Finite Group Databasebeta

← #10877200

Subgroups — #10877200

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#938C211yesmaximal, characteristic
2D3×#384C211yesmaximal, characteristic, complemented
3C3×#1802C211yesmaximal, characteristic, complemented
4#71604C211yesmaximal, characteristic, complemented
5#70475C211yesmaximal, characteristic, complemented
6#70454C211yesmaximal, characteristic, complemented
7#70200C212yesmaximal
8#70184C212yesmaximal, complemented
9#69568C211yesmaximal, characteristic, complemented
10#68875C212yesmaximal, complemented
11#68811C212yesmaximal, complemented
12#27678—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#938C2211yescharacteristic
15C2×C4×#67—21no—
16C2×D3×C4⋊C4—21no—
17C2×D3×C4⋊C4—21no—
18C2×D3×C4⋊C4C2212yes—
19C2×C42×D3C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#102C2212yes—
22C3×#384C2211yescharacteristic
23#5221C2211yescharacteristic
24C2×#832C2212yes—
25C4×#217—21no—
26C2×(C2×D3)⋊C8C2212yes—
27C2×#817C2212yes—
28#4987C2211yescharacteristic
29#4975—21no—
30#4975—21no—
31#4975C2212yes—
32#4972C2211yescharacteristic
33C2×#808C2212yes—
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—
45C2×#384—31no—
46#2100—32no—
47#2094—32no—
48#1948—31no—
49#1948—31no—
50#1874—32no—
51#1853—32no—
52#1802—31no—
53#1802D311yescharacteristic, complemented
54#1797—31no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C22×#67—21no—
58C22×C4×D3—22no—
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3C2311yescharacteristic
62C22×C3×C8C2312yes—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#217—21no—
72C2×#217—21no—
73C2×#217—21no—
74#938—22no—
75#938Q814yes—
76C4×#67—44no—
77C2×(C2×D3)⋊C4—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×C4×Dic3—21no—
88C2×C4×Dic3D411yescharacteristic
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3C2311yescharacteristic
91C22×C3⋊C8C2312yes—
92D3×C4⋊C4—42no—
93D3×C4⋊C4—42no—
94D3×C4⋊C4—44no—
95D3×C4⋊C4—44no—
96C42×D3—22no—
97C42×D3—22no—
98C3×C4⋊C8—22no—
99C3×#102—42no—
100#832—42no—
101C3⋊#95—21no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104(C2×D3)⋊C8—24no—
105#817—42no—
106#808—42no—
107C4⋊(C3⋊C8)—22no—
108C2×#121—31no—
109C2×C4×D4—61no—
110C2×C4×D4C2×D311yescharacteristic
111C22×C4⋊C4—32no—
112C22×C4⋊C4—61no—
113C22×C4⋊C4—61no—
114C22×C42—31no—
115C2×#106—32no—
116C2×#106—32no—
117C2×C4⋊C8—31no—
118C2×C4⋊C8C2×D311yescharacteristic
119C2×#102—32no—
120C2×#102C2×D312yes—
121C2×C22⋊C8—32no—
122#390—31no—
123#389—31no—
124#388—61no—
125#386—61no—
126#384—32no—
127#384—31no—
128#384—61no—
129#384—61no—
130#384C2×D311yescharacteristic
131#380—31no—
132C4×C4⋊C4—31no—
133C4×C22⋊C4—61no—
134#340—32no—
135#340—32no—
136#340—32no—
137#340C2×D312yes—
138#336—32no—
139C24×C3—21no—
140C23×D3C2×D411yescharacteristic
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4C2×Q812yes—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C2×#67—24no—
156C2×#67—42no—
157C2×#67—42no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3C2×D411yescharacteristic
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—21no—
169C2×C4×D3—21no—
170C2×C4×D3—21no—
171C2×C4×D3—21no—
172C2×C4×D3—22no—
173C2×C4×D3—44no—
174C2×C4×D3—44no—
175C2×C4×D3—44no—
176C2×C4×D3SD1614yes—
177C2×C4×D3C2×Q812yes—
178C2×C4×D3#4112yes—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—42no—
188C3×C22⋊C4—42no—
189C3×C42#4112yes—
190C3×C42#4112yes—
191#217—24no—
192#217—42no—
193#217—42no—
194(C2×D3)⋊C4—42no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200C4⋊Dic3—42no—
201C4⋊Dic3—42no—
202C4⋊Dic3C2×Q812yes—
203Dic3⋊C4—44no—
204Dic3⋊C4—44no—
205Dic3⋊C4—44no—
206Dic3⋊C4—42no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C4×Dic3—21no—
210C4×Dic3—21no—
211C4×Dic3—42no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C22×D4—61no—
216C22×D4C22×D311yescharacteristic
217C23×C4—21no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—62no—
222C22×C8—32no—
223C22×C8C22×D312yes—
224#121—34no—
225#121—62no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—124no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—122no—
241C2×C4⋊C4—122no—
242C2×C4⋊C4—124no—
243C2×C4⋊C4—124no—
244C2×C4⋊C4C22×D312yes—
245C2×C4⋊C4C22×D311yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—61no—
249C2×C22⋊C4—61no—
250C2×C22⋊C4—61no—
251C2×C42—31no—
252C2×C42—31no—
253C2×C42—31no—
254C2×C42—62no—
255C2×C42—62no—
256C2×C42—61no—
257C2×C42C22×D311yescharacteristic
258#106—122no—
259#106—122no—
260C4⋊C8—22no—
261C4⋊C8—62no—
262#102—42no—
263#102—122no—
264C22⋊C8—64no—
265#95—61no—
266#95—61no—
267#95—61no—
268C23×C3—22no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3C22×D411yescharacteristic
276C22×D3—21no—
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3C22⋊Q812yes—
281C22×D3C2×SD1612yes—
282C22×D3AGL(1,ℤ/8)12yes—
283C3×D4—22no—
284C3×D4—22no—
285C3×D4—41no—
286C3×D4—41no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—84no—
296C2×C3×C4#12312yes—
297C2×C3×C4#12312yes—
298C2×C3×C4C2×SD1612yes—
299C2×C3×C4AGL(1,ℤ/8)12yes—
300C2×C3×C4C22×Q811yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4H(5,𝔽2)11yescharacteristic
305C2×C3×C4H(5,𝔽2)11yescharacteristic
306#67—48no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—22no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—41no—
322C2×Dic3C22⋊Q812yes—
323C2×Dic3C2×SD1612yes—
324C2×Dic3#13712yes—
325C4×D3—28no—
326C4×D3—22no—
327C4×D3—22no—
328C4×D3—22no—
329C4×D3—22no—
330C4×D3—28no—
331C4×D3—88no—
332C3×C8—42no—
333C3⋊C8—42no—
334C24—21no—
335C24—31no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—64no—
342C2×D4—122no—
343C2×D4—122no—
344C2×D4D3×Q812yes—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—34no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—61no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—124no—
369C22×C4—124no—
370C22×C4—124no—
371C22×C4D3×D411yescharacteristic
372C22×C4D3×D411yescharacteristic
373C22×C4C23×D311yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8—62no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—32no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—61no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—62no—
391C4⋊C4—61no—
392C4⋊C4—122no—
393C4⋊C4—122no—
394C4⋊C4—124no—
395C4⋊C4—124no—
396C4⋊C4—124no—
397C4⋊C4—122no—
398C22⋊C4—42no—
399C22⋊C4—64no—
400C22⋊C4—122no—
401C22⋊C4—122no—
402C22⋊C4—122no—
403C42—62no—
404C42—62no—
405C42—61no—
406C42—61no—
407C42—122no—
408C42#23512yes—
409C42#23712yes—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3—81no—
416C22×C3—81no—
417C22×C3C2×C22⋊Q811yescharacteristic
418C22×C3C2×#12311yescharacteristic
419C22×C3#53511yescharacteristic
420C22×C3C22×SD1611yescharacteristic
421C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
422C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
423C22×C3C2×#13711yescharacteristic
424C2×D3—22no—
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—22no—
430C2×D3#47614yes—
431C3×C4—22no—
432C3×C4—81no—
433C3×C4—82no—
434C3×C4#48114yes—
435C3×C4#55311yescharacteristic
436C3×C4#55311yescharacteristic
437C3×C4#55411yescharacteristic
438C3×C4#55411yescharacteristic
439Dic3—22no—
440Dic3—22no—
441Dic3—22no—
442Dic3—42no—
443Dic3—81no—
444Dic3—82no—
445C23—22no—
446C23—32no—
447C23—32no—
448C23—32no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—61no—
456C23—61no—
457C23—61no—
458C23—61no—
459C23C2×D3×D411yescharacteristic
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—128no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—32no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—68no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—68no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—84no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4—121no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—121no—
498C2×C4—248no—
499C2×C4#88312yes—
500C2×C4#88312yes—
501C2×C4#91112yes—
502C2×C4D3×SD1612yes—
503C2×C4C2×#23511yescharacteristic
504C2×C4C2×#23711yescharacteristic
505C2×C4#100411yescharacteristic
506C2×C4C2×D3×Q811yescharacteristic
507C2×C4D3×#4111yescharacteristic
508C2×C4#100911yescharacteristic
509C8—42no—
510C8—122no—
511C2×C3—82no—
512C2×C3#248711yescharacteristic
513C2×C3C2×#47611yescharacteristic
514C2×C3#307911yescharacteristic
515C2×C3C2×#48111yescharacteristic
516C2×C3#309211yescharacteristic
517C2×C3#320611yescharacteristic
518C2×C3#321711yescharacteristic
519D3—24no—
520C22—34no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—62no—
525C22—62no—
526C22—62no—
527C22—62no—
528C22—62no—
529C22—62no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22C2×#88311yescharacteristic
535C22#560711yescharacteristic
536C22D3×C22⋊Q811yescharacteristic
537C22C2×#91111yescharacteristic
538C22C2×D3×SD1611yescharacteristic
539C22D3×AGL(1,ℤ/8)11yescharacteristic
540C22D3×#13711yescharacteristic
541C4—22no—
542C4—62no—
543C4—62no—
544C4—62no—
545C4—81no—
546C4—82no—
547C4—122no—
548C4—241no—
549C4—242no—
550C4#477714yes—
551C4#551111yescharacteristic
552C4#553011yescharacteristic
553C4#561411yescharacteristic
554C4#563511yescharacteristic
555C3#2767811yescharacteristic, complemented, Sylow
556C2—64no—
557C2—82no—
558C2#7391211yescharacteristic
559C2C2×#477711yescharacteristic
560C2#7889211yescharacteristic
561C2#8033911yescharacteristic
562C2#8043211yescharacteristic
563C2#8096611yescharacteristic
564C2D3×#47611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#1802D3
hypocenterC3#27678
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. D3×#384C2semidirect
3. C3×#1802C2semidirect
4. #71604C2semidirect
5. #70475C2semidirect
6. #70454C2semidirect
8. #70184C2semidirect
9. #69568C2semidirect
10. #68875C2semidirect
11. #68811C2semidirect
53. #1802D3semidirect
555. C3#27678semidirect

Maximal subgroups

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

subgroupindex
C4×#9382
D3×#3842
C3×#18022
#716042
#704752
#704542
#702002
#701842
#695682
#688752
#688112
#276783