Finite Group Databasebeta

← #10847397

Subgroups — #10847397

Order 768.

1194 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic, complemented
2C3×#2854C211yesmaximal, characteristic
3C2×#5041C212yesmaximal
4C2×#4524C211yesmaximal, characteristic, complemented
5#69661C211yesmaximal, characteristic, complemented
6#69660C212yesmaximal, complemented
7#69638C211yesmaximal, characteristic, complemented
8#69633C211yesmaximal, characteristic, complemented
9#69632C212yesmaximal, complemented
10#69605C211yesmaximal, characteristic, complemented
11#69514C212yesmaximal, complemented
12#25497—31nomaximal, Sylow
13C2×C3×#123—21no—
14C2×C3×#121C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×#214C2212yes—
17C2×#926—21no—
18C2×#924—21no—
19C2×#924C2212yes—
20C23×C3⋊C8C2212yes—
21#5470—28no—
22C2×#870—21no—
23C2×#869C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6—21no—
26C2×C4×Dic6C2212yes—
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#389C2211yescharacteristic
31C3×#388—21no—
32C3×#388—21no—
33C3×#386—21no—
34C3×#380C2211yescharacteristic
35C2×#832C2212yes—
36#5041—28no—
37C2×#810C2212yes—
38C2×#810C2212yes—
39C2×#810C2212yes—
40C2×#808C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#533—31no—
47C2×#465—32no—
48#2854D311yescharacteristic
49#2058—31no—
50#2043—31no—
51#2023—31no—
52#2013—31no—
53#1892—32no—
54#1880—32no—
55#1865—32no—
56C2×#336—31no—
57C22×C3×D4—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×Dic6—21no—
63C22×Dic6C2312yes—
64C3×#123—44no—
65C3×#121—24no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C42—24no—
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#217—22no—
83C2×#214—24no—
84C2×#214—24no—
85C2×#214—42no—
86#926—28no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3C2312yes—
91#924—28no—
92#924—24no—
93#924D418yes—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×C4×Dic3—22no—
97C22×C3⋊C8—24no—
98C22×C3⋊C8—24no—
99C22×C3⋊C8—42no—
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2312yes—
102#870—28no—
103#869—24no—
104#869—24no—
105C4×Dic6—28no—
106C4×Dic6—24no—
107C4×Dic6—24no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111#832—24no—
112#810—24no—
113#810—24no—
114#810—42no—
115#808—24no—
116C22×SD16—32no—
117C23×C8—32no—
118#533—68no—
119C2×C4⋊Q8—31no—
120C2×C4⋊Q8—31no—
121C2×#125—61no—
122C2×#123—21no—
123C2×C22⋊Q8—32no—
124C2×C22⋊Q8—61no—
125C2×#121C2×D311yescharacteristic
126C2×C4×Q8—32no—
127C2×C4×Q8—61no—
128C2×#117—61no—
129C22×C42C2×D311yescharacteristic
130#465—68no—
131C2×#106—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#102—32no—
136#390C2×D311yescharacteristic
137#390C2×D311yescharacteristic
138#389C2×D311yescharacteristic
139#389C2×D311yescharacteristic
140#388—21no—
141#388—21no—
142#386—21no—
143#380C2×D311yescharacteristic
144#340—32no—
145#340—32no—
146#336—31no—
147#336—32no—
148#336—31no—
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—41no—
169C22×C3×C4SD1614yescomplemented
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C22×Dic3—22no—
178C22×Dic3—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—24no—
184C2×Dic6—22no—
185C2×Dic6—42no—
186C2×Dic6—42no—
187C2×Dic6C2×D414yes—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—42no—
194C3×C22⋊C4—42no—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—42no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—42no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—22no—
204#217—28no—
205#214—48no—
206C4⋊Dic3—24no—
207C4⋊Dic3—24no—
208C4⋊Dic3—24no—
209C4⋊Dic3—22no—
210C4⋊Dic3C2×D414yes—
211Dic3⋊C4—28no—
212Dic3⋊C4—28no—
213C4×Dic3—28no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—48no—
220C2×C3⋊C8#4114yes—
221C22×Q8—32no—
222C22×Q8—61no—
223C22×D4—21no—
224C22×D4C22×D311yescharacteristic
225C23×C4#6711yescharacteristic
226C23×C4#6711yescharacteristic
227C23×C4C22×D311yescharacteristic
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—122no—
231C22×C8—32no—
232C22×C8—32no—
233C22×C8—64no—
234C22×C8—64no—
235C22×C8—122no—
236C4⋊Q8—64no—
237C4⋊Q8—64no—
238#125—68no—
239#123—44no—
240C22⋊Q8—38no—
241C22⋊Q8—68no—
242C22⋊Q8—64no—
243#121—24no—
244C4×Q8—64no—
245C4×Q8—64no—
246C4×Q8—68no—
247#117—68no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—62no—
266C2×C42—24no—
267C2×C42—62no—
268C2×C42#6711yescharacteristic
269C2×C42#6711yescharacteristic
270C2×C42C22×D311yescharacteristic
271C2×C42C22×D311yescharacteristic
272#106—64no—
273#103—64no—
274#103—64no—
275#103—122no—
276#102—64no—
277#95—21no—
278#95—21no—
279#95—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3#12012yes—
292C23×C3C2×SD1612yes—
293C23×C3#13712yes—
294C23×C3C22×D411yescharacteristic
295C3×D4—42no—
296C3×D4—42no—
297C3×D4—84no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—28no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—44no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4#12012yes—
323C2×C3×C4C2×SD1612yes—
324C2×C3×C4C2×SD1612yes—
325C2×C3×C4#13512yes—
326C2×C3×C4C22×D412yes—
327C2×C3×C4C2×#4112yes—
328C2×C3×C4H(5,𝔽2)11yescharacteristic
329C2×C3×C42−1+411yescharacteristic
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336Dic6—24no—
337Dic6—24no—
338Dic6—48no—
339Dic6—42no—
340Dic6—42no—
341C3⋊C8—24no—
342C3⋊C8—42no—
343C24—21no—
344C24C2×#6711yescharacteristic
345C2×Q8—34no—
346C2×Q8—64no—
347C2×Q8—62no—
348C2×Q8—62no—
349C2×Q8—62no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—122no—
353C2×Q8—122no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—41no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4#21414yescomplemented
375C22×C4D3×D414yes—
376C22×C4C2×#6711yescharacteristic
377C22×C4C2×#6711yescharacteristic
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6711yescharacteristic
381C22×C4C23×D311yescharacteristic
382SD16—128no—
383C2×C8—34no—
384C2×C8—62no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—128no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—34no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—64no—
398C4⋊C4—64no—
399C4⋊C4—68no—
400C4⋊C4—64no—
401C4⋊C4—68no—
402C4⋊C4—62no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—68no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—68no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3#44814yes—
429C22×C3C2×#12011yescharacteristic
430C22×C3#53211yescharacteristic
431C22×C3#53711yescharacteristic
432C22×C3C22×SD1611yescharacteristic
433C22×C3C2×#13511yescharacteristic
434C22×C3C2×#13711yescharacteristic
435C22×C3#57511yescharacteristic
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—41no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4—81no—
444C3×C4#54612yes—
445C3×C4#54712yes—
446Dic3—44no—
447Dic3—44no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23C2×#21412yes—
460C23#93712yes—
461C23#95112yes—
462C23C22×#6711yescharacteristic
463Q8—64no—
464Q8—64no—
465Q8—128no—
466Q8—122no—
467Q8—122no—
468D4—42no—
469D4—42no—
470D4—84no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—28no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4C2×#21412yes—
502C2×C4C2×#21412yes—
503C2×C4#93712yes—
504C2×C4#95012yes—
505C2×C4C2×D3×D412yes—
506C2×C4D3×#4112yes—
507C2×C4#100911yescharacteristic
508C2×C4#101011yescharacteristic
509C8—64no—
510C8—122no—
511C2×C3—24no—
512C2×C3—161no—
513C2×C3#243011yescharacteristic
514C2×C3C2×#44811yescharacteristic
515C2×C3#300211yescharacteristic
516C2×C3#316711yescharacteristic
517C2×C3#318111yescharacteristic
518C2×C3#324611yescharacteristic
519C2×C3#324811yescharacteristic
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22#504014yes—
534C22C22×#21411yescharacteristic
535C22C2×#93711yescharacteristic
536C22C2×#95011yescharacteristic
537C22#582211yescharacteristic
538C22C2×#95111yescharacteristic
539C22#583211yescharacteristic
540C22#583311yescharacteristic
541C4—22no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—124no—
550C4—124no—
551C4#565412yes—
552C4#565512yes—
553C3#2549711yescharacteristic, complemented, Sylow
554C2—24no—
555C2—161no—
556C2C2×#504011yescharacteristic
557C2#8284811yescharacteristic
558C2#8292611yescharacteristic
559C2#8313811yescharacteristic
560C2#8315511yescharacteristic
561C2#8329011yescharacteristic
562C2#8332011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2854C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2854D3
hypocenterC3#25497
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#5470C2semidirect
4. C2×#4524C2semidirect
5. #69661C2semidirect
6. #69660C2semidirect
7. #69638C2semidirect
8. #69633C2semidirect
9. #69632C2semidirect
10. #69605C2semidirect
11. #69514C2semidirect
169. C22×C3×C4SD16semidirect
374. C22×C4#214semidirect
553. C3#25497semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C3×#28542
C2×#50412
C2×#45242
#696612
#696602
#696382
#696332
#696322
#696052
#695142
#254973