Finite Group Databasebeta

← #10877216

Subgroups — #10877216

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
1#74318C211yesmaximal, characteristic, complemented
2C4×#895C211yesmaximal, characteristic
3C3×#1947C211yesmaximal, characteristic, complemented
4#70640C212yesmaximal, complemented
5#70633C212yesmaximal, complemented
6#70479C211yesmaximal, characteristic, complemented
7#70475C211yesmaximal, characteristic, complemented
8#69584C211yesmaximal, characteristic, complemented
9#69467C211yesmaximal, characteristic, complemented
10#68881C212yesmaximal
11#68846C212yesmaximal, complemented
12#27679—31nomaximal, Sylow
13C2×#895C2211yescharacteristic
14C2×#892C2212yes—
15C2×C4×D12—21no—
16C2×C4×D12C2211yescharacteristic
17C2×C42×D3C2211yescharacteristic
18C2×C3×#107C2212yes—
19C2×C3×C4⋊C8C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C3×C4×C4⋊C4C2211yescharacteristic
22C2×#821C2212yes—
23C2×(C2×D3)⋊C8C2212yes—
24#4988C2211yescharacteristic
25#4987—21no—
26#4987—21no—
27#4987C2211yescharacteristic
28#4984C2211yescharacteristic
29#4977C2212yes—
30Dic3×C4⋊C4—21no—
31Dic3×C4⋊C4—21no—
32C2×#809C2212yes—
33C2×#808C2212yes—
34C4×(C2×D3)⋊C4—21no—
35#4935C2211yescharacteristic
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37#4670—21no—
38#4665—21no—
39C3×#340C2212yes—
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C4×#121—31no—
45#2285—31no—
46#2101—32no—
47#2092—32no—
48#1948—31no—
49#1947D311yescharacteristic, complemented
50#1875—32no—
51#1854—32no—
52#1802—31no—
53#1802—31no—
54#1797—31no—
55C22×D12—21no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3C2311yescharacteristic
61C22×C3×C8C2312yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C42—21no—
68C2×C3×C42C2311yescharacteristic
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4×Dic3—22no—
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C22×C3⋊C8C2312yes—
85#895—22no—
86#895Q814yes—
87#892—42no—
88#892—42no—
89C4×D12—22no—
90C4×D12—22no—
91C4×D12—44no—
92C42×D3—22no—
93C42×D3—22no—
94C3×#107—42no—
95C3×C4⋊C8—22no—
96C3×#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#821—42no—
102(C2×D3)⋊C8—24no—
103#809—42no—
104#808—42no—
105C4⋊(C3⋊C8)—22no—
106C2×#121—31no—
107C2×C4×D4—31no—
108C2×C4×D4—61no—
109C2×#117—32no—
110C22×C42—31no—
111C2×#107C2×D312yes—
112C2×#106—32no—
113C2×C4⋊C8—31no—
114C2×C4⋊C8C2×D311yescharacteristic
115C2×#102—32no—
116C2×#102—32no—
117C2×C22⋊C8—32no—
118#390—31no—
119#389—31no—
120#388—61no—
121#386—61no—
122#384—31no—
123#384C2×D311yescharacteristic
124#382—32no—
125#380—31no—
126#380—61no—
127#380—61no—
128C4×C4⋊C4—61no—
129C4×C4⋊C4—61no—
130C4×C4⋊C4C2×D311yescharacteristic
131C4×C22⋊C4—61no—
132#340—32no—
133#340—32no—
134#340—32no—
135#340C2×D312yes—
136#336—32no—
137C23×D3—21no—
138C23×D3C2×D411yescharacteristic
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—22no—
146C22×Dic3—21no—
147C22×Dic3C2×D411yescharacteristic
148C22×Dic3C2×D411yescharacteristic
149C22×Dic3C2×D411yescharacteristic
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—21no—
153C2×D12—21no—
154C2×D12—24no—
155C2×D12—21no—
156C2×D12—42no—
157C2×D12—42no—
158C2×D12C2×Q812yes—
159C2×C4×D3—21no—
160C2×C4×D3—21no—
161C2×C4×D3—21no—
162C2×C4×D3—21no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—41no—
168C2×C4×D3—41no—
169C2×C4×D3—42no—
170C2×C4×D3—42no—
171C2×C4×D3—42no—
172C2×C4×D3C2×Q812yes—
173C2×C4×D3#4112yes—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4C2×Q812yes—
185C3×C42—42no—
186C3×C42#4112yes—
187C3×C42#4112yes—
188(C2×D3)⋊C4—24no—
189(C2×D3)⋊C4—42no—
190(C2×D3)⋊C4—42no—
191(C2×D3)⋊C4—42no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—42no—
194C4⋊Dic3—21no—
195C4⋊Dic3—21no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3—42no—
199C4⋊Dic3—42no—
200C4⋊Dic3—42no—
201C4×Dic3—22no—
202C4×Dic3—22no—
203C4×Dic3—21no—
204C4×Dic3—21no—
205C4×Dic3—44no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C22×D4—31no—
210C22×D4—61no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—31no—
214C23×C4—61no—
215C22×C8—32no—
216C22×C8C22×D312yes—
217#121—34no—
218#121—62no—
219C4×D4—62no—
220C4×D4—62no—
221C4×D4—124no—
222#117—122no—
223#117—122no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4C22×D312yes—
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C22⋊C4—61no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C42—21no—
241C2×C42—31no—
242C2×C42—31no—
243C2×C42—31no—
244C2×C42—62no—
245C2×C42—62no—
246C2×C42—62no—
247C2×C42C22×D311yescharacteristic
248#107—42no—
249#106—122no—
250C4⋊C8—22no—
251C4⋊C8—62no—
252#102—122no—
253#102—122no—
254C22⋊C8—64no—
255#95—21no—
256#95—61no—
257#95—61no—
258#95—62no—
259#95—62no—
260C23×C3C22×D411yescharacteristic
261C22×D3—21no—
262C22×D3—21no—
263C22×D3—21no—
264C22×D3—21no—
265C22×D3—21no—
266C22×D3—21no—
267C22×D3—22no—
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—41no—
271C22×D3—41no—
272C22×D3—41no—
273C22×D3—41no—
274C22×D3C22⋊Q812yes—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4#12312yes—
287C2×C3×C4#12312yes—
288C2×C3×C4#13512yes—
289C2×C3×C4AGL(1,ℤ/8)12yes—
290C2×C3×C4C22×Q811yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C4H(5,𝔽2)11yescharacteristic
295C2×C3×C4H(5,𝔽2)11yescharacteristic
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—41no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3C22⋊Q812yes—
311D12—22no—
312D12—22no—
313D12—48no—
314D12—41no—
315D12—41no—
316C4×D3—22no—
317C4×D3—22no—
318C4×D3—22no—
319C4×D3—22no—
320C4×D3—44no—
321C4×D3—44no—
322C4×D3—84no—
323C3×C8—42no—
324C3⋊C8—42no—
325C24—31no—
326C24—61no—
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—22no—
339C22×C4—31no—
340C22×C4—32no—
341C22×C4—32no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—121no—
355C22×C4—121no—
356C22×C4—122no—
357C22×C4—122no—
358C22×C4—122no—
359C22×C4D3×D411yescharacteristic
360C22×C4D3×D411yescharacteristic
361C22×C4C23×D311yescharacteristic
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—62no—
366C2×C8—62no—
367C2×C8—62no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—61no—
376C4⋊C4—62no—
377C4⋊C4—61no—
378C4⋊C4—62no—
379C4⋊C4—122no—
380C4⋊C4—122no—
381C4⋊C4—122no—
382C4⋊C4D3×Q812yes—
383C22⋊C4—64no—
384C22⋊C4—122no—
385C22⋊C4—122no—
386C22⋊C4—124no—
387C22⋊C4—122no—
388C22⋊C4—122no—
389C42—42no—
390C42—62no—
391C42—62no—
392C42—61no—
393C42—61no—
394C42—124no—
395C42#23512yes—
396C42#23912yes—
397C22×C3C2×C22⋊Q811yescharacteristic
398C22×C3C2×#12311yescharacteristic
399C22×C3#53511yescharacteristic
400C22×C3C2×#13511yescharacteristic
401C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
402C22×C3#57511yescharacteristic
403C22×C3#57511yescharacteristic
404C2×D3—22no—
405C2×D3—22no—
406C2×D3—22no—
407C2×D3—41no—
408C2×D3—41no—
409C2×D3—41no—
410C2×D3—41no—
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—42no—
420C3×C4—82no—
421C3×C4—81no—
422C3×C4#48214yes—
423C3×C4#55311yescharacteristic
424C3×C4#55311yescharacteristic
425C3×C4#55411yescharacteristic
426C3×C4#55411yescharacteristic
427Dic3—22no—
428Dic3—41no—
429Dic3—41no—
430Dic3—81no—
431Dic3—82no—
432C23—32no—
433C23—61no—
434C23—61no—
435C23—61no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—62no—
440C23—121no—
441C23—121no—
442C23—121no—
443C23—121no—
444C23—121no—
445C23—121no—
446C23C2×D3×D411yescharacteristic
447D4—62no—
448D4—62no—
449D4—128no—
450D4—121no—
451D4—121no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—32no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—124no—
477C2×C4—124no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—244no—
485C2×C4#89412yes—
486C2×C4#89412yes—
487C2×C4#91112yes—
488C2×C4#91912yes—
489C2×C4C2×#23511yescharacteristic
490C2×C4#100411yescharacteristic
491C2×C4C2×D3×Q811yescharacteristic
492C2×C4C2×#23911yescharacteristic
493C2×C4D3×#4111yescharacteristic
494C2×C4#100911yescharacteristic
495C8—42no—
496C8—122no—
497C2×C3#248711yescharacteristic
498C2×C3#307611yescharacteristic
499C2×C3#307711yescharacteristic
500C2×C3C2×#48211yescharacteristic
501C2×C3#309311yescharacteristic
502C2×C3#321211yescharacteristic
503C2×C3#321411yescharacteristic
504D3—42no—
505D3—82no—
506C22—62no—
507C22—62no—
508C22—62no—
509C22—121no—
510C22—121no—
511C22—121no—
512C22—121no—
513C22—122no—
514C22—122no—
515C22—122no—
516C22—241no—
517C22—241no—
518C22—241no—
519C22—241no—
520C22C2×#89411yescharacteristic
521C22#560711yescharacteristic
522C22D3×C22⋊Q811yescharacteristic
523C22C2×#91111yescharacteristic
524C22C2×#91911yescharacteristic
525C22#577311yescharacteristic
526C22#577711yescharacteristic
527C4—22no—
528C4—42no—
529C4—62no—
530C4—81no—
531C4—82no—
532C4—121no—
533C4—121no—
534C4—241no—
535C4—242no—
536C4#488514yes—
537C4#555811yescharacteristic
538C4#557911yescharacteristic
539C4#569511yescharacteristic
540C4#573011yescharacteristic
541C3#2767911yescharacteristic, complemented, Sylow
542C2—122no—
543C2—242no—
544C2#7623211yescharacteristic
545C2C2×#488511yescharacteristic
546C2#7921611yescharacteristic
547C2#8033811yescharacteristic
548C2#8048111yescharacteristic
549C2#8096511yescharacteristic
550C2#8110711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1947C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1947D3
hypocenterC3#27679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
1. #74318C2semidirect
3. C3×#1947C2semidirect
4. #70640C2semidirect
5. #70633C2semidirect
6. #70479C2semidirect
7. #70475C2semidirect
8. #69584C2semidirect
9. #69467C2semidirect
11. #68846C2semidirect
49. #1947D3semidirect
541. C3#27679semidirect

Maximal subgroups

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

subgroupindex
#743182
C4×#8952
C3×#19472
#706402
#706332
#704792
#704752
#695842
#694672
#688812
#688462
#276793