Finite Group Databasebeta

← #10822296

Subgroups — #10822296

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×C3×#536C211yesmaximal, characteristic
2C22×#951C211yesmaximal, characteristic
3C23×#216C211yesmaximal, characteristic, complemented
4C2×#5233C211yesmaximal, characteristic
5C2×#5225C211yesmaximal, characteristic, complemented
6#82230C211yesmaximal, characteristic, complemented
7#71793C214yesmaximal, complemented
8#71729C214yesmaximal, complemented
9#71579C211yesmaximal, characteristic, complemented
10#23425—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C3×#536—24no—
15C3×#536—24no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×#124C2212yes—
18C2×C3×#123—22no—
19C2×C3×C22⋊Q8—22no—
20C2×C3×#117C2211yescharacteristic
21C2×#951—24no—
22C2×#951—24no—
23C2×#951—24no—
24C22×#216—24no—
25C22×#216—24no—
26C22×#216—24no—
27C22×#216C2214yes—
28C22×#216C2212yes—
29C22×#214C2212yes—
30C2×#926C2211yescharacteristic
31C22×#208C2211yescharacteristic
32C23×C3⋊C8C2211yescharacteristic
33#5233—24no—
34#5225—24no—
35C2×#835C2214yes—
36C2×#835C2212yes—
37C2×#832C2212yes—
38#4973—22no—
39#4943—22no—
40#4929C2214yes—
41#4524C2214yes—
42C22×#137—31no—
43C23×Q16—31no—
44C2×#536D311yescharacteristic
45C2×#419—31no—
46C2×#416—31no—
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×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8D414yes—
64C22×C3×Q8C2312yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C23×Dic3—21no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—21no—
77C22×Dic6—22no—
78C22×Dic6D414yes—
79C22×Dic6C2312yes—
80C3×C4⋊Q8—24no—
81C3×C4⋊Q8—24no—
82C3×#124—24no—
83C3×#124—24no—
84C3×#123—216no—
85C3×C22⋊Q8—216no—
86C3×#117—22no—
87C3×#117—22no—
88C2×C3×C4⋊C4—24no—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4D412yes—
92C2×C3×C42C2312yes—
93C2×#217—21no—
94#951—232no—
95C2×#216—232no—
96C2×#216—28no—
97C2×#216—24no—
98C2×#216—28no—
99C2×#216—24no—
100C2×#216—28no—
101C2×#216—24no—
102C2×#214—24no—
103C2×#214—24no—
104C2×#214—24no—
105#926—22no—
106#926—22no—
107C2×C4⋊Dic3C2312yes—
108C2×C4×Dic3—21no—
109C2×#208—22no—
110C2×#208—22no—
111C2×#208D414yes—
112C22×C3⋊C8—22no—
113C22×C3⋊C8—22no—
114C22×C3⋊C8D414yes—
115C22×C3⋊C8C2312yes—
116C22×C3⋊C8C2312yes—
117#835—28no—
118#835—24no—
119#832—24no—
120C3⋊#95—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8—31no—
123C23×Q8C2×D311yescharacteristic
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C22×Q16—34no—
128C22×Q16—32no—
129C22×Q16—64no—
130C22×Q16—64no—
131C22×Q16—64no—
132C22×SD16—32no—
133C22×M16—31no—
134C23×C8—31no—
135#536—24no—
136#536—24no—
137C2×C4⋊Q8C2×D312yes—
138C2×#124C2×D312yes—
139C2×#123—22no—
140C2×C22⋊Q8—22no—
141C2×#117—31no—
142C2×#117C2×D311yescharacteristic
143#419—64no—
144#416—64no—
145C2×#103—32no—
146C2×#103—34no—
147C2×#102—32no—
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—24no—
159C2×C3×Q8—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—28no—
162C2×C3×Q8—28no—
163C2×C3×Q8—22no—
164C2×C3×Q8—21no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—21no—
171C2×C3×Q8—44no—
172C2×C3×Q8—42no—
173C2×C3×Q8—44no—
174C2×C3×Q8—42no—
175C2×C3×Q8C2×D414yes—
176C2×C3×Q8C2×D412yes—
177C2×C3×D4—24no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4C2×D412yes—
186C22×C3×C4—24no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—24no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—24no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—42no—
207C22×Dic3—42no—
208C2×Dic6—28no—
209C2×Dic6—28no—
210C2×Dic6—22no—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C2×Dic6—22no—
214C2×Dic6—24no—
215C2×Dic6—22no—
216C2×Dic6—42no—
217C2×Dic6—44no—
218C2×Dic6—44no—
219C2×Dic6—42no—
220C2×Dic6C2×D414yes—
221C3×C4⋊C4—216no—
222C3×C4⋊C4—24no—
223C3×C22⋊C4—216no—
224C3×C22⋊C4—24no—
225C3×C42—22no—
226C3×C42—22no—
227#217—42no—
228#216—232no—
229#216—216no—
230#214—216no—
231C4⋊Dic3—22no—
232C4⋊Dic3—22no—
233C4×Dic3—42no—
234#208—28no—
235C2×C3⋊C8—28no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8C2×D414yes—
241C2×C3⋊C8C2×D414yes—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41#6714yes—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—34no—
255C22×Q8—32no—
256C22×Q8—62no—
257C22×Q8—61no—
258C22×Q8—62no—
259C22×Q8—62no—
260C22×Q8—62no—
261C22×Q8—62no—
262C22×Q8#6714yes—
263C22×Q8C22×D312yes—
264C22×Q8C22×D311yescharacteristic
265C22×D4—21no—
266C22×D4C22×D311yescharacteristic
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271#137—632no—
272C2×Q16—68no—
273C2×Q16—64no—
274C2×Q16—68no—
275C2×Q16—64no—
276C2×Q16—632no—
277C2×Q16—68no—
278C2×Q16—64no—
279C2×SD16—64no—
280C2×SD16—64no—
281C2×SD16—64no—
282C2×M16—34no—
283C2×M16—62no—
284C2×M16—62no—
285C22×C8—34no—
286C22×C8—32no—
287C22×C8—32no—
288C22×C8—62no—
289C22×C8—62no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292#124—24no—
293#124—24no—
294#123—216no—
295C22⋊Q8—216no—
296#117—22no—
297#117—22no—
298#117—62no—
299#117—62no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4#6712yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4#6712yes—
306C2×C42—61no—
307C2×C42C22×D312yes—
308#103—64no—
309#103—68no—
310#102—64no—
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—24no—
322C3×Q8—22no—
323C3×Q8—24no—
324C3×Q8—22no—
325C3×Q8—42no—
326C3×Q8—41no—
327C3×Q8—42no—
328C3×Q8—41no—
329C3×Q8—48no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—48no—
333C3×D4—41no—
334C3×D4—41no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—28no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—48no—
344C2×C3×C4—48no—
345C2×C3×C4#12014yes—
346C2×C3×C4#12014yes—
347C2×C3×C4#12012yes—
348C2×C3×C4#12112yes—
349C2×C3×C4#12114yes—
350C2×C3×C4#12114yes—
351C2×C3×C4#12712yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D411yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×Dic3—24no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—48no—
362C2×Dic3—42no—
363C2×Dic3—44no—
364Dic6—24no—
365Dic6—24no—
366Dic6—42no—
367Dic6—42no—
368Dic6—48no—
369C3⋊C8—24no—
370C3⋊C8—24no—
371C24—21no—
372C24C2×#6711yescharacteristic
373#41—28no—
374#41—28no—
375#41—44no—
376#41—44no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—34no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—64no—
396C2×Q8—62no—
397C2×Q8—68no—
398C2×Q8—68no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×Q8—62no—
403C2×Q8—124no—
404C2×Q8—124no—
405C2×Q8—122no—
406C2×Q8—122no—
407C2×Q8C2×#6714yes—
408C2×Q8C2×#6712yes—
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×#6712yes—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—64no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4D3×D412yes—
432C22×C4D3×D412yes—
433C22×C4D3×D412yes—
434C22×C4#23712yes—
435C22×C4C2×#6714yes—
436C22×C4C2×#6712yes—
437C22×C4C2×#6712yes—
438C22×C4C2×#6711yescharacteristic
439C22×C4C23×D311yescharacteristic
440Q16—632no—
441Q16—616no—
442SD16—616no—
443M16—68no—
444C2×C8—34no—
445C2×C8—34no—
446C2×C8—68no—
447C2×C8—62no—
448C2×C8—62no—
449C2×C8—62no—
450C2×C8—62no—
451C4⋊C4—216no—
452C4⋊C4—24no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C22⋊C4—216no—
456C22⋊C4—24no—
457C22⋊C4—122no—
458C42—22no—
459C42—22no—
460C42—122no—
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—44no—
475C3×C4—44no—
476C3×C4#39214yes—
477C3×C4C2×#12011yescharacteristic
478C3×C4C2×#12011yescharacteristic
479C3×C4C2×#12111yescharacteristic
480C3×C4C2×#12111yescharacteristic
481Dic3—44no—
482Dic3—82no—
483C23—24no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—44no—
488C23#93812yes—
489C23#94012yes—
490C23#95312yes—
491C23C22×#6711yescharacteristic
492Q8—24no—
493Q8—22no—
494Q8—24no—
495Q8—22no—
496Q8—48no—
497Q8—42no—
498Q8—41no—
499Q8—42no—
500Q8—41no—
501Q8—64no—
502Q8—64no—
503Q8—128no—
504Q8—122no—
505Q8—122no—
506D4—22no—
507D4—22no—
508D4—48no—
509D4—41no—
510D4—41no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—48no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—48no—
521C2×C4—64no—
522C2×C4—128no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—124no—
527C2×C4#93714yes—
528C2×C4#93714yes—
529C2×C4#93812yes—
530C2×C4#93914yes—
531C2×C4#93914yes—
532C2×C4#94012yes—
533C2×C4#95312yes—
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×D3×D411yescharacteristic
536C2×C4C2×D3×D411yescharacteristic
537C2×C4C2×#23711yescharacteristic
538C2×C4C22×#6711yescharacteristic
539C2×C4C22×#6711yescharacteristic
540C8—64no—
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#522314yes—
556C22C2×#93811yescharacteristic
557C22C2×#94011yescharacteristic
558C22#583814yes—
559C22C2×#95311yescharacteristic
560C4—42no—
561C4—44no—
562C4—44no—
563C4—124no—
564C4—242no—
565C4#522314yes—
566C4C2×#93711yescharacteristic
567C4C2×#93711yescharacteristic
568C4C2×#93911yescharacteristic
569C4C2×#93911yescharacteristic
570C3#2342511yescharacteristic, complemented, Sylow
571C2—24no—
572C2—82no—
573C2#8340312yes—
574C2#8341212yes—
575C2C2×#522311yescharacteristic
576C2#8411712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#536C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#536D3
hypocenterC3#23425
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
3. C23×#216C2semidirect
5. C2×#5225C2semidirect
6. #82230C2semidirect
7. #71793C2semidirect
8. #71729C2semidirect
9. #71579C2semidirect
570. C3#23425semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C22×#9512
C23×#2162
C2×#52332
C2×#52252
#822302
#717932
#717292
#715792
#234253