Finite Group Databasebeta

← #10824327

Subgroups — #10824327

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D24C212yesmaximal, complemented
2C4×#874C211yesmaximal, characteristic, complemented
3#73299C211yesmaximal, characteristic, complemented
4C3×C4×C4⋊C8C211yesmaximal, characteristic, complemented
5#70484C212yesmaximal, complemented
6#70481C212yesmaximal, complemented
7C4×#821C212yesmaximal, complemented
8#68674C211yesmaximal, characteristic
9#68639C211yesmaximal, characteristic, complemented
10#68635C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#23676—31nomaximal, Sylow
13C22×D24C2212yes—
14C2×#874C2211yescharacteristic
15C2×C4×D12—21no—
16C2×C4×D12—21no—
17C2×C4×D12C2212yes—
18C2×C4×D12C2211yescharacteristic
19C2×#872—21no—
20C2×C3×C4⋊C8C2212yes—
21C2×C3×C4×C8C2212yes—
22C3×C43C2211yescharacteristic
23C2×#821C2212yes—
24C2×#821C2212yes—
25C2×#821C2212yes—
26C2×#821C2212yes—
27C2×#818C2212yes—
28#4940—21no—
29#4940—21no—
30#4939—21no—
31#4939—21no—
32#4939C2211yescharacteristic
33#4939C2211yescharacteristic
34#4939C2212yes—
35C4×(C2×D3)⋊C4—21no—
36#4935C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38#4928C2211yescharacteristic
39#4928C2211yescharacteristic
40C4×D24—28no—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43#4551C2212yes—
44#4551C2212yes—
45C2×C4×D8—32no—
46C4×#127—31no—
47#2293—31no—
48#2083—31no—
49#2071—31no—
50#2067—32no—
51#2054—32no—
52#2041—31no—
53#2011—31no—
54C4×C4⋊C8D311yescharacteristic, complemented
55C4×#102—32no—
56C22×D12—21no—
57C22×D12—21no—
58C22×D12C2311yescharacteristic
59C22×D12C2312yes—
60C22×C4×D3—21no—
61C22×C4×D3—22no—
62C22×C3×C8C2312yes—
63C22×C3×C8C2312yes—
64C2×C3×C42D411yescharacteristic
65C2×C3×C42D411yescharacteristic
66C2×C3×C42D411yescharacteristic
67C2×C3×C42D411yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×C4×Dic3—21no—
88C2×D24—24no—
89C2×D24—24no—
90C2×D24—24no—
91#874—24no—
92C4×D12—22no—
93C4×D12—24no—
94C4×D12—24no—
95C4×D12—44no—
96C4×D12—44no—
97C4×D12D414yescomplemented
98#872—44no—
99C3×C4⋊C8—24no—
100C3×C4×C8—24no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104#821—24no—
105#821—24no—
106#821—24no—
107#821—24no—
108#818—24no—
109C22×D8—32no—
110C2×#127—31no—
111C2×C4×D4—31no—
112C2×C4×D4—32no—
113C2×C4×D4—61no—
114C2×C4×D4—61no—
115C2×#117—61no—
116C4×D8—68no—
117C2×#107—32no—
118C2×C4⋊C8C2×D312yes—
119C2×#102—32no—
120C2×#102—32no—
121C2×#102—32no—
122C2×#102—32no—
123C2×C4×C8C2×D312yes—
124#390—32no—
125#390—31no—
126#390—31no—
127#390—61no—
128#390—61no—
129#389—31no—
130#389—31no—
131#388—61no—
132#388—61no—
133#384—31no—
134C4×C4⋊C4—31no—
135C4×C22⋊C4—61no—
136C43C2×D311yescharacteristic
137#340—32no—
138#340—32no—
139#336C2×D311yescharacteristic
140#336C2×D311yescharacteristic
141C23×D3—21no—
142C23×D3—22no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2411yescharacteristic
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—24no—
161C2×D12—24no—
162C2×D12—21no—
163C2×D12—22no—
164C2×D12—42no—
165C2×D12—42no—
166C2×D12—42no—
167C2×D12—42no—
168C2×D12C2×D412yes—
169C2×D12#4114yes—
170C2×C4×D3—24no—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×C4×D3—44no—
174C2×C4×D3—44no—
175C2×C4×D3—44no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8#4114yes—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—41no—
187C3×C42D814yescomplemented
188C3×C42C2×D412yes—
189C3×C42#4112yes—
190(C2×D3)⋊C4—24no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—42no—
193(C2×D3)⋊C4—44no—
194(C2×D3)⋊C4—42no—
195(C2×D3)⋊C4—42no—
196(C2×D3)⋊C4—44no—
197(C2×D3)⋊C4—42no—
198C4⋊Dic3—21no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—42no—
204C4⋊Dic3—42no—
205C4⋊Dic3C2×D412yes—
206Dic3⋊C4—42no—
207Dic3⋊C4—42no—
208C4×Dic3—42no—
209D24—216no—
210C22×D4—32no—
211C22×D4—31no—
212C22×D4—61no—
213C22×D4—61no—
214C23×C4—61no—
215C23×C4—62no—
216C2×D8—64no—
217C2×D8—64no—
218C2×D8—64no—
219C22×C8C22×D312yes—
220C22×C8C22×D312yes—
221#127—64no—
222C4×D4—34no—
223C4×D4—64no—
224C4×D4—62no—
225C4×D4—64no—
226C4×D4—124no—
227C4×D4—124no—
228#117—124no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—61no—
245C2×C42—61no—
246C2×C42D1211yescharacteristic
247C2×C42D1211yescharacteristic
248C2×C42D1211yescharacteristic
249C2×C42D1211yescharacteristic
250C2×C42C22×D311yescharacteristic
251C2×C42C22×D311yescharacteristic
252C2×C42C22×D311yescharacteristic
253#107—64no—
254C4⋊C8—24no—
255#102—64no—
256#102—64no—
257#102—64no—
258#102—64no—
259C4×C8—24no—
260#95—61no—
261#95—61no—
262#95—61no—
263C23×C3C22×D411yescharacteristic
264C22×D3—22no—
265C22×D3—24no—
266C22×D3—42no—
267C22×D3—41no—
268C22×D3—42no—
269C22×D3—41no—
270C22×D3—41no—
271C22×D3—42no—
272C22×D3—41no—
273C22×D3—42no—
274C22×D3—42no—
275C22×D3—41no—
276C22×D3—41no—
277C22×D3—42no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4#12112yes—
290C2×C3×C4#12112yes—
291C2×C3×C4C2×D812yes—
292C2×C3×C4C2×D812yes—
293C2×C3×C4C2×D812yes—
294C2×C3×C4#13512yes—
295C2×C3×C4AGL(1,ℤ/8)12yes—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4112yes—
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4112yes—
300C2×Dic3—22no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312D12—24no—
313D12—24no—
314D12—24no—
315D12—48no—
316D12—48no—
317D12—42no—
318D12—42no—
319D12—42no—
320C4×D3—48no—
321C4×D3—88no—
322C3×C8—24no—
323C3×C8—42no—
324C24—61no—
325C24—62no—
326C2×D4—32no—
327C2×D4—34no—
328C2×D4—64no—
329C2×D4—64no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—62no—
335C2×D4—61no—
336C2×D4—62no—
337C2×D4—62no—
338C2×D4—122no—
339C2×D4—122no—
340C2×D4—122no—
341C2×D4—122no—
342C22×C4—61no—
343C22×C4—64no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—62no—
347C22×C4—122no—
348C22×C4—122no—
349C22×C4—124no—
350C22×C4—124no—
351C22×C4—124no—
352C22×C4C2×D1211yescharacteristic
353C22×C4C2×D1211yescharacteristic
354C22×C4C2×D1211yescharacteristic
355C22×C4C2×D1211yescharacteristic
356C22×C4C2×D1211yescharacteristic
357C22×C4C2×D1211yescharacteristic
358C22×C4C23×D311yescharacteristic
359D8—616no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8#23514yes—
366C4⋊C4—32no—
367C4⋊C4—61no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—62no—
372C4⋊C4—122no—
373C4⋊C4—122no—
374C4⋊C4—122no—
375C4⋊C4—122no—
376C22⋊C4—64no—
377C22⋊C4—124no—
378C22⋊C4—122no—
379C22⋊C4—124no—
380C22⋊C4—122no—
381C22⋊C4—122no—
382C22⋊C4—124no—
383C22⋊C4—122no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—41no—
389C42—122no—
390C42D2414yescomplemented
391C42D3×D412yes—
392C42#23912yes—
393C22×C3C2×#12111yescharacteristic
394C22×C3#52511yescharacteristic
395C22×C3#53211yescharacteristic
396C22×C3C22×D811yescharacteristic
397C22×C3C2×#13511yescharacteristic
398C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
399C22×C3#57511yescharacteristic
400C2×D3—44no—
401C2×D3—44no—
402C2×D3—42no—
403C2×D3—44no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—82no—
410C2×D3—82no—
411C2×D3—81no—
412C2×D3—82no—
413C2×D3—82no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—41no—
419C3×C4#45914yes—
420C3×C4#52512yes—
421C3×C4#52912yes—
422Dic3—42no—
423Dic3—81no—
424Dic3—81no—
425Dic3—82no—
426C23—62no—
427C23—64no—
428C23—122no—
429C23—121no—
430C23—121no—
431C23—122no—
432C23—121no—
433C23—122no—
434C23—122no—
435C23—121no—
436C23—121no—
437C23—122no—
438C23—122no—
439C23—121no—
440C23C22×D1211yescharacteristic
441D4—64no—
442D4—64no—
443D4—64no—
444D4—128no—
445D4—128no—
446D4—122no—
447D4—122no—
448D4—122no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—62no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—128no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—122no—
468C2×C4—121no—
469C2×C4—122no—
470C2×C4—121no—
471C2×C4—122no—
472C2×C4—121no—
473C2×C4—248no—
474C2×C4#89512yes—
475C2×C4#89512yes—
476C2×C4C2×D2412yes—
477C2×C4C2×D2412yes—
478C2×C4C2×D2412yes—
479C2×C4#90412yes—
480C2×C4#90812yes—
481C2×C4C2×#23512yes—
482C2×C4C2×D3×D411yescharacteristic
483C2×C4C2×#23911yescharacteristic
484C2×C4D3×#4112yes—
485C8—24no—
486C8—42no—
487C2×C3#239211yescharacteristic
488C2×C3C2×#45911yescharacteristic
489C2×C3#303511yescharacteristic
490C2×C3#310011yescharacteristic
491C2×C3#310511yescharacteristic
492C2×C3#316911yescharacteristic
493C2×C3#317111yescharacteristic
494D3—84no—
495D3—82no—
496C22—122no—
497C22—124no—
498C22—122no—
499C22—122no—
500C22—124no—
501C22—124no—
502C22—241no—
503C22—242no—
504C22—242no—
505C22—242no—
506C22—241no—
507C22—242no—
508C22—241no—
509C22—241no—
510C22#547711yescharacteristic
511C22C2×#89511yescharacteristic
512C22#552411yescharacteristic
513C22C22×D2411yescharacteristic
514C22C2×#90411yescharacteristic
515C22C2×#90811yescharacteristic
516C22#574811yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—41no—
522C4—122no—
523C4—241no—
524C4—242no—
525C4—241no—
526C4#484314yes—
527C4#554712yes—
528C4#558012yes—
529C3#2367611yescharacteristic, complemented, Sylow
530C2—242no—
531C2—244no—
532C2#7377611yescharacteristic
533C2#7847011yescharacteristic
534C2#7853211yescharacteristic
535C2C2×#484311yescharacteristic
536C2#7902111yescharacteristic
537C2#7929711yescharacteristic
538C2#7930011yescharacteristic
subgroupG/H
centerC22×C4C2×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊C8C2
socleC23×C3C22×D4
prime radical#68674C2
hypercenterC4×C4⋊C8D3
hypocenterC3#23676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×C4×D24C2semidirect
2. C4×#874C2semidirect
3. #73299C2semidirect
4. C3×C4×C4⋊C8C2semidirect
5. #70484C2semidirect
6. #70481C2semidirect
7. C4×#821C2semidirect
9. #68639C2semidirect
10. #68635C2semidirect
11. #68629C2semidirect
54. C4×C4⋊C8D3semidirect
97. C4×D12D4semidirect
187. C3×C42D8semidirect
390. C42D24semidirect
529. C3#23676semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D242
C4×#8742
#732992
C3×C4×C4⋊C82
#704842
#704812
C4×#8212
#686742
#686392
#686352
#686292
#236763