Finite Group Databasebeta

← #10845415

Subgroups — #10845415

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2537C211yesmaximal, characteristic
2#82382C211yesmaximal, characteristic, complemented
3C2×#5012C211yesmaximal, characteristic, complemented
4C2×#5007C211yesmaximal, characteristic, complemented
5#71682C212yesmaximal, complemented
6#71680C212yesmaximal, complemented
7C4×#832C211yesmaximal, characteristic, complemented
8#69582C211yesmaximal, characteristic, complemented
9#69568C211yesmaximal, characteristic, complemented
10#69519C212yesmaximal, complemented
11#69513C212yesmaximal, complemented
12#25347—31nomaximal, Sylow
13C2×C3×#121—21no—
14C2×C3×#121C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#936—21no—
18C22×#208C2212yes—
19C2×#869C2211yescharacteristic
20C3×#394—22no—
21C3×#384C2211yescharacteristic
22C3×#380—21no—
23C3×#380C2212yes—
24C2×#832C2211yescharacteristic
25C2×#832C2212yes—
26C2×#832C2211yescharacteristic
27#5212C2212yes—
28#5212C2212yes—
29C2×#830C2212yes—
30#5012—24no—
31#5007—24no—
32#4973—21no—
33C2×#808C2212yes—
34C2×#807C2212yes—
35#4953—22no—
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#474—31no—
45C2×#436—31no—
46#2547—31no—
47#2537D311yescharacteristic
48#2076—32no—
49#2034—32no—
50#1990—31no—
51#1948—31no—
52#1895—32no—
53#1809—32no—
54C4×#102—31no—
55C22×C3×D4—22no—
56C22×C3×D4D411yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4C2312yes—
60C22×Dic6—21no—
61C3×#121—24no—
62C3×#121—42no—
63C3×#121—44no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4D412yes—
77C2×C3×C42C2311yescharacteristic
78C2×#217—22no—
79C2×#217—22no—
80#936—44no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×C4×Dic3—21no—
86C2×#208—24no—
87C2×#208—24no—
88C2×#208—42no—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92#869—22no—
93#869—22no—
94C3×#95—22no—
95#832—22no—
96#832—24no—
97#832—22no—
98C3⋊#95—21no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#830—24no—
102#808—42no—
103#807—24no—
104C4⋊(C3⋊C8)D414yescomplemented
105C4×C3⋊C8—22no—
106C22×M16—32no—
107C2×C4⋊Q8—31no—
108C2×#124—61no—
109C2×#121—21no—
110C2×#121C2×D312yes—
111C2×C4×D4C2×D311yescharacteristic
112C2×C4×D4C2×D311yescharacteristic
113#474—64no—
114#436—64no—
115C2×#107—32no—
116C2×#106—32no—
117C2×C4⋊C8—31no—
118C2×#102—31no—
119C2×#102—32no—
120C2×#102—31no—
121C2×C22⋊C8—32no—
122C2×C4×C8—31no—
123#397—62no—
124#394—22no—
125#391—61no—
126#384C2×D311yescharacteristic
127#380—21no—
128#380—32no—
129#380—32no—
130#380C2×D312yes—
131C4×C4⋊C4—31no—
132#340—31no—
133#340—32no—
134#340—31no—
135#340—32no—
136#340—32no—
137C24×C3—21no—
138C24×C3C2×D412yes—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—44no—
148C2×C3×D4—44no—
149C2×C3×D4—82no—
150C2×C3×D4#4112yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C2×Dic6—42no—
170C2×Dic6—42no—
171C2×Dic6—42no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—81no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—42no—
183C3×C42C2×D412yes—
184C3×C42#4112yes—
185#217—44no—
186#217—44no—
187C4⋊Dic3—22no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—22no—
191C4⋊Dic3—21no—
192C4⋊Dic3—21no—
193C4⋊Dic3—42no—
194C4×Dic3—42no—
195#208—48no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8C2×D414yes—
204C2×C3⋊C8#4112yes—
205C22×Q8—61no—
206C22×D4—22no—
207C22×D4#6711yescharacteristic
208C22×D4C22×D311yescharacteristic
209C23×C4—21no—
210C23×C4C22×D312yes—
211C2×M16—64no—
212C2×M16—64no—
213C2×M16—122no—
214C22×C8—31no—
215C22×C8—32no—
216C22×C8—31no—
217C4⋊Q8—62no—
218C4⋊Q8—62no—
219#124—124no—
220#121—24no—
221#121—42no—
222#121—44no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4#6711yescharacteristic
234C2×C4⋊C4C22×D311yescharacteristic
235C2×C4⋊C4C22×D312yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—62no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4#6712yes—
242C2×C42—61no—
243C2×C42C22×D311yescharacteristic
244#107—64no—
245#106—122no—
246C4⋊C8—34no—
247#102—62no—
248#102—64no—
249#102—62no—
250C22⋊C8—64no—
251C4×C8—62no—
252#95—22no—
253#95—61no—
254#95—62no—
255#95—62no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3C22×D411yescharacteristic
271C3×D4—22no—
272C3×D4—22no—
273C3×D4—41no—
274C3×D4—41no—
275C3×D4—44no—
276C3×D4—88no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—84no—
293C2×C3×C4#12112yes—
294C2×C3×C4#12112yes—
295C2×C3×C4AGL(1,ℤ/8)12yes—
296C2×C3×C4AGL(1,ℤ/8)12yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4H(5,𝔽2)11yescharacteristic
302C2×C3×C42−1+411yescharacteristic
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312Dic6—84no—
313C3⋊C8—24no—
314C3⋊C8—22no—
315C3⋊C8—41no—
316C24—21no—
317C24C2×#6712yes—
318C2×Q8—122no—
319C2×Q8—122no—
320C2×Q8—122no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—82no—
332C2×D4#23712yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—62no—
347C22×C4C2×#6712yes—
348C22×C4C2×#6711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350C22×C4C23×D311yescharacteristic
351M16—128no—
352C2×C8—34no—
353C2×C8—32no—
354C2×C8—61no—
355C2×C8—62no—
356C2×C8—61no—
357C2×C8—62no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8—61no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4—62no—
368C4⋊C4—61no—
369C4⋊C4—62no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—61no—
373C4⋊C4—81no—
374C4⋊C4—122no—
375C22⋊C4—24no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—124no—
380C22⋊C4—124no—
381C42—122no—
382C42D3×D412yes—
383C42#23712yes—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3C2×#12111yescharacteristic
399C22×C3#54611yescharacteristic
400C22×C3#54711yescharacteristic
401C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
402C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
403C22×C3#57711yescharacteristic
404C22×C3#57811yescharacteristic
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—81no—
408C3×C4—82no—
409C3×C4#46914yes—
410C3×C4#52911yescharacteristic
411C3×C4#54211yescharacteristic
412C3×C4#54611yescharacteristic
413C3×C4#54811yescharacteristic
414Dic3—81no—
415Dic3—81no—
416Dic3—82no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23C22×#6711yescharacteristic
432Q8—244no—
433D4—22no—
434D4—22no—
435D4—44no—
436D4—41no—
437D4—41no—
438D4—88no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—84no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4#93212yes—
465C2×C4#93912yes—
466C2×C4#93912yes—
467C2×C4#94912yes—
468C2×C4C2×D3×D411yescharacteristic
469C2×C4C2×#23711yescharacteristic
470C2×C4C2×#23711yescharacteristic
471C2×C4#100411yescharacteristic
472C2×C4D3×#4111yescharacteristic
473C2×C4#101011yescharacteristic
474C8—62no—
475C8—64no—
476C8—121no—
477C2×C3—82no—
478C2×C3—82no—
479C2×C3#242711yescharacteristic
480C2×C3C2×#46911yescharacteristic
481C2×C3#306311yescharacteristic
482C2×C3#332411yescharacteristic
483C2×C3#333011yescharacteristic
484C2×C3#335311yescharacteristic
485C2×C3#335411yescharacteristic
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22C2×#93211yescharacteristic
501C22C2×#93911yescharacteristic
502C22#580611yescharacteristic
503C22C2×#94911yescharacteristic
504C22#582411yescharacteristic
505C22#583611yescharacteristic
506C22#583811yescharacteristic
507C4—22no—
508C4—42no—
509C4—82no—
510C4—81no—
511C4—241no—
512C4—242no—
513C4—241no—
514C4#503914yes—
515C4#559811yescharacteristic
516C4#560011yescharacteristic
517C4#560311yescharacteristic
518C4#561211yescharacteristic
519C3#2534711yescharacteristic, complemented, Sylow
520C2—82no—
521C2—82no—
522C2C2×#503911yescharacteristic
523C2#8264711yescharacteristic
524C2#8342311yescharacteristic
525C2#8348611yescharacteristic
526C2#8351411yescharacteristic
527C2#8352011yescharacteristic
528C2#8365011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2537C2
socleC23×C3C22×D4
hypercenter#2537D3
hypocenterC3#25347
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. #82382C2semidirect
3. C2×#5012C2semidirect
4. C2×#5007C2semidirect
5. #71682C2semidirect
6. #71680C2semidirect
7. C4×#832C2semidirect
8. #69582C2semidirect
9. #69568C2semidirect
10. #69519C2semidirect
11. #69513C2semidirect
104. C4⋊(C3⋊C8)D4semidirect
519. C3#25347semidirect

Maximal subgroups

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

subgroupindex
C3×#25372
#823822
C2×#50122
C2×#50072
#716822
#716802
C4×#8322
#695822
#695682
#695192
#695132
#253473