Finite Group Databasebeta

← C2×#69497 #10822583

Subgroups — C2×#69497 #10822583

Order 768.

2818 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5483C211yesmaximal, characteristic, complemented
2C2×C3×#379C211yesmaximal, characteristic
3C2×#5247C212yesmaximal
4C2×#4527C212yesmaximal, complemented
5C2×#4526C211yesmaximal, characteristic, complemented
6#69497C218yesmaximal, direct product factor
7C2×#2029—31nomaximal, Sylow
8C26×C3C2211yescharacteristic
9C22×C3×C22⋊C4—22no—
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×#217—24no—
13C22×#217C2212yes—
14C22×#217C2212yes—
15C2×#924—22no—
16#5483—22no—
17#5483—24no—
18#5483C2214yescomplemented
19C2×#878—24no—
20C3×#379—22no—
21C3×#379C2214yes—
22#5247—24no—
23#5247C2218yes—
24#4527C2218yescomplemented
25#4526C2214yescomplemented
26C2×#543—31no—
27C2×#379—32no—
28C2×#379D311yescharacteristic
29#2029—38no—
30C2×#342—31no—
31C2×#342—32no—
32C25×C3—22no—
33C25×C3—21no—
34C25×C3—22no—
35C25×C3—21no—
36C25×C3—22no—
37C25×C3—24no—
38C25×C3—48no—
39C25×C3Q814yescomplemented
40C25×C3C2312yes—
41C25×C3C2311yescharacteristic
42C23×C3×C4—21no—
43C23×C3×C4D412yes—
44C23×Dic3—22no—
45C23×Dic3D414yes—
46C22×Dic6—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—24no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—28no—
52C2×C3×C22⋊C4—48no—
53C2×C3×C22⋊C4D412yes—
54C2×C3×C22⋊C4C2312yes—
55C2×C3×C22⋊C4C2312yes—
56C2×#217—22no—
57C2×#217—22no—
58C2×#217—22no—
59C2×#217—22no—
60C2×#217—24no—
61C2×#217—24no—
62C2×#217—22no—
63C2×#217—216no—
64C2×#217—416no—
65C2×#217D414yes—
66C2×#217C2314yes—
67C2×#217C2314yes—
68C2×C4⋊Dic3—22no—
69#924—28no—
70#924—28no—
71C2×Dic3⋊C4—24no—
72#878—216no—
73#878—216no—
74C26C2×D311yescharacteristic
75#543—34no—
76#543—64no—
77#543—62no—
78C2×C22⋊Q8—64no—
79C2×C22⋊Q8—62no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—32no—
82C22×C22⋊C4—32no—
83C22×C22⋊C4—64no—
84C22×C22⋊C4C2×D311yescharacteristic
85C22×C22⋊C4C2×D311yescharacteristic
86#379—22no—
87#379—38no—
88#379—64no—
89#379C2×D314yes—
90#342—34no—
91#342—38no—
92C24×C3—216no—
93C24×C3—28no—
94C24×C3—28no—
95C24×C3—24no—
96C24×C3—24no—
97C24×C3—22no—
98C24×C3—28no—
99C24×C3—24no—
100C24×C3—22no—
101C24×C3—24no—
102C24×C3—416no—
103C24×C3—48no—
104C24×C3—44no—
105C24×C3—48no—
106C24×C3—44no—
107C24×C3—416no—
108C24×C3—48no—
109C24×C3—416no—
110C24×C3—432no—
111C24×C3—41no—
112C24×C3—42no—
113C24×C3—41no—
114C24×C3—44no—
115C24×C3—44no—
116C24×C3—42no—
117C24×C3—42no—
118C24×C3C2×D412yes—
119C24×C3C2×D412yes—
120C24×C3C2×D411yescharacteristic
121C24×C3C2×D411yescharacteristic
122C24×C3C2×Q812yes—
123C24×C3C2×Q814yescomplemented
124C24×C3#4112yes—
125C24×C3#4114yes—
126C24×C3C2411yescharacteristic
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—44no—
131C22×C3×C4—44no—
132C22×C3×C4—42no—
133C22×C3×C4C2×D414yes—
134C22×Dic3—24no—
135C22×Dic3—24no—
136C22×Dic3—24no—
137C22×Dic3—48no—
138C22×Dic3—48no—
139C22×Dic3—44no—
140C22×Dic3C2×D418yes—
141C2×Dic6—28no—
142C2×Dic6—84no—
143C3×C22⋊C4—24no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—24no—
146C3×C22⋊C4—416no—
147C3×C22⋊C4—42no—
148#217—24no—
149#217—24no—
150#217—24no—
151#217—24no—
152#217—24no—
153#217—24no—
154#217—432no—
155#217—44no—
156C4⋊Dic3—28no—
157Dic3⋊C4—216no—
158C25—22no—
159C25—21no—
160C25—22no—
161C25—22no—
162C25—24no—
163C25—21no—
164C25—48no—
165C25Dic614yescomplemented
166C25C22×D312yes—
167C25C22×D311yescharacteristic
168C22×Q8—62no—
169C23×C4—21no—
170C23×C4—34no—
171C23×C4—62no—
172C23×C4#6712yes—
173C22⋊Q8—616no—
174C22⋊Q8—616no—
175C22⋊Q8—68no—
176C22⋊Q8—68no—
177C2×C4⋊C4—64no—
178C2×C4⋊C4—62no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—34no—
185C2×C22⋊C4—34no—
186C2×C22⋊C4—34no—
187C2×C22⋊C4—48no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—64no—
193C2×C22⋊C4—64no—
194C2×C22⋊C4—616no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—1216no—
197C2×C22⋊C4#6712yes—
198C2×C22⋊C4C22×D312yes—
199C2×C22⋊C4C22×D312yes—
200C23×C3—28no—
201C23×C3—24no—
202C23×C3—216no—
203C23×C3—28no—
204C23×C3—232no—
205C23×C3—216no—
206C23×C3—28no—
207C23×C3—24no—
208C23×C3—44no—
209C23×C3—42no—
210C23×C3—44no—
211C23×C3—48no—
212C23×C3—44no—
213C23×C3—48no—
214C23×C3—42no—
215C23×C3—48no—
216C23×C3—416no—
217C23×C3—44no—
218C23×C3—48no—
219C23×C3—432no—
220C23×C3—416no—
221C23×C3—416no—
222C23×C3—48no—
223C23×C3—432no—
224C23×C3—416no—
225C23×C3—48no—
226C23×C3—416no—
227C23×C3—432no—
228C23×C3—416no—
229C23×C3—42no—
230C23×C3—44no—
231C23×C3—42no—
232C23×C3—48no—
233C23×C3—44no—
234C23×C3—44no—
235C23×C3—48no—
236C23×C3#12011yescharacteristic
237C23×C3#12012yes—
238C23×C3#12011yescharacteristic
239C23×C3C22⋊Q814yes—
240C23×C3C22⋊Q814yes—
241C23×C3C22⋊Q812yes—
242C23×C3C22⋊Q812yes—
243C23×C3#12414yes—
244C23×C3#12414yes—
245C23×C3#12412yes—
246C23×C3#12412yes—
247C23×C3C22×D412yes—
248C23×C3C22×D411yescharacteristic
249C23×C3C22×Q811yescharacteristic
250C23×C3C2×#4111yescharacteristic
251C23×C3C2×#4112yes—
252C2×C3×C4—24no—
253C2×C3×C4—44no—
254C2×C3×C4—44no—
255C2×C3×C4—44no—
256C2×C3×C4—42no—
257C2×C3×C4—82no—
258C2×C3×C4—84no—
259C2×Dic3—28no—
260C2×Dic3—48no—
261C2×Dic3—48no—
262C2×Dic3—48no—
263C2×Dic3—44no—
264C2×Dic3—84no—
265C2×Dic3—88no—
266Dic6—88no—
267C24—28no—
268C24—216no—
269C24—28no—
270C24—24no—
271C24—24no—
272C24—22no—
273C24—28no—
274C24—24no—
275C24—22no—
276C24—24no—
277C24—416no—
278C24—48no—
279C24—44no—
280C24—44no—
281C24—48no—
282C24—48no—
283C24—416no—
284C24—416no—
285C24—432no—
286C24—44no—
287C24—44no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—41no—
292C24—41no—
293C24C2×Dic612yes—
294C24C2×Dic614yescomplemented
295C24#23512yes—
296C24D3×D412yes—
297C24D3×D412yes—
298C24#23714yes—
299C24C2×#6711yescharacteristic
300C24C2×#6711yescharacteristic
301C24C23×D311yescharacteristic
302C2×Q8—68no—
303C2×Q8—244no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—38no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—64no—
312C22×C4—64no—
313C22×C4—64no—
314C22×C4—128no—
315C22×C4—128no—
316C22×C4—124no—
317C22×C4C2×#6714yes—
318C4⋊C4—616no—
319C4⋊C4—68no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—416no—
324C22⋊C4—42no—
325C22⋊C4—64no—
326C22⋊C4—64no—
327C22⋊C4—64no—
328C22⋊C4—64no—
329C22⋊C4—64no—
330C22⋊C4—64no—
331C22⋊C4—1232no—
332C22⋊C4—124no—
333C22×C3—24no—
334C22×C3—24no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—28no—
338C22×C3—24no—
339C22×C3—28no—
340C22×C3—24no—
341C22×C3—216no—
342C22×C3—28no—
343C22×C3—44no—
344C22×C3—42no—
345C22×C3—44no—
346C22×C3—42no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—42no—
350C22×C3—416no—
351C22×C3—48no—
352C22×C3—48no—
353C22×C3—44no—
354C22×C3—432no—
355C22×C3—416no—
356C22×C3—44no—
357C22×C3—48no—
358C22×C3—416no—
359C22×C3#39312yes—
360C22×C3#39314yes—
361C22×C3#39312yes—
362C22×C3B(4,𝔽2)14yes—
363C22×C3C2×#12011yescharacteristic
364C22×C3C2×C22⋊Q812yes—
365C22×C3C2×C22⋊Q811yescharacteristic
366C22×C3C2×#12412yes—
367C22×C3C2×#12411yescharacteristic
368C3×C4—44no—
369C3×C4—82no—
370Dic3—48no—
371Dic3—84no—
372C23—24no—
373C23—28no—
374C23—28no—
375C23—216no—
376C23—232no—
377C23—216no—
378C23—28no—
379C23—24no—
380C23—44no—
381C23—48no—
382C23—44no—
383C23—44no—
384C23—48no—
385C23—42no—
386C23—42no—
387C23—416no—
388C23—432no—
389C23—416no—
390C23—48no—
391C23—44no—
392C23—48no—
393C23—48no—
394C23—416no—
395C23—432no—
396C23—416no—
397C23—48no—
398C23—416no—
399C23—432no—
400C23—416no—
401C23—48no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—48no—
408C23#87814yes—
409C23#87814yes—
410C23#88514yes—
411C23#88514yes—
412C23#92412yes—
413C23#92412yes—
414C23#93612yes—
415C23#93612yes—
416C23#93712yes—
417C23#93711yescharacteristic
418C23#93711yescharacteristic
419C23C22×Dic611yescharacteristic
420C23C2×#23511yescharacteristic
421C23C2×D3×D412yes—
422C23C2×#23712yes—
423C23C22×#6711yescharacteristic
424Q8—248no—
425C2×C4—24no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—68no—
431C2×C4—82no—
432C2×C4—84no—
433C2×C4—128no—
434C2×C4—128no—
435C2×C4—128no—
436C2×C4—124no—
437C2×C4—244no—
438C2×C4—248no—
439C2×C3—22no—
440C2×C3—21no—
441C2×C3—22no—
442C2×C3—21no—
443C2×C3—24no—
444C2×C3—22no—
445C2×C3—48no—
446C2×C3#202914yescomplemented
447C2×C3C2×#39311yescharacteristic
448C2×C3C2×B(4,𝔽2)12yes—
449C22—24no—
450C22—28no—
451C22—28no—
452C22—24no—
453C22—22no—
454C22—22no—
455C22—24no—
456C22—24no—
457C22—216no—
458C22—28no—
459C22—44no—
460C22—44no—
461C22—42no—
462C22—42no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—48no—
467C22—416no—
468C22—48no—
469C22—44no—
470C22—416no—
471C22—432no—
472C22—416no—
473C22—48no—
474C22—44no—
475C22#494714yes—
476C22#494712yes—
477C22#494712yes—
478C22#503214yes—
479C22C2×#87812yes—
480C22C2×#88512yes—
481C22C2×#92411yescharacteristic
482C22C2×#93611yescharacteristic
483C22C2×#93711yescharacteristic
484C4—44no—
485C4—82no—
486C4—128no—
487C4—244no—
488C3C2×#202911yescharacteristic, complemented, Sylow
489C2—21no—
490C2—22no—
491C2—22no—
492C2—21no—
493C2—22no—
494C2—24no—
495C2—48no—
496C2#6949714yesdirect product factor
497C2C2×#494711yescharacteristic
498C2C2×#503212yes—
subgroupG/H
centerC23#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#379C2
socleC23×C3#120
prime radicalC26×C3C22
hypercenterC2×#379D3
hypocenterC3C2×#2029
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C2×#5483C2semidirect
4. C2×#4527C2semidirect
5. C2×#4526C2semidirect
6. #69497C2direct (N × H)
18. #5483C22semidirect
24. #4527C22semidirect
25. #4526C22semidirect
39. C25×C3Q8semidirect
123. C24×C3C2×Q8semidirect
165. C25Dic6semidirect
294. C24C2×Dic6semidirect
446. C2×C3#2029semidirect
488. C3C2×#2029semidirect
496. C2#69497direct (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×#3792
C2×#52472
C2×#45272
C2×#45262
#694972
C2×#20293