Finite Group Databasebeta

← #10694024

Subgroups — #10694024

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4990C211yesmaximal, characteristic
2C2×C3×#324C211yesmaximal, characteristic, complemented
3C2×#4528C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic
5C2×#4468C211yesmaximal, characteristic
6#69908C211yesmaximal, characteristic, complemented
7#69906C211yesmaximal, characteristic, complemented
8#9652—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×(C2×D3)⋊C4C411yescharacteristic
11C22×(C2×D3)⋊C4C411yescharacteristic
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C22×C4×Dic3C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3⋊#95C411yescharacteristic
17C2×C3⋊#95C411yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#815—21no—
21C2×Dic3⋊C8—21no—
22#4990—21no—
23#4990—21no—
24#4990—21no—
25#4990—21no—
26C3×#324—22no—
27#4550—21no—
28#4550—21no—
29#4528—22no—
30#4468—22no—
31#4468—22no—
32C2×#388—31no—
33#1797—31no—
34#1749—31no—
35C2×#324—31no—
36C2×#324D311yescharacteristic, complemented
37C2×#323—31no—
38C2×#323—31no—
39C24×D3C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2×C411yescharacteristic
44C23×Dic3C2×C411yescharacteristic
45C23×Dic3C2×C411yescharacteristic
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—42no—
67C2×C4×Dic3—42no—
68C2×C4×Dic3D411yescharacteristic
69C2×C4×Dic3D411yescharacteristic
70C2×C4×Dic3D411yescharacteristic
71C2×C4×Dic3D411yescharacteristic
72C22×C3⋊C8—21no—
73C22×C3⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yes—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85#830D412yes—
86#815—42no—
87Dic3⋊C8—42no—
88C22×C4⋊C4C2×D311yescharacteristic
89C22×C22⋊C4—31no—
90C22×C22⋊C4—31no—
91C22×C22⋊C4—31no—
92C22×C42—31no—
93C2×C4⋊C8—61no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8C2×D311yescharacteristic
97C2×C22⋊C8C2×D311yescharacteristic
98C2×#97—61no—
99#388—61no—
100#388—61no—
101#388—61no—
102#388—61no—
103C2×#95—31no—
104C2×#95—31no—
105#336—61no—
106#336—61no—
107#324—22no—
108#324—62no—
109#323—62no—
110#323—62no—
111C24×C3C22×C411yescharacteristic
112C23×D3—21no—
113C23×D3—21no—
114C23×D3—21no—
115C23×D3—21no—
116C23×D3—21no—
117C23×D3—21no—
118C23×D3—41no—
119C23×D3—41no—
120C23×D3—41no—
121C23×D3—41no—
122C23×D3M1612yes—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4M1612yes—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—22no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3C22⋊C411yescharacteristic
157C22×Dic3C22⋊C411yescharacteristic
158C22×Dic3C22⋊C411yescharacteristic
159C22×Dic3C22⋊C411yescharacteristic
160C2×C3×C8—22no—
161C2×C3×C8—41no—
162C2×C3×C8—41no—
163C2×C3×C8—41no—
164C2×C3×C8—41no—
165C2×C3×C8—41no—
166C3×C4⋊C4—41no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—41no—
169C3×C4⋊C4—41no—
170(C2×D3)⋊C4—24no—
171(C2×D3)⋊C4—24no—
172(C2×D3)⋊C4—42no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—41no—
177(C2×D3)⋊C4—42no—
178C4×Dic3—22no—
179C4×Dic3—22no—
180C4×Dic3—41no—
181C4×Dic3—41no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—41no—
184C2×C3⋊C8—41no—
185C2×C3⋊C8—41no—
186C2×C3⋊C8—41no—
187C2×C3⋊C8—41no—
188C25—31no—
189C23×C4—31no—
190C23×C4—31no—
191C23×C4—31no—
192C23×C4C4×D311yescharacteristic
193C23×C4C4×D311yescharacteristic
194C23×C4C22×D311yescharacteristic
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—61no—
198C22×C8—61no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—122no—
218C2×C42—31no—
219C2×C42—31no—
220C2×C42—31no—
221C2×C42—31no—
222C2×C42—122no—
223C4⋊C8—122no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—32no—
228C22⋊C8—61no—
229C22⋊C8—61no—
230C22⋊C8—61no—
231C22⋊C8#6712yes—
232#97—122no—
233#95—61no—
234#95—61no—
235#95—61no—
236#95—61no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3C2×C22⋊C411yescharacteristic
242C23×C3C4×D411yescharacteristic
243C23×C3C4×D411yescharacteristic
244C23×C3C2×M1611yescharacteristic
245C23×C3C2×M1611yescharacteristic
246C23×C3#13111yescharacteristic
247C23×C3#13111yescharacteristic
248C22×D3—22no—
249C22×D3—22no—
250C22×D3—22no—
251C22×D3—42no—
252C22×D3—42no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—42no—
262C22×D3—42no—
263C22×D3—81no—
264C22×D3—81no—
265C22×D3—81no—
266C22×D3—81no—
267C22×D3—81no—
268C22×D3—81no—
269C22×D3—81no—
270C22×D3—81no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—81no—
290C2×C3×C4—81no—
291C2×C3×C4#12111yescharacteristic
292C2×C3×C4#12111yescharacteristic
293C2×C3×C4#12411yescharacteristic
294C2×C3×C4#12711yescharacteristic
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—81no—
314C2×Dic3—81no—
315C2×Dic3#9912yes—
316C3×C8—42no—
317C3×C8—81no—
318C3⋊C8—42no—
319C3⋊C8—81no—
320C24—32no—
321C24—61no—
322C24—61no—
323C24—61no—
324C24—61no—
325C24—61no—
326C24—61no—
327C24—121no—
328C24—121no—
329C24—121no—
330C24—121no—
331C24C2×C4×D311yescharacteristic
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4—121no—
360C22×C4—121no—
361C22×C4—121no—
362C22×C4ΓL(1,𝔽25)12yes—
363C22×C4#23511yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4D3×D411yescharacteristic
366C22×C4D3×D411yescharacteristic
367C22×C4#23711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C2×C8—22no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—62no—
376C2×C8—121no—
377C2×C8—121no—
378C2×C8—121no—
379C2×C8—121no—
380C2×C8—121no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C22⋊C4—64no—
386C22⋊C4—64no—
387C22⋊C4—122no—
388C22⋊C4—122no—
389C22⋊C4—121no—
390C22⋊C4—121no—
391C22⋊C4—121no—
392C22⋊C4—121no—
393C42—62no—
394C42—62no—
395C42—121no—
396C42—121no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3#39011yescharacteristic
410C22×C3#40711yescharacteristic
411C22×C3#40811yescharacteristic
412C22×C3C2×#9911yescharacteristic
413C22×C3#41011yescharacteristic
414C22×C3#41311yescharacteristic
415C22×C3#41311yescharacteristic
416C22×C3#43511yescharacteristic
417C22×C3#43511yescharacteristic
418C22×C3#43511yescharacteristic
419C22×C3#43511yescharacteristic
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—82no—
424C2×D3—82no—
425C2×D3—81no—
426C2×D3—81no—
427C2×D3—81no—
428C2×D3—81no—
429C2×D3—82no—
430C2×D3—82no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—42no—
436C3×C4—81no—
437Dic3—22no—
438Dic3—41no—
439Dic3—41no—
440Dic3—41no—
441Dic3—81no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—62no—
447C23—62no—
448C23—62no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23—122no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—241no—
462C23—241no—
463C23—241no—
464C23—241no—
465C23—241no—
466C23—241no—
467C23—241no—
468C23—241no—
469C23D3×C22⋊C411yescharacteristic
470C23#67⋊C411yescharacteristic
471C23C2×ΓL(1,𝔽25)11yescharacteristic
472C23#90111yescharacteristic
473C23D3×M1611yescharacteristic
474C23#90711yescharacteristic
475C23C4×#6711yescharacteristic
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—32no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—241no—
516C2×C4—241no—
517C2×C4#88411yescharacteristic
518C2×C4#88511yescharacteristic
519C2×C4#93911yescharacteristic
520C2×C4#94011yescharacteristic
521C8—42no—
522C8—81no—
523C8—122no—
524C8—241no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3#146411yescharacteristic
530C2×C3#146511yescharacteristic
531C2×C3#146811yescharacteristic
532C2×C3#147311yescharacteristic
533C2×C3#195511yescharacteristic
534C2×C3#196411yescharacteristic
535C2×C3#196711yescharacteristic
536D3—82no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22—242no—
553C22—242no—
554C22—242no—
555C22—242no—
556C22—241no—
557C22—241no—
558C22—241no—
559C22—241no—
560C22#472611yescharacteristic
561C22#472711yescharacteristic
562C22#472811yescharacteristic
563C22#473011yescharacteristic
564C22#474011yescharacteristic
565C22D3×#9911yescharacteristic
566C22#475111yescharacteristic
567C22#475111yescharacteristic
568C22#495711yescharacteristic
569C22#511111yescharacteristic
570C22#512711yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—62no—
577C4—81no—
578C4—121no—
579C4—121no—
580C4—121no—
581C4—241no—
582C3#965211yescharacteristic, complemented, Sylow
583C2—21no—
584C2—21no—
585C2—21no—
586C2—21no—
587C2—242no—
588C2#6812611yescharacteristic
589C2#6812911yescharacteristic
590C2#6814511yescharacteristic
591C2#6814611yescharacteristic
592C2#6991711yescharacteristic
593C2#7003011yescharacteristic
594C2#7009911yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC23×C3C2×C22⋊C4
prime radicalC24×D3C2×C4
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9652
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×C3×#324C2semidirect
3. C2×#4528C2semidirect
6. #69908C2semidirect
7. #69906C2semidirect
36. C2×#324D3semidirect
582. C3#9652semidirect

Maximal subgroups

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

subgroupindex
C2×#49902
C2×C3×#3242
C2×#45282
C2×#44682
#699082
#699062
#96523