Finite Group Databasebeta

← #10690340

Subgroups — #10690340

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#480C211yesmaximal, characteristic
2#82634C211yesmaximal, characteristic
3C2×#4537C211yesmaximal, characteristic
4#71569C211yesmaximal, characteristic, complemented
5#71553C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9043—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×D4×Dic3—21no—
12C3×#480—22no—
13C3×#480C414yescomplemented
14C2×C3×#107—21no—
15C2×C3×#102—21no—
16C2×C3×C22⋊C8C2211yescharacteristic
17#5220—21no—
18#5220—21no—
19#5212C2211yescharacteristic
20C2×C3⋊#95—21no—
21C2×C3⋊#95—21no—
22C2×C3⋊#95C2211yescharacteristic
23C2×C3⋊#95C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#4980—21no—
26#4951—21no—
27#4950—21no—
28#4537—22no—
29#4529—21no—
30#4529—21no—
31C2×#480D311yescharacteristic
32#2325—31no—
33#1823—31no—
34C2×#327—31no—
35#1445—31no—
36#1294—31no—
37#1280—31no—
38C22×C3×D4—21no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C23×Dic3D411yescharacteristic
46C22×C3×C8—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121C2×C412yes—
50C3×#121D412yes—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4C2×C412yescomplemented
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—21no—
63D4×Dic3—42no—
64D4×Dic3—42no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D411yescharacteristic
67C2×C4⋊Dic3D411yescharacteristic
68C2×C4×Dic3—21no—
69C22×C3⋊C8—21no—
70C3×#107—24no—
71C3×#102—24no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8C2×C412yescomplemented
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—42no—
82C3⋊#95—42no—
83#830—21no—
84#830—21no—
85C2×#121C2×D311yescharacteristic
86C2×C4×D4—61no—
87C22×C4⋊C4C2×D311yescharacteristic
88#480—22no—
89#480Dic314yescomplemented
90C2×#107—21no—
91C2×#102—21no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8C2×D311yescharacteristic
94#388—61no—
95#387—61no—
96#386—61no—
97#386—61no—
98#385—61no—
99#380—31no—
100C2×#95—31no—
101C2×#95—31no—
102C2×#95—61no—
103C2×#95—61no—
104#340—61no—
105#340—61no—
106#327—62no—
107C24×C3—21no—
108C24×C3C2×D411yescharacteristic
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4C22⋊C412yes—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—42no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3—81no—
155C22×Dic3—81no—
156C22×Dic3D812yes—
157C2×C3×C8—22no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—41no—
165C3×C4⋊C4C22⋊C412yes—
166C3×C22⋊C4—22no—
167C3×C22⋊C4—22no—
168#217—42no—
169#217—41no—
170#217—41no—
171#217—81no—
172#217—81no—
173C4⋊Dic3—42no—
174C4⋊Dic3—41no—
175C4⋊Dic3—41no—
176C4×Dic3—42no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C22×D4—21no—
181C22×D4#6711yescharacteristic
182C23×C4—21no—
183C23×C4—31no—
184C23×C4—31no—
185C23×C4—61no—
186C23×C4—61no—
187C23×C4C22×D311yescharacteristic
188C22×C8—21no—
189C22×C8—61no—
190#121—21no—
191#121—21no—
192#121C2×Dic312yes—
193#121#6712yes—
194C4×D4—122no—
195C4×D4—122no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4C2×Dic312yescomplemented
205C2×C4⋊C4#6711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C42—61no—
212#107—24no—
213#102—24no—
214C22⋊C8—21no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8C2×Dic312yescomplemented
218#95—61no—
219#95—61no—
220#95—61no—
221#95—61no—
222#95—61no—
223#95—61no—
224#95—61no—
225#95—122no—
226#95—122no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—22no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3C4×D411yescharacteristic
239C23×C3#12011yescharacteristic
240C23×C3#12111yescharacteristic
241C23×C3C2×D811yescharacteristic
242C23×C3#13511yescharacteristic
243C23×C3AGL(1,ℤ/8)11yescharacteristic
244C23×C3AGL(1,ℤ/8)11yescharacteristic
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—81no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—81no—
266C2×C3×C4—81no—
267C2×C3×C4—81no—
268C2×C3×C4—81no—
269C2×C3×C4C2×C22⋊C411yescharacteristic
270C2×C3×C4C4×D411yescharacteristic
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12311yescharacteristic
273C2×Dic3—22no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—81no—
291C2×Dic3—84no—
292C2×Dic3—81no—
293C2×Dic3—81no—
294C2×Dic3—81no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297C3×C8—42no—
298C3⋊C8—81no—
299C24—21no—
300C24D3×D411yescharacteristic
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4#21712yes—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—32no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—62no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4—122no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—121no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—241no—
342C22×C4—241no—
343C22×C4D3×D411yescharacteristic
344C22×C4#23711yescharacteristic
345C22×C4#23711yescharacteristic
346C22×C4C22×Dic311yescharacteristic
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C2×C8—22no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—121no—
353C2×C8—121no—
354C2×C8—121no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—121no—
361C4⋊C4—121no—
362C4⋊C4—122no—
363C4⋊C4#21712yes—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—122no—
367C22⋊C4—121no—
368C22⋊C4—121no—
369C22⋊C4—241no—
370C22⋊C4—241no—
371C42—122no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3#38611yescharacteristic
392C22×C3C4×D811yescharacteristic
393C22×C3#44211yescharacteristic
394C22×C3#44711yescharacteristic
395C22×C3#44911yescharacteristic
396C22×C3#45511yescharacteristic
397C22×C3#45611yescharacteristic
398C22×C3#45811yescharacteristic
399C22×C3#45811yescharacteristic
400C22×C3#45911yescharacteristic
401C22×C3#46311yescharacteristic
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—42no—
407C3×C4—82no—
408Dic3—42no—
409Dic3—82no—
410Dic3—81no—
411Dic3—81no—
412Dic3—81no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23D3×D811yescharacteristic
425C23#91111yescharacteristic
426C23#91111yescharacteristic
427C23#91211yescharacteristic
428C23D4×Dic311yescharacteristic
429C23#93711yescharacteristic
430C23#93911yescharacteristic
431D4—42no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—81no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—62no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—241no—
473C2×C4—244no—
474C2×C4—241no—
475C2×C4—241no—
476C2×C4—241no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4D4×Dic311yescharacteristic
480C2×C4#93511yescharacteristic
481C2×C4#93911yescharacteristic
482C2×C4C2×#21711yescharacteristic
483C8—42no—
484C8—241no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—82no—
490C2×C3#160711yescharacteristic
491C2×C3#160911yescharacteristic
492C2×C3#161011yescharacteristic
493C2×C3#161211yescharacteristic
494C2×C3#187911yescharacteristic
495C2×C3#188211yescharacteristic
496C2×C3#189011yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22#505311yescharacteristic
517C22#505311yescharacteristic
518C22Dic3×D811yescharacteristic
519C22#515011yescharacteristic
520C22#515211yescharacteristic
521C22#515311yescharacteristic
522C22#515611yescharacteristic
523C22#515811yescharacteristic
524C22#520211yescharacteristic
525C22#520511yescharacteristic
526C22#522011yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—42no—
532C4—82no—
533C4—122no—
534C4—242no—
535C4—241no—
536C4—241no—
537C4—241no—
538C3#904311yescharacteristic, complemented, Sylow
539C2—21no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—82no—
544C2#7116411yescharacteristic
545C2#7116511yescharacteristic
546C2#7116611yescharacteristic
547C2#7116711yescharacteristic
548C2#7170411yescharacteristic
549C2#7203711yescharacteristic
550C2#7210011yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#480C2
socleC23×C3#120
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenterC2×#480D3
hypocenterC3#9043
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
4. #71569C2semidirect
5. #71553C2semidirect
6. #67675C2semidirect
7. #67674C2semidirect
13. C3×#480C4semidirect
56. C2×C3×C4⋊C4C2×C4semidirect
73. C3×C22⋊C8C2×C4semidirect
89. #480Dic3semidirect
204. C2×C4⋊C4C2×Dic3semidirect
217. C22⋊C8C2×Dic3semidirect
538. C3#9043semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4802
#826342
C2×#45372
#715692
#715532
#676752
#676742
#90433