Finite Group Databasebeta

← #10883511

Subgroups — #10883511

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3036C211yesmaximal, characteristic, complemented
2#83297C211yesmaximal, characteristic, complemented
3#83276C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#72095C212yesmaximal, complemented
6#72091C212yesmaximal, complemented
7#71617C211yesmaximal, characteristic, complemented
8#70527C211yesmaximal, characteristic, complemented
9#70517C211yesmaximal, characteristic, complemented
10#70303C212yesmaximal, complemented
11#70302C212yesmaximal, complemented
12#28177—31nomaximal, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C3×#532C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#949C2212yes—
17C2×#948C2212yes—
18C22×#213C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C3×#463—22no—
21C3×#461—21no—
22C3×#459—21no—
23C3×#423C2211yescharacteristic
24C3×#417C2212yes—
25#5234C2212yes—
26#5232C2212yes—
27C2×#833C2212yes—
28C2×#832C2211yescharacteristic
29C2×#823C2212yes—
30#5083—21no—
31#5072—21no—
32#5057—22no—
33#5014—22no—
34#5014—22no—
35#5011—22no—
36#5011—22no—
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4924C2211yescharacteristic
40#4924C2211yescharacteristic
41C2×#803C2211yescharacteristic
42#4558C2212yes—
43#4558C2212yes—
44#4554C2212yes—
45#4530C2212yes—
46#3036—31no—
47#3036D311yescharacteristic, complemented
48C2×#442—31no—
49#2932—31no—
50#2079—32no—
51#2018—32no—
52#1907—32no—
53#1907—32no—
54#1838—31no—
55#1800—31no—
56#1800—31no—
57C2×C3×#41C2312yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C3×AGL(1,ℤ/8)—24no—
63C3×AGL(1,ℤ/8)—42no—
64C2×C3×SD16—22no—
65C2×C3×D8—22no—
66C2×C3×M16C2312yes—
67C3×C4⋊Q8D411yescharacteristic
68C3×#127D411yescharacteristic
69C3×#124D412yes—
70C3×#121—21no—
71C3×#121—21no—
72C3×C4×D4—21no—
73C3×C4×D4—21no—
74C3×C4×D4—21no—
75C3×C4×D4—21no—
76C3×C4×D4—21no—
77C3×C4×D4—21no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C42C2311yescharacteristic
81#949—24no—
82#949—42no—
83#948—42no—
84#948—42no—
85C2×#215—22no—
86C2×#213—22no—
87C2×#213—22no—
88C2×#213—22no—
89C2×#213—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×#208—22no—
93C2×#208C2312yes—
94C2×#208C2312yes—
95C22×C3⋊C8—22no—
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98C4×D12—21no—
99C4×D12—21no—
100C4×D12—21no—
101C4×D12—21no—
102C3×C4⋊C8—21no—
103C3×C4⋊C8—21no—
104C3×#103—22no—
105C3×#102—22no—
106#835—22no—
107#833—42no—
108#832—22no—
109#832—22no—
110#830—24no—
111#823—42no—
112#809—22no—
113#808—22no—
114C4⋊(C3⋊C8)—21no—
115C4⋊(C3⋊C8)—21no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118#803D412yes—
119#803D412yes—
120C2×AGL(1,ℤ/8)—32no—
121C2×AGL(1,ℤ/8)C2×D312yes—
122C22×D8—31no—
123C2×#131—32no—
124#532C2×D311yescharacteristic
125C2×C4×D4—31no—
126C2×C4×D4C2×D311yescharacteristic
127#463—22no—
128#463—62no—
129#461—21no—
130#461—61no—
131#459—21no—
132#459—61no—
133#442—62no—
134#442—62no—
135#435—62no—
136#435—62no—
137C2×#106—31no—
138#423—31no—
139#423—31no—
140#423C2×D311yescharacteristic
141#417—32no—
142#417C2×D312yes—
143C2×#102—31no—
144C2×#102—31no—
145C2×#100—32no—
146C2×#100—32no—
147#408—32no—
148C2×#97—31no—
149#343—32no—
150#343—32no—
151#341—32no—
152#341—32no—
153C24×C3—21no—
154C23×D3—21no—
155C3×#41—22no—
156C3×#41—22no—
157C3×#41—22no—
158C3×#41—22no—
159C2×C3×Q8C2×D412yes—
160C2×C3×D4—22no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4C2×D412yes—
169C2×C3×D4#4112yes—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—41no—
173C22×C3×C4—41no—
174C22×C3×C4—82no—
175C22×C3×C4#4111yescharacteristic
176C22×C3×C4#4111yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C22×Dic3—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12#4112yes—
185C2×C4×D3—41no—
186C2×C4×D3—41no—
187C2×C4×D3—82no—
188C3×SD16—44no—
189C3×D8—44no—
190C3×M16—22no—
191C3×M16—22no—
192C2×C3×C8—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—41no—
200C3×C22⋊C4—41no—
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205#215—44no—
206#213—28no—
207#213—44no—
208(C2×D3)⋊C4—41no—
209(C2×D3)⋊C4—41no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212#208—22no—
213#208—22no—
214#208—22no—
215#208—22no—
216#208—44no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—44no—
224C2×C3⋊C8C2×D412yes—
225C2×C3⋊C8C2×D412yes—
226C2×#41C22×D312yes—
227C22×D4—31no—
228C22×D4C22×D311yescharacteristic
229C23×C4—21no—
230C23×C4—61no—
231AGL(1,ℤ/8)—24no—
232AGL(1,ℤ/8)—42no—
233AGL(1,ℤ/8)—64no—
234AGL(1,ℤ/8)—122no—
235C2×SD16—22no—
236C2×SD16—62no—
237C2×D8—22no—
238C2×D8—62no—
239C2×D8—62no—
240C2×D8—62no—
241C2×D8—62no—
242#131—122no—
243#131—122no—
244C2×M16—32no—
245C2×M16—32no—
246C2×M16—62no—
247C2×M16C22×D312yes—
248C22×C8—31no—
249C22×C8—31no—
250C22×C8—62no—
251C4⋊Q8#6711yescharacteristic
252#127#6711yescharacteristic
253#124#6712yes—
254#121—21no—
255#121—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—61no—
263C4×D4—61no—
264C4×D4—61no—
265C4×D4—61no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—61no—
270C2×C42C22×D311yescharacteristic
271#106—62no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—61no—
275C4⋊C8—61no—
276C4⋊C8—61no—
277C4⋊C8—61no—
278#103—22no—
279#103—62no—
280#102—22no—
281#102—62no—
282#102—62no—
283#102—62no—
284#100—122no—
285#100—122no—
286C22⋊C8—64no—
287#97—32no—
288#97—32no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—81no—
294C23×C3—81no—
295C23×C3C2×#4111yescharacteristic
296C22×D3—22no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—81no—
300C22×D3—81no—
301C3×Q8—22no—
302C3×Q8—22no—
303C3×D4—22no—
304C3×D4—22no—
305C3×D4—22no—
306C3×D4—22no—
307C3×D4—42no—
308C3×D4—42no—
309C3×D4—41no—
310C3×D4—41no—
311C3×D4—81no—
312C3×D4—81no—
313C2×C3×C4—22no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—82no—
320C2×C3×C4—82no—
321C2×C3×C4—81no—
322C2×C3×C4#12111yescharacteristic
323C2×C3×C4#12111yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4H(5,𝔽2)11yescharacteristic
331C2×C3×C4H(5,𝔽2)11yescharacteristic
332C2×Dic3—41no—
333C2×Dic3—81no—
334D12—22no—
335D12—22no—
336D12—41no—
337D12—41no—
338C4×D3—82no—
339C4×D3—82no—
340C3×C8—42no—
341C3⋊C8—22no—
342C3⋊C8—22no—
343C3⋊C8—42no—
344C3⋊C8—42no—
345C24—21no—
346C24—61no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351C2×Q8C2×#6712yes—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—32no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—61no—
366C2×D4#23712yes—
367C2×D4C2×#6712yes—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—61no—
373C22×C4—82no—
374C22×C4—121no—
375C22×C4—121no—
376C22×C4—242no—
377C22×C4#23711yescharacteristic
378C22×C4#23711yescharacteristic
379C22×C4C23×D311yescharacteristic
380SD16—44no—
381SD16—124no—
382D8—44no—
383D8—68no—
384D8—124no—
385M16—22no—
386M16—22no—
387M16—62no—
388M16—62no—
389M16—62no—
390M16—62no—
391M16—124no—
392C2×C8—22no—
393C2×C8—32no—
394C2×C8—32no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—61no—
398C2×C8—61no—
399C2×C8—62no—
400C2×C8—62no—
401C2×C8—124no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—121no—
413C22⋊C4—121no—
414C42D3×D411yescharacteristic
415C42D3×D411yescharacteristic
416C42C2×#6711yescharacteristic
417C42C2×#6711yescharacteristic
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3C2×#12111yescharacteristic
429C22×C3#55311yescharacteristic
430C22×C3#55311yescharacteristic
431C2×D3—42no—
432C2×D3—82no—
433C2×D3—81no—
434C2×D3—81no—
435C2×D3—82no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—42no—
441C3×C4—81no—
442C3×C4#52511yescharacteristic
443C3×C4#53511yescharacteristic
444C3×C4#54611yescharacteristic
445C3×C4#54611yescharacteristic
446Dic3—81no—
447C23—22no—
448C23—22no—
449C23—41no—
450C23—41no—
451C23—62no—
452C23—81no—
453C23—81no—
454C23—121no—
455C23—121no—
456C23—241no—
457C23—241no—
458C23C2×#23711yescharacteristic
459Q8—22no—
460Q8—22no—
461D4—22no—
462D4—22no—
463D4—22no—
464D4—22no—
465D4—42no—
466D4—42no—
467D4—41no—
468D4—41no—
469D4—62no—
470D4—62no—
471D4—81no—
472D4—81no—
473D4—121no—
474D4—121no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—121no—
485C2×C4—242no—
486C2×C4—242no—
487C2×C4—241no—
488C2×C4#93911yescharacteristic
489C2×C4#93911yescharacteristic
490C2×C4#93911yescharacteristic
491C2×C4#93911yescharacteristic
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×#23711yescharacteristic
494C2×C4#100411yescharacteristic
495C2×C4D3×#4111yescharacteristic
496C2×C4#100911yescharacteristic
497C2×C4C22×#6711yescharacteristic
498C8—42no—
499C8—62no—
500C8—62no—
501C8—122no—
502C8—122no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—82no—
506C2×C3—82no—
507C2×C3#253511yescharacteristic
508C2×C3#306811yescharacteristic
509C2×C3#306811yescharacteristic
510D3—82no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—82no—
518C22—82no—
519C22—81no—
520C22—81no—
521C22—122no—
522C22—242no—
523C22—242no—
524C22—241no—
525C22—241no—
526C22#561411yescharacteristic
527C22#561811yescharacteristic
528C22C2×#93911yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—42no—
534C4—81no—
535C4—241no—
536C4#560611yescharacteristic
537C4#560711yescharacteristic
538C4#580611yescharacteristic
539C4#582911yescharacteristic
540C3#2817711yescharacteristic, complemented, Sylow
541C2—21no—
542C2—21no—
543C2—82no—
544C2—82no—
545C2—242no—
546C2#8266611yescharacteristic
547C2#8394311yescharacteristic
548C2#8400211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3036C2
socleC22×C3C2×#121
hypercenter#3036D3
hypocenterC3#28177
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. C3×#3036C2semidirect
2. #83297C2semidirect
3. #83276C2semidirect
4. C2×#5014C2semidirect
5. #72095C2semidirect
6. #72091C2semidirect
7. #71617C2semidirect
8. #70527C2semidirect
9. #70517C2semidirect
10. #70303C2semidirect
11. #70302C2semidirect
47. #3036D3semidirect
540. C3#28177semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#832972
#832762
C2×#50142
#720952
#720912
#716172
#705272
#705172
#703032
#703022
#281773