Finite Group Databasebeta

← C2×#73955 #10962252

Subgroups — C2×#73955 #10962252

Order 768.

2614 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#926C211yesmaximal, characteristic
2C22×#890C211yesmaximal, characteristic, complemented
3C2×C3×#380C211yesmaximal, characteristic
4C2×#5212C211yesmaximal, characteristic
5C2×#4984C211yesmaximal, characteristic, complemented
6C2×#4972C211yesmaximal, characteristic, complemented
7C2×#4975C211yesmaximal, characteristic, complemented
8C2×#4948C212yesmaximal
9C2×#4948C212yesmaximal
10#73955C2116yesmaximal, direct product factor
11C2×#4661C212yesmaximal, complemented
12C2×#4657C212yesmaximal, complemented
13C2×#2629—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×#217C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×#217C2212yes—
21C2×#926—28no—
22C2×#926C2218yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C22×C4⋊Dic3C2212yes—
28C22×C4⋊Dic3C2212yes—
29C22×Dic3⋊C4C2212yes—
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C22×C4×Dic3C2211yescharacteristic
35C2×#890—28no—
36C2×#890C2218yescomplemented
37C3×#380C2218yes—
38C2×C3×#95C2212yes—
39#5212C2218yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4984C2218yescomplemented
44#4975C2218yescomplemented
45#4972C2218yescomplemented
46#4948C22116yes—
47#4948C22116yes—
48#4661C22116yescomplemented
49#4657C22116yescomplemented
50C22×#125—31no—
51C22×#117—31no—
52#2629—316no—
53C2×#398—32no—
54C2×#398—32no—
55C2×#396—32no—
56C2×#396—32no—
57C2×#389—31no—
58C2×#389—31no—
59C2×#384—31no—
60C2×#380—31no—
61C2×#380D311yescharacteristic
62C25×C3C2311yescharacteristic
63C23×C3×C4—24no—
64C23×C3×C4—24no—
65C23×C3×C4Q818yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2314yes—
70C23×C3×C4C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2312yes—
74C23×Dic3C2312yes—
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2311yescharacteristic
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4C2314yes—
80C2×C3×C4⋊C4C2314yes—
81C2×C3×C22⋊C4—44no—
82C2×C3×C22⋊C4C2318yes—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—44no—
86C2×#217C2314yes—
87C2×#217C2314yes—
88C2×#217C2318yes—
89#926—232no—
90C2×C4⋊Dic3—24no—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—28no—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—44no—
96C2×C4⋊Dic3C2314yes—
97C2×C4⋊Dic3C2314yes—
98C2×C4⋊Dic3C2318yes—
99C2×C4⋊Dic3C2314yes—
100C2×C4⋊Dic3C2314yes—
101C2×C4⋊Dic3C2318yes—
102C2×Dic3⋊C4—28no—
103C2×Dic3⋊C4—44no—
104C2×Dic3⋊C4—44no—
105C2×Dic3⋊C4—44no—
106C2×Dic3⋊C4C2318yes—
107C2×Dic3⋊C4C2318yes—
108C2×Dic3⋊C4C2318yes—
109C2×Dic3⋊C4C2318yes—
110C2×C4×Dic3—24no—
111C2×C4×Dic3D418yes—
112C2×C4×Dic3C2314yes—
113C2×C4×Dic3C2314yes—
114#890—232no—
115C3×#95C2318yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C3⋊#95C2318yes—
119C24×C4C2×D311yescharacteristic
120C2×#125—38no—
121C2×#125—68no—
122C2×#117—38no—
123C2×#117—68no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—31no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C4⋊C4C2×D311yescharacteristic
136C22×C22⋊C4—31no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4C2×D312yes—
140C22×C42—31no—
141C22×C42—31no—
142#398—316no—
143#398—316no—
144#396—316no—
145#396—316no—
146#389—38no—
147#389—38no—
148#384—38no—
149#380—38no—
150#380C2×D318yes—
151C2×#95—32no—
152C2×#95—32no—
153C2×#95—32no—
154C2×#95C2×D312yes—
155C24×C3—22no—
156C24×C3—22no—
157C24×C3—22no—
158C24×C3—22no—
159C24×C3—22no—
160C24×C3—22no—
161C24×C3C2×Q814yes—
162C24×C3C2412yes—
163C24×C3C2411yescharacteristic
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—232no—
167C22×C3×C4—28no—
168C22×C3×C4—28no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2×Q814yes—
177C22×C3×C4C2×Q8116yes—
178C22×C3×C4C2×Q814yes—
179C22×C3×C4#4114yes—
180C22×C3×C4C2414yes—
181C22×C3×C4C2414yes—
182C22×C3×C4C2412yes—
183C22×C3×C4C2412yes—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—24no—
191C22×Dic3—24no—
192C22×Dic3—22no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—22no—
196C22×Dic3—24no—
197C22×Dic3—22no—
198C22×Dic3—24no—
199C22×Dic3—22no—
200C22×Dic3C2×D414yes—
201C22×Dic3C2×D414yes—
202C22×Dic3C2412yes—
203C22×Dic3C2412yes—
204C22×Dic3C2414yes—
205C22×Dic3C2414yes—
206C22×Dic3C2412yes—
207C22×Dic3C2412yes—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—28no—
210C3×C22⋊C4—48no—
211#217—28no—
212#217—28no—
213#217—48no—
214C4⋊Dic3—28no—
215C4⋊Dic3—28no—
216C4⋊Dic3—216no—
217C4⋊Dic3—28no—
218C4⋊Dic3—28no—
219C4⋊Dic3—48no—
220Dic3⋊C4—216no—
221Dic3⋊C4—48no—
222Dic3⋊C4—48no—
223Dic3⋊C4—48no—
224C4×Dic3—28no—
225C4×Dic3C2×D4116yes—
226C25C22×D311yescharacteristic
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—31no—
230C23×C4—31no—
231C23×C4—32no—
232C23×C4—32no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4Dic618yes—
236C23×C4C22×D312yes—
237C23×C4C22×D312yes—
238C23×C4C22×D314yes—
239C23×C4C22×D311yescharacteristic
240C23×C4C22×D311yescharacteristic
241#125—632no—
242#117—632no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—34no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—38no—
249C2×C4⋊C4—38no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—38no—
252C2×C4⋊C4—38no—
253C2×C4⋊C4—38no—
254C2×C4⋊C4—38no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—68no—
258C2×C4⋊C4—68no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—124no—
262C2×C4⋊C4—124no—
263C2×C4⋊C4—124no—
264C2×C4⋊C4—124no—
265C2×C4⋊C4C22×D314yes—
266C2×C4⋊C4C22×D314yes—
267C2×C22⋊C4—34no—
268C2×C22⋊C4—34no—
269C2×C22⋊C4—38no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—124no—
274C2×C22⋊C4C22×D318yes—
275C2×C42—38no—
276C2×C42—34no—
277C2×C42—34no—
278C2×C42—64no—
279#95—38no—
280#95—38no—
281#95—38no—
282#95C22×D318yes—
283C23×C3—28no—
284C23×C3—28no—
285C23×C3—28no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—44no—
293C23×C3—44no—
294C23×C3—44no—
295C23×C3—44no—
296C23×C3C22×D412yes—
297C23×C3C22×Q814yes—
298C23×C3C22×Q812yes—
299C23×C3C2×#4112yes—
300C23×C3H(5,𝔽2)12yes—
301C23×C3H(5,𝔽2)12yes—
302C23×C32−1+412yes—
303C23×C32−1+412yes—
304C23×C3C2511yescharacteristic
305C2×C3×C4—28no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—232no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—48no—
316C2×C3×C4—48no—
317C2×C3×C4C22⋊Q8116yes—
318C2×C3×C4C22×D414yes—
319C2×C3×C4C22×Q814yes—
320C2×C3×C4C22×Q814yes—
321C2×C3×C4C2×#4114yes—
322C2×Dic3—28no—
323C2×Dic3—28no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—22no—
336C2×Dic3—24no—
337C2×Dic3—22no—
338C2×Dic3—24no—
339C2×Dic3—22no—
340C2×Dic3—44no—
341C2×Dic3—44no—
342C2×Dic3—48no—
343C2×Dic3—48no—
344C2×Dic3C22×D414yes—
345C2×Dic3C22×D414yes—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24C2×Dic614yes—
353C24C23×D312yes—
354C24C23×D311yescharacteristic
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—232no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—34no—
367C22×C4—34no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—34no—
373C22×C4—34no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—64no—
381C22×C4—64no—
382C22×C4—62no—
383C22×C4—64no—
384C22×C4—64no—
385C22×C4—62no—
386C22×C4—64no—
387C22×C4—64no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4C2×Dic614yes—
391C22×C4C2×Dic6116yes—
392C22×C4C2×Dic614yes—
393C22×C4D3×D414yes—
394C22×C4#23714yes—
395C22×C4C23×D314yes—
396C22×C4C23×D314yes—
397C22×C4C23×D312yes—
398C22×C4C23×D312yes—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—68no—
402C4⋊C4—68no—
403C4⋊C4—616no—
404C4⋊C4—616no—
405C4⋊C4—68no—
406C4⋊C4—68no—
407C4⋊C4—128no—
408C4⋊C4—128no—
409C4⋊C4—128no—
410C4⋊C4—128no—
411C22⋊C4—48no—
412C22⋊C4—68no—
413C22⋊C4—68no—
414C22⋊C4—128no—
415C42—316no—
416C42—68no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—416no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3C2×C22⋊Q814yes—
426C22×C3#53714yes—
427C22×C3#53714yes—
428C22×C3#54014yes—
429C22×C3#54114yes—
430C22×C3#54414yes—
431C22×C3#54414yes—
432C22×C3C23×D411yescharacteristic
433C22×C3C23×Q811yescharacteristic
434C22×C3C22×#4111yescharacteristic
435C22×C3C2×H(5,𝔽2)11yescharacteristic
436C22×C3C2×H(5,𝔽2)11yescharacteristic
437C22×C3C2×2−1+411yescharacteristic
438C22×C3C2×2−1+411yescharacteristic
439C3×C4—24no—
440C3×C4—44no—
441C3×C4—44no—
442C3×C4C2×C22⋊Q818yes—
443Dic3—24no—
444Dic3—24no—
445Dic3—42no—
446Dic3—42no—
447Dic3—44no—
448Dic3—44no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23C22×Dic614yes—
463C23C22×Dic612yes—
464C23C2×D3×D412yes—
465C23C2×#23712yes—
466C23#100412yes—
467C23#100712yes—
468C23#100912yes—
469C23#101012yes—
470C23C24×D311yescharacteristic
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—232no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—34no—
482C2×C4—34no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—68no—
490C2×C4—68no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—64no—
494C2×C4—64no—
495C2×C4—64no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—124no—
504C2×C4—128no—
505C2×C4—124no—
506C2×C4—128no—
507C2×C4#878116yes—
508C2×C4C22×Dic614yes—
509C2×C4C22×Dic614yes—
510C2×C4C2×D3×D414yes—
511C2×C4C2×#23714yes—
512C2×C3—44no—
513C2×C3#262918yescomplemented
514C2×C3C22×C22⋊Q811yescharacteristic
515C2×C3C2×#53711yescharacteristic
516C2×C3C2×#53711yescharacteristic
517C2×C3C2×#54011yescharacteristic
518C2×C3C2×#54111yescharacteristic
519C2×C3C2×#54411yescharacteristic
520C2×C3C2×#54411yescharacteristic
521C22—24no—
522C22—24no—
523C22—24no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—416no—
529C22C2×#87814yes—
530C22#550014yes—
531C22#551914yes—
532C22#560814yes—
533C22#561914yes—
534C22#562514yes—
535C22#562914yes—
536C22C23×Dic611yescharacteristic
537C22C22×D3×D411yescharacteristic
538C22C22×#23711yescharacteristic
539C22C2×#100411yescharacteristic
540C22C2×#100711yescharacteristic
541C22C2×#100911yescharacteristic
542C22C2×#101011yescharacteristic
543C4—24no—
544C4—44no—
545C4—44no—
546C4—64no—
547C4—64no—
548C4—124no—
549C4—122no—
550C4—122no—
551C4—124no—
552C4C2×#87818yes—
553C3C2×#262911yescharacteristic, complemented, Sylow
554C2—44no—
555C2#7395518yesdirect product factor
556C2C22×#87811yescharacteristic
557C2C2×#550011yescharacteristic
558C2C2×#551911yescharacteristic
559C2C2×#561911yescharacteristic
560C2C2×#560811yescharacteristic
561C2C2×#562511yescharacteristic
562C2C2×#562911yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#380C2
socleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#380D3
hypocenterC3C2×#2629
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#890C2semidirect
5. C2×#4984C2semidirect
6. C2×#4972C2semidirect
7. C2×#4975C2semidirect
10. #73955C2direct (N × H)
11. C2×#4661C2semidirect
12. C2×#4657C2semidirect
36. C2×#890C22semidirect
43. #4984C22semidirect
44. #4975C22semidirect
45. #4972C22semidirect
48. #4661C22semidirect
49. #4657C22semidirect
513. C2×C3#2629semidirect
553. C3C2×#2629semidirect
555. C2#73955direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9262
C22×#8902
C2×C3×#3802
C2×#52122
C2×#49842
C2×#49722
C2×#49752
C2×#49482
#739552
C2×#46612
C2×#46572
C2×#26293