Finite Group Databasebeta

← #6506567

Subgroups — #6506567

Order 512.

1716 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37386C211yesmaximal, characteristic
2#35574C211yesmaximal, characteristic
3#29341C212yesmaximal, complemented
4#29332C212yesmaximal, complemented
5#27985C212yesmaximal, complemented
6#27722C214yesmaximal, complemented
7#27698C214yesmaximal, complemented
8#27277C214yesmaximal, complemented
9#27258C214yesmaximal, complemented
10#25902C212yesmaximal, complemented
11#21771C212yesmaximal
12#20913C212yesmaximal
13#20530C211yesmaximal, characteristic
14C2×#536C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C2×#520C2211yescharacteristic
17C2×#487C2214yes—
18C2×#419C2212yes—
19C22×#103C2212yes—
20C2×#408C2212yes—
21#2857C2212yes—
22#2734—22no—
23#2724—22no—
24#2717—22no—
25#2717—22no—
26#2621C2212yes—
27#2579—22no—
28#2499C2214yes—
29#2452—22no—
30#2451—22no—
31#2440C2214yes—
32#2439C2214yes—
33#2434C2214yes—
34#2409—22no—
35#2317C2211yescharacteristic
36#2316C2211yescharacteristic
37#2311C2211yescharacteristic
38#2297C2211yescharacteristic
39#2296C2211yescharacteristic
40#2101C2214yes—
41#2099C2212yes—
42#2099C2212yes—
43#2074C2214yes—
44#2072C2214yes—
45#2043C2212yes—
46#2043C2214yes—
47#2043C2212yes—
48#2024C2214yes—
49#2024C2214yes—
50#1990C2214yes—
51#1960C2212yes—
52#1956C2212yes—
53#1935C2214yes—
54#1933C2214yes—
55#1933C2214yes—
56#1933C2214yes—
57#1894C2214yes—
58#1892C2214yes—
59#1873C2214yes—
60#1873C2214yes—
61#1873C2214yes—
62#1872C2214yes—
63#1872C2214yes—
64#1872C2214yes—
65#1836C2212yes—
66#1835C2212yes—
67#1825C2214yes—
68#1787C2214yes—
69#1753C2214yes—
70C8×C22⋊C4C2214yes—
71#1750C2212yes—
72C4×C22⋊C8C2212yes—
73#1736C2212yes—
74#1735C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8—21no—
77C23×Q8C2312yes—
78C22×M16C2312yes—
79C23×C8C2312yes—
80#536—28no—
81#536—28no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—44no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2314yes—
88C2×#124—22no—
89C2×#123—22no—
90C2×#123—22no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8C2314yes—
93C2×C22⋊Q8C2314yes—
94#520—44no—
95C2×C4×Q8—21no—
96C2×C4×D4—22no—
97C2×C4×D4—21no—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#487—48no—
109#419—44no—
110C2×#103—44no—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119#408—28no—
120C2×C22⋊C8C2314yes—
121C2×C22⋊C8C2314yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124C2×C4×C8C2314yes—
125#398—24no—
126#398—24no—
127#397—24no—
128#397—24no—
129#397—24no—
130#396—24no—
131#395—24no—
132#395—24no—
133#395—24no—
134#393—24no—
135#391—21no—
136#391—24no—
137#391D412yes—
138#391C2314yes—
139#391C2314yes—
140#390—21no—
141#390D412yes—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389C2314yes—
147#389C2312yes—
148#389C2312yes—
149#387—24no—
150#387—24no—
151#387—24no—
152#386—22no—
153#386D412yes—
154#386D412yes—
155#385—21no—
156#385—21no—
157#385—22no—
158#385—24no—
159#385—24no—
160#385—24no—
161#384C2312yes—
162#384C2312yes—
163#382C2312yes—
164#382C2312yes—
165#380—22no—
166#380—22no—
167#380—22no—
168#380C2314yes—
169C4×C4⋊C4—22no—
170C4×C22⋊C4C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2314yes—
180#336C2312yes—
181#336C2312yes—
182C2×#41—42no—
183C2×#41—41no—
184C2×#41—41no—
185C2×#41#4118yes—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—44no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8—84no—
196C22×Q8C2×D411yescharacteristic
197C22×Q8C2414yes—
198C22×D4C2×D411yescharacteristic
199C22×D4C2×D412yes—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C2×M16—24no—
209C2×M16—42no—
210C2×M16—42no—
211C22×C8—24no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8C2414yes—
215C22×C8C2414yes—
216C4⋊Q8—28no—
217C4⋊Q8—44no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—44no—
221C4⋊Q8—48no—
222C4⋊Q8—48no—
223#124—48no—
224#123—48no—
225#123—48no—
226C22⋊Q8—28no—
227C22⋊Q8—28no—
228C22⋊Q8—28no—
229C22⋊Q8—28no—
230C22⋊Q8—88no—
231C4×Q8—44no—
232C4×D4—44no—
233C4×D4—48no—
234#117—24no—
235#117—24no—
236#117—24no—
237#117—41no—
238#117—41no—
239#117—41no—
240#117—41no—
241#117—42no—
242#117—42no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2411yescharacteristic
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2411yescharacteristic
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—42no—
285C2×C42C2×D412yes—
286C2×C42C2412yes—
287C2×C42C2412yes—
288#103—28no—
289#103—28no—
290#103—44no—
291#103—44no—
292#103—44no—
293#103—44no—
294#103—44no—
295#103—44no—
296C22⋊C8—28no—
297C22⋊C8—28no—
298#97—42no—
299C4×C8—42no—
300C4×C8—44no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95C2×D414yes—
312#95C2×D414yes—
313C24C22×D412yes—
314C24C22×D411yescharacteristic
315#41—48no—
316#41—48no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—44no—
329C2×Q8—48no—
330C2×Q8—82no—
331C2×Q8—82no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×Q8—84no—
335C2×Q8—816no—
336C2×Q8#12414yes—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4#12414yes—
342C2×D4#12418yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—84no—
370C22×C4C22×D414yes—
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C2×#4114yes—
376C22×C4C2×#4114yes—
377C22×C4C2×#4114yes—
378C22×C4C2511yescharacteristic
379M16—84no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—84no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—42no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—84no—
411C4⋊C4H(5,𝔽2)14yes—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—84no—
422C42—24no—
423C42—22no—
424C42—22no—
425C42—42no—
426C42—44no—
427C42—42no—
428C42—42no—
429C42—44no—
430C23—21no—
431C23—22no—
432C23—21no—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23C2×#12412yes—
439C23C2×#12414yes—
440C23C23×D411yescharacteristic
441Q8—44no—
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—44no—
446Q8—816no—
447Q8—88no—
448D4—44no—
449D4—48no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—84no—
487C2×C4—88no—
488C2×C4—84no—
489C2×C4—164no—
490C2×C4C2×#12414yes—
491C2×C4C2×#12412yes—
492C2×C4#57714yes—
493C2×C4#57814yes—
494C2×C4C22×#4112yes—
495C2×C4C2×H(5,𝔽2)12yes—
496C2×C4#58511yescharacteristic
497C2×C4#58511yescharacteristic
498C8—44no—
499C8—44no—
500C22—24no—
501C22—22no—
502C22—28no—
503C22—24no—
504C22—42no—
505C22—44no—
506C22—41no—
507C22—42no—
508C22—41no—
509C22—42no—
510C22C22×#12411yescharacteristic
511C22#348411yescharacteristic
512C22#348511yescharacteristic
513C22C2×#57712yes—
514C22C2×#57812yes—
515C4—22no—
516C4—24no—
517C4—24no—
518C4—24no—
519C4—44no—
520C4—44no—
521C4—84no—
522C4—82no—
523C4—82no—
524C4—81no—
525C4—81no—
526C4—84no—
527C4#350512yes—
528C4#350512yes—
529C2—44no—
530C2—42no—
531C2#3537711yescharacteristic
532C2#6062312yes—
533C2#6163111yescharacteristic
534C2#6163411yescharacteristic
535C2#6178912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29341C2semidirect
4. #29332C2semidirect
5. #27985C2semidirect
6. #27722C2semidirect
7. #27698C2semidirect
8. #27277C2semidirect
9. #27258C2semidirect
10. #25902C2semidirect

Maximal subgroups

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

subgroupindex
#373862
#355742
#293412
#293322
#279852
#277222
#276982
#272772
#272582
#259022
#217712
#209132
#205302