Finite Group Databasebeta

← #229746

Subgroups — #229746

Order 512.

727 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23680C211yesmaximal, characteristic
2#23204C211yesmaximal, characteristic, complemented
3#22651C211yesmaximal, characteristic, complemented
4C2×#1583C211yesmaximal, characteristic, complemented
5C2×#1580C211yesmaximal, characteristic
6#9646C211yesmaximal, characteristic, complemented
7#9602C211yesmaximal, characteristic, complemented
8#9558C211yesmaximal, characteristic, complemented
9#9555C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic
11#8945C211yesmaximal, characteristic, complemented
12#8846C211yesmaximal, characteristic, complemented
13#8845C211yesmaximal, characteristic, complemented
14#8263C211yesmaximal, characteristic, complemented
15#8181C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#424—21no—
23C4×#124C2211yescharacteristic
24#2293C2211yescharacteristic
25#2085C411yescharacteristic
26#2073C411yescharacteristic
27#2068C411yescharacteristic
28#2065C411yescharacteristic
29#2055C411yescharacteristic
30#2054C411yescharacteristic
31#2042C411yescharacteristic
32#2012C411yescharacteristic
33#1984C2211yescharacteristic
34#1954C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1923—21no—
38#1923—21no—
39#1851—21no—
40#1850—21no—
41#1850—21no—
42#1842C2211yescharacteristic
43#1798—21no—
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C8×C22⋊C4—21no—
50C4×#97C2211yescharacteristic
51#1583—24no—
52#1580—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74C2×#124C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C8×Q8—22no—
80C8×Q8—41no—
81C4×SD16—22no—
82C4×SD16—22no—
83C4×D8—22no—
84C4×D8—22no—
85#436—22no—
86#436—41no—
87C8×D4—22no—
88C8×D4—22no—
89C8×D4—41no—
90C8×D4—41no—
91C2×#107C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93#424—44no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2×C411yescharacteristic
100C2×C4⋊C8C2×C411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#391C2×C411yescharacteristic
131#390C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#390C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#382C2×C411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C43C2311yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336C2×C411yescharacteristic
156#336C2×C411yescharacteristic
157#336D411yescharacteristic
158#336D411yescharacteristic
159#334—22no—
160#334—22no—
161Q8⋊C8—22no—
162Q8⋊C8—41no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—41no—
166D4⋊C8D414yes—
167#322—22no—
168#322—22no—
169C22×Q8C22×C411yescharacteristic
170C22×D4C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C2×SD16—22no—
175C2×SD16—22no—
176C2×SD16—41no—
177C2×D8—22no—
178C2×D8—22no—
179C2×D8—22no—
180C2×M16—42no—
181C2×M16—42no—
182C2×M16—81no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—81no—
194C22×C8—81no—
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C22×C411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203#124—22no—
204#124—22no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4C2×D412yes—
216C4×D4#4112yes—
217#117—42no—
218#117—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42C2411yescharacteristic
246#107—22no—
247#106—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C4⋊C8—42no—
263C4⋊C8—81no—
264C4⋊C8C2×D412yes—
265#103—22no—
266#103—22no—
267#103—22no—
268#102—22no—
269#102—22no—
270#102—22no—
271#102—22no—
272#102—41no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—81no—
278#97—42no—
279#97—42no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—42no—
292C4×C8—81no—
293C4×C8C2×D412yes—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301C24—21no—
302C24—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8#13112yes—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4C4×D412yes—
320C2×D4#13112yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346SD16—44no—
347D8—28no—
348M16—84no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—84no—
390C2×C8—84no—
391C2×C8#13112yes—
392C2×C8AGL(1,ℤ/8)12yes—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4C4×D412yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—81no—
418C22⋊C4—81no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—22no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42C22×D411yescharacteristic
434C42C2×#4111yescharacteristic
435C42H(5,𝔽2)11yescharacteristic
436C422−1+411yescharacteristic
437C23—22no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C2×C4×D411yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—41no—
455Q8—41no—
456D4—22no—
457D4—22no—
458D4—22no—
459D4—22no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—41no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
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—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4#44412yes—
504C2×C4C2×C4×D411yescharacteristic
505C2×C4#52011yescharacteristic
506C2×C4D4211yescharacteristic
507C2×C4#54711yescharacteristic
508C2×C4#54811yescharacteristic
509C2×C4#55311yescharacteristic
510C2×C4C2×#13111yescharacteristic
511C2×C4C2×#13111yescharacteristic
512C2×C4C2×#13111yescharacteristic
513C2×C4#56811yescharacteristic
514C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
515C2×C4#57511yescharacteristic
516C2×C4#57711yescharacteristic
517C2×C4#57711yescharacteristic
518C8—22no—
519C8—22no—
520C8—42no—
521C8—42no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—42no—
529C8—81no—
530C8—81no—
531C8—81no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22#236111yescharacteristic
547C22#292911yescharacteristic
548C22#293511yescharacteristic
549C22C4×AGL(1,ℤ/8)11yescharacteristic
550C22#295211yescharacteristic
551C22C2×#44411yescharacteristic
552C22#295611yescharacteristic
553C4—22no—
554C4—21no—
555C4—21no—
556C4—41no—
557C4—41no—
558C4—42no—
559C4—42no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4#292911yescharacteristic
566C4#293311yescharacteristic
567C4#299211yescharacteristic
568C4#299411yescharacteristic
569C4#326511yescharacteristic
570C4#326711yescharacteristic
571C4#337311yescharacteristic
572C4#337611yescharacteristic
573C2—82no—
574C2—82no—
575C2#1791711yescharacteristic
576C2#1792711yescharacteristic
577C2#1807811yescharacteristic
578C2#1809311yescharacteristic
579C2#2299111yescharacteristic
580C2#2300711yescharacteristic
581C2#2301611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23204C2semidirect
3. #22651C2semidirect
4. C2×#1583C2semidirect
6. #9646C2semidirect
7. #9602C2semidirect
8. #9558C2semidirect
9. #9555C2semidirect
11. #8945C2semidirect
12. #8846C2semidirect
13. #8845C2semidirect
14. #8263C2semidirect
15. #8181C2semidirect

Maximal subgroups

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

subgroupindex
#236802
#232042
#226512
C2×#15832
C2×#15802
#96462
#96022
#95582
#95552
#89512
#89452
#88462
#88452
#82632
#81812