Finite Group Databasebeta

← #10853174

Subgroups — #10853174

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2344C211yesmaximal, characteristic
2#73494C211yesmaximal, characteristic
3#71558C211yesmaximal, characteristic
4#71554C211yesmaximal, characteristic
5#69622C211yesmaximal, characteristic, complemented
6#69612C211yesmaximal, characteristic, complemented
7#69610C211yesmaximal, characteristic, complemented
8#69606C211yesmaximal, characteristic, complemented
9#69513C211yesmaximal, characteristic
10#69506C211yesmaximal, characteristic
11#69491C211yesmaximal, characteristic, complemented
12#69467C211yesmaximal, characteristic, complemented
13#69461C211yesmaximal, characteristic, complemented
14#69458C211yesmaximal, characteristic, complemented
15#69450C211yesmaximal, characteristic
16#26014—31nomaximal, Sylow
17C2×C3×#117—21no—
18C22×C3×C4⋊C4C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×#208C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#387—21no—
25C3×#387—21no—
26C3×#381—21no—
27C3×#380C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C22⋊C4C2211yescharacteristic
31#5212C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#808C2211yescharacteristic
34C2×#808C2211yescharacteristic
35C2×#807C2211yescharacteristic
36C2×#807C2211yescharacteristic
37#4949—21no—
38#4948—21no—
39#4935C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4654—21no—
45#4647—21no—
46#4529C2211yescharacteristic
47#4529C2211yescharacteristic
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2441—31no—
61#2344D311yescharacteristic
62#2100—31no—
63#2088—31no—
64#2086—31no—
65#2059—31no—
66#1989—31no—
67#1948—31no—
68#1921—31no—
69#1918—31no—
70#1854—31no—
71#1851—31no—
72#1824—31no—
73#1809—31no—
74#1736—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C3×#117—44no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4Q814yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96#926—22no—
97#926—41no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×#208—22no—
110C2×#208—22no—
111C2×#208—41no—
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#870—22no—
117#870—41no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95D411yescharacteristic
124C3×#95D411yescharacteristic
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#830—22no—
129#808—41no—
130#808—41no—
131#807—22no—
132#807—22no—
133C4⋊(C3⋊C8)—22no—
134C4⋊(C3⋊C8)—41no—
135C22×M16—31no—
136C2×#125—31no—
137C2×#117—21no—
138C2×#117—31no—
139C22×C4⋊C4C2×D311yescharacteristic
140C2×#107—31no—
141C2×#107—31no—
142C2×#106—31no—
143C2×#106—31no—
144C2×C4⋊C8—31no—
145C2×C4⋊C8—31no—
146C2×C22⋊C8—31no—
147#399—61no—
148#396—61no—
149#389—31no—
150#389—31no—
151#389—61no—
152#389C2×D311yescharacteristic
153#389C2×D311yescharacteristic
154#387—21no—
155#387—21no—
156#387—61no—
157#384—31no—
158#381—21no—
159#380—31no—
160#380C2×D311yescharacteristic
161C4×C4⋊C4C2×D311yescharacteristic
162C4×C4⋊C4C2×D311yescharacteristic
163C4×C22⋊C4C2×D311yescharacteristic
164#340—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336—31no—
177#336—31no—
178C24×C3C2×D411yescharacteristic
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—24no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×Q812yes—
196C22×C3×C4#4112yes—
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—42no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4C2×Q812yes—
212C3×C4⋊C4C2×Q812yes—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—41no—
215C3×C22⋊C4—42no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—42no—
221C3×C42—42no—
222#217—42no—
223#217—81no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—42no—
232C4⋊Dic3—81no—
233C4⋊Dic3#4112yes—
234Dic3⋊C4—42no—
235Dic3⋊C4—42no—
236Dic3⋊C4—42no—
237Dic3⋊C4—81no—
238C4×Dic3—42no—
239#208—44no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8#4112yes—
252C23×C4—21no—
253C23×C4C22×D311yescharacteristic
254C2×M16—62no—
255C2×M16—62no—
256C2×M16—121no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—31no—
260C22×C8—31no—
261#125—62no—
262#125—121no—
263#117—44no—
264#117—62no—
265#117—121no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4Dic614yes—
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4#6711yescharacteristic
289C2×C22⋊C4#6711yescharacteristic
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—61no—
293C2×C42C22×D311yescharacteristic
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#107—62no—
297#106—121no—
298#106—121no—
299C4⋊C8—62no—
300C4⋊C8—121no—
301C22⋊C8—62no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95#6711yescharacteristic
311#95#6711yescharacteristic
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3C22⋊Q812yes—
319C23×C3C22×D411yescharacteristic
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—48no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4C22⋊Q812yes—
351C2×C3×C4AGL(1,ℤ/8)12yes—
352C2×C3×C4C22×Q811yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358C2×Dic3—22no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C3⋊C8—22no—
371C3⋊C8—41no—
372C3⋊C8—41no—
373C3⋊C8—41no—
374C24C2×#6711yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4C2×Dic612yes—
391C22×C4#23512yes—
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C23×D311yescharacteristic
398M16—124no—
399C2×C8—32no—
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—61no—
411C4⋊C4—24no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—32no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—122no—
429C4⋊C4—122no—
430C4⋊C4—122no—
431C4⋊C4—122no—
432C4⋊C4—241no—
433C4⋊C4—241no—
434C4⋊C4C2×Dic612yes—
435C4⋊C4C2×Dic612yes—
436C22⋊C4—22no—
437C22⋊C4—41no—
438C22⋊C4—42no—
439C22⋊C4—122no—
440C22⋊C4—241no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—42no—
446C42—42no—
447C42—122no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3C2×C22⋊Q811yescharacteristic
456C22×C3#54611yescharacteristic
457C22×C3#54711yescharacteristic
458C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
459C22×C3#57511yescharacteristic
460C22×C3#57711yescharacteristic
461C22×C3#57811yescharacteristic
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—41no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4#53311yescharacteristic
472C3×C4#54411yescharacteristic
473C3×C4#55111yescharacteristic
474C3×C4#55211yescharacteristic
475Dic3—42no—
476Dic3—81no—
477Dic3—81no—
478Dic3—81no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23#92412yes—
486C23C22×#6711yescharacteristic
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—48no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—62no—
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—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4#92412yes—
530C2×C4#93212yes—
531C2×C4C22×Dic611yescharacteristic
532C2×C4C2×#23511yescharacteristic
533C2×C4D3×#4111yescharacteristic
534C2×C4D3×#4111yescharacteristic
535C2×C4#100911yescharacteristic
536C2×C4#101011yescharacteristic
537C8—62no—
538C8—121no—
539C8—121no—
540C8—121no—
541C2×C3—42no—
542C2×C3#243811yescharacteristic
543C2×C3#307711yescharacteristic
544C2×C3#308211yescharacteristic
545C2×C3#331711yescharacteristic
546C2×C3#333011yescharacteristic
547C2×C3#334511yescharacteristic
548C2×C3#335211yescharacteristic
549C22—22no—
550C22—22no—
551C22—22no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22C2×#92411yescharacteristic
557C22C2×#93211yescharacteristic
558C22#582211yescharacteristic
559C22#582411yescharacteristic
560C22#582911yescharacteristic
561C22#583711yescharacteristic
562C22#583811yescharacteristic
563C4—22no—
564C4—22no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—122no—
573C4—241no—
574C4—241no—
575C4—241no—
576C4#551711yescharacteristic
577C4#551911yescharacteristic
578C4#552711yescharacteristic
579C4#553111yescharacteristic
580C3#2601411yescharacteristic, complemented, Sylow
581C2—42no—
582C2#8224711yescharacteristic
583C2#8229611yescharacteristic
584C2#8255011yescharacteristic
585C2#8351911yescharacteristic
586C2#8352711yescharacteristic
587C2#8359711yescharacteristic
588C2#8360411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2344C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2344D3
hypocenterC3#26014
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
5. #69622C2semidirect
6. #69612C2semidirect
7. #69610C2semidirect
8. #69606C2semidirect
11. #69491C2semidirect
12. #69467C2semidirect
13. #69461C2semidirect
14. #69458C2semidirect
580. C3#26014semidirect

Maximal subgroups

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

subgroupindex
C3×#23442
#734942
#715582
#715542
#696222
#696122
#696102
#696062
#695132
#695062
#694912
#694672
#694612
#694582
#694502
#260143