Finite Group Databasebeta

← #10877215

Subgroups — #10877215

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
1C4×#938C211yesmaximal, characteristic
2#74318C211yesmaximal, characteristic, complemented
3C3×#1802C211yesmaximal, characteristic, complemented
4#71604C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic, complemented
6#70454C211yesmaximal, characteristic, complemented
7#70201C212yesmaximal
8#70182C212yesmaximal, complemented
9#69567C211yesmaximal, characteristic, complemented
10#68876C212yesmaximal, complemented
11#68844C212yesmaximal, complemented
12#27679—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#938C2211yescharacteristic
15C2×C4×#67—21no—
16C2×#892C2212yes—
17C2×C42×D3C2211yescharacteristic
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×#102C2212yes—
20C3×#384C2211yescharacteristic
21#5221C2211yescharacteristic
22C2×#832C2212yes—
23C4×#217—21no—
24C2×(C2×D3)⋊C8C2212yes—
25C2×#817C2212yes—
26#4987—21no—
27#4987—21no—
28#4987C2211yescharacteristic
29#4977C2212yes—
30Dic3×C4⋊C4—21no—
31Dic3×C4⋊C4—21no—
32#4972C2211yescharacteristic
33C2×#807C2212yes—
34#4956—21no—
35#4944—21no—
36#4935C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
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#1947—31no—
50#1875—32no—
51#1854—32no—
52#1802—31no—
53#1802D311yescharacteristic, complemented
54#1797—31no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C22×#67—21no—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3C2311yescharacteristic
61C22×C3×C8C2312yes—
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×C3×C42C2311yescharacteristic
70C2×#217—21no—
71C2×#217—21no—
72C2×#217—21no—
73#938—22no—
74#938Q814yes—
75C4×#67—44no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—21no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3C2311yescharacteristic
89C22×C3⋊C8C2312yes—
90#892—42no—
91#892—42no—
92C42×D3—22no—
93C42×D3—22no—
94C3×C4⋊C8—22no—
95C3×#102—42no—
96#832—42no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102(C2×D3)⋊C8—24no—
103#817—42no—
104#807—42no—
105C4⋊(C3⋊C8)—22no—
106C2×#121—31no—
107C2×C4×D4—61no—
108C2×C4×D4C2×D311yescharacteristic
109C2×#117—32no—
110C22×C42—31no—
111C2×#107—32no—
112C2×#106—32no—
113C2×C4⋊C8—31no—
114C2×C4⋊C8C2×D311yescharacteristic
115C2×#102—32no—
116C2×#102C2×D312yes—
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—31no—
129C4×C4⋊C4—61no—
130C4×C4⋊C4—61no—
131C4×C22⋊C4—61no—
132#340—32no—
133#340—32no—
134#340—32no—
135#340C2×D312yes—
136#336—32no—
137C24×C3—21no—
138C23×D3C2×D411yescharacteristic
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4C2×Q812yes—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2411yescharacteristic
153C2×#67—24no—
154C2×#67—42no—
155C2×#67—42no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—21no—
159C22×Dic3C2×D411yescharacteristic
160C22×Dic3C2×D411yescharacteristic
161C22×Dic3C2×D411yescharacteristic
162C2×C4×D3—21no—
163C2×C4×D3—21no—
164C2×C4×D3—21no—
165C2×C4×D3—21no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×C4×D3—22no—
170C2×C4×D3—41no—
171C2×C4×D3—41no—
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—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C22⋊C4—42no—
184C3×C42#4112yes—
185C3×C42#4112yes—
186#217—24no—
187#217—42no—
188#217—42no—
189(C2×D3)⋊C4—44no—
190(C2×D3)⋊C4—42no—
191C4⋊Dic3—21no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—42no—
197C4⋊Dic3—42no—
198C4⋊Dic3C2×Q812yes—
199Dic3⋊C4—42no—
200C4×Dic3—22no—
201C4×Dic3—22no—
202C4×Dic3—21no—
203C4×Dic3—21no—
204C4×Dic3—44no—
205C4×Dic3—42no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C22×D4—61no—
210C22×D4C22×D311yescharacteristic
211C23×C4—21no—
212C23×C4—31no—
213C23×C4—31no—
214C23×C4—31no—
215C22×C8—32no—
216C22×C8C22×D312yes—
217#121—34no—
218#121—62no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—124no—
222#117—122no—
223#117—122no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4C22×D312yes—
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C42—31no—
241C2×C42—31no—
242C2×C42—31no—
243C2×C42—62no—
244C2×C42—62no—
245C2×C42—62no—
246C2×C42—61no—
247C2×C42C22×D311yescharacteristic
248#107—122no—
249#106—122no—
250C4⋊C8—22no—
251C4⋊C8—62no—
252#102—42no—
253#102—122no—
254C22⋊C8—64no—
255#95—61no—
256#95—61no—
257#95—62no—
258#95—62no—
259#95—61no—
260C23×C3—22no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3C22×D411yescharacteristic
268C22×D3—21no—
269C22×D3—21no—
270C22×D3—21no—
271C22×D3—21no—
272C22×D3—21no—
273C22×D3—21no—
274C22×D3C22⋊Q812yes—
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—41no—
278C3×D4—41no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—84no—
288C2×C3×C4#12312yes—
289C2×C3×C4#12312yes—
290C2×C3×C4#13512yes—
291C2×C3×C4AGL(1,ℤ/8)12yes—
292C2×C3×C4C22×Q811yescharacteristic
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4H(5,𝔽2)11yescharacteristic
297C2×C3×C4H(5,𝔽2)11yescharacteristic
298#67—48no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—22no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—41no—
314C2×Dic3—42no—
315C2×Dic3—41no—
316C2×Dic3C22⋊Q812yes—
317C4×D3—22no—
318C4×D3—22no—
319C4×D3—22no—
320C4×D3—22no—
321C4×D3—44no—
322C4×D3—44no—
323C3×C8—42no—
324C3⋊C8—42no—
325C24—21no—
326C24—31no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—64no—
333C2×D4—122no—
334C2×D4—122no—
335C2×D4D3×Q812yes—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—31no—
339C22×C4—32no—
340C22×C4—32no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—62no—
357C22×C4—121no—
358C22×C4—121no—
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—32no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—62no—
378C4⋊C4—62no—
379C4⋊C4—61no—
380C4⋊C4—122no—
381C4⋊C4—122no—
382C4⋊C4—122no—
383C22⋊C4—42no—
384C22⋊C4—64no—
385C22⋊C4—124no—
386C22⋊C4—122no—
387C22⋊C4—122no—
388C22⋊C4—122no—
389C42—62no—
390C42—62no—
391C42—61no—
392C42—61no—
393C42—124no—
394C42—122no—
395C42#23512yes—
396C42#23712yes—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
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—22no—
412C2×D3—22no—
413C2×D3—22no—
414C2×D3—41no—
415C2×D3—41no—
416C2×D3—41no—
417C2×D3—41no—
418C3×C4—22no—
419C3×C4—81no—
420C3×C4—82no—
421C3×C4#48214yes—
422C3×C4#55311yescharacteristic
423C3×C4#55311yescharacteristic
424C3×C4#55411yescharacteristic
425C3×C4#55411yescharacteristic
426Dic3—22no—
427Dic3—42no—
428Dic3—41no—
429Dic3—41no—
430Dic3—82no—
431Dic3—81no—
432C23—22no—
433C23—32no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—61no—
441C23—61no—
442C23—61no—
443C23—61no—
444C23—61no—
445C23—61no—
446C23C2×D3×D411yescharacteristic
447D4—22no—
448D4—22no—
449D4—41no—
450D4—41no—
451D4—128no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—32no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—62no—
474C2×C4—84no—
475C2×C4—124no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—124no—
479C2×C4—121no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4#88312yes—
486C2×C4#88312yes—
487C2×C4#91112yes—
488C2×C4#91612yes—
489C2×C4C2×#23511yescharacteristic
490C2×C4C2×#23711yescharacteristic
491C2×C4#100411yescharacteristic
492C2×C4C2×D3×Q811yescharacteristic
493C2×C4D3×#4111yescharacteristic
494C2×C4#100911yescharacteristic
495C8—42no—
496C8—122no—
497C2×C3—82no—
498C2×C3#248711yescharacteristic
499C2×C3#307611yescharacteristic
500C2×C3#307711yescharacteristic
501C2×C3C2×#48211yescharacteristic
502C2×C3#309311yescharacteristic
503C2×C3#321211yescharacteristic
504C2×C3#321411yescharacteristic
505D3—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—62no—
510C22—62no—
511C22—62no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—121no—
517C22—121no—
518C22—121no—
519C22—121no—
520C22C2×#88311yescharacteristic
521C22#560711yescharacteristic
522C22D3×C22⋊Q811yescharacteristic
523C22C2×#91111yescharacteristic
524C22C2×#91611yescharacteristic
525C22#577311yescharacteristic
526C22#577711yescharacteristic
527C4—22no—
528C4—62no—
529C4—81no—
530C4—82no—
531C4—121no—
532C4—122no—
533C4—121no—
534C4—242no—
535C4—241no—
536C4#477914yes—
537C4#551111yescharacteristic
538C4#553011yescharacteristic
539C4#561411yescharacteristic
540C4#563511yescharacteristic
541C3#2767911yescharacteristic, complemented, Sylow
542C2—82no—
543C2—122no—
544C2#7391211yescharacteristic
545C2C2×#477911yescharacteristic
546C2#7889311yescharacteristic
547C2#8033811yescharacteristic
548C2#8043311yescharacteristic
549C2#8096511yescharacteristic
550C2#8105911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#1802D3
hypocenterC3#27679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #74318C2semidirect
3. C3×#1802C2semidirect
4. #71604C2semidirect
5. #70475C2semidirect
6. #70454C2semidirect
8. #70182C2semidirect
9. #69567C2semidirect
10. #68876C2semidirect
11. #68844C2semidirect
53. #1802D3semidirect
541. C3#27679semidirect

Maximal subgroups

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

subgroupindex
C4×#9382
#743182
C3×#18022
#716042
#704752
#704542
#702012
#701822
#695672
#688762
#688442
#276793