Finite Group Databasebeta

← #7195464

Subgroups — #7195464

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic
2#59821C211yesmaximal, characteristic
3#32014C214yesmaximal, complemented
4#31347C214yesmaximal, complemented
5#29273C211yesmaximal, characteristic, complemented
6#29259C211yesmaximal, characteristic, complemented
7#28608C212yesmaximal, complemented
8#26299C212yesmaximal, complemented
9#26234C212yesmaximal, complemented
10#26144C214yesmaximal, complemented
11#24108C214yesmaximal, complemented
12#23837C211yesmaximal, characteristic, complemented
13#22665C212yesmaximal, complemented
14#22264C211yesmaximal, characteristic
15#22221C211yesmaximal, characteristic, complemented
16C2×#556C2212yes—
17C2×#487C2211yescharacteristic
18C2×#483C2214yes—
19C2×#436C2212yes—
20C2×#428C2214yes—
21#2804—22no—
22#2727—22no—
23#2580—22no—
24#2580—22no—
25#2571—21no—
26#2563C2212yes—
27#2563C2212yes—
28#2548—21no—
29#2542C2212yes—
30#2538C2212yes—
31#2512—22no—
32#2487C2214yes—
33#2448—22no—
34#2415C2214yes—
35#2366—21no—
36#2354C2212yes—
37Q8×C4⋊C4—21no—
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2099C2212yes—
46#2095C2212yes—
47#2084C2212yes—
48#2084C2212yes—
49#2077C2214yes—
50#2075C2214yes—
51#2058C2212yes—
52#2058C2212yes—
53#2058C2212yes—
54#2035C2214yes—
55#2035C2214yes—
56#2035C2214yes—
57#2035C2214yes—
58#2024C2212yes—
59#1990C2211yescharacteristic
60#1987C2212yes—
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1954C2212yes—
64#1948C2212yes—
65#1947C2212yes—
66#1928C2214yes—
67#1919C2214yes—
68#1894C2214yes—
69#1894C2214yes—
70#1872C2214yes—
71#1870C2214yes—
72#1870C2214yes—
73#1863C2214yes—
74#1863C2214yes—
75#1854C2214yes—
76#1854C2214yes—
77#1852C2214yes—
78#1845C2211yescharacteristic
79#1831C2211yescharacteristic
80#1825C2214yes—
81#1809C2214yes—
82C4×#106C2212yes—
83C4×#106C2212yes—
84#1744C2211yescharacteristic
85C22×M16C2314yes—
86#556—48no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127D411yescharacteristic
92C2×#126—24no—
93C2×#124—22no—
94C2×C22⋊Q8C2314yes—
95C2×C22⋊Q8C2314yes—
96C2×C4×Q8—22no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8C2312yes—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117C2314yes—
107C22×C4⋊C4C2314yes—
108#487—24no—
109#483—48no—
110#436—28no—
111#428—48no—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#106C2312yes—
115C2×#106C2314yes—
116C2×#106C2312yes—
117C2×#106C2312yes—
118C2×#106C2312yes—
119C2×C4⋊C8C2312yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#400—24no—
130#399—24no—
131#399—24no—
132#397—24no—
133#396—24no—
134#394—24no—
135#393—24no—
136#393—24no—
137#391—21no—
138#391—21no—
139#391—21no—
140#391C2312yes—
141#390—22no—
142#390—22no—
143#390—22no—
144#390—22no—
145#390—22no—
146#390D412yes—
147#390D412yes—
148#390D412yes—
149#389—22no—
150#389—22no—
151#389—24no—
152#389—22no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#388—24no—
162#388—24no—
163#386—24no—
164#385—24no—
165#385—24no—
166#385—24no—
167#384—22no—
168#384D411yescharacteristic
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#382—22no—
174#382—22no—
175#380C2314yes—
176#380C2314yes—
177#380C2314yes—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C43C2311yescharacteristic
189#340C2312yes—
190#340C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2312yes—
195#340C2314yes—
196#340C2314yes—
197#340C2312yes—
198#336C2312yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8C2412yes—
206C22×D4C2×D412yes—
207C22×D4C2×D412yes—
208C22×D4C2×D412yes—
209C23×C4—24no—
210C23×C4C2414yes—
211C2×M16—28no—
212C2×M16—44no—
213C2×M16—44no—
214C22×C8C2414yes—
215C22×C8C2412yes—
216C22×C8C2412yes—
217C4⋊Q8—24no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—42no—
223#127—24no—
224#126—416no—
225#124—48no—
226C22⋊Q8—28no—
227C22⋊Q8—44no—
228C22⋊Q8—44no—
229C22⋊Q8—44no—
230C4×Q8—44no—
231C4×Q8—48no—
232C4×Q8—44no—
233C4×Q8—82no—
234C4×Q8—82no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—84no—
240C4×D4—84no—
241#117—28no—
242#117—44no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C22⋊C4C2×D414yes—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297#107—44no—
298#107—44no—
299#106—24no—
300#106—28no—
301#106—24no—
302#106—42no—
303#106—42no—
304C4⋊C8—24no—
305#103—28no—
306#103—44no—
307#103—44no—
308#103—44no—
309C22⋊C8—28no—
310C22⋊C8—44no—
311C22⋊C8—44no—
312C4×C8—22no—
313C4×C8#4114yes—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326C24C22×D414yes—
327C2×Q8—24no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—81no—
337C2×Q8—81no—
338C2×Q8—81no—
339C2×Q8—81no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D414yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2511yescharacteristic
374M16—416no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8C2×#4114yes—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—48no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—81no—
411C4⋊C4—81no—
412C4⋊C4—84no—
413C4⋊C4—81no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—81no—
417C4⋊C4—84no—
418C4⋊C4H(5,𝔽2)14yes—
419C22⋊C4—28no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—41no—
433C42—41no—
434C42—48no—
435C42—44no—
436C42—44no—
437C42—44no—
438C42—44no—
439C42#13714yes—
440C42C2×#4112yes—
441C422−1+412yes—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23C23×D411yescharacteristic
450Q8—48no—
451Q8—44no—
452Q8—44no—
453Q8—84no—
454Q8—84no—
455D4—48no—
456D4—48no—
457D4—84no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—168no—
502C2×C4#54712yes—
503C2×C4#54712yes—
504C2×C4C2×#13712yes—
505C2×C4C2×#13712yes—
506C2×C4C2×#13712yes—
507C2×C4#57712yes—
508C2×C4C22×#4111yescharacteristic
509C2×C4C2×H(5,𝔽2)12yes—
510C2×C4C2×2−1+411yescharacteristic
511C2×C4#58512yes—
512C8—24no—
513C8—44no—
514C8—42no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22C2×#54711yescharacteristic
523C22#348411yescharacteristic
524C22#348511yescharacteristic
525C22C22×#13711yescharacteristic
526C22C2×#57711yescharacteristic
527C22#358511yescharacteristic
528C22#358611yescharacteristic
529C4—22no—
530C4—22no—
531C4—22no—
532C4—44no—
533C4—41no—
534C4—44no—
535C4—41no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4#335614yes—
544C4#350612yes—
545C4#355412yes—
546C2—84no—
547C2#4535411yescharacteristic
548C2C2×#335611yescharacteristic
549C2#6129111yescharacteristic
550C2#6158611yescharacteristic
551C2#6168211yescharacteristic
552C2#6171911yescharacteristic
553C2#6180311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32014C2semidirect
4. #31347C2semidirect
5. #29273C2semidirect
6. #29259C2semidirect
7. #28608C2semidirect
8. #26299C2semidirect
9. #26234C2semidirect
10. #26144C2semidirect
11. #24108C2semidirect
12. #23837C2semidirect
13. #22665C2semidirect
15. #22221C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#598212
#320142
#313472
#292732
#292592
#286082
#262992
#262342
#261442
#241082
#238372
#226652
#222642
#222212