Finite Group Databasebeta

← #10822297

Subgroups — #10822297

Order 768.

1938 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5825C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic
3C23×Dic12C211yesmaximal, characteristic, complemented
4C2×C3×#416C211yesmaximal, characteristic
5#82230C211yesmaximal, characteristic, complemented
6C2×#5004C211yesmaximal, characteristic, complemented
7#71965C211yesmaximal, characteristic, complemented
8#70283C214yesmaximal, complemented
9#70215C214yesmaximal, complemented
10#23425—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16#5825—24no—
17#5825—24no—
18C2×#951—24no—
19C2×#951—24no—
20C2×#951—24no—
21C2×#944C2212yes—
22C22×#216C2212yes—
23C2×#936C2212yes—
24C2×#935—22no—
25C22×#214C2212yes—
26C2×#926C2211yescharacteristic
27C2×#924—22no—
28C22×#208C2211yescharacteristic
29C22×Dic12—24no—
30C22×Dic12—24no—
31C22×Dic12—24no—
32C22×Dic12C2214yes—
33C3×#416—24no—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36#5004—24no—
37#4973C2214yes—
38C2×#810C2214yes—
39#4943—22no—
40#4929—22no—
41#4550C2214yes—
42C22×#137—31no—
43C23×Q16—31no—
44C2×#536—31no—
45C2×#419—31no—
46C2×#416D311yescharacteristic
47#2310—31no—
48#2016—34no—
49#2015—34no—
50#1729—31no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41D414yes—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C23×Dic3—21no—
63C23×Dic3—21no—
64C22×Dic6—22no—
65C22×Dic6—22no—
66C22×Dic6—22no—
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—22no—
72C22×Dic6—21no—
73C22×Dic6—21no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6D414yes—
77C22×Dic6D414yes—
78C22×Dic6C2312yes—
79C22×Dic6C2312yes—
80C22×C3×C8—22no—
81C22×C3×C8—22no—
82C22×C3×C8D414yes—
83C22×C3×C8C2312yes—
84C3×#117—22no—
85C3×#117—22no—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×#217—24no—
90C2×#217D412yes—
91#951—232no—
92#944—24no—
93#944—24no—
94C2×#216—24no—
95C2×#216—24no—
96C2×#216—24no—
97#936—24no—
98#936—24no—
99#935—216no—
100C2×#214—24no—
101C2×#214—24no—
102C2×#214—24no—
103#926—22no—
104#926—22no—
105C2×C4⋊Dic3D412yes—
106#924—216no—
107C2×Dic3⋊C4—24no—
108C2×C4×Dic3C2312yes—
109C2×#208—22no—
110C2×#208—22no—
111C2×#208D414yes—
112C22×C3⋊C8C2312yes—
113C2×Dic12—232no—
114C2×Dic12—28no—
115C2×Dic12—28no—
116C2×Dic12—28no—
117C3×#103—24no—
118C3×#102—24no—
119C3⋊#95—28no—
120#810—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8—31no—
123C23×Q8—31no—
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C22×Q16—32no—
128C22×Q16—34no—
129C22×Q16—64no—
130C22×Q16—64no—
131C22×Q16—64no—
132C22×SD16—32no—
133C22×M16—31no—
134C23×C8C2×D311yescharacteristic
135#536—64no—
136#536—64no—
137C2×C4⋊Q8—32no—
138C2×#124—32no—
139C2×#123—62no—
140C2×C22⋊Q8—62no—
141C2×#117—31no—
142C2×#117C2×D311yescharacteristic
143#419—64no—
144#416—24no—
145C2×#103—34no—
146C2×#103C2×D312yes—
147C2×#102C2×D312yes—
148#391—34no—
149#391—62no—
150#381—62no—
151#336—34no—
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C3×#41—28no—
155C3×#41—28no—
156C3×#41—44no—
157C3×#41—44no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8C2×D412yes—
164C2×C3×D4—24no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4C2×D412yes—
173C22×C3×C4—21no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4#4112yes—
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—
194C22×Dic3C2×D414yes—
195C2×Dic6—28no—
196C2×Dic6—28no—
197C2×Dic6—28no—
198C2×Dic6—28no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—22no—
207C2×Dic6—24no—
208C2×Dic6—24no—
209C2×Dic6—22no—
210C2×Dic6—22no—
211C2×Dic6—44no—
212C2×Dic6—42no—
213C2×Dic6—44no—
214C2×Dic6—42no—
215C2×Dic6—44no—
216C2×Dic6—44no—
217C2×Dic6—42no—
218C2×Dic6—42no—
219C2×Dic6C2×D414yes—
220C2×Dic6C2×D414yes—
221C2×C3×C8—28no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8C2×D414yes—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C22⋊C4—42no—
228C3×C42—42no—
229#217—216no—
230#217—24no—
231#216—216no—
232#214—216no—
233C4⋊Dic3—24no—
234Dic3⋊C4—216no—
235C4×Dic3—22no—
236C4×Dic3—22no—
237#208—28no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8C2×D414yes—
241Dic12—232no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41D1214yes—
248C22×Q8—34no—
249C22×Q8—34no—
250C22×Q8—32no—
251C22×Q8—32no—
252C22×Q8—62no—
253C22×Q8—62no—
254C22×Q8—61no—
255C22×Q8—61no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8—62no—
260C22×Q8—62no—
261C22×Q8—62no—
262C22×Q8—62no—
263C22×Q8—62no—
264C22×Q8C22×D311yescharacteristic
265C22×D4—21no—
266C22×D4C22×D311yescharacteristic
267C23×C4—21no—
268C23×C4—61no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271#137—632no—
272C2×Q16—68no—
273C2×Q16—68no—
274C2×Q16—632no—
275C2×Q16—64no—
276C2×Q16—64no—
277C2×Q16—64no—
278C2×Q16—68no—
279C2×SD16—64no—
280C2×SD16—64no—
281C2×SD16—64no—
282C2×M16—34no—
283C2×M16—62no—
284C2×M16—62no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—32no—
288C22×C8#6714yes—
289C22×C8C22×D312yes—
290C4⋊Q8—64no—
291C4⋊Q8—64no—
292#124—64no—
293#124—64no—
294#123—616no—
295C22⋊Q8—616no—
296#117—22no—
297#117—22no—
298#117—62no—
299#117—62no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—64no—
302C2×C4⋊C4C22×D312yes—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—32no—
305C2×C22⋊C4—64no—
306C2×C42—21no—
307C2×C42—32no—
308#103—24no—
309#103—68no—
310#102—24no—
311#95—68no—
312C23×C3—24no—
313C23×C3—22no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—44no—
317C23×C3#12012yes—
318C23×C3#12112yes—
319C23×C3#12712yes—
320C23×C3C22×D411yescharacteristic
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—41no—
324C3×Q8—41no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—48no—
328C3×D4—41no—
329C3×D4—41no—
330C2×C3×C4—22no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—48no—
334C2×C3×C4—42no—
335C2×C3×C4—44no—
336C2×C3×C4#12014yes—
337C2×C3×C4#12012yes—
338C2×C3×C4#12014yes—
339C2×C3×C4#12114yes—
340C2×C3×C4#12112yes—
341C2×C3×C4#12114yes—
342C2×C3×C4#12712yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—28no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—48no—
358C2×Dic3—48no—
359Dic6—24no—
360Dic6—24no—
361Dic6—24no—
362Dic6—24no—
363Dic6—42no—
364Dic6—42no—
365Dic6—42no—
366Dic6—42no—
367Dic6—48no—
368Dic6—48no—
369C3×C8—24no—
370C3⋊C8—24no—
371C24—21no—
372C24D3×D411yescharacteristic
373#41—28no—
374#41—28no—
375#41—44no—
376#41—44no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—34no—
383C2×Q8—34no—
384C2×Q8—64no—
385C2×Q8—64no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8—68no—
389C2×Q8—68no—
390C2×Q8—68no—
391C2×Q8—68no—
392C2×Q8—62no—
393C2×Q8—62no—
394C2×Q8—62no—
395C2×Q8—62no—
396C2×Q8—62no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×Q8—62no—
400C2×Q8—124no—
401C2×Q8—122no—
402C2×Q8—124no—
403C2×Q8—122no—
404C2×Q8—124no—
405C2×Q8—124no—
406C2×Q8—122no—
407C2×Q8—122no—
408C2×Q8C2×D1212yes—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C2×D1212yes—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—34no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4C2×D1212yes—
433C22×C4#23512yes—
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D412yes—
436C22×C4D3×D412yes—
437C22×C4D3×D412yes—
438C22×C4C2×#6712yes—
439C22×C4C23×D311yescharacteristic
440Q16—632no—
441Q16—616no—
442SD16—616no—
443M16—68no—
444C2×C8—28no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—34no—
448C2×C8—62no—
449C2×C8—62no—
450C2×C8C2×#6714yes—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—616no—
454C4⋊C4—64no—
455C22⋊C4—42no—
456C22⋊C4—616no—
457C22⋊C4—64no—
458C42—42no—
459C42—62no—
460C42—62no—
461C22×C3—24no—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3—42no—
465C22×C3—44no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3#39214yes—
469C22×C3C2×#12011yescharacteristic
470C22×C3C2×#12111yescharacteristic
471C22×C3C2×#12711yescharacteristic
472C22×C3#57814yes—
473C3×C4—42no—
474C3×C4—82no—
475C3×C4#39214yes—
476C3×C4C2×#12011yescharacteristic
477C3×C4C2×#12011yescharacteristic
478C3×C4C2×#12111yescharacteristic
479C3×C4C2×#12111yescharacteristic
480Dic3—44no—
481Dic3—44no—
482Dic3—44no—
483C23—24no—
484C23—22no—
485C23—41no—
486C23—44no—
487C23—41no—
488C23#88212yes—
489C23#88412yes—
490C23#94012yes—
491C23C2×D3×D411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496Q8—64no—
497Q8—64no—
498Q8—64no—
499Q8—64no—
500Q8—128no—
501Q8—128no—
502Q8—122no—
503Q8—122no—
504Q8—122no—
505Q8—122no—
506D4—22no—
507D4—22no—
508D4—48no—
509D4—41no—
510D4—41no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—68no—
521C2×C4—128no—
522C2×C4—128no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4#88212yes—
528C2×C4#88214yes—
529C2×C4#88412yes—
530C2×C4#88414yes—
531C2×C4#93014yes—
532C2×C4#93714yes—
533C2×C4#94012yes—
534C2×C4C22×D1211yescharacteristic
535C2×C4C2×#23511yescharacteristic
536C2×C4C2×D3×D411yescharacteristic
537C2×C4C2×D3×D411yescharacteristic
538C2×C4C2×D3×D411yescharacteristic
539C2×C4C22×#6711yescharacteristic
540C8—24no—
541C8—64no—
542C2×C3—24no—
543C2×C3—82no—
544C2×C3C2×#39211yescharacteristic
545C2×C3#301112yes—
546C2×C3#304412yes—
547C2×C3#315612yes—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—42no—
552C22—44no—
553C22—82no—
554C22—82no—
555C22#496014yes—
556C22C2×#88211yescharacteristic
557C22C2×#88411yescharacteristic
558C22#577114yes—
559C22C2×#94011yescharacteristic
560C4—42no—
561C4—82no—
562C4—124no—
563C4—124no—
564C4—124no—
565C4#496014yes—
566C4C2×#88211yescharacteristic
567C4C2×#88411yescharacteristic
568C4C2×#93011yescharacteristic
569C4C2×#93711yescharacteristic
570C3#2342511yescharacteristic, complemented, Sylow
571C2—24no—
572C2—82no—
573C2#7883712yes—
574C2#7886512yes—
575C2C2×#496011yescharacteristic
576C2#8390712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#416C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#416D3
hypocenterC3#23425
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
3. C23×Dic12C2semidirect
5. #82230C2semidirect
6. C2×#5004C2semidirect
7. #71965C2semidirect
8. #70283C2semidirect
9. #70215C2semidirect
570. C3#23425semidirect

Maximal subgroups

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

subgroupindex
C2×#58252
C22×#9512
C23×Dic122
C2×C3×#4162
#822302
C2×#50042
#719652
#702832
#702152
#234253