Finite Group Databasebeta

← C2×#69594 #10823317

Subgroups — C2×#69594 #10823317

Order 768.

1614 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5483C211yesmaximal, characteristic
2C2×C3×#385C211yesmaximal, characteristic
3C2×#5210C212yesmaximal
4C2×#4527C212yesmaximal
5C2×#4526C211yesmaximal, characteristic
6#69594C218yesmaximal, direct product factor
7C2×#2062—31nomaximal, Sylow
8C24×C3×C4C2211yescharacteristic
9C22×C3×C4⋊C4—21no—
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×#217C2212yes—
13C22×#217C2212yes—
14C2×#924—21no—
15C2×#924—21no—
16C22×Dic3⋊C4—22no—
17#5483—22no—
18#5483—24no—
19#5483C2214yes—
20C2×#878—22no—
21C2×#878—22no—
22C3×#385—22no—
23C3×#385C2214yes—
24C2×C3×#95—21no—
25#5210—24no—
26#5210C2218yes—
27C2×C3⋊#95—22no—
28#4527C2218yes—
29#4526C2214yes—
30C2×#543—31no—
31C2×#385—32no—
32C2×#385D311yescharacteristic
33#2062—38no—
34C2×#342—31no—
35C2×#342—32no—
36C25×C3C2311yescharacteristic
37C23×C3×C4—22no—
38C23×C3×C4—21no—
39C23×C3×C4—21no—
40C23×C3×C4—22no—
41C23×C3×C4—21no—
42C23×C3×C4—44no—
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2312yes—
46C23×Dic3—22no—
47C23×Dic3D412yes—
48C23×Dic3D412yes—
49C22×Dic6—21no—
50C22×Dic6—21no—
51C2×C3×C4⋊C4—24no—
52C2×C3×C4⋊C4—44no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4D412yes—
58C2×C3×C22⋊C4Q814yescomplemented
59C2×C3×C22⋊C4C2312yes—
60C2×C3×C22⋊C4C2312yes—
61C2×#217—22no—
62C2×#217—22no—
63C2×#217—22no—
64C2×#217—22no—
65C2×#217—22no—
66C2×#217—24no—
67C2×#217D414yes—
68C2×#217Q818yescomplemented
69C2×#217C2314yes—
70C2×#217C2314yes—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73#924—24no—
74#924—24no—
75#924—24no—
76#924—24no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—28no—
80C2×Dic3⋊C4—48no—
81#878—28no—
82#878—28no—
83#878—28no—
84#878—28no—
85C3×#95—24no—
86C3⋊#95—28no—
87C24×C4C2×D311yescharacteristic
88#543—34no—
89#543—64no—
90#543—62no—
91C2×C22⋊Q8—62no—
92C2×C22⋊Q8—62no—
93C2×C22⋊Q8—61no—
94C2×C22⋊Q8—61no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—62no—
97C22×C22⋊C4—32no—
98C22×C22⋊C4—32no—
99C22×C22⋊C4C2×D311yescharacteristic
100C22×C22⋊C4C2×D311yescharacteristic
101#385—22no—
102#385—38no—
103#385—64no—
104#385C2×D314yes—
105C2×#95—21no—
106C2×#95—62no—
107#342—34no—
108#342—38no—
109C24×C3—42no—
110C24×C3—42no—
111C24×C3C2×D412yes—
112C24×C3C2×D412yes—
113C24×C3C2×D411yescharacteristic
114C24×C3C2×D411yescharacteristic
115C24×C3C2×Q812yes—
116C24×C3C2×Q814yes—
117C24×C3#4112yes—
118C24×C3C2411yescharacteristic
119C22×C3×C4—28no—
120C22×C3×C4—24no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—44no—
130C22×C3×C4—48no—
131C22×C3×C4—44no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—44no—
135C22×C3×C4—42no—
136C22×C3×C4—41no—
137C22×C3×C4—42no—
138C22×C3×C4—41no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—28no—
146C22×Dic3—24no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—44no—
150C22×Dic3—44no—
151C22×Dic3—44no—
152C22×Dic3—44no—
153C22×Dic3C2×D414yes—
154C22×Dic3C2×D414yes—
155C2×Dic6—24no—
156C2×Dic6—24no—
157C2×Dic6—82no—
158C2×Dic6—82no—
159C3×C4⋊C4—48no—
160C3×C22⋊C4—24no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—22no—
164C3×C22⋊C4C2×Q814yescomplemented
165#217—24no—
166#217—24no—
167#217—28no—
168#217—24no—
169#217—24no—
170#217—24no—
171#217C2×Q818yescomplemented
172C4⋊Dic3—24no—
173C4⋊Dic3—24no—
174Dic3⋊C4—28no—
175Dic3⋊C4—28no—
176Dic3⋊C4—416no—
177C25C22×D311yescharacteristic
178C22×Q8—61no—
179C22×Q8—61no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—32no—
186C23×C4—32no—
187C23×C4—44no—
188C23×C4—62no—
189C23×C4#6711yescharacteristic
190C23×C4#6711yescharacteristic
191C23×C4C22×D312yes—
192C22⋊Q8—68no—
193C22⋊Q8—68no—
194C22⋊Q8—68no—
195C22⋊Q8—68no—
196C22⋊Q8—64no—
197C22⋊Q8—64no—
198C22⋊Q8—64no—
199C22⋊Q8—64no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—68no—
207C2×C4⋊C4—128no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—38no—
213C2×C22⋊C4—34no—
214C2×C22⋊C4—34no—
215C2×C22⋊C4—34no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4—62no—
221C2×C22⋊C4—62no—
222C2×C22⋊C4Dic614yescomplemented
223C2×C22⋊C4#6712yes—
224C2×C22⋊C4C22×D312yes—
225C2×C22⋊C4C22×D312yes—
226#95—24no—
227#95—68no—
228C23×C3—24no—
229C23×C3—22no—
230C23×C3—24no—
231C23×C3—22no—
232C23×C3—42no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—42no—
236C23×C3—41no—
237C23×C3—44no—
238C23×C3—48no—
239C23×C3—44no—
240C23×C3—41no—
241C23×C3#12011yescharacteristic
242C23×C3#12012yes—
243C23×C3#12011yescharacteristic
244C23×C3C22⋊Q814yes—
245C23×C3C22⋊Q814yes—
246C23×C3C22⋊Q812yes—
247C23×C3C22⋊Q812yes—
248C23×C3C4⋊Q814yes—
249C23×C3C4⋊Q814yes—
250C23×C3C4⋊Q812yes—
251C23×C3C4⋊Q812yes—
252C23×C3C22×D412yes—
253C23×C3C22×D411yescharacteristic
254C23×C3C22×Q811yescharacteristic
255C23×C3C22×Q812yes—
256C23×C3C2×#4111yescharacteristic
257C2×C3×C4—28no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—24no—
262C2×C3×C4—42no—
263C2×C3×C4—48no—
264C2×C3×C4—42no—
265C2×C3×C4—44no—
266C2×C3×C4—48no—
267C2×C3×C4—42no—
268C2×C3×C4—44no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—44no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—81no—
276C2×C3×C4—81no—
277C2×Dic3—24no—
278C2×Dic3—24no—
279C2×Dic3—28no—
280C2×Dic3—44no—
281C2×Dic3—416no—
282C2×Dic3—44no—
283C2×Dic3—44no—
284C2×Dic3—44no—
285C2×Dic3—44no—
286C2×Dic3—44no—
287C2×Dic3—82no—
288C2×Dic3—82no—
289Dic6—84no—
290Dic6—84no—
291C24—42no—
292C24—42no—
293C24C2×Dic612yes—
294C24#23512yes—
295C24D3×D412yes—
296C24D3×D412yes—
297C24D3×Q814yes—
298C24C2×#6711yescharacteristic
299C24C2×#6711yescharacteristic
300C24C23×D311yescharacteristic
301C2×Q8—64no—
302C2×Q8—64no—
303C2×Q8—242no—
304C2×Q8—242no—
305C22×C4—24no—
306C22×C4—28no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—34no—
316C22×C4—34no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—48no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—68no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—64no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—124no—
336C22×C4—124no—
337C22×C4—124no—
338C22×C4—124no—
339C22×C4C2×#6712yes—
340C22×C4C2×#6712yes—
341C4⋊C4—48no—
342C4⋊C4—68no—
343C4⋊C4—68no—
344C4⋊C4—64no—
345C4⋊C4—64no—
346C4⋊C4—1216no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—22no—
350C22⋊C4—24no—
351C22⋊C4—38no—
352C22⋊C4—68no—
353C22⋊C4—64no—
354C22⋊C4—64no—
355C22⋊C4—64no—
356C22⋊C4—64no—
357C22⋊C4—64no—
358C22⋊C4C2×Dic614yescomplemented
359C22×C3—28no—
360C22×C3—24no—
361C22×C3—28no—
362C22×C3—24no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—44no—
372C22×C3—42no—
373C22×C3—44no—
374C22×C3—42no—
375C22×C3—41no—
376C22×C3—42no—
377C22×C3—41no—
378C22×C3#39512yes—
379C22×C3#39514yes—
380C22×C3#39512yes—
381C22×C3B(4,𝔽2)12yes—
382C22×C3#45812yes—
383C22×C3C2×#12011yescharacteristic
384C22×C3C2×C22⋊Q812yes—
385C22×C3C2×C22⋊Q811yescharacteristic
386C22×C3C2×C4⋊Q812yes—
387C22×C3C2×C4⋊Q811yescharacteristic
388C3×C4—44no—
389C3×C4—42no—
390C3×C4—44no—
391C3×C4—42no—
392Dic3—48no—
393Dic3—44no—
394Dic3—44no—
395C23—22no—
396C23—24no—
397C23—22no—
398C23—24no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—44no—
403C23—44no—
404C23—42no—
405C23—48no—
406C23—41no—
407C23—41no—
408C23#88814yes—
409C23#88814yes—
410C23#89614yes—
411C23#89614yes—
412C23#92412yes—
413C23#92412yes—
414C23#93712yes—
415C23#93711yescharacteristic
416C23#93711yescharacteristic
417C23#94412yes—
418C23#94412yes—
419C23C22×Dic611yescharacteristic
420C23C2×#23511yescharacteristic
421C23C2×D3×D412yes—
422C23C2×D3×Q812yes—
423C23C22×#6711yescharacteristic
424Q8—244no—
425Q8—244no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—42no—
437C2×C4—48no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—68no—
445C2×C4—64no—
446C2×C4—64no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—124no—
450C2×C4—124no—
451C2×C4—1216no—
452C2×C4—124no—
453C2×C4—124no—
454C2×C4—124no—
455C2×C4—124no—
456C2×C4—242no—
457C2×C4—242no—
458C2×C3—22no—
459C2×C3—21no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—24no—
463C2×C3—22no—
464C2×C3#206214yescomplemented
465C2×C3C2×#39511yescharacteristic
466C2×C3C2×B(4,𝔽2)11yescharacteristic
467C2×C3C2×#45811yescharacteristic
468C22—21no—
469C22—22no—
470C22—21no—
471C22—22no—
472C22—21no—
473C22—21no—
474C22—22no—
475C22—22no—
476C22—24no—
477C22—28no—
478C22—28no—
479C22—24no—
480C22—44no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—41no—
485C22—41no—
486C22—42no—
487C22#497914yes—
488C22#497912yes—
489C22#497912yes—
490C22#503212yes—
491C22#505312yes—
492C22C2×#88812yes—
493C22C2×#89612yes—
494C22C2×#92411yescharacteristic
495C22C2×#93711yescharacteristic
496C22C2×#94411yescharacteristic
497C4—44no—
498C4—44no—
499C4—42no—
500C4—42no—
501C4—128no—
502C4—124no—
503C4—124no—
504C3C2×#206211yescharacteristic, complemented, Sylow
505C2—21no—
506C2—22no—
507C2—22no—
508C2—24no—
509C2—22no—
510C2—21no—
511C2#6959414yesdirect product factor
512C2C2×#497911yescharacteristic
513C2C2×#503211yescharacteristic
514C2C2×#505311yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#385C2
socleC23×C3#120
prime radicalC25×C3C23
hypercenterC2×#385D3
hypocenterC3C2×#2062
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
6. #69594C2direct (N × H)
58. C2×C3×C22⋊C4Q8semidirect
68. C2×#217Q8semidirect
164. C3×C22⋊C4C2×Q8semidirect
171. #217C2×Q8semidirect
222. C2×C22⋊C4Dic6semidirect
358. C22⋊C4C2×Dic6semidirect
464. C2×C3#2062semidirect
504. C3C2×#2062semidirect
511. C2#69594direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#54832
C2×C3×#3852
C2×#52102
C2×#45272
C2×#45262
#695942
C2×#20623