Finite Group Databasebeta

← #6484976

Subgroups — #6484976

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#33100C214yesmaximal
4#29913C214yesmaximal
5#26317C214yesmaximal, complemented
6#26277C214yesmaximal, complemented
7#25061C214yesmaximal, complemented
8#24995C214yesmaximal, complemented
9#21981C211yesmaximal, characteristic
10#21860C211yesmaximal, characteristic
11#21818C211yesmaximal, characteristic
12#21385C211yesmaximal, characteristic
13#21358C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2855C2214yes—
19#2627—22no—
20#2582C2212yes—
21#2582C2212yes—
22#2581—22no—
23#2568C2212yes—
24#2567C2212yes—
25#2565—22no—
26#2355—22no—
27#2355—22no—
28#2355C2214yes—
29#2355C2214yes—
30#2317—21no—
31#2310—21no—
32C2×#391—21no—
33C2×#387—22no—
34#2285C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2212yes—
37C2×C4×C4⋊C4—21no—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40#2101C2218yes—
41#2097C2218yes—
42#2085C2218yes—
43#2074C2218yes—
44#1990C2212yes—
45#1987C2212yes—
46#1984C2214yes—
47#1984C2214yes—
48#1981C2212yes—
49#1956C2212yes—
50#1948C2214yes—
51#1947C2212yes—
52#1947C2212yes—
53#1928C2218yes—
54#1927C2218yes—
55#1919C2218yes—
56#1918C2218yes—
57#1872C2212yes—
58#1872C2214yes—
59#1872C2212yes—
60#1853C2214yes—
61#1852C2212yes—
62#1852C2212yes—
63#1844C2212yes—
64#1834C2212yes—
65#1817C2214yes—
66#1803C2214yes—
67C4×#107C2214yes—
68C4×#103C2214yes—
69C23×Q8D411yescharacteristic
70#533—24no—
71#533—24no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×#125—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8C2312yes—
77C2×C4×Q8—21no—
78C2×C4×Q8—24no—
79C2×C4×Q8—24no—
80C2×C4×Q8D412yes—
81C2×C4×Q8D412yes—
82C2×C4×Q8C2312yes—
83C2×C4×Q8C2312yes—
84C2×C4×Q8C2312yes—
85C2×#117—21no—
86C2×#117—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4D412yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C42—21no—
95C22×C42—22no—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98#431—24no—
99#431—24no—
100C2×#107C2314yes—
101C2×#107C2314yes—
102C2×#106C2318yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×#103C2314yes—
106C2×#103C2318yes—
107C2×#103C2314yes—
108C2×C22⋊C8C2312yes—
109C2×C22⋊C8C2312yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2312yes—
112#400—24no—
113#398—24no—
114#398—24no—
115#395—24no—
116#391—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#391—24no—
121#391—24no—
122#391C2314yes—
123#391C2314yes—
124#389—22no—
125#389—22no—
126#389C2314yes—
127#389C2314yes—
128#387—28no—
129#387—28no—
130#387—24no—
131#387—24no—
132#385—24no—
133#384—22no—
134#384—21no—
135#384—21no—
136#384C2312yes—
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2312yes—
140#382—22no—
141#382—21no—
142#382—24no—
143#382C2318yes—
144#381—24no—
145#380—21no—
146#380C2312yes—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4C2312yes—
154C4×C4⋊C4C2318yes—
155C4×C4⋊C4C2312yes—
156C4×C4⋊C4C2312yes—
157C2×#95—22no—
158C2×#95—21no—
159C2×#95—22no—
160C2×#95—22no—
161#340C2318yes—
162#340C2314yes—
163#340C2318yes—
164#340C2314yes—
165#340C2314yes—
166#340C2314yes—
167#336C2318yes—
168#336C2318yes—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—21no—
173C22×Q8C2×D412yes—
174C22×Q8C2×D412yes—
175C22×Q8C2412yes—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8C2414yes—
185C22×C8C2414yes—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190#125—44no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—44no—
194C4×Q8—28no—
195C4×Q8—28no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—24no—
202C4×Q8—44no—
203#117—44no—
204#117—44no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2412yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—28no—
249C2×C42—44no—
250C2×C42—84no—
251C2×C42C2×D414yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42#4114yes—
255C2×C42#4114yes—
256C2×C42C2414yes—
257C2×C42C2411yescharacteristic
258C2×C42C2411yescharacteristic
259#107—28no—
260#107—44no—
261#106—48no—
262C4⋊C8—24no—
263C4⋊C8—24no—
264#103—28no—
265#103—44no—
266#103—48no—
267C22⋊C8—24no—
268C22⋊C8—24no—
269C4×C8—24no—
270C4×C8—24no—
271#95—28no—
272#95—22no—
273#95—22no—
274#95—28no—
275#95—28no—
276#95—22no—
277#95—22no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284C24C22×D411yescharacteristic
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—42no—
296C2×Q8—416no—
297C2×Q8—81no—
298C2×Q8—81no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4#12314yes—
326C22×C4#12314yes—
327C22×C4#13514yes—
328C22×C4#13714yes—
329C22×C4C22×D414yes—
330C22×C4C22×D414yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4C2×#4112yes—
338C22×C4C2511yescharacteristic
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C42−1+414yes—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C42—28no—
379C42—24no—
380C42—24no—
381C42—44no—
382C42—44no—
383C42—44no—
384C42—44no—
385C42—88no—
386C42#12314yes—
387C42#12314yes—
388C42C2×#4112yes—
389C42C2×#4112yes—
390C42C2×#4112yes—
391C42C2×#4112yes—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23C2×#12312yes—
397C23C2×#13512yes—
398C23C2×#13712yes—
399C23C23×D411yescharacteristic
400Q8—48no—
401Q8—48no—
402Q8—44no—
403Q8—44no—
404Q8—84no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—84no—
445C2×C4—88no—
446C2×C4#485116yes—
447C2×C4C2×#12312yes—
448C2×C4C2×#12312yes—
449C2×C4C2×#12312yes—
450C2×C4C2×#12312yes—
451C2×C4C2×#12312yes—
452C2×C4C2×#13512yes—
453C2×C4C2×#13512yes—
454C2×C4C2×#13712yes—
455C2×C4C2×#13712yes—
456C2×C4C22×#4111yescharacteristic
457C2×C4C22×#4111yescharacteristic
458C2×C4C2×2−1+412yes—
459C2×C4#58512yes—
460C8—44no—
461C8—44no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22C2×#48514yes—
469C22C22×#12311yescharacteristic
470C22#348611yescharacteristic
471C22#348711yescharacteristic
472C22C22×#13511yescharacteristic
473C22C22×#13711yescharacteristic
474C22#358511yescharacteristic
475C22#358611yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—44no—
480C4—44no—
481C4—44no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4—84no—
488C4C2×#48514yes—
489C4C2×#48514yes—
490C4#346112yes—
491C4#350712yes—
492C2—24no—
493C2#3518211yescharacteristic
494C2C22×#48511yescharacteristic
495C2#6051711yescharacteristic
496C2#6181911yescharacteristic
497C2#6182911yescharacteristic
498C2#6188211yescharacteristic
499C2#6189211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #26317C2semidirect
6. #26277C2semidirect
7. #25061C2semidirect
8. #24995C2semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#23552
#331002
#299132
#263172
#262772
#250612
#249952
#219812
#218602
#218182
#213852
#213582