Finite Group Databasebeta

← C2×#10595434 #11901873

Subgroups — C2×#10595434 #11901873

Order 1152.

2753 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64155C211yesmaximal, characteristic, complemented
2C2×C3×#946C212yesmaximal, complemented
3#10595434C2116yesmaximal, direct product factor
4C2×#63921C211yesmaximal, characteristic
5C2×D3×(C2×D3)⋊C4C212yesmaximal
6C2×#63899C212yesmaximal, complemented
7C2×#63895C212yesmaximal, complemented
8C2×#63893C212yesmaximal, complemented
9C2×#63865C211yesmaximal, characteristic, complemented
10C2×#63851C212yesmaximal, complemented
11C2×#5634—32nomaximal
12C23×D32C2211yescharacteristic
13C22×C32×Q8C2211yescharacteristic, complemented
14C22×C4×C3⋊D3C2211yescharacteristic
15C22×C3×C4×D3C2212yes—
16C22×#627C2211yescharacteristic, complemented
17C22×#627C2212yescomplemented
18C22×#624C2211yescharacteristic
19C22×D3×Dic3C2212yes—
20#64155C2218yescomplemented
21C2×#3752C2212yes—
22C2×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
23C2×#3750C2212yescomplemented
24C3×#946C22116yescomplemented
25C2×C3×(C2×D3)⋊C4C2212yes—
26C2×C3×(C2×D3)⋊C4C2212yes—
27C2×C3×C4⋊Dic3C2212yescomplemented
28C2×C3×Dic3⋊C4C2212yescomplemented
29C2×C3×Dic3⋊C4C2212yescomplemented
30C2×#3724C2211yescharacteristic, complemented
31C2×Dic3⋊Dic3C2212yescomplemented
32C2×#3722C2212yes—
33C2×(C2×D3)⋊Dic3C2212yes—
34C2×(C2×D3)⋊Dic3C2212yes—
35#63921C2218yes—
36D3×(C2×D3)⋊C4C22116yes—
37#63899C22116yescomplemented
38#63895C22116yescomplemented
39#63893C22116yescomplemented
40#63865C2218yescomplemented
41#63851C22116yescomplemented
42C2×C3×C22⋊Q8—32no—
43C2×#946—32no—
44C2×#946—61no—
45C2×#946D312yescomplemented
46C22×(C2×D3)⋊C4—32no—
47C2×#924—32no—
48#5634—332no—
49C2×#897—32no—
50C2×#897—32no—
51C2×#896—32no—
52C2×#896—32no—
53C2×#896—32no—
54C2×#896—32no—
55C2×D3×C22⋊C4—32no—
56C2×D3×C22⋊C4—32no—
57C2×#878—32no—
58C2×#878—32no—
59C23×C3⋊D3C2311yescharacteristic
60C23×C3×D3C2312yes—
61C22×D32—42no—
62C22×D32—42no—
63C22×D32Q818yescomplemented
64C22×D32C2314yes—
65C2×C32×Q8—42no—
66C2×C32×Q8C2314yescomplemented
67C22×C32×C4C2311yescharacteristic
68C22×C32×C4C2312yes—
69C22×C3⋊Dic3C2312yes—
70C22×C3⋊Dic3C2311yescharacteristic
71C2×C4×C3⋊D3—42no—
72C2×C4×C3⋊D3C2314yes—
73C22×C3×Dic3C2312yes—
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2312yes—
76C2×C3×C4×D3—44no—
77C2×C3×C4×D3C2318yes—
78C2×#627—44no—
79C2×#627—42no—
80C2×#627C2314yescomplemented
81C2×#627C2318yescomplemented
82C2×#624—42no—
83C2×#624C2314yes—
84C2×D3×Dic3—44no—
85C2×D3×Dic3C2318yes—
86#3752C2318yes—
87C4⋊(C3⋊Dic3)C2314yescomplemented
88#3750C2318yescomplemented
89C3×(C2×D3)⋊C4C2318yes—
90C3×(C2×D3)⋊C4C2318yes—
91C3×C4⋊Dic3C2318yescomplemented
92C3×Dic3⋊C4C2318yescomplemented
93C3×Dic3⋊C4C2318yescomplemented
94#3724C2314yescomplemented
95Dic3⋊Dic3C2318yescomplemented
96#3722C2318yes—
97(C2×D3)⋊Dic3C2318yes—
98(C2×D3)⋊Dic3C2318yes—
99C2×#543—91noSylow
100C24×D3—32no—
101C22×C3×Q8—21no—
102C22×C3×Q8C2×D312yescomplemented
103C23×C3×C4—32no—
104C23×Dic3—32no—
105C22×C4×D3—32no—
106C22×C4×D3—32no—
107C22×C4×D3—32no—
108C22×C4×D3—61no—
109C22×C4×D3C2×D312yes—
110C22×Dic6—32no—
111C22×Dic6—32no—
112C22×Dic6—32no—
113C3×C22⋊Q8—316no—
114C2×C3×C4⋊C4—32no—
115C2×C3×C4⋊C4—32no—
116C2×C3×C4⋊C4—32no—
117C2×C3×C22⋊C4—32no—
118C2×C3×C22⋊C4—32no—
119C2×#217—32no—
120C2×#217—32no—
121#946—316no—
122#946—68no—
123#946C2×D3116yescomplemented
124C2×(C2×D3)⋊C4—32no—
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—316no—
131C2×(C2×D3)⋊C4—62no—
132C2×(C2×D3)⋊C4C2×D312yes—
133C2×(C2×D3)⋊C4C2×D312yes—
134C2×C4⋊Dic3—32no—
135C2×C4⋊Dic3—32no—
136C2×C4⋊Dic3—61no—
137C2×C4⋊Dic3C2×D312yescomplemented
138#924—316no—
139C2×Dic3⋊C4—32no—
140C2×Dic3⋊C4—32no—
141C2×Dic3⋊C4—32no—
142C2×Dic3⋊C4—32no—
143C2×Dic3⋊C4—62no—
144C2×Dic3⋊C4C2×D312yescomplemented
145C2×Dic3⋊C4C2×D312yescomplemented
146#897—316no—
147#897—316no—
148#896—316no—
149#896—316no—
150#896—316no—
151#896—316no—
152D3×C22⋊C4—316no—
153D3×C22⋊C4—316no—
154#878—316no—
155#878—316no—
156C23×C32C2411yescharacteristic
157C22×C3⋊D3—22no—
158C22×C3⋊D3—22no—
159C22×C3⋊D3C2×Q814yes—
160C22×C3⋊D3C2412yes—
161C22×C3×D3—24no—
162C22×C3×D3—24no—
163C22×C3×D3C2×Q818yes—
164C22×C3×D3C2414yes—
165C2×D32—416no—
166C2×D32—44no—
167C2×D32—44no—
168C2×D32C2×Q8116yescomplemented
169C32×Q8—44no—
170C2×C32×C4—22no—
171C2×C32×C4—24no—
172C2×C32×C4C2412yes—
173C2×C32×C4C2414yes—
174C2×C3⋊Dic3—24no—
175C2×C3⋊Dic3—22no—
176C2×C3⋊Dic3C2412yes—
177C2×C3⋊Dic3C2414yes—
178C4×C3⋊D3—44no—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3C2414yes—
183C2×C3×Dic3C2414yes—
184C2×C3×Dic3C2414yes—
185C3×C4×D3—48no—
186#627—48no—
187#627—44no—
188#624—44no—
189D3×Dic3—48no—
190#543—916no—
191C2×C22⋊Q8—32no—
192C2×C22⋊Q8—92no—
193C2×C22⋊Q8—92no—
194C2×C22⋊Q8—91no—
195C2×C22⋊Q8—92no—
196C2×C22⋊Q8—92no—
197C2×C22⋊Q8—91no—
198C22×C22⋊C4—91no—
199C22×C22⋊C4—92no—
200C24×C3—32no—
201C23×D3—32no—
202C23×D3—316no—
203C23×D3—38no—
204C23×D3—61no—
205C23×D3—124no—
206C23×D3—124no—
207C23×D3C22×D312yes—
208C2×C3×Q8—24no—
209C2×C3×Q8—44no—
210C2×C3×Q8—44no—
211C2×C3×Q8C22×D318yescomplemented
212C22×C3×C4—21no—
213C22×C3×C4—22no—
214C22×C3×C4—32no—
215C22×C3×C4—32no—
216C22×C3×C4—32no—
217C22×C3×C4—38no—
218C22×C3×C4—124no—
219C22×C3×C4C22×D312yes—
220C22×C3×C4C22×D312yes—
221C22×C3×C4C22×D312yes—
222C22×Dic3—32no—
223C22×Dic3—32no—
224C22×Dic3—32no—
225C22×Dic3—38no—
226C22×Dic3—61no—
227C22×Dic3—62no—
228C22×Dic3—124no—
229C22×Dic3C22×D312yes—
230C22×Dic3C22×D312yes—
231C22×Dic3C22×D312yes—
232C2×C4×D3—38no—
233C2×C4×D3—38no—
234C2×C4×D3—38no—
235C2×C4×D3—44no—
236C2×C4×D3—64no—
237C2×C4×D3—124no—
238C2×C4×D3—124no—
239C2×C4×D3—124no—
240C2×C4×D3—124no—
241C2×C4×D3C22×D318yes—
242C2×Dic6—38no—
243C2×Dic6—38no—
244C2×Dic6—38no—
245C2×Dic6—124no—
246C2×Dic6—124no—
247C2×Dic6—124no—
248C3×C4⋊C4—38no—
249C3×C4⋊C4—38no—
250C3×C4⋊C4—38no—
251C3×C22⋊C4—38no—
252C3×C22⋊C4—38no—
253#217—38no—
254#217—38no—
255(C2×D3)⋊C4—38no—
256(C2×D3)⋊C4—38no—
257(C2×D3)⋊C4—38no—
258(C2×D3)⋊C4—38no—
259(C2×D3)⋊C4—38no—
260(C2×D3)⋊C4—38no—
261(C2×D3)⋊C4—68no—
262(C2×D3)⋊C4C22×D318yes—
263(C2×D3)⋊C4C22×D318yes—
264C4⋊Dic3—38no—
265C4⋊Dic3—38no—
266C4⋊Dic3—64no—
267C4⋊Dic3C22×D318yescomplemented
268Dic3⋊C4—38no—
269Dic3⋊C4—38no—
270Dic3⋊C4—38no—
271Dic3⋊C4—38no—
272Dic3⋊C4—68no—
273Dic3⋊C4C22×D318yescomplemented
274Dic3⋊C4C22×D318yescomplemented
275C22×C32C22×Q812yes—
276C22×C32H(5,𝔽2)12yes—
277C22×C32H(5,𝔽2)12yes—
278C22×C32C2511yescharacteristic
279C2×C3⋊D3—28no—
280C2×C3⋊D3—22no—
281C2×C3⋊D3—22no—
282C2×C3⋊D3C22×Q814yes—
283C2×C3×D3—216no—
284C2×C3×D3—24no—
285C2×C3×D3—24no—
286C2×C3×D3C22×Q818yes—
287D32—416no—
288C32×C4—22no—
289C32×C4—24no—
290C3⋊Dic3—24no—
291C3⋊Dic3—22no—
292C3×Dic3—24no—
293C3×Dic3—24no—
294C3×Dic3—24no—
295C25—91no—
296C22×Q8—92no—
297C22×Q8—91no—
298C22×Q8D3211yescharacteristic, complemented
299C23×C4—32no—
300C23×C4—91no—
301C23×C4—92no—
302C23×C4—91no—
303C22⋊Q8—316no—
304C22⋊Q8—916no—
305C22⋊Q8—916no—
306C22⋊Q8—98no—
307C22⋊Q8—916no—
308C22⋊Q8—916no—
309C22⋊Q8—98no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—91no—
314C2×C4⋊C4—92no—
315C2×C4⋊C4—92no—
316C2×C4⋊C4—91no—
317C2×C22⋊C4—32no—
318C2×C22⋊C4—32no—
319C2×C22⋊C4—98no—
320C2×C22⋊C4—916no—
321C2×C22⋊C4—92no—
322C2×C22⋊C4—92no—
323C2×C22⋊C4—92no—
324C2×C22⋊C4—92no—
325C23×C3—21no—
326C23×C3—38no—
327C23×C3—34no—
328C23×C3—64no—
329C23×C3—64no—
330C23×C3C23×D312yes—
331C22×D3—24no—
332C22×D3—24no—
333C22×D3—38no—
334C22×D3—34no—
335C22×D3—332no—
336C22×D3—64no—
337C22×D3—62no—
338C22×D3—64no—
339C22×D3—64no—
340C22×D3—122no—
341C22×D3—122no—
342C22×D3—1232no—
343C22×D3—128no—
344C22×D3—128no—
345C22×D3D3×Q818yes—
346C22×D3C23×D314yes—
347C3×Q8—48no—
348C3×Q8—48no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—34no—
356C2×C3×C4—34no—
357C2×C3×C4—34no—
358C2×C3×C4—44no—
359C2×C3×C4—64no—
360C2×C3×C4—64no—
361C2×C3×C4—64no—
362C2×C3×C4—128no—
363C2×C3×C4C23×D314yes—
364C2×C3×C4C23×D314yes—
365C2×C3×C4C23×D314yes—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—34no—
370C2×Dic3—34no—
371C2×Dic3—34no—
372C2×Dic3—62no—
373C2×Dic3—64no—
374C2×Dic3—64no—
375C2×Dic3—64no—
376C2×Dic3—64no—
377C2×Dic3—122no—
378C2×Dic3—124no—
379C2×Dic3—128no—
380C2×Dic3C23×D314yes—
381C2×Dic3C23×D314yes—
382C2×Dic3C23×D314yes—
383C4×D3—48no—
384C4×D3—128no—
385C4×D3—128no—
386C4×D3—128no—
387C4×D3—128no—
388Dic6—128no—
389Dic6—128no—
390Dic6—128no—
391C2×C32#54314yescomplemented
392C2×C32C23×Q811yescharacteristic
393C2×C32C2×H(5,𝔽2)11yescharacteristic
394C2×C32C2×H(5,𝔽2)11yescharacteristic
395C3⋊D3—24no—
396C3×D3—28no—
397C24—32no—
398C24—94no—
399C24—98no—
400C24—91no—
401C24—362no—
402C24—362no—
403C2×Q8—42no—
404C2×Q8—98no—
405C2×Q8—94no—
406C2×Q8—364no—
407C2×Q8—362no—
408C2×Q8C2×D3214yescomplemented
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—38no—
413C22×C4—94no—
414C22×C4—98no—
415C22×C4—92no—
416C22×C4—91no—
417C22×C4—94no—
418C22×C4—124no—
419C22×C4—362no—
420C22×C4—362no—
421C22×C4—364no—
422C22×C4C2×D3211yescharacteristic
423C22×C4C2×D3212yes—
424C4⋊C4—38no—
425C4⋊C4—38no—
426C4⋊C4—38no—
427C4⋊C4—94no—
428C4⋊C4—98no—
429C4⋊C4—98no—
430C4⋊C4—94no—
431C22⋊C4—38no—
432C22⋊C4—38no—
433C22⋊C4—98no—
434C22⋊C4—98no—
435C22⋊C4—98no—
436C22⋊C4—98no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—21no—
441C22×C3—38no—
442C22×C3—616no—
443C22×C3—64no—
444C22×C3—64no—
445C22×C3#100414yes—
446C22×C3C2×D3×Q814yes—
447C22×C3#100914yes—
448C22×C3C24×D312yes—
449C2×D3—216no—
450C2×D3—24no—
451C2×D3—24no—
452C2×D3—38no—
453C2×D3—64no—
454C2×D3—616no—
455C2×D3—64no—
456C2×D3—64no—
457C2×D3—122no—
458C2×D3—122no—
459C2×D3—128no—
460C2×D3—1232no—
461C2×D3C2×D3×Q818yes—
462C3×C4—24no—
463C3×C4—24no—
464C3×C4—24no—
465C3×C4—24no—
466C3×C4—44no—
467C3×C4—64no—
468C3×C4—64no—
469C3×C4—64no—
470Dic3—24no—
471Dic3—24no—
472Dic3—24no—
473Dic3—64no—
474Dic3—64no—
475Dic3—64no—
476Dic3—122no—
477Dic3—124no—
478C32C2×#54311yescharacteristic, complemented, Sylow
479C23—38no—
480C23—34no—
481C23—64no—
482C23—64no—
483C23—916no—
484C23—94no—
485C23—92no—
486C23—182no—
487C23—182no—
488C23—364no—
489C23—364no—
490C23—3616no—
491C23C22×D3211yescharacteristic
492Q8—44no—
493Q8—368no—
494Q8—364no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—34no—
498C2×C4—34no—
499C2×C4—34no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—94no—
504C2×C4—92no—
505C2×C4—128no—
506C2×C4—184no—
507C2×C4—182no—
508C2×C4—364no—
509C2×C4—364no—
510C2×C4—368no—
511C2×C4C22×D3212yes—
512C2×C4C22×D3214yes—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—24no—
517C2×C3—68no—
518C2×C3#563418yescomplemented
519C2×C3C2×#100412yes—
520C2×C3C22×D3×Q812yes—
521C2×C3C2×#100912yes—
522D3—28no—
523D3—68no—
524D3—124no—
525C22—38no—
526C22—64no—
527C22—64no—
528C22—616no—
529C22—94no—
530C22—182no—
531C22—182no—
532C22—188no—
533C22—3616no—
534C22D32×Q812yes—
535C22#6432012yes—
536C22#6433012yes—
537C22C23×D3211yescharacteristic
538C4—24no—
539C4—22no—
540C4—64no—
541C4—64no—
542C4—64no—
543C4—184no—
544C4—182no—
545C3—21no—
546C3C2×#563412yescomplemented
547C2—68no—
548C2—184no—
549C2#1059543414yesdirect product factor
550C2C2×D32×Q811yescharacteristic
551C2C2×#6432011yescharacteristic
552C2C2×#6433011yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC22×C32×Q8C22
socleC23×C32C24
prime radicalC23×D32C22
hypercenterC22×Q8D32
hypocenterC32C2×#543
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#64155C2semidirect
2. C2×C3×#946C2semidirect
3. #10595434C2direct (N × H)
6. C2×#63899C2semidirect
7. C2×#63895C2semidirect
8. C2×#63893C2semidirect
9. C2×#63865C2semidirect
10. C2×#63851C2semidirect
13. C22×C32×Q8C22semidirect
16. C22×#627C22semidirect
17. C22×#627C22semidirect
20. #64155C22semidirect
22. C2×C4⋊(C3⋊Dic3)C22semidirect
23. C2×#3750C22semidirect
24. C3×#946C22semidirect
27. C2×C3×C4⋊Dic3C22semidirect
28. C2×C3×Dic3⋊C4C22semidirect
29. C2×C3×Dic3⋊C4C22semidirect
30. C2×#3724C22semidirect
31. C2×Dic3⋊Dic3C22semidirect
37. #63899C22semidirect
38. #63895C22semidirect
39. #63893C22semidirect
40. #63865C22semidirect
41. #63851C22semidirect
45. C2×#946D3semidirect
63. C22×D32Q8semidirect
66. C2×C32×Q8C23semidirect
80. C2×#627C23semidirect
81. C2×#627C23semidirect
87. C4⋊(C3⋊Dic3)C23semidirect
88. #3750C23semidirect
91. C3×C4⋊Dic3C23semidirect
92. C3×Dic3⋊C4C23semidirect
93. C3×Dic3⋊C4C23semidirect
94. #3724C23semidirect
95. Dic3⋊Dic3C23semidirect
102. C22×C3×Q8C2×D3semidirect
123. #946C2×D3semidirect
137. C2×C4⋊Dic3C2×D3semidirect
144. C2×Dic3⋊C4C2×D3semidirect
145. C2×Dic3⋊C4C2×D3semidirect
168. C2×D32C2×Q8semidirect
211. C2×C3×Q8C22×D3semidirect
267. C4⋊Dic3C22×D3semidirect
273. Dic3⋊C4C22×D3semidirect
274. Dic3⋊C4C22×D3semidirect
298. C22×Q8D32semidirect
391. C2×C32#543semidirect
408. C2×Q8C2×D32semidirect
478. C32C2×#543semidirect
518. C2×C3#5634semidirect
546. C3C2×#5634semidirect
549. C2#10595434direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#641552
C2×C3×#9462
#105954342
C2×#639212
C2×D3×(C2×D3)⋊C42
C2×#638992
C2×#638952
C2×#638932
C2×#638652
C2×#638512
C2×#56343