—122no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—34no—
358C2×C3×C4—34no—
359C2×C3×C4—34no—
360C2×C3×C4—34no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—64no—
364C2×C3×C4—64no—
365C2×C3×C4—64no—
366C2×C3×C4—64no—
367C2×C3×C4C23×D314yes—
368C2×C3×C4C23×D314yes—
369#67—616no—
370#67—616no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—38no—
376C2×Dic3—38no—
377C2×Dic3—34no—
378C2×Dic3—34no—
379C2×Dic3—34no—
380C2×Dic3—64no—
381C2×Dic3—64no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—616no—
386C2×Dic3—64no—
387C2×Dic3—128no—
388C2×Dic3—122no—
389C2×Dic3C23×D314yes—
390C2×Dic3C23×D314yes—
391C2×Dic3C23×D314yes—
392C2×Dic3C23×D314yes—
393C4×D3—616no—
394C4×D3—128no—
395Dic6—616no—
396Dic6—616no—
397Dic6—616no—
398C2×C32—42no—
399C2×C32#55614yescomplemented
400C2×C32C22×#4111yescharacteristic
401C2×C32C22×#4111yescharacteristic
402C2×C32C2×2−1+411yescharacteristic
403C3⋊D3—42no—
404C24—91no—
405C24C2×D3211yescharacteristic
406C2×Q8—94no—
407C2×Q8—98no—
408C2×Q8—188no—
409C2×Q8—184no—
410C2×D4—94no—
411C2×D4—184no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4—94no—
419C22×C4—91no—
420C22×C4—94no—
421C22×C4—184no—
422C22×C4—184no—
423C22×C4C2×D3211yescharacteristic
424C22×C4C2×D3211yescharacteristic
425C4⋊C4—38no—
426C4⋊C4—38no—
427C4⋊C4—38no—
428C4⋊C4—94no—
429C4⋊C4—94no—
430C4⋊C4—94no—
431C4⋊C4—98no—
432C4⋊C4—94no—
433C22⋊C4—38no—
434C22⋊C4—38no—
435C22⋊C4—98no—
436C22⋊C4—98no—
437C22⋊C4—94no—
438C22⋊C4C2×D3214yes—
439C42—38no—
440C42—98no—
441C42—98no—
442C42—94no—
443C22×C3—24no—
444C22×C3—24no—
445C22×C3—24no—
446C22×C3—24no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—21no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—48no—
454C22×C3—48no—
455C22×C3D3×#4114yes—
456C22×C3D3×#4114yes—
457C22×C3#101014yes—
458C22×C3C24×D312yes—
459C2×D3—64no—
460C2×D3—64no—
461C2×D3—64no—
462C2×D3—64no—
463C2×D3—122no—
464C2×D3—122no—
465C2×D3—128no—
466C2×D3—128no—
467C3×C4—24no—
468C3×C4—24no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—64no—
472C3×C4—64no—
473C3×C4—64no—
474C3×C4—64no—
475Dic3—24no—
476Dic3—24no—
477Dic3—24no—
478Dic3—24no—
479Dic3—38no—
480Dic3—38no—
481Dic3—64no—
482Dic3—64no—
483Dic3—64no—
484Dic3—122no—
485C32C2×#55611yescharacteristic, complemented, Sylow
486C23—22no—
487C23—22no—
488C23—22no—
489C23—92no—
490C23—182no—
491C23—182no—
492C23—182no—
493C23C22×D3212yes—
494C23C22×D3211yescharacteristic
495Q8—1816no—
496Q8—188no—
497D4—188no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—34no—
501C2×C4—34no—
502C2×C4—34no—
503C2×C4—34no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—94no—
509C2×C4—94no—
510C2×C4—92no—
511C2×C4—92no—
512C2×C4—92no—
513C2×C4—188no—
514C2×C4—188no—
515C2×C4—182no—
516C2×C4C22×D3212yes—
517C2×C4C22×D3212yes—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—24no—
522C2×C3—44no—
523C2×C3—44no—
524C2×C3#570218yescomplemented
525C2×C3C2×D3×#4112yes—
526C2×C3C2×D3×#4112yes—
527C2×C3C2×#101012yes—
528D3—124no—
529D3—124no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—44no—
534C22—182no—
535C22—182no—
536C22—182no—
537C22—364no—
538C22#6430612yes—
539C22#6430912yes—
540C22#6431212yes—
541C22C23×D3211yescharacteristic
542C4—22no—
543C4—22no—
544C4—64no—
545C4—64no—
546C4—64no—
547C4—64no—
548C4—94no—
549C4—94no—
550C4—182no—
551C3—21no—
552C3C2×#570212yescomplemented
553C2—42no—
554C2—362no—
555C2#1059425014yesdirect product factor
556C2C2×#6430611yescharacteristic
557C2C2×#6430911yescharacteristic
558C2C2×#6431211yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC2×C32×C22⋊C4C22
socleC23×C32C24
prime radicalC22×#638C22
hypercenterC2×C22⋊C4D32
hypocenterC32C2×#556
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#879C2semidirect
3. C2×#63982C2semidirect
4. C2×#63957C2semidirect
6. C2×#63919C2semidirect
7. C2×#63874C2semidirect
8. C2×#63865C2semidirect
9. C2×#63848C2semidirect
10. C2×#63838C2semidirect
11. C2×#63837C2semidirect
12. #10594250C2direct (N × H)
30. C3×#879C22semidirect
31. #63982C22semidirect
38. #63957C22semidirect
42. #63919C22semidirect
43. #63874C22semidirect
44. #63865C22semidirect
45. #63848C22semidirect
46. #63838C22semidirect
47. #63837C22semidirect
60. C2×#879D3semidirect
158. #879C2×D3semidirect
399. C2×C32#556semidirect
485. C32C2×#556semidirect
524. C2×C3#5702semidirect
552. C3C2×#5702semidirect
555. C2#10594250direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#640902
C2×C3×#8792
C2×#639822
C2×#639572
C2×#639532
C2×#639192
C2×#638742
C2×#638652
C2×#638482
C2×#638382
C2×#638372
#105942502
C2×#57023