Finite Group Databasebeta

← #10877207

Subgroups — #10877207

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×#2285C211yesmaximal, characteristic, complemented
2C4×#930C211yesmaximal, characteristic
3#71586C212yesmaximal
4#71576C212yesmaximal, complemented
5#71530C211yesmaximal, characteristic, complemented
6#69584C211yesmaximal, characteristic, complemented
7#69584C211yesmaximal, characteristic, complemented
8#69568C211yesmaximal, characteristic, complemented
9#69567C211yesmaximal, characteristic, complemented
10#69558C212yesmaximal, complemented
11#69542C212yesmaximal, complemented
12#27679—31nomaximal, Sylow
13C2×C3×#117C2212yes—
14C22×C3×C42C2211yescharacteristic
15C2×#930C2211yescharacteristic
16C2×C4×#67—21no—
17C2×C4×D12C2211yescharacteristic
18C3×#384C2211yescharacteristic
19C3×#384C2211yescharacteristic
20C3×#382C2212yes—
21C3×#380—21no—
22C3×#380—21no—
23C3×C4×C4⋊C4—21no—
24C3×C4×C4⋊C4—21no—
25#5213—21no—
26#5212C2211yescharacteristic
27C2×#830C2212yes—
28C2×#809C2212yes—
29C2×#809C2212yes—
30C2×#808C2212yes—
31C2×#807C2212yes—
32#4940—21no—
33#4939C2211yescharacteristic
34C4×(C2×D3)⋊C4—21no—
35C4×C4⋊Dic3C2211yescharacteristic
36#4928C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C4×#121—31no—
45#2285D311yescharacteristic, complemented
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—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C3×#117—42no—
62C3×#117—42no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C22⋊C4—22no—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C3×C42—22no—
73C2×C3×C42D411yescharacteristic
74C2×C3×C42D411yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×#217—21no—
78#930—22no—
79#930Q814yes—
80C4×#67—44no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×Dic3⋊C4—21no—
89C2×C4×Dic3—21no—
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2312yes—
92C4×D12—22no—
93C4×D12—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99#830—24no—
100#809—42no—
101#809—42no—
102#808—42no—
103#807—42no—
104C4⋊(C3⋊C8)—22no—
105C4⋊(C3⋊C8)—22no—
106C2×#121—31no—
107C2×C4×D4—31no—
108C2×C4×D4—61no—
109C2×#117C2×D312yes—
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#390—31no—
119#389—31no—
120#388—61no—
121#386—61no—
122#384C2×D311yescharacteristic
123#384C2×D311yescharacteristic
124#382C2×D312yes—
125#380—21no—
126#380—21no—
127#380—31no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—31no—
131C4×C22⋊C4—61no—
132#340—32no—
133#340—32no—
134#340—32no—
135#340—32no—
136#336—32no—
137C24×C3C2×D411yescharacteristic
138C23×D3—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×Q812yes—
157C22×C3×C4#4112yes—
158C22×C3×C4C2411yescharacteristic
159C2×#67—24no—
160C2×#67—42no—
161C2×#67—42no—
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12C2×Q812yes—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×C4×D3—42no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C22⋊C4—44no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—21no—
186C3×C42—21no—
187C3×C42—44no—
188C3×C42#4112yes—
189C3×C42#4112yes—
190#217—42no—
191(C2×D3)⋊C4—24no—
192(C2×D3)⋊C4—42no—
193(C2×D3)⋊C4—42no—
194(C2×D3)⋊C4—42no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3C2×Q812yes—
201Dic3⋊C4—42no—
202C4×Dic3—42no—
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—31no—
210C22×D4—61no—
211C23×C4—61no—
212C23×C4#6711yescharacteristic
213C23×C4#6711yescharacteristic
214C23×C4C22×D311yescharacteristic
215C22×C8—32no—
216C22×C8—32no—
217#121—34no—
218#121—62no—
219C4×D4—62no—
220C4×D4—62no—
221C4×D4—124no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4C22×D312yes—
233C2×C4⋊C4C22×D312yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—61no—
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—22no—
256#95—22no—
257#95—61no—
258#95—61no—
259#95—61no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3C22⋊Q812yes—
267C23×C3C22×D411yescharacteristic
268C22×D3—22no—
269C22×D3—41no—
270C22×D3—41no—
271C22×D3—41no—
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—41no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—44no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4C22⋊Q812yes—
298C2×C3×C4#12312yes—
299C2×C3×C4#12312yes—
300C2×C3×C4#13512yes—
301C2×C3×C4AGL(1,ℤ/8)12yes—
302C2×C3×C4C22×Q811yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4H(5,𝔽2)11yescharacteristic
307C2×C3×C4H(5,𝔽2)11yescharacteristic
308#67—48no—
309C2×Dic3—22no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318D12—22no—
319D12—22no—
320D12—41no—
321D12—41no—
322C4×D3—84no—
323C3⋊C8—42no—
324C3⋊C8—42no—
325C24—61no—
326C24C2×#6711yescharacteristic
327C2×D4—32no—
328C2×D4—64no—
329C2×D4—61no—
330C2×D4—61no—
331C2×D4—61no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—122no—
335C2×D4—122no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4—122no—
354C22×C4D3×Q812yes—
355C22×C4#23912yes—
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—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—32no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—122no—
383C22⋊C4—44no—
384C22⋊C4—64no—
385C22⋊C4—122no—
386C22⋊C4—122no—
387C22⋊C4—122no—
388C22⋊C4—122no—
389C42—22no—
390C42—22no—
391C42—21no—
392C42—21no—
393C42—44no—
394C42—122no—
395C42#23512yes—
396C42#23512yes—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3C2×C22⋊Q811yescharacteristic
405C22×C3C2×#12311yescharacteristic
406C22×C3#53511yescharacteristic
407C22×C3C2×#13511yescharacteristic
408C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
409C22×C3#57511yescharacteristic
410C22×C3#57511yescharacteristic
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—42no—
414C2×D3—81no—
415C2×D3—81no—
416C2×D3—81no—
417C2×D3—81no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—82no—
423C3×C4—82no—
424C3×C4#48214yes—
425C3×C4#55311yescharacteristic
426C3×C4#55311yescharacteristic
427C3×C4#55411yescharacteristic
428C3×C4#55411yescharacteristic
429Dic3—42no—
430Dic3—81no—
431Dic3—81no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—62no—
439C23—121no—
440C23—121no—
441C23—121no—
442C23—121no—
443C23—121no—
444C23—121no—
445C23#94612yes—
446C23C22×#6711yescharacteristic
447D4—62no—
448D4—62no—
449D4—128no—
450D4—121no—
451D4—121no—
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—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—62no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—244no—
484C2×C4#92912yes—
485C2×C4#92912yes—
486C2×C4#94612yes—
487C2×C4#94912yes—
488C2×C4#95012yes—
489C2×C4C2×#23511yescharacteristic
490C2×C4C2×#23511yescharacteristic
491C2×C4#100411yescharacteristic
492C2×C4#100411yescharacteristic
493C2×C4C2×D3×Q811yescharacteristic
494C2×C4C2×#23911yescharacteristic
495C8—122no—
496C8—122no—
497C2×C3—42no—
498C2×C3#248711yescharacteristic
499C2×C3#307611yescharacteristic
500C2×C3#307711yescharacteristic
501C2×C3C2×#48211yescharacteristic
502C2×C3#309311yescharacteristic
503C2×C3#321211yescharacteristic
504C2×C3#321411yescharacteristic
505D3—82no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—122no—
514C22—122no—
515C22—122no—
516C22—241no—
517C22—241no—
518C22—241no—
519C22—241no—
520C22C2×#92911yescharacteristic
521C22#580511yescharacteristic
522C22C2×#94611yescharacteristic
523C22C2×#94911yescharacteristic
524C22C2×#95011yescharacteristic
525C22#582211yescharacteristic
526C22#582211yescharacteristic
527C4—22no—
528C4—22no—
529C4—41no—
530C4—41no—
531C4—82no—
532C4—82no—
533C4—122no—
534C4—241no—
535C4—241no—
536C4#500314yes—
537C4#551111yescharacteristic
538C4#551111yescharacteristic
539C4#551411yescharacteristic
540C4#551411yescharacteristic
541C3#2767911yescharacteristic, complemented, Sylow
542C2—42no—
543C2—242no—
544C2#8219711yescharacteristic
545C2C2×#500311yescharacteristic
546C2#8221611yescharacteristic
547C2#8330511yescharacteristic
548C2#8330711yescharacteristic
549C2#8334911yescharacteristic
550C2#8335011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2285C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2285D3
hypocenterC3#27679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
1. C3×#2285C2semidirect
4. #71576C2semidirect
5. #71530C2semidirect
6. #69584C2semidirect
7. #69584C2semidirect
8. #69568C2semidirect
9. #69567C2semidirect
10. #69558C2semidirect
11. #69542C2semidirect
45. #2285D3semidirect
541. C3#27679semidirect

Maximal subgroups

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

subgroupindex
C3×#22852
C4×#9302
#715862
#715762
#715302
#695842
#695682
#695672
#695582
#695422
#276793