Finite Group Databasebeta

← #10697488

Subgroups — #10697488

Order 768.

744 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic, complemented
3#82376C211yesmaximal, characteristic, complemented
4#71764C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C2×#4484C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10229—31nomaximal, Sylow
9C22×C3×D8—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C22×#214—21no—
13C2×#924—21no—
14C2×C4×Dic6C2211yescharacteristic
15C3×#459—22no—
16C3×#459—22no—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#102—21no—
19#5221—21no—
20C2×#832—21no—
21C2×#832—22no—
22#5212—21no—
23#5075—22no—
24#5075—22no—
25#4972—21no—
26#4948—22no—
27#4935C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29C2×C4×C3⋊C8C2211yescharacteristic
30D4⋊(C3⋊C8)—22no—
31#4484—22no—
32C2×#460—31no—
33C2×#459D311yescharacteristic
34#2538—31no—
35#1957—31no—
36C2×Q8⋊C8—31no—
37C2×D4⋊C8—31no—
38#1296—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C22×Dic6D411yescharacteristic
45C2×C3×D8—24no—
46C2×C3×D8—42no—
47C2×C3×D8—42no—
48C22×C3×C8—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#127—21no—
52C3×#127D412yes—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#217—22no—
61C2×#217—22no—
62C2×#214—24no—
63C2×#214—42no—
64C2×#214—42no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3D411yescharacteristic
68C2×C4⋊Dic3D412yes—
69#924—42no—
70#924—42no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—21no—
73C2×C4×Dic3—21no—
74C22×C3⋊C8—21no—
75C22×C3⋊C8—21no—
76C4×Dic6—21no—
77C4×Dic6—21no—
78C4×Dic6—21no—
79C4×Dic6—21no—
80C3×C4⋊C8—21no—
81C3×C4⋊C8D412yes—
82C3×#102—24no—
83#832—24no—
84#832—44no—
85C3⋊#95—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88C4×C3⋊C8—21no—
89C4×C3⋊C8—21no—
90C22×SD16—61no—
91C22×D8—21no—
92C2×#127C2×D311yescharacteristic
93C2×C22⋊Q8—61no—
94C2×C4×Q8—31no—
95C2×C4×D4C2×D311yescharacteristic
96#460—62no—
97#460—62no—
98#459—22no—
99#459—22no—
100C2×C4⋊C8—31no—
101C2×C4⋊C8C2×D311yescharacteristic
102C2×#102—21no—
103C2×#102—61no—
104C2×#102—62no—
105C2×C4×C8—31no—
106#396—62no—
107#390—61no—
108#389—61no—
109#384—31no—
110#380—61no—
111Q8⋊C8—62no—
112D4⋊C8—62no—
113C24×C3—21no—
114C24×C3—41no—
115C2×C3×D4—21no—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—21no—
119C2×C3×D4—22no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C2×C3×D4—41no—
136C2×C3×D4SD1612yescomplemented
137C22×C3×C4—21no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C2×Dic6—21no—
151C2×Dic6—21no—
152C2×Dic6—21no—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C2×Dic6D812yescomplemented
156C3×D8—48no—
157C2×C3×C8—22no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42#4111yescharacteristic
168#217—44no—
169#217—82no—
170#214—48no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C4⋊Dic3—41no—
174C4⋊Dic3—41no—
175C4⋊Dic3—41no—
176C4⋊Dic3—41no—
177C4⋊Dic3—42no—
178Dic3⋊C4—44no—
179Dic3⋊C4—41no—
180Dic3⋊C4—41no—
181C4×Dic3—41no—
182C4×Dic3—41no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—41no—
186C2×C3⋊C8—41no—
187C2×C3⋊C8—41no—
188C2×C3⋊C8—41no—
189C22×Q8—31no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4#6711yescharacteristic
194C23×C4—21no—
195C2×SD16—64no—
196C2×SD16—122no—
197C2×SD16—122no—
198C2×D8—24no—
199C2×D8—42no—
200C2×D8—42no—
201C22×C8—21no—
202C22×C8—61no—
203C22×C8—61no—
204#127—21no—
205#127—21no—
206#127—21no—
207#127#6712yes—
208C22⋊Q8—122no—
209C22⋊Q8—122no—
210C4×Q8—61no—
211C4×Q8—61no—
212C4×Q8—61no—
213C4×Q8—61no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4#6711yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C42—61no—
229C2×C42C22×D311yescharacteristic
230C4⋊C8—21no—
231C4⋊C8—32no—
232C4⋊C8—61no—
233C4⋊C8#6712yes—
234#102—24no—
235#102—64no—
236#102—124no—
237C4×C8—61no—
238C4×C8—61no—
239#95—62no—
240C23×C3—22no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—42no—
248C23×C3—81no—
249C23×C3—81no—
250C23×C3—81no—
251C23×C3—81no—
252C23×C3—81no—
253C23×C3—81no—
254C23×C3#12011yescharacteristic
255C3×D4—22no—
256C3×D4—22no—
257C3×D4—44no—
258C3×D4—42no—
259C3×D4—42no—
260C3×D4—42no—
261C3×D4—41no—
262C3×D4—42no—
263C3×D4—41no—
264C3×D4—82no—
265C3×D4—81no—
266C3×D4—81no—
267C3×D4—81no—
268C3×D4—81no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4—81no—
277C2×C3×C4—81no—
278C2×C3×C4#12011yescharacteristic
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12111yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12711yescharacteristic
284C2×C3×C4C2×D811yescharacteristic
285C2×C3×C4C2×SD1611yescharacteristic
286C2×C3×C4#13511yescharacteristic
287C2×C3×C4#13511yescharacteristic
288C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
289C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
290C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
291C2×C3×C4#13711yescharacteristic
292C2×Dic3—22no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301Dic6—22no—
302Dic6—22no—
303Dic6—41no—
304Dic6—41no—
305C3×C8—42no—
306C3⋊C8—42no—
307C3⋊C8—42no—
308C24—21no—
309C24—41no—
310C2×Q8—32no—
311C2×Q8—61no—
312C2×Q8—61no—
313C2×Q8—61no—
314C2×Q8—61no—
315C2×Q8—61no—
316C2×D4—21no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4#21412yescomplemented
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—62no—
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D411yescharacteristic
350C22×C4C2×#6711yescharacteristic
351SD16—128no—
352D8—48no—
353C2×C8—22no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—62no—
357C2×C8—62no—
358C2×C8—121no—
359C2×C8—121no—
360C2×C8—121no—
361C2×C8—121no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—122no—
367C4⋊C4—124no—
368C4⋊C4—121no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C4⋊C4—121no—
373C4⋊C4—121no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—124no—
377C22⋊C4—242no—
378C42—121no—
379C42—121no—
380C42D3×D411yescharacteristic
381C42#23711yescharacteristic
382C42C2×#6711yescharacteristic
383C42C2×#6711yescharacteristic
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—82no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—161no—
395C22×C3—161no—
396C22×C3—161no—
397C22×C3—161no—
398C22×C3#39211yescharacteristic
399C22×C3#44711yescharacteristic
400C22×C3#44811yescharacteristic
401C22×C3#44911yescharacteristic
402C22×C3#44911yescharacteristic
403C22×C3#45311yescharacteristic
404C22×C3#45411yescharacteristic
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—81no—
408C3×C4#44711yescharacteristic
409C3×C4#44911yescharacteristic
410C3×C4#45011yescharacteristic
411C3×C4#45211yescharacteristic
412C3×C4#46511yescharacteristic
413C3×C4#46611yescharacteristic
414C3×C4#46811yescharacteristic
415C3×C4#46911yescharacteristic
416Dic3—42no—
417Dic3—82no—
418Dic3—81no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23#93711yescharacteristic
434Q8—62no—
435Q8—62no—
436Q8—121no—
437Q8—121no—
438D4—22no—
439D4—22no—
440D4—44no—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—41no—
445D4—42no—
446D4—41no—
447D4—82no—
448D4—81no—
449D4—81no—
450D4—81no—
451D4—81no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—62no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—122no—
470C2×C4D3×D811yescharacteristic
471C2×C4#91111yescharacteristic
472C2×C4#91111yescharacteristic
473C2×C4#91211yescharacteristic
474C2×C4#93211yescharacteristic
475C2×C4C2×#21411yescharacteristic
476C2×C4#93711yescharacteristic
477C2×C4#93811yescharacteristic
478C2×C4#93911yescharacteristic
479C2×C4#93911yescharacteristic
480C2×C4#94011yescharacteristic
481C2×C4#95011yescharacteristic
482C2×C4#95111yescharacteristic
483C2×C4#95311yescharacteristic
484C8—42no—
485C8—122no—
486C8—122no—
487C2×C3—82no—
488C2×C3—162no—
489C2×C3#162111yescharacteristic
490C2×C3#162211yescharacteristic
491C2×C3#162711yescharacteristic
492C2×C3#162811yescharacteristic
493C2×C3#199911yescharacteristic
494C2×C3#200011yescharacteristic
495C2×C3#200611yescharacteristic
496C22—42no—
497C22—42no—
498C22—42no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22—161no—
510C22#503711yescharacteristic
511C22#504011yescharacteristic
512C22#515611yescharacteristic
513C22#515811yescharacteristic
514C22#519811yescharacteristic
515C22#520111yescharacteristic
516C22#522311yescharacteristic
517C4—21no—
518C4—21no—
519C4—81no—
520C4—122no—
521C4—242no—
522C4—241no—
523C4#504111yescharacteristic
524C4#504211yescharacteristic
525C4#515611yescharacteristic
526C4#515711yescharacteristic
527C4#515811yescharacteristic
528C4#515911yescharacteristic
529C4#521811yescharacteristic
530C4#523911yescharacteristic
531C3#1022911yescharacteristic, complemented, Sylow
532C2—82no—
533C2—162no—
534C2#7101311yescharacteristic
535C2#7101411yescharacteristic
536C2#7101611yescharacteristic
537C2#7101911yescharacteristic
538C2#7168511yescharacteristic
539C2#7204311yescharacteristic
540C2#7209811yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3#120
hypercenterC2×#459D3
hypocenterC3#10229
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. C2×#5075C2semidirect
3. #82376C2semidirect
4. #71764C2semidirect
5. C2×D4⋊(C3⋊C8)C2semidirect
6. C2×#4484C2semidirect
7. #67702C2semidirect
136. C2×C3×D4SD16semidirect
155. C2×Dic6D8semidirect
337. C2×D4#214semidirect
531. C3#10229semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
C2×#50752
#823762
#717642
C2×D4⋊(C3⋊C8)2
C2×#44842
#677022
#102293