Finite Group Databasebeta

← #6531997

Subgroups — #6531997

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#27617C214yesmaximal
4#27589C214yesmaximal
5#25063C214yesmaximal, complemented
6#25061C214yesmaximal, complemented
7#24997C214yesmaximal, complemented
8#24995C214yesmaximal, complemented
9#21982C211yesmaximal, characteristic
10#21981C211yesmaximal, characteristic
11#21818C211yesmaximal, characteristic
12#20734C211yesmaximal, characteristic
13#20734C211yesmaximal, characteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2582—22no—
19#2582—22no—
20#2582C2214yes—
21#2582C2214yes—
22C2×#398—22no—
23#2356—22no—
24#2356—22no—
25#2355C2212yes—
26#2355C2212yes—
27#2355C2212yes—
28#2355C2212yes—
29#2313—22no—
30#2310—21no—
31C2×#391—21no—
32C2×#391C2212yes—
33C2×#389—21no—
34C2×#389C2212yes—
35#2285C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38#2283C2214yes—
39#2274—21no—
40#2101C2218yes—
41#2097C2218yes—
42#2085C2218yes—
43#2074C2218yes—
44#1984C2214yes—
45#1984C2214yes—
46#1948C2214yes—
47#1947C2214yes—
48#1928C2218yes—
49#1927C2218yes—
50#1919C2218yes—
51#1918C2218yes—
52#1853C2214yes—
53#1852C2214yes—
54#1845C2212yes—
55#1844C2212yes—
56#1834C2212yes—
57#1834C2212yes—
58#1824C2212yes—
59#1824C2212yes—
60#1817C2214yes—
61#1803C2212yes—
62#1803C2212yes—
63#1803C2212yes—
64#1803C2212yes—
65#1787C2212yes—
66#1787C2212yes—
67C4×#103C2214yes—
68C4×#103C2214yes—
69C23×Q8D411yescharacteristic
70C23×Q8C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—24no—
76C2×C4×Q8—24no—
77C2×C4×Q8D412yes—
78C2×C4×Q8D412yes—
79C2×C4×Q8C2312yes—
80C2×C4×Q8C2312yes—
81C2×C4×Q8C2312yes—
82C2×C4×Q8C2312yes—
83C2×#117—21no—
84C2×#117—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4D412yes—
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C42—21no—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96#431—24no—
97#431—24no—
98C2×#107C2318yes—
99C2×#106C2318yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×#103C2314yes—
103C2×#103C2314yes—
104C2×#103C2314yes—
105C2×#103C2314yes—
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2312yes—
108C2×C4×C8C2312yes—
109C2×C4×C8C2312yes—
110#398—28no—
111#398—28no—
112#391—22no—
113#391—22no—
114#391—22no—
115#391—22no—
116#391—24no—
117#391—24no—
118#391—24no—
119#391C2318yes—
120#389—24no—
121#389—24no—
122#389—24no—
123#389C2318yes—
124#387—24no—
125#387—24no—
126#387—24no—
127#387—24no—
128#384—21no—
129#384—21no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2312yes—
135#384C2312yes—
136#383—24no—
137#382—21no—
138#382—21no—
139#382C2314yes—
140#382C2314yes—
141#381—24no—
142#380—21no—
143#380—21no—
144#380C2312yes—
145#380C2312yes—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2314yes—
153C4×C4⋊C4C2314yes—
154C4×C4⋊C4C2312yes—
155C4×C4⋊C4C2312yes—
156C4×C22⋊C4—24no—
157#376—24no—
158C2×#95—22no—
159C2×#95—22no—
160C2×#95—22no—
161#340C2314yes—
162#340C2314yes—
163#340C2314yes—
164#340C2314yes—
165#340C2318yes—
166#340C2318yes—
167#336C2318yes—
168#336C2318yes—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—42no—
177C22×Q8C2×D412yes—
178C22×Q8C2×D412yes—
179C22×Q8C2412yes—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C22×C8C2414yes—
189C22×C8C2414yes—
190C4×Q8—28no—
191C4×Q8—28no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—44no—
201C4×Q8—44no—
202#117—44no—
203#117—44no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
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—28no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4—84no—
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—24no—
249C2×C42—24no—
250C2×C42—44no—
251C2×C42C2×D414yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42#4114yes—
255C2×C42#4114yes—
256C2×C42C2414yes—
257C2×C42C2411yescharacteristic
258C2×C42C2411yescharacteristic
259#107—48no—
260#106—48no—
261C4⋊C8—24no—
262C4⋊C8—24no—
263#103—28no—
264#103—28no—
265#103—44no—
266#103—44no—
267C22⋊C8—24no—
268C22⋊C8—24no—
269C4×C8—24no—
270C4×C8—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—22no—
275#95—22no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—22no—
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—24no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—48no—
298C2×Q8—48no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—416no—
302C2×Q82−1+414yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4#12314yes—
330C22×C4#12314yes—
331C22×C4#13514yes—
332C22×C4#13714yes—
333C22×C4C22×D414yes—
334C22×C4C22×D414yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4C2511yescharacteristic
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—88no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C42—28no—
380C42—24no—
381C42—24no—
382C42—44no—
383C42—44no—
384C42—44no—
385C42—44no—
386C42—84no—
387C42—84no—
388C42#12314yes—
389C42#12314yes—
390C42C2×#4112yes—
391C42C2×#4112yes—
392C42C2×#4112yes—
393C42C2×#4112yes—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23C2×#12312yes—
399C23C2×#13512yes—
400C23C2×#13712yes—
401C23C23×D411yescharacteristic
402Q8—24no—
403Q8—24no—
404Q8—48no—
405Q8—48no—
406Q8—48no—
407Q8—42no—
408Q8—42no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—88no—
450C2×C4#485116yes—
451C2×C4C2×#12312yes—
452C2×C4C2×#12312yes—
453C2×C4C2×#12312yes—
454C2×C4C2×#12312yes—
455C2×C4C2×#12312yes—
456C2×C4C2×#13512yes—
457C2×C4C2×#13512yes—
458C2×C4C2×#13712yes—
459C2×C4C2×#13712yes—
460C2×C4C22×#4111yescharacteristic
461C2×C4C22×#4111yescharacteristic
462C2×C4C2×2−1+412yes—
463C2×C4#58512yes—
464C8—44no—
465C8—44no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22C2×#48514yes—
473C22C22×#12311yescharacteristic
474C22#348611yescharacteristic
475C22#348711yescharacteristic
476C22C22×#13511yescharacteristic
477C22C22×#13711yescharacteristic
478C22#358511yescharacteristic
479C22#358611yescharacteristic
480C4—22no—
481C4—22no—
482C4—22no—
483C4—44no—
484C4—44no—
485C4—44no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—84no—
491C4—82no—
492C4C2×#48514yes—
493C4C2×#48514yes—
494C4#349512yes—
495C4#349512yes—
496C2—24no—
497C2#3567211yescharacteristic
498C2C22×#48511yescharacteristic
499C2#6052011yescharacteristic
500C2#6182211yescharacteristic
501C2#6183011yescharacteristic
502C2#6188511yescharacteristic
503C2#6189311yescharacteristic
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. #25063C2semidirect
6. #25061C2semidirect
7. #24997C2semidirect
8. #24995C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
#341872
#276172
#275892
#250632
#250612
#249972
#249952
#219822
#219812
#218182
#207342