Finite Group Databasebeta

← #10826950

Subgroups — #10826950

Order 768.

1908 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C22×#924C211yesmaximal, characteristic
3C2×#5228C212yesmaximal, complemented
4C2×#5050C212yesmaximal, complemented
5#72126C211yesmaximal, characteristic
6#71700C214yesmaximal, complemented
7#71692C212yesmaximal, complemented
8#71559C212yesmaximal, complemented
9#23921—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C2×C3×C22⋊Q8—21no—
14C2×C3×C22⋊Q8—21no—
15C2×C3×C22⋊Q8—28no—
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×C22⋊Q8C2212yes—
18C22×C3×C4⋊C4—21no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4—21no—
21C22×#217—21no—
22C22×#216C2214yescomplemented
23C22×C4⋊Dic3C2211yescharacteristic
24C2×#924—21no—
25C2×#924—21no—
26C2×#924—28no—
27C2×#924C2212yes—
28C2×#924C2212yes—
29C22×Dic3⋊C4—21no—
30#5228—28no—
31C2×#835C2214yescomplemented
32C2×#830C2212yes—
33C2×#830C2212yes—
34C2×#830C2212yes—
35#5050—28no—
36C2×#810C2214yescomplemented
37#4529C2214yescomplemented
38C22×C22⋊Q8—31no—
39C22×C22⋊Q8D311yescharacteristic
40C2×#451—32no—
41C2×#451—32no—
42#2001—34no—
43#1872—32no—
44#1872—32no—
45#1779—31no—
46C25×C3D411yescharacteristic
47C22×C3×Q8—24no—
48C22×C3×Q8—21no—
49C22×C3×Q8—24no—
50C22×C3×Q8—42no—
51C22×C3×Q8C2312yes—
52C23×C3×C4—21no—
53C23×C3×C4—21no—
54C23×C3×C4—24no—
55C23×C3×C4—24no—
56C23×C3×C4—42no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—21no—
64C23×Dic3—21no—
65C22×Dic6—21no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6—42no—
69C22×Dic6C2312yes—
70C3×C22⋊Q8—28no—
71C3×C22⋊Q8—28no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8D418yes—
76C2×C3×C4⋊C4—28no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—28no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×#217—28no—
86C2×#217—22no—
87C2×#217—22no—
88C2×#216—28no—
89C2×#216—28no—
90C2×#216—44no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3C2312yes—
94#924—28no—
95#924—28no—
96#924—24no—
97#924—24no—
98#924—24no—
99#924D418yes—
100C2×Dic3⋊C4—28no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C22×C3⋊C8C2314yes—
104#835—28no—
105#830—24no—
106#830—24no—
107#830—42no—
108#810—28no—
109C23×Q8—31no—
110C23×Q8C2×D311yescharacteristic
111C24×C4C2×D311yescharacteristic
112C22×Q16—34no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—28no—
116C2×C22⋊Q8—32no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—68no—
119C2×C22⋊Q8—61no—
120C2×C22⋊Q8—61no—
121C2×C22⋊Q8C2×D312yes—
122C2×C22⋊Q8C2×D312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4—61no—
126C22×C4⋊C4C2×D311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—61no—
129#451—68no—
130#451—68no—
131C2×#103—34no—
132C2×#103—34no—
133C2×C22⋊C8—32no—
134C2×C22⋊C8—32no—
135C2×C22⋊C8—32no—
136#340—34no—
137C24×C3—22no—
138C24×C3—22no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—41no—
142C24×C3—41no—
143C24×C3Q1614yescomplemented
144C24×C3C2×D411yescharacteristic
145C24×C3C2×D411yescharacteristic
146C24×C3C2×D411yescharacteristic
147C2×C3×Q8—24no—
148C2×C3×Q8—22no—
149C2×C3×Q8—24no—
150C2×C3×Q8—24no—
151C2×C3×Q8—416no—
152C2×C3×Q8—42no—
153C2×C3×Q8—42no—
154C2×C3×Q8C2×D414yes—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—41no—
173C22×C3×C4—41no—
174C22×C3×C4—41no—
175C22×C3×C4—416no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D414yes—
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—24no—
187C22×Dic3—24no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—42no—
191C22×Dic3—42no—
192C22×Dic3—42no—
193C22×Dic3—42no—
194C2×Dic6—24no—
195C2×Dic6—24no—
196C2×Dic6—24no—
197C2×Dic6—22no—
198C2×Dic6—416no—
199C2×Dic6—42no—
200C2×Dic6—42no—
201C2×Dic6C2×D414yes—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4C2×D414yes—
207C3×C22⋊C4—28no—
208C3×C22⋊C4—28no—
209#217—28no—
210#217—28no—
211#216—416no—
212C4⋊Dic3—24no—
213C4⋊Dic3—22no—
214C4⋊Dic3C2×D414yes—
215Dic3⋊C4—28no—
216Dic3⋊C4—28no—
217C2×C3⋊C8—24no—
218C2×C3⋊C8—24no—
219C2×C3⋊C8—24no—
220C25#6711yescharacteristic
221C22×Q8—24no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—32no—
225C22×Q8—42no—
226C22×Q8—64no—
227C22×Q8—61no—
228C22×Q8—64no—
229C22×Q8—122no—
230C22×Q8C22×D312yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—42no—
236C23×C4—61no—
237C23×C4—61no—
238C23×C4#6711yescharacteristic
239C23×C4#6711yescharacteristic
240C23×C4#6711yescharacteristic
241C23×C4C22×D311yescharacteristic
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244C2×Q16—68no—
245C2×Q16—68no—
246C2×Q16—124no—
247C22×C8—34no—
248C22⋊Q8—28no—
249C22⋊Q8—28no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—38no—
254C22⋊Q8—64no—
255C22⋊Q8—64no—
256C22⋊Q8—68no—
257C22⋊Q8—68no—
258C22⋊Q8—64no—
259C22⋊Q8#6718yes—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—64no—
267C2×C4⋊C4—68no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4C22×D312yes—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—68no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278#103—68no—
279#103—68no—
280C22⋊C8—64no—
281C22⋊C8—64no—
282C22⋊C8—122no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—28no—
298C23×C3—28no—
299C23×C3—44no—
300C23×C3—44no—
301C23×C3—44no—
302C23×C3—44no—
303C23×C3—44no—
304C23×C3#12012yes—
305C23×C3#12012yes—
306C23×C3C2×Q1612yes—
307C23×C3C2×Q1612yes—
308C23×C3C2×Q1612yes—
309C23×C3#13712yes—
310C23×C3#13712yes—
311C23×C3C22×D411yescharacteristic
312C3×Q8—28no—
313C3×Q8—48no—
314C3×Q8—44no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—28no—
320C2×C3×C4—28no—
321C2×C3×C4—21no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—44no—
327C2×C3×C4—44no—
328C2×C3×C4—44no—
329C2×C3×C4—48no—
330C2×C3×C4—48no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—44no—
336C2×C3×C4#12012yes—
337C2×C3×C4#12012yes—
338C2×C3×C4C22×D412yes—
339C2×C3×C4C22×D412yes—
340C2×C3×C4H(5,𝔽2)12yes—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—48no—
344C2×Dic3—48no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349Dic6—28no—
350Dic6—48no—
351Dic6—44no—
352C3⋊C8—44no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—41no—
358C24—41no—
359C24#21614yescomplemented
360C24C2×#6711yescharacteristic
361C24C2×#6711yescharacteristic
362C24C2×#6711yescharacteristic
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—34no—
368C2×Q8—416no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—64no—
372C2×Q8—62no—
373C2×Q8—64no—
374C2×Q8—64no—
375C2×Q8—1216no—
376C2×Q8—122no—
377C2×Q8—122no—
378C2×Q8C2×#6714yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—416no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—122no—
409C22×C4—122no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4D3×D414yes—
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6711yescharacteristic
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6714yes—
417C22×C4C23×D311yescharacteristic
418Q16—1216no—
419C2×C8—64no—
420C2×C8—64no—
421C2×C8—64no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—34no—
427C4⋊C4—68no—
428C4⋊C4—64no—
429C4⋊C4—68no—
430C4⋊C4—62no—
431C4⋊C4C2×#6714yes—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—68no—
435C22⋊C4—68no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—416no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3#45114yescomplemented
457C22×C3#45114yescomplemented
458C22×C3C2×#12011yescharacteristic
459C22×C3C2×#12011yescharacteristic
460C22×C3#53411yescharacteristic
461C22×C3C22×Q1611yescharacteristic
462C22×C3C2×#13711yescharacteristic
463C22×C3C2×#13711yescharacteristic
464C22×C3#57511yescharacteristic
465C3×C4—22no—
466C3×C4—22no—
467C3×C4—41no—
468C3×C4—44no—
469C3×C4—44no—
470C3×C4D4214yes—
471Dic3—44no—
472Dic3—44no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—28no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—28no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23#93712yes—
495C23#94212yes—
496C23C2×#21612yes—
497C23C2×#21612yes—
498C23C2×#21612yes—
499C23#95112yes—
500C23#95312yes—
501C23C22×#6711yescharacteristic
502Q8—28no—
503Q8—48no—
504Q8—44no—
505Q8—68no—
506Q8—128no—
507Q8—124no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—128no—
532C2×C4—128no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4#93712yes—
538C2×C4#95312yes—
539C2×C4C2×D3×D412yes—
540C2×C4#100412yes—
541C2×C4C22×#6712yes—
542C8—124no—
543C2×C3—24no—
544C2×C3—24no—
545C2×C3—42no—
546C2×C3#240311yescharacteristic
547C2×C3C2×#45111yescharacteristic
548C2×C3C2×#45111yescharacteristic
549C2×C3#300711yescharacteristic
550C2×C3#300711yescharacteristic
551C2×C3#318911yescharacteristic
552C2×C3#319511yescharacteristic
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—416no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#505014yescomplemented
574C22#522814yescomplemented
575C22C2×#93711yescharacteristic
576C22#580411yescharacteristic
577C22C2×#94211yescharacteristic
578C22C22×#21611yescharacteristic
579C22#582211yescharacteristic
580C22C2×#95111yescharacteristic
581C22C2×#95311yescharacteristic
582C4—22no—
583C4—22no—
584C4—41no—
585C4—44no—
586C4—44no—
587C4—124no—
588C4—124no—
589C4D4×#6714yes—
590C3#2392111yescharacteristic, complemented, Sylow
591C2—24no—
592C2—24no—
593C2—42no—
594C2#8273511yescharacteristic
595C2C2×#505011yescharacteristic
596C2#8274111yescharacteristic
597C2#8274211yescharacteristic
598C2#8405211yescharacteristic
599C2C2×#522811yescharacteristic
600C2#8408411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC25×C3D4
hypercenterC22×C22⋊Q8D3
hypocenterC3#23921
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#5228C2semidirect
4. C2×#5050C2semidirect
6. #71700C2semidirect
7. #71692C2semidirect
8. #71559C2semidirect
22. C22×#216C22semidirect
31. C2×#835C22semidirect
36. C2×#810C22semidirect
37. #4529C22semidirect
143. C24×C3Q16semidirect
359. C24#216semidirect
456. C22×C3#451semidirect
457. C22×C3#451semidirect
573. C22#5050semidirect
574. C22#5228semidirect
590. C3#23921semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C22×#9242
C2×#52282
C2×#50502
#721262
#717002
#716922
#715592
#239213