Finite Group Databasebeta

← #10823194

Subgroups — #10823194

Order 768.

1674 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#908C211yesmaximal, characteristic
2C2×#5118C211yesmaximal, characteristic, complemented
3#82228C211yesmaximal, characteristic, complemented
4C2×#5001C211yesmaximal, characteristic, complemented
5C2×#4988C211yesmaximal, characteristic, complemented
6C2×#4966C211yesmaximal, characteristic, complemented
7C3×#1736C211yesmaximal, characteristic, complemented
8#68863C214yesmaximal, complemented
9#68862C214yesmaximal, complemented
10#23546—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4—22no—
17C22×C4×Dic3C2211yescharacteristic
18C22×#208C2211yescharacteristic
19C2×#908—24no—
20C2×#908—24no—
21C2×#908—24no—
22C22×D24C2212yes—
23C22×#204C2212yes—
24#5118—24no—
25C2×Dic3⋊C8C2214yescomplemented
26#5001—24no—
27#4988—21no—
28#4988—21no—
29#4988C2214yes—
30#4988C2212yescomplemented
31#4973C2212yescomplemented
32C2×#810C2212yescomplemented
33#4966—24no—
34C2×#809C2214yes—
35C2×#809C2212yescomplemented
36#4958—22no—
37C4×(C2×D3)⋊C4—21no—
38C4×(C2×D3)⋊C4—21no—
39C3×#340C2214yescomplemented
40C22×AGL(1,ℤ/8)—31no—
41C2×#423—31no—
42C2×#417—31no—
43C2×#418—31no—
44#2297—31no—
45C2×#390—31no—
46#2055—34no—
47#2054—34no—
48#1736D311yescharacteristic, complemented
49C24×D3—21no—
50C23×C3×C4—21no—
51C23×C3×C4C2311yescharacteristic
52C22×#67—21no—
53C23×Dic3D412yes—
54C2×#235—22no—
55C2×#235—22no—
56C2×#235—22no—
57C2×#235—22no—
58C2×#235—22no—
59C2×#235D414yes—
60C22×D12—22no—
61C22×D12—22no—
62C22×D12—22no—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12—22no—
66C22×D12D414yes—
67C22×D12C2312yes—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×Dic6C2311yescharacteristic
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16D414yes—
74C22×C3×C8C2312yes—
75C3×#117—22no—
76C3×#117—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42—21no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—48no—
89C2×C4×Dic3—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3C2312yes—
94C2×#208—22no—
95C2×#208—22no—
96C2×#208—22no—
97C22×C3⋊C8C2312yes—
98#908—232no—
99C2×D24—24no—
100C2×D24—24no—
101C2×D24—24no—
102C2×#204—24no—
103C2×#204—24no—
104C2×#204—24no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107Dic3⋊C8—28no—
108#810—24no—
109#809—28no—
110#809—24no—
111C22×#41—31no—
112C23×D4—31no—
113C2×AGL(1,ℤ/8)—64no—
114C2×AGL(1,ℤ/8)—64no—
115C2×AGL(1,ℤ/8)—64no—
116C22×SD16—32no—
117C22×D8—32no—
118C22×M16—31no—
119C22×M16C2×D311yescharacteristic
120C2×#117C2×D311yescharacteristic
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C22⋊C4—62no—
123C22×C42—31no—
124#423—64no—
125C2×C4⋊C8—34no—
126#418—64no—
127#417—64no—
128C2×#103—32no—
129C2×#102—32no—
130C2×#102—34no—
131#391—32no—
132#390—32no—
133#390—34no—
134#390—61no—
135#390—61no—
136#386—62no—
137C4×C22⋊C4—61no—
138C4×C22⋊C4—61no—
139#340C2×D314yescomplemented
140C24×C3C2×D411yescharacteristic
141C23×D3—24no—
142C23×D3—21no—
143C23×D3—22no—
144C23×D3—44no—
145C23×D3—44no—
146C23×D3—42no—
147C23×D3—42no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—44no—
151C22×C3×C4—42no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4#4112yes—
157C22×C3×C4#4112yes—
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C2×#67—24no—
161C2×#67—42no—
162C2×#67—42no—
163C22×Dic3—24no—
164C22×Dic3—21no—
165C22×Dic3—24no—
166C22×Dic3—42no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169#235—28no—
170#235—28no—
171#235—44no—
172#235—44no—
173C2×D12—28no—
174C2×D12—28no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—24no—
184C2×D12—22no—
185C2×D12—21no—
186C2×D12—42no—
187C2×D12—42no—
188C2×D12—44no—
189C2×D12—44no—
190C2×D12C2×D414yes—
191C2×D12C2×D412yes—
192C2×C4×D3—24no—
193C2×C4×D3—42no—
194C2×C4×D3—42no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C2×Dic6—21no—
200C2×Dic6C2×D412yes—
201C3×M16—28no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8C2×D414yes—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—42no—
210C3×C22⋊C4—42no—
211C3×C42—42no—
212(C2×D3)⋊C4—44no—
213(C2×D3)⋊C4—48no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—48no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220#208—44no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224D24—216no—
225#204—216no—
226C25—61no—
227C2×#41—34no—
228C2×#41—62no—
229C2×#41—62no—
230C2×#41—62no—
231C2×#41—62no—
232C2×#41—62no—
233C22×Q8—31no—
234C22×D4—34no—
235C22×D4—31no—
236C22×D4—32no—
237C22×D4—61no—
238C22×D4—62no—
239C22×D4—62no—
240C22×D4—62no—
241C22×D4—62no—
242C22×D4—61no—
243C22×D4—62no—
244C23×C4—21no—
245C23×C4—32no—
246C23×C4—61no—
247C23×C4C22×D311yescharacteristic
248AGL(1,ℤ/8)—632no—
249C2×SD16—64no—
250C2×SD16—64no—
251C2×SD16—64no—
252C2×D8—64no—
253C2×D8—64no—
254C2×D8—64no—
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—62no—
258C2×M16—62no—
259C2×M16—62no—
260C2×M16D1214yes—
261C22×C8—32no—
262C22×C8C22×D312yes—
263#117—22no—
264#117—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C22×D312yes—
269C2×C4⋊C4C22×D312yes—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—128no—
276C2×C42—21no—
277C2×C42—31no—
278C2×C42—31no—
279C2×C42—32no—
280C2×C42—62no—
281C2×C42—62no—
282C4⋊C8—68no—
283#103—64no—
284#102—64no—
285#102—68no—
286#95—62no—
287#95—62no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3#12112yes—
291C23×C3#12112yes—
292C23×C3#12112yes—
293C23×C3C22×D411yescharacteristic
294C22×D3—22no—
295C22×D3—24no—
296C22×D3—416no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—44no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—44no—
305C22×D3—48no—
306C22×D3—84no—
307C22×D3—84no—
308C22×D3—88no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—44no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—84no—
315C2×C3×C4#12014yes—
316C2×C3×C4#12112yes—
317C2×C3×C4#12112yes—
318C2×C3×C4#12112yes—
319C2×C3×C4#12314yes—
320C2×C3×C4#12314yes—
321C2×C3×C4#12414yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328#67—48no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—48no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3AGL(1,ℤ/8)14yescomplemented
339D12—24no—
340D12—22no—
341D12—24no—
342D12—22no—
343D12—42no—
344D12—41no—
345D12—42no—
346D12—41no—
347D12—48no—
348C4×D3—48no—
349Dic6—22no—
350Dic6—22no—
351Dic6—41no—
352Dic6—41no—
353C3×C8—24no—
354C3⋊C8—42no—
355C24—64no—
356C24—61no—
357C24—62no—
358C24—124no—
359C24—124no—
360C24—122no—
361C24—122no—
362C24D3×D411yescharacteristic
363#41—68no—
364#41—68no—
365#41—124no—
366#41—124no—
367C2×Q8—32no—
368C2×Q8—61no—
369C2×Q8—61no—
370C2×Q8—61no—
371C2×Q8—61no—
372C2×Q8—61no—
373C2×D4—32no—
374C2×D4—34no—
375C2×D4—64no—
376C2×D4—68no—
377C2×D4—64no—
378C2×D4—68no—
379C2×D4—61no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—61no—
383C2×D4—62no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—61no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—124no—
392C2×D4—122no—
393C2×D4—124no—
394C2×D4—122no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—61no—
404C22×C4—64no—
405C22×C4—122no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4C2×D1212yes—
409C22×C4#23512yes—
410C22×C4#23512yes—
411C22×C4D3×D412yes—
412C22×C4D3×D412yes—
413C22×C4D3×D411yescharacteristic
414C22×C4#23912yes—
415C22×C4C23×D311yescharacteristic
416SD16—616no—
417D8—616no—
418M16—28no—
419M16—124no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C2×C8C2×D1214yes—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—128no—
433C22⋊C4—124no—
434C22⋊C4—128no—
435C22⋊C4—124no—
436C42—42no—
437C42—62no—
438C42—62no—
439C42—62no—
440C42—62no—
441C22×C3—24no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—24no—
445C22×C3#39414yes—
446C22×C3C2×#12111yescharacteristic
447C22×C3C2×#12111yescharacteristic
448C22×C3C2×#12111yescharacteristic
449C22×C3C2×AGL(1,ℤ/8)12yes—
450C22×C3#57512yes—
451C2×D3—42no—
452C2×D3—44no—
453C2×D3—48no—
454C2×D3—42no—
455C2×D3—42no—
456C2×D3—816no—
457C2×D3—82no—
458C2×D3—82no—
459C2×D3—82no—
460C2×D3—82no—
461C2×D3—84no—
462C3×C4—82no—
463C3×C4—82no—
464C3×C4#39414yes—
465C3×C4C2×#12011yescharacteristic
466C3×C4C2×#12311yescharacteristic
467C3×C4C2×#12311yescharacteristic
468C3×C4C2×#12411yescharacteristic
469Dic3—24no—
470Dic3—42no—
471Dic3—42no—
472C23—22no—
473C23—22no—
474C23—62no—
475C23—64no—
476C23—1216no—
477C23—128no—
478C23—121no—
479C23—122no—
480C23—121no—
481C23—122no—
482C23—124no—
483C23—122no—
484C23—122no—
485C23—124no—
486C23—244no—
487C23—248no—
488C23—244no—
489C23#88412yes—
490C23#88412yes—
491C23#89512yes—
492C23C2×D3×D411yescharacteristic
493Q8—62no—
494Q8—62no—
495Q8—121no—
496Q8—121no—
497D4—62no—
498D4—64no—
499D4—64no—
500D4—62no—
501D4—128no—
502D4—128no—
503D4—122no—
504D4—121no—
505D4—122no—
506D4—121no—
507C2×C4—34no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—84no—
520C2×C4—128no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—128no—
524C2×C4#87514yes—
525C2×C4#88214yes—
526C2×C4#88412yes—
527C2×C4#88412yes—
528C2×C4#89414yes—
529C2×C4#89414yes—
530C2×C4#89512yes—
531C2×C4C22×D1211yescharacteristic
532C2×C4C2×#23511yescharacteristic
533C2×C4C2×#23511yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×D3×D411yescharacteristic
536C2×C4C2×#23911yescharacteristic
537C8—24no—
538C8—122no—
539C2×C3—24no—
540C2×C3C2×#39411yescharacteristic
541C2×C3#303612yes—
542C2×C3#303811yescharacteristic
543C2×C3#304011yescharacteristic
544C2×C3#305512yes—
545D3—82no—
546D3—84no—
547C22—24no—
548C22—22no—
549C22—24no—
550C22—22no—
551C22—128no—
552C22—122no—
553C22—122no—
554C22—124no—
555C22—122no—
556C22—2416no—
557C22—242no—
558C22—242no—
559C22—244no—
560C22—242no—
561C22—242no—
562C22#467214yes—
563C22C2×#88411yescharacteristic
564C22C2×#88411yescharacteristic
565C22C2×#89511yescharacteristic
566C22D3×AGL(1,ℤ/8)12yes—
567C22#577712yes—
568C4—64no—
569C4—82no—
570C4—82no—
571C4—122no—
572C4—122no—
573C4#467214yes—
574C4C2×#87511yescharacteristic
575C4C2×#88211yescharacteristic
576C4C2×#89411yescharacteristic
577C4C2×#89411yescharacteristic
578C3#2354611yescharacteristic, complemented, Sylow
579C2—24no—
580C2—244no—
581C2—242no—
582C2C2×#467211yescharacteristic
583C2#7891012yes—
584C2#7894211yescharacteristic
585C2#7897511yescharacteristic
586C2#7921112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1736C2
socleC23×C3C22×D4
prime radicalC22×#908C2
hypercenter#1736D3
hypocenterC3#23546
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#5118C2semidirect
3. #82228C2semidirect
4. C2×#5001C2semidirect
5. C2×#4988C2semidirect
6. C2×#4966C2semidirect
7. C3×#1736C2semidirect
8. #68863C2semidirect
9. #68862C2semidirect
25. C2×Dic3⋊C8C22semidirect
30. #4988C22semidirect
31. #4973C22semidirect
32. C2×#810C22semidirect
35. C2×#809C22semidirect
39. C3×#340C22semidirect
48. #1736D3semidirect
139. #340C2×D3semidirect
338. C2×Dic3AGL(1,ℤ/8)semidirect
578. C3#23546semidirect

Maximal subgroups

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

subgroupindex
C22×#9082
C2×#51182
#822282
C2×#50012
C2×#49882
C2×#49662
C3×#17362
#688632
#688622
#235463