Finite Group Databasebeta

← #10822597

Subgroups — #10822597

Order 768.

1506 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5483C211yesmaximal, characteristic, complemented
2C3×#2308C211yesmaximal, characteristic
3#84043C212yesmaximal
4C2×#4527C212yesmaximal, complemented
5C2×#4526C211yesmaximal, characteristic, complemented
6#69536C214yesmaximal, complemented
7#69497C214yesmaximal, complemented
8#23468—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×#217C2212yes—
13C22×#217C2212yes—
14C2×#924—22no—
15#5483—22no—
16#5483—24no—
17#5483C2214yes—
18C2×#878—24no—
19C3×#386—22no—
20C3×#386—22no—
21C3×#381C2212yes—
22C3×#379C2212yes—
23#5247C2214yes—
24#5220—24no—
25#5220—24no—
26#5211C2214yes—
27#4527C2218yes—
28#4526C2214yes—
29C2×#543—31no—
30#2308—32no—
31#2308D311yescharacteristic
32#2030—34no—
33#2029—34no—
34C2×#342—31no—
35C2×#342—32no—
36C25×C3C2311yescharacteristic
37C25×C3C2311yescharacteristic
38C22×C3×D4—22no—
39C22×C3×D4—22no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4—22no—
43C22×C3×D4—24no—
44C22×C3×D4Q814yescomplemented
45C23×C3×C4—21no—
46C23×C3×C4D412yes—
47C23×C3×C4C2311yescharacteristic
48C23×Dic3—22no—
49C23×Dic3D414yes—
50C22×Dic6—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—24no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C22⋊C4D412yes—
57C2×C3×C22⋊C4C2312yes—
58C2×C3×C22⋊C4C2312yes—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—22no—
63C2×#217—24no—
64C2×#217—24no—
65C2×#217—22no—
66C2×#217—28no—
67C2×#217D414yes—
68C2×#217C2314yes—
69C2×#217C2314yes—
70C2×C4⋊Dic3—22no—
71#924—28no—
72#924—28no—
73C2×Dic3⋊C4—24no—
74#878—216no—
75#878—216no—
76C3×#95—24no—
77C3⋊#95—28no—
78C23×D4C2×D311yescharacteristic
79#543—34no—
80#543—64no—
81#543—62no—
82C2×C22⋊Q8—64no—
83C2×C22⋊Q8—62no—
84C22×C22⋊C4—32no—
85C22×C22⋊C4—32no—
86C22×C22⋊C4C2×D311yescharacteristic
87C22×C22⋊C4C2×D311yescharacteristic
88#386—22no—
89#386—22no—
90#386—64no—
91#386—64no—
92#381—34no—
93#381C2×D312yes—
94#379—34no—
95#379C2×D312yes—
96#342—34no—
97#342—38no—
98C24×C3—22no—
99C24×C3—21no—
100C24×C3—21no—
101C24×C3—22no—
102C24×C3—82no—
103C24×C3—82no—
104C24×C3C2×D412yes—
105C24×C3C2×D412yes—
106C24×C3C2×D411yescharacteristic
107C24×C3C2×D411yescharacteristic
108C24×C3C2×Q812yes—
109C24×C3C2×Q812yescomplemented
110C24×C3#4112yes—
111C24×C3#4114yes—
112C24×C3C2411yescharacteristic
113C2×C3×D4—28no—
114C2×C3×D4—28no—
115C2×C3×D4—24no—
116C2×C3×D4—24no—
117C2×C3×D4—41no—
118C2×C3×D4—42no—
119C2×C3×D4—41no—
120C2×C3×D4—44no—
121C2×C3×D4—44no—
122C2×C3×D4—42no—
123C2×C3×D4—42no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—42no—
132C22×C3×C4—82no—
133C22×C3×C4—82no—
134C22×C3×C4C2×D414yes—
135C22×C3×C4C2×Q812yescomplemented
136C22×Dic3—24no—
137C22×Dic3—24no—
138C22×Dic3—24no—
139C22×Dic3—44no—
140C22×Dic3—84no—
141C22×Dic3—84no—
142C22×Dic3C2×D418yes—
143C2×Dic6—28no—
144C2×Dic6—84no—
145C3×C22⋊C4—24no—
146C3×C22⋊C4—24no—
147C3×C22⋊C4—24no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—84no—
150#217—24no—
151#217—24no—
152#217—24no—
153#217—24no—
154#217—24no—
155#217—24no—
156#217—44no—
157#217—88no—
158C4⋊Dic3—28no—
159Dic3⋊C4—216no—
160C25C22×D311yescharacteristic
161C25C22×D311yescharacteristic
162C22×Q8—62no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—24no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4Dic614yescomplemented
170C23×C4—21no—
171C23×C4—34no—
172C23×C4—62no—
173C23×C4#6712yes—
174C23×C4C22×D311yescharacteristic
175C22⋊Q8—616no—
176C22⋊Q8—616no—
177C22⋊Q8—68no—
178C22⋊Q8—68no—
179C2×C4⋊C4—64no—
180C2×C4⋊C4—62no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—34no—
187C2×C22⋊C4—34no—
188C2×C22⋊C4—34no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—64no—
194C2×C22⋊C4—64no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—68no—
197C2×C22⋊C4#6712yes—
198C2×C22⋊C4C22×D312yes—
199C2×C22⋊C4C22×D312yes—
200#95—24no—
201#95—68no—
202C23×C3—24no—
203C23×C3—22no—
204C23×C3—44no—
205C23×C3—44no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—48no—
210C23×C3—44no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—82no—
215C23×C3—82no—
216C23×C3—82no—
217C23×C3—84no—
218C23×C3—82no—
219C23×C3—84no—
220C23×C3#12011yescharacteristic
221C23×C3#12012yes—
222C23×C3#12011yescharacteristic
223C23×C3C22⋊Q814yes—
224C23×C3C22⋊Q814yes—
225C23×C3C22⋊Q812yes—
226C23×C3C22⋊Q812yes—
227C23×C3#12414yes—
228C23×C3#12414yes—
229C23×C3#12412yes—
230C23×C3#12412yes—
231C23×C3C22×D412yes—
232C23×C3C22×D411yescharacteristic
233C23×C3C22×Q811yescharacteristic
234C23×C3C2×#4111yescharacteristic
235C23×C3C2×#4112yes—
236C3×D4—44no—
237C3×D4—48no—
238C3×D4—44no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—44no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—44no—
247C2×C3×C4—44no—
248C2×C3×C4—42no—
249C2×C3×C4—82no—
250C2×C3×C4—84no—
251C2×Dic3—28no—
252C2×Dic3—48no—
253C2×Dic3—48no—
254C2×Dic3—48no—
255C2×Dic3—44no—
256C2×Dic3—84no—
257C2×Dic3—88no—
258Dic6—88no—
259C24—22no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—82no—
264C24—82no—
265C24C2×Dic612yes—
266C24C2×Dic612yescomplemented
267C24#23512yes—
268C24D3×D412yes—
269C24D3×D412yes—
270C24#23714yes—
271C24C2×#6711yescharacteristic
272C24C2×#6711yescharacteristic
273C24C23×D311yescharacteristic
274C2×Q8—68no—
275C2×Q8—244no—
276C2×D4—28no—
277C2×D4—28no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—38no—
295C22×C4—42no—
296C22×C4—64no—
297C22×C4—64no—
298C22×C4—64no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—124no—
302C22×C4—244no—
303C22×C4—244no—
304C22×C4C2×Dic612yescomplemented
305C22×C4C2×#6714yes—
306C4⋊C4—616no—
307C4⋊C4—68no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—42no—
312C22⋊C4—64no—
313C22⋊C4—64no—
314C22⋊C4—64no—
315C22⋊C4—64no—
316C22⋊C4—64no—
317C22⋊C4—64no—
318C22⋊C4—84no—
319C22⋊C4—124no—
320C22⋊C4—248no—
321C22×C3—22no—
322C22×C3—24no—
323C22×C3—24no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—44no—
327C22×C3—42no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—44no—
332C22×C3—44no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—44no—
336C22×C3—42no—
337C22×C3—48no—
338C22×C3—44no—
339C22×C3—42no—
340C22×C3—84no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3#39312yes—
344C22×C3#39314yes—
345C22×C3#39312yes—
346C22×C3C2×#12011yescharacteristic
347C22×C3C2×C22⋊Q812yes—
348C22×C3C2×C22⋊Q811yescharacteristic
349C22×C3C2×#12412yes—
350C22×C3C2×#12411yescharacteristic
351C3×C4—42no—
352C3×C4—44no—
353C3×C4—82no—
354Dic3—48no—
355Dic3—84no—
356C23—24no—
357C23—22no—
358C23—42no—
359C23—42no—
360C23—44no—
361C23—44no—
362C23—42no—
363C23—44no—
364C23—48no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—84no—
372C23—82no—
373C23—84no—
374C23#87814yes—
375C23#87814yes—
376C23#88514yes—
377C23#88514yes—
378C23#92412yes—
379C23#92412yes—
380C23#93612yes—
381C23#93612yes—
382C23#93712yes—
383C23#93711yescharacteristic
384C23#93711yescharacteristic
385C23C22×Dic611yescharacteristic
386C23C2×#23511yescharacteristic
387C23C2×D3×D412yes—
388C23C2×#23712yes—
389C23C22×#6711yescharacteristic
390Q8—248no—
391D4—44no—
392D4—48no—
393D4—44no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—68no—
405C2×C4—82no—
406C2×C4—84no—
407C2×C4—128no—
408C2×C4—128no—
409C2×C4—128no—
410C2×C4—124no—
411C2×C4—244no—
412C2×C4—248no—
413C2×C3—22no—
414C2×C3—21no—
415C2×C3—22no—
416C2×C3—21no—
417C2×C3—22no—
418C2×C3—24no—
419C2×C3—22no—
420C2×C3—82no—
421C2×C3C2×#39311yescharacteristic
422C2×C3#302612yes—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—24no—
427C22—24no—
428C22—42no—
429C22—44no—
430C22—41no—
431C22—41no—
432C22—42no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—48no—
440C22—44no—
441C22—42no—
442C22—82no—
443C22—84no—
444C22—82no—
445C22#494712yes—
446C22#494714yes—
447C22#494712yes—
448C22C2×#87812yes—
449C22C2×#88512yes—
450C22C2×#92411yescharacteristic
451C22C2×#93611yescharacteristic
452C22C2×#93711yescharacteristic
453C4—44no—
454C4—42no—
455C4—82no—
456C4—128no—
457C4—244no—
458C3#2346811yescharacteristic, complemented, Sylow
459C2—21no—
460C2—22no—
461C2—22no—
462C2—21no—
463C2—22no—
464C2—22no—
465C2—24no—
466C2—82no—
467C2C2×#494711yescharacteristic
468C2#8337512yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2308C2
socleC22×C3C2×#120
prime radicalC23×C3×D4C22
hypercenter#2308D3
hypocenterC3#23468
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #2308

Complements

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

NH ≅ G/Ntype
1. C2×#5483C2semidirect
4. C2×#4527C2semidirect
5. C2×#4526C2semidirect
6. #69536C2semidirect
7. #69497C2semidirect
44. C22×C3×D4Q8semidirect
109. C24×C3C2×Q8semidirect
135. C22×C3×C4C2×Q8semidirect
169. C22×D4Dic6semidirect
266. C24C2×Dic6semidirect
304. C22×C4C2×Dic6semidirect
458. C3#23468semidirect

Maximal subgroups

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

subgroupindex
C2×#54832
C3×#23082
#840432
C2×#45272
C2×#45262
#695362
#694972
#234683