Finite Group Databasebeta

← #10836655

Subgroups — #10836655

Order 768.

706 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×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#82515C211yesmaximal, characteristic, complemented
3#82508C211yesmaximal, characteristic, complemented
4C2×#5008C211yesmaximal, characteristic, complemented
5#72093C212yesmaximal
6#71617C211yesmaximal, characteristic, complemented
7#70668C212yesmaximal, complemented
8#70318C212yesmaximal, complemented
9#70302C212yesmaximal, complemented
10#69879C211yesmaximal, characteristic, complemented
11#69875C211yesmaximal, characteristic, complemented
12#24721—31nomaximal, Sylow
13C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
14C3×C4×#41C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#869C2211yescharacteristic
17C3×#442—21no—
18C3×#440—21no—
19C3×C4×SD16—21no—
20C3×C4×D8—21no—
21C3×#428C2211yescharacteristic
22C3×#418C2211yescharacteristic
23C3×#417C2211yescharacteristic
24C3×C4×M16C2211yescharacteristic
25#5234C2212yes—
26#5232C2212yes—
27C2×#833C2212yes—
28C2×#832C2212yes—
29C2×#822C2212yes—
30#5021—21no—
31#5020—21no—
32#5008—21no—
33#5008—21no—
34#5008—21no—
35#5008—22no—
36#5007—21no—
37#5000—22no—
38#4997C2212yes—
39#4995C2212yes—
40C2×#808C2212yes—
41#4926—21no—
42#4925—21no—
43#4924C2211yescharacteristic
44#4924C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4558C2212yes—
47#4554C2212yes—
48#4554C2212yes—
49#4530C2212yes—
50#3079—31no—
51C2×#476—31no—
52C4×AGL(1,ℤ/8)D311yescharacteristic
53#2925—31no—
54#2079—32no—
55#2051—32no—
56#1997—31no—
57#1951—32no—
58#1950—31no—
59#1931—32no—
60#1800—31no—
61C2×C3×#41C2311yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C22×Dic6—21no—
65C3×AGL(1,ℤ/8)—22no—
66C3×AGL(1,ℤ/8)—22no—
67C2×C3×SD16—21no—
68C2×C3×D8—21no—
69C2×C3×M16C2311yescharacteristic
70C2×C3×M16C2311yescharacteristic
71C3×C4×Q8D411yescharacteristic
72C3×C4×Q8D411yescharacteristic
73C3×C4×D4—21no—
74C3×C4×D4—21no—
75C3×C4×D4—21no—
76C3×C4×D4—21no—
77C3×C4×D4—21no—
78C3×C4×D4—21no—
79C3×C4×D4D411yescharacteristic
80C3×C4×D4D411yescharacteristic
81C3×#117—21no—
82C3×#117C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2312yes—
89C2×#208C2312yes—
90C2×#208C2312yes—
91C22×C3⋊C8C2312yes—
92#869—21no—
93#869—21no—
94#869—21no—
95#869—21no—
96C3×#107—21no—
97C3×#106—21no—
98C3×#103—21no—
99C3×#102—21no—
100C3×#102—21no—
101C3×#102—21no—
102C3×#97—21no—
103C3×C4×C8—21no—
104#835—22no—
105#833—24no—
106#832—22no—
107#832—22no—
108#832—22no—
109#830—22no—
110#830—22no—
111#822—24no—
112#808—22no—
113#808—22no—
114#808—22no—
115#807—22no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118C4⋊(C3⋊C8)—21no—
119C4⋊(C3⋊C8)—21no—
120C4⋊(C3⋊C8)D411yescharacteristic
121C4⋊(C3⋊C8)D411yescharacteristic
122C4⋊(C3⋊C8)D412yes—
123#803—22no—
124C4×C3⋊C8—22no—
125C2×AGL(1,ℤ/8)C2×D311yescharacteristic
126C2×C4⋊Q8—31no—
127C4×#41C2×D311yescharacteristic
128C2×C4×D4C2×D311yescharacteristic
129#477—61no—
130#476—61no—
131#476—61no—
132#476—62no—
133#476—61no—
134#475—61no—
135#474—61no—
136#442—21no—
137#440—21no—
138C4×SD16—21no—
139C4×D8—21no—
140#433—62no—
141#432—61no—
142#431—61no—
143C2×#108—32no—
144#428—32no—
145#428C2×D311yescharacteristic
146C2×#106—32no—
147#423—31no—
148#423—31no—
149C2×C4⋊C8—31no—
150#418C2×D311yescharacteristic
151#417—32no—
152#417C2×D311yescharacteristic
153C2×#102—32no—
154C2×#100—32no—
155#408—32no—
156#405—32no—
157C4×M16C2×D311yescharacteristic
158#343—32no—
159#343—32no—
160#343—32no—
161#341—32no—
162C24×C3—21no—
163C3×#41—21no—
164C3×#41—21no—
165C3×#41—21no—
166C3×#41—21no—
167C2×C3×Q8C2×D411yescharacteristic
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4C2×D411yescharacteristic
175C2×C3×D4#4112yes—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—41no—
181C22×C3×C4—41no—
182C22×C3×C4—82no—
183C22×C3×C4#4111yescharacteristic
184C22×C3×C4#4111yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C22×Dic3—22no—
187C2×Dic6—41no—
188C2×Dic6—41no—
189C2×Dic6—82no—
190C3×SD16—24no—
191C3×D8—24no—
192C3×M16—21no—
193C3×M16—21no—
194C3×M16—21no—
195C3×M16#4112yes—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4C2×D411yescharacteristic
203C3×C4⋊C4C2×D411yescharacteristic
204C3×C22⋊C4—21no—
205C3×C22⋊C4—21no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215C3×C42C2×D411yescharacteristic
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—41no—
219C4⋊Dic3—41no—
220#208—22no—
221#208—22no—
222#208—22no—
223#208—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8C2×D412yes—
229C2×C3⋊C8C2×D412yes—
230C2×#41C22×D311yescharacteristic
231C22×Q8—61no—
232C22×D4C22×D311yescharacteristic
233C23×C4—21no—
234AGL(1,ℤ/8)—22no—
235AGL(1,ℤ/8)—22no—
236C2×SD16—21no—
237C2×D8—21no—
238C2×M16—32no—
239C2×M16—32no—
240C2×M16C22×D311yescharacteristic
241C2×M16C22×D311yescharacteristic
242C22×C8—32no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247C4×Q8#6711yescharacteristic
248C4×Q8#6711yescharacteristic
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4#6711yescharacteristic
256C4×D4#6711yescharacteristic
257#117—21no—
258#117C22×D311yescharacteristic
259C2×C4⋊C4—32no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C42—21no—
264C2×C42C22×D311yescharacteristic
265#108—64no—
266#107—21no—
267#107—62no—
268#106—21no—
269#106—62no—
270#106—62no—
271#106—62no—
272C4⋊C8—31no—
273C4⋊C8—31no—
274C4⋊C8—32no—
275C4⋊C8—61no—
276C4⋊C8—61no—
277C4⋊C8—61no—
278C4⋊C8—61no—
279#103—21no—
280#103—62no—
281#102—21no—
282#102—21no—
283#102—21no—
284#102—62no—
285#102—62no—
286#102—62no—
287#100—64no—
288C22⋊C8—62no—
289C22⋊C8—62no—
290#97—21no—
291#97—62no—
292C4×C8—21no—
293C4×C8—62no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—81no—
299C23×C3—81no—
300C23×C3C2×#4111yescharacteristic
301C3×Q8—21no—
302C3×Q8—21no—
303C3×D4—21no—
304C3×D4—22no—
305C3×D4—21no—
306C3×D4—22no—
307C3×D4—41no—
308C3×D4—41no—
309C3×D4—41no—
310C3×D4—41no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—81no—
321C2×C3×C4—82no—
322C2×C3×C4—82no—
323C2×C3×C4—81no—
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4#12111yescharacteristic
327C2×C3×C4#12111yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C2×#4111yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C42−1+411yescharacteristic
333C2×C3×C42−1+411yescharacteristic
334C2×Dic3—42no—
335C2×Dic3—82no—
336Dic6—82no—
337Dic6—82no—
338C3×C8—22no—
339C3×C8—41no—
340C3⋊C8—22no—
341C3⋊C8—22no—
342C3⋊C8—42no—
343C3⋊C8—42no—
344C24—21no—
345#41—21no—
346#41—21no—
347#41—21no—
348#41—21no—
349C2×Q8—121no—
350C2×Q8—121no—
351C2×Q8—242no—
352C2×Q8C2×#6711yescharacteristic
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4#23512yes—
360C2×D4C2×#6711yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—62no—
368C22×C4—82no—
369C22×C4#23711yescharacteristic
370C22×C4#23711yescharacteristic
371C22×C4C23×D311yescharacteristic
372SD16—24no—
373D8—24no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—62no—
378M16—62no—
379M16—62no—
380M16—62no—
381M16#23512yes—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—32no—
385C2×C8—32no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—62no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4—121no—
397C4⋊C4—121no—
398C4⋊C4C2×#6711yescharacteristic
399C4⋊C4C2×#6711yescharacteristic
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42D3×D411yescharacteristic
409C42D3×D411yescharacteristic
410C42C2×#6711yescharacteristic
411C42C2×#6711yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—41no—
415C22×C3—42no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3—81no—
421C22×C3—81no—
422C22×C3C2×#12111yescharacteristic
423C22×C3#55211yescharacteristic
424C22×C3#55611yescharacteristic
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—81no—
432C3×C4#47112yes—
433C3×C4#52511yescharacteristic
434C3×C4#53611yescharacteristic
435C3×C4#54711yescharacteristic
436C3×C4#54811yescharacteristic
437Dic3—82no—
438C23—21no—
439C23—22no—
440C23—41no—
441C23—41no—
442C23—81no—
443C23—81no—
444C23C2×#23711yescharacteristic
445Q8—21no—
446Q8—21no—
447Q8—242no—
448Q8—242no—
449D4—21no—
450D4—22no—
451D4—22no—
452D4—21no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—122no—
471C2×C4—242no—
472C2×C4#93911yescharacteristic
473C2×C4#93911yescharacteristic
474C2×C4#93911yescharacteristic
475C2×C4#93911yescharacteristic
476C2×C4C2×#23511yescharacteristic
477C2×C4C2×#23511yescharacteristic
478C2×C4C2×D3×D411yescharacteristic
479C2×C4#101011yescharacteristic
480C2×C4#101011yescharacteristic
481C2×C4C22×#6711yescharacteristic
482C8—22no—
483C8—41no—
484C8—62no—
485C8—62no—
486C8—122no—
487C8—122no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—81no—
491C2×C3—82no—
492C2×C3#241511yescharacteristic
493C2×C3C2×#47111yescharacteristic
494C2×C3#306811yescharacteristic
495C22—21no—
496C22—21no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—41no—
501C22—82no—
502C22—82no—
503C22—81no—
504C22—81no—
505C22#554111yescharacteristic
506C22#554211yescharacteristic
507C22C2×#93911yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—81no—
515C4—242no—
516C4#520012yes—
517C4#555111yescharacteristic
518C4#555211yescharacteristic
519C4#579111yescharacteristic
520C4#582511yescharacteristic
521C3#2472111yescharacteristic, complemented, Sylow
522C2—21no—
523C2—21no—
524C2—81no—
525C2—82no—
526C2#8235111yescharacteristic
527C2C2×#520011yescharacteristic
528C2#8400211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×AGL(1,ℤ/8)C2
socleC22×C3C2×#121
prime radicalC2×C3×AGL(1,ℤ/8)C22
hypercenterC4×AGL(1,ℤ/8)D3
hypocenterC3#24721
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. #82515C2semidirect
3. #82508C2semidirect
4. C2×#5008C2semidirect
6. #71617C2semidirect
7. #70668C2semidirect
8. #70318C2semidirect
9. #70302C2semidirect
10. #69879C2semidirect
11. #69875C2semidirect
521. C3#24721semidirect

Maximal subgroups

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

subgroupindex
C3×C4×AGL(1,ℤ/8)2
#825152
#825082
C2×#50082
#720932
#716172
#706682
#703182
#703022
#698792
#698752
#247213