Finite Group Databasebeta

← #10867913

Subgroups — #10867913

Order 768.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2360C211yesmaximal, characteristic
2#73511C211yesmaximal, characteristic
3#71695C211yesmaximal, characteristic
4#71694C211yesmaximal, characteristic
5C4×#835C211yesmaximal, characteristic, complemented
6#71628C211yesmaximal, characteristic, complemented
7#71555C211yesmaximal, characteristic
8#71553C211yesmaximal, characteristic
9#69683C211yesmaximal, characteristic, complemented
10#69671C211yesmaximal, characteristic, complemented
11#69619C211yesmaximal, characteristic, complemented
12#69602C211yesmaximal, characteristic, complemented
13#69468C211yesmaximal, characteristic, complemented
14#69458C211yesmaximal, characteristic, complemented
15#69451C211yesmaximal, characteristic
16#27035—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C3×#391C2211yescharacteristic
21C3×#389—21no—
22C3×#389C2211yescharacteristic
23C3×#387—21no—
24C3×#385—21no—
25C3×C4×C4⋊C4C2211yescharacteristic
26C3×C4×C22⋊C4—21no—
27C3×C4×C22⋊C4C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4952—21no—
41#4944—21no—
42#4936C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4658—21no—
49#4651—21no—
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2495—31no—
61#2360D311yescharacteristic
62#2074—31no—
63#2068—31no—
64#2036—31no—
65#2035—31no—
66#1949—31no—
67#1942—31no—
68#1923—31no—
69#1920—31no—
70#1918—31no—
71#1863—31no—
72#1852—31no—
73#1823—31no—
74C4×#103—31no—
75C22×C3×Q8C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—41no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—41no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C22⋊C4D411yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95D411yescharacteristic
116C3×#95D411yescharacteristic
117#835—22no—
118#835—22no—
119#835—22no—
120#835—41no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#830—22no—
129#830—41no—
130#808—22no—
131#807—22no—
132#807—22no—
133#807—41no—
134C4⋊(C3⋊C8)—22no—
135C4×C3⋊C8—41no—
136C2×C22⋊Q8C2×D311yescharacteristic
137C2×C4×Q8C2×D311yescharacteristic
138C22×C4⋊C4C2×D311yescharacteristic
139C2×#107—31no—
140C2×#107—31no—
141C2×#107—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×C4×C8—31no—
151#400—61no—
152#397—61no—
153#391C2×D311yescharacteristic
154#389—21no—
155#389—31no—
156#389—31no—
157#389C2×D311yescharacteristic
158#388—61no—
159#387—21no—
160#385—21no—
161#383—61no—
162#382—31no—
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4C2×D311yescharacteristic
167C4×C22⋊C4—21no—
168C4×C22⋊C4C2×D311yescharacteristic
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336—31no—
178#336—31no—
179C24×C3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4—81no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—81no—
217C3×C4⋊C4#4112yes—
218C3×C4⋊C4#4112yes—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—42no—
227C3×C42—81no—
228#217—42no—
229#217—81no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238Dic3⋊C4—81no—
239Dic3⋊C4—81no—
240C4×Dic3—42no—
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—21no—
252C2×C3⋊C8#4112yes—
253C22×Q8C22×D311yescharacteristic
254C23×C4—21no—
255C23×C4C22×D311yescharacteristic
256C22×C8—31no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—31no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C4×Q8—22no—
263C4×Q8—41no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4#6711yescharacteristic
283C2×C22⋊C4#6711yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—61no—
287C2×C42C22×D311yescharacteristic
288C2×C42C22×D311yescharacteristic
289#107—62no—
290#107—62no—
291#107—121no—
292#106—62no—
293C4⋊C8—62no—
294#103—62no—
295#103—62no—
296#103—62no—
297#103—121no—
298C22⋊C8—62no—
299C22⋊C8—121no—
300C4×C8—121no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95#6711yescharacteristic
313#95#6711yescharacteristic
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3#13512yes—
321C23×C3C22×D411yescharacteristic
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—41no—
325C3×Q8—41no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
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—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
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×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—84no—
357C2×C3×C4#13712yes—
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C42−1+411yescharacteristic
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C3⋊C8—22no—
377C3⋊C8—41no—
378C3⋊C8—41no—
379C3⋊C8—41no—
380C24C2×#6711yescharacteristic
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8#23512yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—81no—
404C22×C4C2×#6711yescharacteristic
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C2×#6711yescharacteristic
408C22×C4C2×#6711yescharacteristic
409C22×C4C23×D311yescharacteristic
410C2×C8—32no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C4⋊C4—24no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—81no—
438C4⋊C4—241no—
439C4⋊C4—241no—
440C4⋊C4#23512yes—
441C4⋊C4#23712yes—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—122no—
446C22⋊C4—241no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—42no—
452C42—81no—
453C42—122no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3#52511yescharacteristic
462C22×C3#54611yescharacteristic
463C22×C3#54711yescharacteristic
464C22×C3C2×#13511yescharacteristic
465C22×C3C2×#13711yescharacteristic
466C22×C3#57611yescharacteristic
467C22×C3#57811yescharacteristic
468C3×C4—22no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—42no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—42no—
476C3×C4—81no—
477C3×C4#52911yescharacteristic
478C3×C4#53911yescharacteristic
479C3×C4#55311yescharacteristic
480C3×C4#55611yescharacteristic
481Dic3—81no—
482Dic3—81no—
483Dic3—81no—
484Dic3—81no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23#95012yes—
492C23C22×#6711yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—84no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4#94212yes—
541C2×C4C2×#23511yescharacteristic
542C2×C4C2×#23511yescharacteristic
543C2×C4C2×#23711yescharacteristic
544C2×C4#100411yescharacteristic
545C2×C4D3×#4111yescharacteristic
546C2×C4#101011yescharacteristic
547C8—62no—
548C8—121no—
549C8—121no—
550C8—121no—
551C2×C3—42no—
552C2×C3#246611yescharacteristic
553C2×C3#310811yescharacteristic
554C2×C3#312411yescharacteristic
555C2×C3#329711yescharacteristic
556C2×C3#331211yescharacteristic
557C2×C3#333611yescharacteristic
558C2×C3#335711yescharacteristic
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22#579111yescharacteristic
567C22#580611yescharacteristic
568C22C2×#94211yescharacteristic
569C22C2×#95011yescharacteristic
570C22#582411yescharacteristic
571C22#583511yescharacteristic
572C22#583811yescharacteristic
573C4—22no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—81no—
582C4—241no—
583C4—241no—
584C4—241no—
585C4—241no—
586C4#553811yescharacteristic
587C4#554211yescharacteristic
588C4#555811yescharacteristic
589C4#556411yescharacteristic
590C3#2703511yescharacteristic, complemented, Sylow
591C2—42no—
592C2#8236911yescharacteristic
593C2#8244311yescharacteristic
594C2#8256911yescharacteristic
595C2#8349511yescharacteristic
596C2#8355211yescharacteristic
597C2#8359011yescharacteristic
598C2#8362711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2360C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2360D3
hypocenterC3#27035
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
5. C4×#835C2semidirect
6. #71628C2semidirect
9. #69683C2semidirect
10. #69671C2semidirect
11. #69619C2semidirect
12. #69602C2semidirect
13. #69468C2semidirect
14. #69458C2semidirect
590. C3#27035semidirect

Maximal subgroups

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

subgroupindex
C3×#23602
#735112
#716952
#716942
C4×#8352
#716282
#715552
#715532
#696832
#696712
#696192
#696022
#694682
#694582
#694512
#270353