Finite Group Databasebeta

← #10830085

Subgroups — #10830085

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic, complemented
2C3×#2580C211yesmaximal, characteristic, complemented
3C2×#5019C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic
5#71635C211yesmaximal, characteristic, complemented
6#69660C212yesmaximal, complemented
7#69659C212yesmaximal
8#69632C212yesmaximal, complemented
9#69628C212yesmaximal, complemented
10#69556C211yesmaximal, characteristic, complemented
11#69552C211yesmaximal, characteristic, complemented
12#24197—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#946—21no—
16C22×#214C2212yes—
17C22×#213C2211yescharacteristic
18#5584—24no—
19#5584—24no—
20C2×#898—22no—
21C2×#892—21no—
22C2×#875C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2212yes—
26C3×#399—21no—
27C3×#393—21no—
28C3×#390C2212yes—
29C3×#389C2212yes—
30C3×#385—22no—
31C3×#382C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5019—24no—
34#5014—24no—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2211yescharacteristic
39C2×#809C2212yes—
40C2×#808C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×#442—31no—
49#2580D311yescharacteristic, complemented
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1960—31no—
54#1880—32no—
55#1865—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×Dic6C2312yes—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#946—28no—
78C2×#214—24no—
79C2×#214—24no—
80C2×#214—42no—
81C2×#213—22no—
82C2×#213—22no—
83C2×#213—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—22no—
91C2×C4×Dic3—22no—
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94C22×C3⋊C8C2311yescharacteristic
95#898—216no—
96#892—28no—
97#875—24no—
98#875—24no—
99C4×D12—22no—
100C4×D12D414yes—
101#869—22no—
102#869D414yes—
103C4×Dic6—24no—
104C4×Dic6—24no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—22no—
108#832—22no—
109#810—24no—
110#810—24no—
111#810—42no—
112#809—22no—
113#809—42no—
114#808—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—32no—
118C22×D8—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—32no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—61no—
130#461—64no—
131#442—64no—
132C2×#106—31no—
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97—31no—
141#399—21no—
142#393—21no—
143#390C2×D312yes—
144#389C2×D312yes—
145#385—22no—
146#382C2×D311yescharacteristic
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C23×D3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—42no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4#4112yes—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—42no—
163C22×C3×C4—82no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—21no—
170C22×Dic3—22no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12C2×D412yes—
177C2×C4×D3—24no—
178C2×C4×D3—44no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6C2×D414yes—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—82no—
192C3×C22⋊C4—44no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—82no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#214—48no—
200#213—28no—
201(C2×D3)⋊C4—28no—
202(C2×D3)⋊C4—28no—
203(C2×D3)⋊C4—24no—
204C4⋊Dic3—24no—
205C4⋊Dic3—24no—
206C4⋊Dic3—21no—
207C4⋊Dic3C2×D412yes—
208Dic3⋊C4—28no—
209C4×Dic3—28no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8#4112yes—
219C22×Q8—32no—
220C22×Q8#6711yescharacteristic
221C22×D4—31no—
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4—61no—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C2×D8—62no—
229C2×D8—62no—
230C2×D8—62no—
231C22×C8—31no—
232C22×C8—32no—
233C22×C8—31no—
234C4⋊Q8—34no—
235C4⋊Q8—62no—
236#126—616no—
237#124—64no—
238#124—64no—
239C22⋊Q8—68no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—64no—
243C4×Q8—64no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—34no—
247C4×D4—62no—
248#117—68no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4#6711yescharacteristic
255C2×C4⋊C4#6712yes—
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—62no—
264C2×C42—62no—
265C2×C42C22×D312yes—
266C2×C42C22×D311yescharacteristic
267#106—62no—
268C4⋊C8—62no—
269#103—64no—
270#103—64no—
271#103—122no—
272#102—62no—
273#102—62no—
274#102—122no—
275#97—62no—
276#95—21no—
277#95—21no—
278#95—22no—
279C23×C3—22no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—42no—
289C22×D3—42no—
290C3×Q8—44no—
291C3×D4—22no—
292C3×D4—22no—
293C3×D4—41no—
294C3×D4—41no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—84no—
309C2×C3×C4#12012yes—
310C2×C3×C4#12012yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317C2×Dic3—22no—
318C2×Dic3—24no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322D12—22no—
323D12—22no—
324D12—41no—
325D12—41no—
326C4×D3—48no—
327Dic6—24no—
328Dic6—24no—
329Dic6—42no—
330Dic6—42no—
331C3⋊C8—22no—
332C3⋊C8—41no—
333C3⋊C8—42no—
334C24—21no—
335C24—61no—
336C2×Q8—22no—
337C2×Q8—34no—
338C2×Q8—42no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—32no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4#23912yes—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—42no—
359C22×C4—61no—
360C22×C4—62no—
361C22×C4—64no—
362C22×C4—82no—
363C22×C4—124no—
364C22×C4C2×#6712yes—
365C22×C4C2×#6712yes—
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368C22×C4C23×D311yescharacteristic
369SD16—128no—
370D8—68no—
371C2×C8—32no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8—62no—
376C2×C8—62no—
377C2×C8—61no—
378C2×C8—62no—
379C2×C8—61no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—32no—
387C4⋊C4—68no—
388C4⋊C4—64no—
389C4⋊C4—64no—
390C4⋊C4—61no—
391C4⋊C4—82no—
392C22⋊C4—44no—
393C22⋊C4—42no—
394C22⋊C4—68no—
395C22⋊C4—68no—
396C22⋊C4—64no—
397C22⋊C4—82no—
398C42—22no—
399C42—22no—
400C42—68no—
401C42D3×D412yes—
402C42D3×D412yes—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—82no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3#55011yescharacteristic
411C22×C3#57512yes—
412C22×C3#57712yes—
413C2×D3—42no—
414C2×D3—42no—
415C2×D3—42no—
416C2×D3—82no—
417C2×D3—82no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4—82no—
423C3×C4—81no—
424C3×C4#52511yescharacteristic
425C3×C4#53711yescharacteristic
426C3×C4#54611yescharacteristic
427C3×C4#54711yescharacteristic
428Dic3—44no—
429Dic3—42no—
430C23—22no—
431C23—41no—
432C23—42no—
433C23—41no—
434C23—42no—
435C23—62no—
436C23—121no—
437C23—122no—
438C23—121no—
439C23—122no—
440C23C22×#6711yescharacteristic
441Q8—44no—
442Q8—64no—
443Q8—64no—
444Q8—122no—
445Q8—122no—
446D4—22no—
447D4—22no—
448D4—41no—
449D4—41no—
450D4—62no—
451D4—62no—
452D4—121no—
453D4—121no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—62no—
468C2×C4—64no—
469C2×C4—84no—
470C2×C4—128no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4#93712yes—
475C2×C4#93712yes—
476C2×C4C2×D3×D411yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C2×C4C2×#23911yescharacteristic
479C2×C4#100711yescharacteristic
480C2×C4D3×#4111yescharacteristic
481C2×C4#100911yescharacteristic
482C8—62no—
483C8—121no—
484C8—122no—
485C2×C3—82no—
486C2×C3#240711yescharacteristic
487C2×C3#300511yescharacteristic
488C2×C3#301011yescharacteristic
489C2×C3#337612yes—
490C2×C3#340112yes—
491D3—82no—
492C22—42no—
493C22—44no—
494C22—82no—
495C22—81no—
496C22—81no—
497C22—122no—
498C22—122no—
499C22—122no—
500C22—242no—
501C22—242no—
502C22C2×#93711yescharacteristic
503C22#581711yescharacteristic
504C22#582212yes—
505C22#583411yescharacteristic
506C22#583612yes—
507C4—22no—
508C4—22no—
509C4—42no—
510C4—42no—
511C4—82no—
512C4—81no—
513C4—124no—
514C4—122no—
515C4#562711yescharacteristic
516C4#562911yescharacteristic
517C4#563011yescharacteristic
518C4#563111yescharacteristic
519C3#2419711yescharacteristic, complemented, Sylow
520C2—82no—
521C2—242no—
522C2#8281211yescharacteristic
523C2#8292811yescharacteristic
524C2#8339011yescharacteristic
525C2#8364512yes—
526C2#8371212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2580C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2580D3
hypocenterC3#24197
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. C2×#5584C2semidirect
2. C3×#2580C2semidirect
3. C2×#5019C2semidirect
5. #71635C2semidirect
6. #69660C2semidirect
8. #69632C2semidirect
9. #69628C2semidirect
10. #69556C2semidirect
11. #69552C2semidirect
49. #2580D3semidirect
519. C3#24197semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#25802
C2×#50192
C2×#50142
#716352
#696602
#696592
#696322
#696282
#695562
#695522
#241973