Finite Group Databasebeta

← #10690312

Subgroups — #10690312

Order 768.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#448C211yesmaximal, characteristic
2#82634C211yesmaximal, characteristic
3#72134C211yesmaximal, characteristic, complemented
4#72132C211yesmaximal, characteristic, complemented
5C2×#4537C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9039—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×SD16—21no—
11C2×C3×#121C2211yescharacteristic
12C2×D4×Dic3—21no—
13C3×#448—22no—
14C3×#448C414yescomplemented
15C2×C3×#103—21no—
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×#835—21no—
18C2×#835—21no—
19#5220—21no—
20#5220—21no—
21#5212C2211yescharacteristic
22C2×C3⋊#95—21no—
23C2×C3⋊#95—21no—
24C2×C3⋊#95C2211yescharacteristic
25C2×C3⋊#95C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#4980—21no—
28#4951—21no—
29#4950—21no—
30#4537—22no—
31C2×#448D311yescharacteristic
32#2325—31no—
33#1787—31no—
34C2×#327—31no—
35#1446—31no—
36#1294—31no—
37#1280—31no—
38C22×C3×Q8—21no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—42no—
44C22×C3×Q8—42no—
45C22×C3×Q8C2×C412yescomplemented
46C22×C3×D4—21no—
47C22×C3×D4D411yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C23×Dic3—21no—
52C23×Dic3D411yescharacteristic
53C23×Dic3D411yescharacteristic
54C2×C3×SD16—24no—
55C2×C3×SD16—42no—
56C2×C3×SD16—42no—
57C22×C3×C8—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121C2×C412yes—
61C3×#121D412yes—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×#217—21no—
65C2×#217—21no—
66C2×#217—21no—
67C2×#217—21no—
68D4×Dic3—42no—
69D4×Dic3—42no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4⋊Dic3D411yescharacteristic
73C2×C4×Dic3—21no—
74C22×C3⋊C8—21no—
75C3×#103—24no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8C2×C412yescomplemented
78#835—42no—
79#835—42no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87C3⋊#95—42no—
88C3⋊#95—42no—
89#830—21no—
90#830—21no—
91C23×Q8C2×D311yescharacteristic
92C22×SD16—21no—
93C2×#121C2×D311yescharacteristic
94C2×C4×D4—61no—
95#448—22no—
96#448Dic314yescomplemented
97C2×#103—21no—
98C2×#103—61no—
99C2×#103—61no—
100C2×C22⋊C8—31no—
101C2×C22⋊C8C2×D311yescharacteristic
102#388—61no—
103#387—61no—
104#386—61no—
105#386—61no—
106#385—61no—
107#380—31no—
108C2×#95—31no—
109C2×#95—31no—
110C2×#95—61no—
111C2×#95—61no—
112#327—62no—
113C24×C3—21no—
114C24×C3C2×D411yescharacteristic
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—44no—
122C2×C3×Q8—41no—
123C2×C3×Q8—41no—
124C2×C3×Q8—41no—
125C2×C3×Q8—41no—
126C2×C3×Q8—44no—
127C2×C3×Q8—44no—
128C2×C3×Q8—44no—
129C2×C3×Q8—41no—
130C2×C3×Q8—41no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—21no—
134C2×C3×D4—41no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4C22⋊C412yes—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4C22×C411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C22×C3×C4#4111yescharacteristic
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—42no—
166C22×Dic3—41no—
167C22×Dic3—41no—
168C22×Dic3—41no—
169C22×Dic3—41no—
170C22×Dic3—41no—
171C22×Dic3—41no—
172C22×Dic3—41no—
173C22×Dic3—41no—
174C22×Dic3—41no—
175C22×Dic3—41no—
176C22×Dic3—81no—
177C22×Dic3—81no—
178C22×Dic3SD1612yescomplemented
179C3×SD16—48no—
180C2×C3×C8—22no—
181C2×C3×C8—41no—
182C2×C3×C8—41no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4C22⋊C412yes—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187#217—42no—
188#217—41no—
189#217—41no—
190#217—81no—
191#217—81no—
192C4⋊Dic3—42no—
193C4⋊Dic3—41no—
194C4⋊Dic3—41no—
195C4×Dic3—42no—
196C2×C3⋊C8—41no—
197C2×C3⋊C8—41no—
198C2×C3⋊C8—41no—
199C22×Q8—21no—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8C2×Dic312yescomplemented
207C22×D4—21no—
208C22×D4#6711yescharacteristic
209C23×C4—21no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—61no—
213C23×C4—61no—
214C23×C4C22×D311yescharacteristic
215C2×SD16—24no—
216C2×SD16—42no—
217C2×SD16—42no—
218C22×C8—21no—
219C22×C8—61no—
220#121—21no—
221#121—21no—
222#121C2×Dic312yes—
223#121#6712yes—
224C4×D4—122no—
225C4×D4—122no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4#6711yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—61no—
233C2×C22⋊C4—61no—
234C2×C22⋊C4—61no—
235C2×C42—61no—
236#103—24no—
237#103—122no—
238#103—122no—
239C22⋊C8—21no—
240C22⋊C8—61no—
241C22⋊C8—61no—
242C22⋊C8C2×Dic312yescomplemented
243#95—61no—
244#95—61no—
245#95—61no—
246#95—61no—
247#95—61no—
248#95—61no—
249#95—61no—
250#95—122no—
251#95—122no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—22no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3C4×D411yescharacteristic
264C23×C3#12011yescharacteristic
265C23×C3#12111yescharacteristic
266C23×C3C2×SD1611yescharacteristic
267C23×C3#13511yescharacteristic
268C23×C3AGL(1,ℤ/8)11yescharacteristic
269C23×C3#13711yescharacteristic
270C3×Q8—24no—
271C3×Q8—24no—
272C3×Q8—42no—
273C3×Q8—42no—
274C3×Q8—42no—
275C3×Q8—42no—
276C3×Q8—42no—
277C3×Q8—42no—
278C3×D4—42no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—42no—
283C3×D4—81no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4C2×C22⋊C411yescharacteristic
303C2×C3×C4C4×D411yescharacteristic
304C2×C3×C4#12111yescharacteristic
305C2×C3×C4#12311yescharacteristic
306C2×Dic3—22no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—81no—
324C2×Dic3—84no—
325C2×Dic3—81no—
326C2×Dic3—81no—
327C2×Dic3—81no—
328C2×Dic3—81no—
329C2×Dic3—81no—
330C3×C8—42no—
331C3⋊C8—81no—
332C24—21no—
333C24D3×D411yescharacteristic
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—44no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4#21712yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—32no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—122no—
380C22×C4—121no—
381C22×C4—121no—
382C22×C4—121no—
383C22×C4—121no—
384C22×C4—121no—
385C22×C4—121no—
386C22×C4—121no—
387C22×C4—121no—
388C22×C4—121no—
389C22×C4—121no—
390C22×C4—241no—
391C22×C4—241no—
392C22×C4D3×D411yescharacteristic
393C22×C4#23711yescharacteristic
394C22×C4#23711yescharacteristic
395C22×C4C22×Dic311yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398SD16—48no—
399C2×C8—22no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—121no—
403C2×C8—121no—
404C2×C8—121no—
405C4⋊C4—21no—
406C4⋊C4—121no—
407C4⋊C4—121no—
408C4⋊C4—122no—
409C4⋊C4#21712yes—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—122no—
413C22⋊C4—121no—
414C22⋊C4—121no—
415C22⋊C4—241no—
416C22⋊C4—241no—
417C42—122no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3#38611yescharacteristic
438C22×C3C4×SD1611yescharacteristic
439C22×C3#44011yescharacteristic
440C22×C3#45011yescharacteristic
441C22×C3#45211yescharacteristic
442C22×C3#45511yescharacteristic
443C22×C3#45611yescharacteristic
444C22×C3B(4,𝔽2)11yescharacteristic
445C22×C3B(4,𝔽2)11yescharacteristic
446C22×C3#46111yescharacteristic
447C22×C3#46311yescharacteristic
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—42no—
453C3×C4—44no—
454Dic3—42no—
455Dic3—82no—
456Dic3—81no—
457Dic3—81no—
458Dic3—81no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23D3×SD1611yescharacteristic
471C23#91411yescharacteristic
472C23#91511yescharacteristic
473C23#91611yescharacteristic
474C23D4×Dic311yescharacteristic
475C23#93711yescharacteristic
476C23#93911yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—81no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—62no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—241no—
527C2×C4—244no—
528C2×C4—241no—
529C2×C4—241no—
530C2×C4—241no—
531C2×C4—241no—
532C2×C4—241no—
533C2×C4D4×Dic311yescharacteristic
534C2×C4#93511yescharacteristic
535C2×C4#93911yescharacteristic
536C2×C4C2×#21711yescharacteristic
537C8—42no—
538C8—241no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—82no—
544C2×C3#160111yescharacteristic
545C2×C3#160311yescharacteristic
546C2×C3#160411yescharacteristic
547C2×C3#160611yescharacteristic
548C2×C3#187711yescharacteristic
549C2×C3#188211yescharacteristic
550C2×C3#188911yescharacteristic
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22#503211yescharacteristic
571C22#503211yescharacteristic
572C22Dic3×SD1611yescharacteristic
573C22#516211yescharacteristic
574C22#516411yescharacteristic
575C22#516511yescharacteristic
576C22#516911yescharacteristic
577C22#517311yescharacteristic
578C22#520211yescharacteristic
579C22#520511yescharacteristic
580C22#522011yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—42no—
586C4—44no—
587C4—122no—
588C4—242no—
589C4—241no—
590C4—241no—
591C4—241no—
592C3#903911yescharacteristic, complemented, Sylow
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—82no—
598C2#7097211yescharacteristic
599C2#7097311yescharacteristic
600C2#7097411yescharacteristic
601C2#7097511yescharacteristic
602C2#7166911yescharacteristic
603C2#7204711yescharacteristic
604C2#7210011yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#448C2
socleC23×C3#120
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenterC2×#448D3
hypocenterC3#9039
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
3. #72134C2semidirect
4. #72132C2semidirect
6. #67675C2semidirect
7. #67674C2semidirect
14. C3×#448C4semidirect
45. C22×C3×Q8C2×C4semidirect
77. C3×C22⋊C8C2×C4semidirect
96. #448Dic3semidirect
178. C22×Dic3SD16semidirect
206. C22×Q8C2×Dic3semidirect
242. C22⋊C8C2×Dic3semidirect
592. C3#9039semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4482
#826342
#721342
#721322
C2×#45372
#676752
#676742
#90393