Finite Group Databasebeta

← #10877213

Subgroups — #10877213

Order 768.

854 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
1C3×C4×#121C211yesmaximal, characteristic
2#82062C211yesmaximal, characteristic, complemented
3#71678C212yesmaximal
4#71604C211yesmaximal, characteristic, complemented
5#71604C211yesmaximal, characteristic, complemented
6#71530C211yesmaximal, characteristic, complemented
7#69654C212yesmaximal, complemented
8#69613C212yesmaximal, complemented
9#69506C212yesmaximal, complemented
10#69467C211yesmaximal, characteristic, complemented
11#69466C211yesmaximal, characteristic, complemented
12#27679—31nomaximal, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C2×#926C2212yes—
18C3×#390C2211yescharacteristic
19C3×#389C2211yescharacteristic
20C3×#388—21no—
21C3×#386—21no—
22C3×#380C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C3×C4×C22⋊C4—21no—
25C2×#832C2212yes—
26C2×#832C2212yes—
27#5212—21no—
28#5212—21no—
29C2×#830C2212yes—
30C2×#808C2212yes—
31C2×#807C2212yes—
32#4936C2212yes—
33#4935C2211yescharacteristic
34#4935C2211yescharacteristic
35C4×C4⋊Dic3—21no—
36C4×C4⋊Dic3—21no—
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C4×#121D311yescharacteristic
45#2285—31no—
46#2101—32no—
47#2092—32no—
48#1948—31no—
49#1947—31no—
50#1875—32no—
51#1854—32no—
52#1802—31no—
53#1802—31no—
54#1797—31no—
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C3×#121—22no—
62C3×#121Q814yes—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C3×C4×D4—44no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42—21no—
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#217—22no—
83#926—42no—
84#926—42no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2312yes—
91C2×C4×Dic3—22no—
92C22×C3⋊C8C2312yes—
93C22×C3⋊C8C2312yes—
94C3×#95—21no—
95C3×#95—21no—
96C3×#95—21no—
97#832—42no—
98#832—42no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#830—24no—
102#808—42no—
103#807—42no—
104C4⋊(C3⋊C8)—22no—
105C4⋊(C3⋊C8)—22no—
106C2×#121C2×D311yescharacteristic
107C2×C4×D4—21no—
108C2×C4×D4C2×D311yescharacteristic
109C2×#117—32no—
110C22×C42C2×D311yescharacteristic
111C2×#107—32no—
112C2×#106—32no—
113C2×C4⋊C8—31no—
114C2×C4⋊C8—31no—
115C2×#102—32no—
116C2×#102—32no—
117C2×C22⋊C8—32no—
118#390C2×D311yescharacteristic
119#389C2×D311yescharacteristic
120#388—21no—
121#386—21no—
122#384—31no—
123#384—31no—
124#382—32no—
125#380—61no—
126#380—61no—
127#380C2×D311yescharacteristic
128C4×C4⋊C4—61no—
129C4×C4⋊C4—61no—
130C4×C4⋊C4C2×D311yescharacteristic
131C4×C22⋊C4—21no—
132#340—32no—
133#340—32no—
134#340—32no—
135#340—32no—
136#336—32no—
137C24×C3—21no—
138C24×C3C2×D411yescharacteristic
139C2×C3×D4—24no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4C2×Q812yes—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—41no—
160C22×C3×C4—41no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×Q812yes—
170C22×C3×C4#4112yes—
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—22no—
173C22×Dic3—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4C2×Q812yes—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—42no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—21no—
189C3×C42—21no—
190C3×C42—42no—
191C3×C42#4112yes—
192C3×C42#4112yes—
193#217—44no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—42no—
199C4⋊Dic3—42no—
200C4⋊Dic3—42no—
201C4⋊Dic3—42no—
202C4×Dic3—44no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C22×D4—21no—
210C22×D4C22×D311yescharacteristic
211C23×C4—21no—
212C23×C4#6711yescharacteristic
213C23×C4#6711yescharacteristic
214C23×C4C22×D311yescharacteristic
215C22×C8—32no—
216C22×C8—32no—
217#121—22no—
218#121Dic614yes—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—44no—
222#117—122no—
223#117—122no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—32no—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4C22×D311yescharacteristic
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—62no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—62no—
244C2×C42#6711yescharacteristic
245C2×C42#6711yescharacteristic
246C2×C42C22×D311yescharacteristic
247C2×C42C22×D311yescharacteristic
248#107—122no—
249#106—122no—
250C4⋊C8—62no—
251C4⋊C8—62no—
252#102—122no—
253#102—122no—
254C22⋊C8—64no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—62no—
259#95—62no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—22no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3C22⋊Q812yes—
274C23×C3C22×D411yescharacteristic
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—48no—
278C3×D4—41no—
279C3×D4—41no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—44no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—44no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—84no—
306C2×C3×C4C22⋊Q812yes—
307C2×C3×C4#12312yes—
308C2×C3×C4#12312yes—
309C2×C3×C4#13512yes—
310C2×C3×C4AGL(1,ℤ/8)12yes—
311C2×C3×C4C22×Q811yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C3⋊C8—42no—
324C3⋊C8—42no—
325C24—21no—
326C24C2×#6711yescharacteristic
327C2×D4—24no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4C2×Dic612yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4C2×Dic612yes—
355C22×C4#23512yes—
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361C22×C4C23×D311yescharacteristic
362C2×C8—62no—
363C2×C8—62no—
364C2×C8—62no—
365C2×C8—62no—
366C2×C8—62no—
367C2×C8—62no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—42no—
374C4⋊C4—62no—
375C4⋊C4—62no—
376C4⋊C4—62no—
377C4⋊C4—62no—
378C4⋊C4—122no—
379C4⋊C4—122no—
380C4⋊C4—122no—
381C4⋊C4—122no—
382C4⋊C4C2×Dic612yes—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—124no—
389C42—22no—
390C42—22no—
391C42—21no—
392C42—21no—
393C42—42no—
394C42—124no—
395C42#23712yes—
396C42#23712yes—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3C2×C22⋊Q811yescharacteristic
412C22×C3C2×#12311yescharacteristic
413C22×C3#53511yescharacteristic
414C22×C3C2×#13511yescharacteristic
415C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
416C22×C3#57511yescharacteristic
417C22×C3#57511yescharacteristic
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—81no—
424C3×C4—81no—
425C3×C4#48214yes—
426C3×C4#55311yescharacteristic
427C3×C4#55311yescharacteristic
428C3×C4#55411yescharacteristic
429C3×C4#55411yescharacteristic
430Dic3—82no—
431Dic3—82no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23#92412yes—
446C23C22×#6711yescharacteristic
447D4—22no—
448D4—22no—
449D4—48no—
450D4—41no—
451D4—41no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—84no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4#92412yes—
485C2×C4#93212yes—
486C2×C4#93512yes—
487C2×C4#93512yes—
488C2×C4#95012yes—
489C2×C4C22×Dic611yescharacteristic
490C2×C4C2×#23511yescharacteristic
491C2×C4C2×#23711yescharacteristic
492C2×C4C2×#23711yescharacteristic
493C2×C4#100911yescharacteristic
494C2×C4#100911yescharacteristic
495C8—122no—
496C8—122no—
497C2×C3—42no—
498C2×C3—82no—
499C2×C3#248711yescharacteristic
500C2×C3#307611yescharacteristic
501C2×C3#307711yescharacteristic
502C2×C3C2×#48211yescharacteristic
503C2×C3#309311yescharacteristic
504C2×C3#321211yescharacteristic
505C2×C3#321411yescharacteristic
506C22—22no—
507C22—22no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22C2×#92411yescharacteristic
521C22C2×#93211yescharacteristic
522C22C2×#93511yescharacteristic
523C22C2×#95011yescharacteristic
524C22#582211yescharacteristic
525C22#582211yescharacteristic
526C22#583011yescharacteristic
527C4—22no—
528C4—22no—
529C4—42no—
530C4—41no—
531C4—41no—
532C4—81no—
533C4—81no—
534C4—242no—
535C4—242no—
536C4#503514yes—
537C4#554311yescharacteristic
538C4#554311yescharacteristic
539C4#556511yescharacteristic
540C4#556511yescharacteristic
541C3#2767911yescharacteristic, complemented, Sylow
542C2—42no—
543C2—82no—
544C2#8229611yescharacteristic
545C2#8237411yescharacteristic
546C2#8250711yescharacteristic
547C2C2×#503511yescharacteristic
548C2#8291511yescharacteristic
549C2#8314511yescharacteristic
550C2#8330811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#121C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC4×#121D3
hypocenterC3#27679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. #82062C2semidirect
4. #71604C2semidirect
5. #71604C2semidirect
6. #71530C2semidirect
7. #69654C2semidirect
8. #69613C2semidirect
9. #69506C2semidirect
10. #69467C2semidirect
11. #69466C2semidirect
541. C3#27679semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1212
#820622
#716782
#716042
#715302
#696542
#696132
#695062
#694672
#694662
#276793