Finite Group Databasebeta

← #10901351

Subgroups — #10901351

Order 768.

1348 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic
2C2×#4759C212yesmaximal, complemented
3#76288C211yesmaximal, characteristic, complemented
4C3×#2041C211yesmaximal, characteristic, complemented
5#71762C211yesmaximal, characteristic, complemented
6#70206C212yesmaximal, complemented
7#70175C212yesmaximal, complemented
8#70153C212yesmaximal, complemented
9#69604C211yesmaximal, characteristic, complemented
10#68662C211yesmaximal, characteristic, complemented
11#68632C211yesmaximal, characteristic, complemented
12#29687—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×#127C2211yescharacteristic
15C2×#938—21no—
16C2×#938C2212yes—
17C2×#936—21no—
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2212yes—
20#5716—28no—
21C2×#896—21no—
22C2×#888C2211yescharacteristic
23C2×C42×D3C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C3×#389C2211yescharacteristic
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31C2×#818C2212yes—
32C2×Dic3⋊C8C2212yes—
33#4987C2211yescharacteristic
34#4987C2211yescharacteristic
35#4976—21no—
36#4975C2211yescharacteristic
37C2×#807C2212yes—
38#4928C2211yescharacteristic
39#4759—28no—
40#4667—21no—
41#4646—21no—
42C3×#336C2211yescharacteristic
43#4551C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C2×#532—31no—
47C2×#474—32no—
48#2857—31no—
49#2057—31no—
50#2057—31no—
51#2041—31no—
52#2041D311yescharacteristic, complemented
53#2033—32no—
54#2033—32no—
55#1802—32no—
56#1797—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×#67—22no—
61C23×Dic3—22no—
62C2×#237—28no—
63C2×#237—44no—
64C2×#237—44no—
65C22×C4×D3D411yescharacteristic
66C22×C4×D3D411yescharacteristic
67C22×C4×D3C2311yescharacteristic
68C22×Dic6—21no—
69C22×Dic6—21no—
70C22×C3×C8C2312yes—
71C3×#127—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×#217—22no—
77C2×#217—22no—
78#938—28no—
79#938—24no—
80#938—24no—
81#936—28no—
82D4×Dic3—28no—
83D4×Dic3—24no—
84D4×Dic3D418yes—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×C4×Dic3D411yescharacteristic
96C2×C4×Dic3D411yescharacteristic
97C2×C4×Dic3C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99#896—44no—
100#888—24no—
101C42×D3—24no—
102#869—24no—
103C3×#102—24no—
104C3×#102—42no—
105C3×#95—21no—
106#832—24no—
107#832—42no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111(C2×D3)⋊C8—42no—
112#818—24no—
113Dic3⋊C8—24no—
114#807—24no—
115C22×#41—61no—
116#532—68no—
117C2×C4⋊Q8—31no—
118C2×C4⋊Q8—31no—
119C2×#127C2×D311yescharacteristic
120C2×#124—61no—
121C2×C22⋊Q8—61no—
122C2×#121—32no—
123C2×#121—61no—
124C2×C4×D4—32no—
125C2×C4×D4—61no—
126C22×C42—31no—
127#474—68no—
128C2×#107—32no—
129C2×#107—32no—
130C2×C4⋊C8—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102C2×D312yes—
134C2×#102C2×D312yes—
135C2×C22⋊C8—32no—
136#391—61no—
137#389—31no—
138#389—61no—
139#389C2×D311yescharacteristic
140#384—31no—
141#380—31no—
142#380—31no—
143#380—61no—
144#340—32no—
145#340—32no—
146#336—31no—
147#336C2×D311yescharacteristic
148C24×C3—22no—
149C23×D3C2×D411yescharacteristic
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—42no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4C2×D414yes—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C2×#67—28no—
168C2×#67—44no—
169C2×#67—44no—
170C22×Dic3—22no—
171C22×Dic3—28no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—44no—
175C22×Dic3—44no—
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178C22×Dic3C2×D411yescharacteristic
179#237—432no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—41no—
189C2×C4×D3D814yes—
190C2×Dic6—24no—
191C2×Dic6—42no—
192C2×Dic6—42no—
193C2×Dic6—42no—
194C2×Dic6—42no—
195C2×Dic6—42no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—42no—
203C3×C42—22no—
204#217—28no—
205#217—28no—
206(C2×D3)⋊C4—42no—
207(C2×D3)⋊C4—42no—
208C4⋊Dic3—24no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3C2×D414yes—
214Dic3⋊C4—42no—
215Dic3⋊C4—42no—
216Dic3⋊C4—42no—
217Dic3⋊C4—42no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220C4×Dic3C2×D414yes—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×#41—68no—
225C2×#41—124no—
226C2×#41—124no—
227C22×Q8—61no—
228C22×Q8—61no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—62no—
232C22×D4C22×D312yes—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—62no—
237C22×C8—32no—
238C22×C8C22×D312yes—
239C4⋊Q8—64no—
240C4⋊Q8—64no—
241#127—24no—
242#124—68no—
243C22⋊Q8—124no—
244#121—64no—
245#121—68no—
246#121—64no—
247C4×D4—38no—
248C4×D4—64no—
249C4×D4—68no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C42—31no—
266C2×C42—31no—
267C2×C42—31no—
268C2×C42—64no—
269C2×C42C22×D311yescharacteristic
270#107—64no—
271#107—64no—
272C4⋊C8—64no—
273#102—24no—
274#102—42no—
275#102—64no—
276#102—122no—
277C22⋊C8—122no—
278#95—21no—
279#95—61no—
280#95—61no—
281#95—61no—
282C23×C3—24no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3C22×D411yescharacteristic
290C22×D3—21no—
291C22×D3—21no—
292C22×D3—21no—
293C22×D3—21no—
294C22×D3—21no—
295C22×D3—21no—
296C22×D3C2×D812yes—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—48no—
300C3×D4—48no—
301C3×D4—42no—
302C3×D4—42no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4C2×D812yes—
314C2×C3×C4#13712yes—
315C2×C3×C4C22×D412yes—
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×C3×C42−1+411yescharacteristic
318C2×C3×C42−1+412yes—
319#67—416no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—24no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—416no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3#12012yes—
336C2×Dic3#12012yes—
337C2×Dic3C2×D812yes—
338C2×Dic3AGL(1,ℤ/8)12yes—
339C4×D3—28no—
340C4×D3—44no—
341C4×D3—42no—
342C4×D3—42no—
343Dic6—48no—
344Dic6—84no—
345C3×C8—42no—
346C3⋊C8—42no—
347C24—22no—
348C24—31no—
349#41—1232no—
350C2×Q8—64no—
351C2×Q8—122no—
352C2×Q8—122no—
353C2×Q8—122no—
354C2×Q8—122no—
355C2×Q8—122no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—68no—
368C2×D4—124no—
369C2×D4—124no—
370C2×D4D3×D414yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—34no—
376C22×C4—31no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—68no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—121no—
390C22×C4—124no—
391C22×C4—124no—
392C22×C4D3×D411yescharacteristic
393C22×C4D3×D411yescharacteristic
394C22×C4C23×D311yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—62no—
399C2×C8—62no—
400C2×C8—62no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—34no—
405C4⋊C4—42no—
406C4⋊C4—64no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—122no—
412C4⋊C4—122no—
413C4⋊C4—122no—
414C4⋊C4—122no—
415C22⋊C4—68no—
416C22⋊C4—68no—
417C22⋊C4—122no—
418C22⋊C4—122no—
419C42—22no—
420C42—34no—
421C42—62no—
422C42—62no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3C2×#12011yescharacteristic
431C22×C3#53611yescharacteristic
432C22×C3#53711yescharacteristic
433C22×C3C22×D811yescharacteristic
434C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
435C22×C3C2×#13711yescharacteristic
436C22×C3#57511yescharacteristic
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—41no—
441C2×D3—41no—
442C2×D3—41no—
443C2×D3—41no—
444C3×C4—22no—
445C3×C4—22no—
446C3×C4—81no—
447C3×C4—81no—
448C3×C4#53612yes—
449C3×C4#53712yes—
450Dic3—22no—
451Dic3—41no—
452Dic3—44no—
453Dic3—81no—
454Dic3—81no—
455Dic3#44714yes—
456C23—24no—
457C23—32no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—61no—
465C23—61no—
466C23—61no—
467C23—61no—
468C23—61no—
469C23—61no—
470C23C2×D3×D411yescharacteristic
471Q8—128no—
472Q8—244no—
473D4—24no—
474D4—24no—
475D4—48no—
476D4—48no—
477D4—42no—
478D4—42no—
479D4—1216no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—32no—
485C2×C4—32no—
486C2×C4—32no—
487C2×C4—32no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—68no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—64no—
500C2×C4—61no—
501C2×C4—124no—
502C2×C4—1216no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4D3×D812yes—
514C2×C4#91512yes—
515C2×C4C2×D3×D412yes—
516C2×C4#100411yescharacteristic
517C2×C4#101011yescharacteristic
518C2×C4#101012yes—
519C8—42no—
520C8—122no—
521C2×C3—84no—
522C2×C3#261611yescharacteristic
523C2×C3C2×#44711yescharacteristic
524C2×C3#300711yescharacteristic
525C2×C3#322611yescharacteristic
526C2×C3#323511yescharacteristic
527C2×C3#324511yescharacteristic
528C2×C3#325411yescharacteristic
529D3—42no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—62no—
534C22—62no—
535C22—62no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—121no—
541C22—121no—
542C22—121no—
543C22—121no—
544C22D3×#12011yescharacteristic
545C22#560811yescharacteristic
546C22#567511yescharacteristic
547C22C2×D3×D811yescharacteristic
548C22C2×#91511yescharacteristic
549C22D3×AGL(1,ℤ/8)11yescharacteristic
550C22#577311yescharacteristic
551C4—22no—
552C4—22no—
553C4—34no—
554C4—62no—
555C4—81no—
556C4—81no—
557C4—124no—
558C4—121no—
559C4—241no—
560C4—241no—
561C4#559712yes—
562C4#564812yes—
563C3#2968711yescharacteristic, complemented, Sylow
564C2—84no—
565C2—122no—
566C2#7697011yescharacteristic
567C2D3×#44711yescharacteristic
568C2#8014711yescharacteristic
569C2#8034211yescharacteristic
570C2#8036711yescharacteristic
571C2#8157111yescharacteristic
572C2#8172211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×#5716C2
hypercenter#2041D3
hypocenterC3#29687
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
2. C2×#4759C2semidirect
3. #76288C2semidirect
4. C3×#2041C2semidirect
5. #71762C2semidirect
6. #70206C2semidirect
7. #70175C2semidirect
8. #70153C2semidirect
9. #69604C2semidirect
10. #68662C2semidirect
11. #68632C2semidirect
52. #2041D3semidirect
563. C3#29687semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×#47592
#762882
C3×#20412
#717622
#702062
#701752
#701532
#696042
#686622
#686322
#296873