Finite Group Databasebeta

← #10822598

Subgroups — #10822598

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
2#82121C211yesmaximal, characteristic
3#82121C212yesmaximal
4C2×C3×#342C211yesmaximal, characteristic, complemented
5C2×#4527C212yesmaximal, complemented
6#68971C214yesmaximal, complemented
7#68956C214yesmaximal, complemented
8#23468—31nomaximal, Sylow
9C23×#67C2211yescharacteristic
10C22×C3×C22⋊C4C2212yes—
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×#217C2211yescharacteristic
13C22×#217C2212yes—
14C2×#924—22no—
15#5483—22no—
16#5483—24no—
17#5483C2214yes—
18C2×#878—24no—
19#5213—22no—
20#5213—24no—
21#4958—24no—
22#4958—22no—
23#4955C2212yes—
24#4955C2214yes—
25#4943C2212yes—
26#4943C2214yes—
27C3×#342C2214yes—
28#4527C2218yes—
29C2×#543—31no—
30#2308—32no—
31#2308—31no—
32#2030—34no—
33#2029—34no—
34C2×#342—32no—
35C2×#342D311yescharacteristic, complemented
36C25×C3C2311yescharacteristic
37C24×D3C2311yescharacteristic
38C23×C3×C4—22no—
39C23×C3×C4D412yes—
40C22×#67—21no—
41C22×#67—22no—
42C22×#67—21no—
43C22×#67—22no—
44C22×#67—24no—
45C22×#67—22no—
46C22×#67Q814yescomplemented
47C23×Dic3—21no—
48C23×Dic3D414yes—
49C23×Dic3C2311yescharacteristic
50C22×Dic6—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—24no—
54C2×C3×C22⋊C4—24no—
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4D412yes—
57C2×C3×C22⋊C4C2314yes—
58C2×C3×C22⋊C4C2312yes—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—22no—
63C2×#217—24no—
64C2×#217—21no—
65C2×#217D414yes—
66C2×#217C2312yes—
67C2×#217C2314yes—
68C2×(C2×D3)⋊C4—24no—
69C2×(C2×D3)⋊C4—28no—
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×D4—31no—
79#543—34no—
80#543—64no—
81#543—62no—
82C2×C22⋊Q8—64no—
83C2×C22⋊Q8—62no—
84C22×C22⋊C4—32no—
85C22×C22⋊C4—31no—
86C22×C22⋊C4C2×D312yes—
87C22×C22⋊C4C2×D311yescharacteristic
88#386—64no—
89#386—64no—
90#386—62no—
91#386—62no—
92#381—34no—
93#381—32no—
94#379—34no—
95#379—32no—
96#342—38no—
97#342C2×D314yes—
98C24×C3—82no—
99C24×C3C2×D411yescharacteristic
100C24×C3C2×D411yescharacteristic
101C24×C3C2×D412yes—
102C24×C3C2×D412yes—
103C24×C3C2×Q812yes—
104C24×C3#4114yes—
105C24×C3#4112yes—
106C24×C3C2411yescharacteristic
107C23×D3—21no—
108C23×D3—22no—
109C23×D3—22no—
110C23×D3—21no—
111C23×D3—82no—
112C23×D3C2×Q812yescomplemented
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—24no—
116C22×C3×C4—44no—
117C22×C3×C4—82no—
118C22×C3×C4—84no—
119C22×C3×C4C2×D414yes—
120C2×#67—28no—
121C2×#67—28no—
122C2×#67—24no—
123C2×#67—24no—
124C2×#67—41no—
125C2×#67—42no—
126C2×#67—41no—
127C2×#67—42no—
128C2×#67—42no—
129C2×#67—44no—
130C2×#67—44no—
131C22×Dic3—24no—
132C22×Dic3—21no—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—21no—
136C22×Dic3—24no—
137C22×Dic3—22no—
138C22×Dic3—42no—
139C22×Dic3—84no—
140C22×Dic3—82no—
141C22×Dic3C2×D418yes—
142C22×Dic3C2×Q812yescomplemented
143C2×Dic6—28no—
144C2×Dic6—84no—
145C3×C22⋊C4—24no—
146C3×C22⋊C4—24no—
147C3×C22⋊C4—24no—
148C3×C22⋊C4—24no—
149C3×C22⋊C4—44no—
150#217—24no—
151#217—24no—
152#217—24no—
153#217—24no—
154#217—24no—
155#217—42no—
156(C2×D3)⋊C4—88no—
157(C2×D3)⋊C4—84no—
158C4⋊Dic3—28no—
159Dic3⋊C4—216no—
160C25—31no—
161C25C22×D311yescharacteristic
162C22×Q8—62no—
163C22×D4—34no—
164C22×D4—61no—
165C22×D4—61no—
166C22×D4—64no—
167C22×D4—62no—
168C22×D4—62no—
169C22×D4—62no—
170C23×C4—22no—
171C23×C4—31no—
172C23×C4—34no—
173C23×C4—61no—
174C23×C4D1212yes—
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—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—34no—
187C2×C22⋊C4—34no—
188C2×C22⋊C4—32no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—64no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—68no—
196C2×C22⋊C4—64no—
197C2×C22⋊C4D1212yes—
198C2×C22⋊C4C22×D314yes—
199C2×C22⋊C4C22×D312yes—
200#95—68no—
201#95—64no—
202C23×C3—42no—
203C23×C3—44no—
204C23×C3—42no—
205C23×C3—44no—
206C23×C3—42no—
207C23×C3—82no—
208C23×C3—84no—
209C23×C3—82no—
210C23×C3#12011yescharacteristic
211C23×C3#12012yes—
212C23×C3#12011yescharacteristic
213C23×C3C22⋊Q814yes—
214C23×C3C22⋊Q814yes—
215C23×C3C22⋊Q812yes—
216C23×C3C22⋊Q812yes—
217C23×C3#12412yes—
218C23×C3#12412yes—
219C23×C3#12414yes—
220C23×C3#12414yes—
221C23×C3C22×D411yescharacteristic
222C23×C3C22×D412yes—
223C23×C3C22×Q811yescharacteristic
224C23×C3C2×#4112yes—
225C23×C3C2×#4111yescharacteristic
226C22×D3—24no—
227C22×D3—22no—
228C22×D3—44no—
229C22×D3—48no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—42no—
233C22×D3—82no—
234C22×D3—84no—
235C22×D3—82no—
236C2×C3×C4—24no—
237C2×C3×C4—44no—
238C2×C3×C4—44no—
239C2×C3×C4—48no—
240C2×C3×C4—44no—
241C2×C3×C4—82no—
242C2×C3×C4—88no—
243#67—44no—
244#67—48no—
245#67—44no—
246C2×Dic3—28no—
247C2×Dic3—24no—
248C2×Dic3—22no—
249C2×Dic3—48no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253C2×Dic3—48no—
254C2×Dic3—44no—
255C2×Dic3—42no—
256C2×Dic3—84no—
257C2×Dic3—84no—
258Dic6—88no—
259C24—32no—
260C24—61no—
261C24—62no—
262C24—62no—
263C24—61no—
264C24—82no—
265C24—242no—
266C24C2×D1211yescharacteristic
267C24C2×D1211yescharacteristic
268C24#23514yes—
269C24D3×D412yes—
270C24D3×D412yes—
271C24#23712yes—
272C24D3×Q812yes—
273C24C23×D311yescharacteristic
274C2×Q8—68no—
275C2×Q8—244no—
276C2×D4—68no—
277C2×D4—68no—
278C2×D4—64no—
279C2×D4—64no—
280C2×D4—122no—
281C2×D4—122no—
282C2×D4—122no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—124no—
286C2×D4—124no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—32no—
291C22×C4—38no—
292C22×C4—44no—
293C22×C4—64no—
294C22×C4—61no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—61no—
298C22×C4—64no—
299C22×C4—62no—
300C22×C4—82no—
301C22×C4—84no—
302C22×C4—122no—
303C22×C4—244no—
304C22×C4—242no—
305C22×C4C2×D1214yes—
306C4⋊C4—616no—
307C4⋊C4—68no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—44no—
313C22⋊C4—64no—
314C22⋊C4—64no—
315C22⋊C4—64no—
316C22⋊C4—64no—
317C22⋊C4—64no—
318C22⋊C4—122no—
319C22⋊C4—244no—
320C22⋊C4—248no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—24no—
324C22×C3—24no—
325C22×C3—22no—
326C22×C3—42no—
327C22×C3—44no—
328C22×C3—41no—
329C22×C3—42no—
330C22×C3—41no—
331C22×C3—44no—
332C22×C3—44no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—44no—
337C22×C3#39312yes—
338C22×C3#39314yes—
339C22×C3#39312yes—
340C22×C3C2×#12011yescharacteristic
341C22×C3C2×C22⋊Q812yes—
342C22×C3C2×C22⋊Q811yescharacteristic
343C22×C3C2×#12411yescharacteristic
344C22×C3C2×#12412yes—
345C2×D3—48no—
346C2×D3—44no—
347C2×D3—42no—
348C2×D3—82no—
349C2×D3—84no—
350C2×D3—82no—
351C3×C4—44no—
352C3×C4—84no—
353Dic3—42no—
354Dic3—48no—
355Dic3—82no—
356C23—42no—
357C23—44no—
358C23—42no—
359C23—44no—
360C23—42no—
361C23—64no—
362C23—62no—
363C23—84no—
364C23—82no—
365C23—82no—
366C23—124no—
367C23—128no—
368C23—121no—
369C23—121no—
370C23—122no—
371C23—242no—
372C23—244no—
373C23—242no—
374C23#87512yes—
375C23#87512yes—
376C23#88212yes—
377C23#88211yescharacteristic
378C23#88211yescharacteristic
379C23#88514yes—
380C23#88514yes—
381C23#89614yes—
382C23#89614yes—
383C23#89712yes—
384C23#89712yes—
385C23C22×D1211yescharacteristic
386C23C2×#23512yes—
387C23C2×D3×D412yes—
388C23C2×#23711yescharacteristic
389C23C2×D3×Q811yescharacteristic
390Q8—248no—
391D4—124no—
392D4—124no—
393D4—128no—
394C2×C4—24no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—68no—
400C2×C4—64no—
401C2×C4—62no—
402C2×C4—82no—
403C2×C4—88no—
404C2×C4—128no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—122no—
408C2×C4—128no—
409C2×C4—124no—
410C2×C4—122no—
411C2×C4—244no—
412C2×C4—244no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—22no—
416C2×C3—21no—
417C2×C3—22no—
418C2×C3—24no—
419C2×C3—22no—
420C2×C3C2×#39311yescharacteristic
421C2×C3#302612yes—
422D3—82no—
423C22—22no—
424C22—22no—
425C22—24no—
426C22—22no—
427C22—24no—
428C22—44no—
429C22—42no—
430C22—42no—
431C22—41no—
432C22—41no—
433C22—42no—
434C22—44no—
435C22—44no—
436C22—44no—
437C22—42no—
438C22—42no—
439C22—128no—
440C22—124no—
441C22—122no—
442C22—244no—
443C22—242no—
444C22—242no—
445C22#467312yes—
446C22#467312yes—
447C22#467314yes—
448C22C2×#87511yescharacteristic
449C22C2×#88211yescharacteristic
450C22C2×#88512yes—
451C22C2×#89612yes—
452C22C2×#89711yescharacteristic
453C4—44no—
454C4—84no—
455C4—128no—
456C4—122no—
457C4—242no—
458C3#2346811yescharacteristic, complemented, Sylow
459C2—22no—
460C2—21no—
461C2—22no—
462C2—21no—
463C2—22no—
464C2—22no—
465C2—24no—
466C2—242no—
467C2C2×#467311yescharacteristic
468C2#7868612yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#342C2
socleC22×C3C2×#120
prime radicalC23×#67C22
hypercenterC2×#342D3
hypocenterC3#23468
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
1. C2×#5483C2semidirect
4. C2×C3×#342C2semidirect
5. C2×#4527C2semidirect
6. #68971C2semidirect
7. #68956C2semidirect
35. C2×#342D3semidirect
46. C22×#67Q8semidirect
112. C23×D3C2×Q8semidirect
142. C22×Dic3C2×Q8semidirect
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
#821212
C2×C3×#3422
C2×#45272
#689712
#689562
#234683