Finite Group Databasebeta

← #7500093

Subgroups — #7500093

Order 512.

1445 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic, complemented
2#39168C211yesmaximal, characteristic
3#32669C212yesmaximal
4#32645C212yesmaximal, complemented
5#30878C214yesmaximal, complemented
6#30760C214yesmaximal, complemented
7#30041C212yesmaximal
8#29371C212yesmaximal, complemented
9#27439C214yesmaximal, complemented
10#27193C214yesmaximal, complemented
11#25936C212yesmaximal, complemented
12#25641C212yesmaximal
13#23365C211yesmaximal, characteristic, complemented
14C2×#559C2211yescharacteristic
15C2×Q82C2211yescharacteristic
16C2×#497C2212yes—
17C2×#487C2212yes—
18C2×#462C2212yes—
19C2×#461C2212yes—
20#2858C2211yescharacteristic
21#2809—21no—
22#2809—21no—
23#2674—24no—
24#2640—21no—
25#2571—21no—
26#2568C2212yes—
27#2568C2212yes—
28#2567C2211yescharacteristic
29#2548C2211yescharacteristic
30#2546C2214yes—
31#2545—21no—
32#2542—21no—
33#2508—24no—
34#2452C2214yes—
35#2448C2214yes—
36#2369—21no—
37#2355C2214yes—
38#2326—21no—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41#2295C2211yescharacteristic
42#2099C2212yes—
43#2096C2212yes—
44#2071C2214yes—
45#2066C2214yes—
46#2055C2214yes—
47#2043C2214yes—
48#2043C2212yes—
49#2024C2214yes—
50#2024C2214yes—
51#2016C2212yes—
52#2015C2214yes—
53#2014C2214yes—
54#2013C2214yes—
55#2013C2214yes—
56#1987C2212yes—
57#1985C2212yes—
58#1984C2212yes—
59#1983C2212yes—
60#1968C2214yes—
61#1935C2214yes—
62#1927C2214yes—
63#1926C2214yes—
64#1920C2214yes—
65#1917C2214yes—
66#1916C2214yes—
67#1880C2214yes—
68#1880C2214yes—
69#1878C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1877C2214yes—
73#1865C2214yes—
74#1839C2212yes—
75#1762C2212yes—
76C4×#103C2212yes—
77C22×Q16C2314yes—
78C22×SD16C2314yes—
79C22×SD16C2314yes—
80#559—42no—
81#559—42no—
82Q82—42no—
83Q82—42no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8D411yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#124—21no—
94C2×#124—21no—
95C2×#124—22no—
96C2×#124C2312yes—
97C2×#123—22no—
98C2×C22⋊Q8—24no—
99C2×C4×Q8—21no—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2311yescharacteristic
105C2×C4×D4—21no—
106C2×C4×D4C2312yes—
107#497—44no—
108#487—44no—
109#462—44no—
110#461—44no—
111C2×#106C2314yes—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2312yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124#400—22no—
125#400—22no—
126#400—22no—
127#400—22no—
128#399—24no—
129#398—24no—
130#398—24no—
131#397—24no—
132#396—24no—
133#395—22no—
134#394—22no—
135#393—24no—
136#391—21no—
137#391—22no—
138#391—22no—
139#391D411yescharacteristic
140#391D411yescharacteristic
141#391C2314yes—
142#391C2314yes—
143#391C2314yes—
144#391C2312yes—
145#390—21no—
146#390—21no—
147#390—21no—
148#390C2314yes—
149#390C2314yes—
150#389—24no—
151#389—24no—
152#389—22no—
153#389—22no—
154#389D414yes—
155#389C2314yes—
156#389C2312yes—
157#388—24no—
158#388—24no—
159#387—22no—
160#387—24no—
161#387—24no—
162#385—24no—
163#384—21no—
164#384D411yescharacteristic
165#384C2312yes—
166#382—22no—
167#382—22no—
168#382—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2314yes—
177#376—21no—
178C43C2311yescharacteristic
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2312yes—
185#336C2312yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C22×Q8—22no—
190C22×Q8—21no—
191C22×Q8C2×D411yescharacteristic
192C22×Q8C2×D411yescharacteristic
193C22×Q8C2414yes—
194C22×Q8C2411yescharacteristic
195C22×D4—21no—
196C22×D4—21no—
197C22×D4C2412yes—
198C23×C4—22no—
199C2×Q16—28no—
200C2×Q16—44no—
201C2×Q16—44no—
202C2×SD16—44no—
203C2×SD16—44no—
204C2×SD16—44no—
205C2×SD16—44no—
206C2×SD16—44no—
207C2×SD16—44no—
208C22×C8C2414yes—
209C22×C8C2414yes—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—42no—
213C4⋊Q8—41no—
214C4⋊Q8—41no—
215C4⋊Q8—44no—
216C4⋊Q8—44no—
217C4⋊Q8—42no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—48no—
222#127—41no—
223#127—41no—
224#124—24no—
225#124—42no—
226#124—44no—
227#124—44no—
228#124—84no—
229#123—48no—
230C22⋊Q8—88no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—24no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8—44no—
239C4×Q8—44no—
240C4×Q8—42no—
241C4×Q8—44no—
242C4×D4—24no—
243C4×D4—42no—
244C4×D4—44no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2411yescharacteristic
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2412yes—
287C2×C42C2411yescharacteristic
288C2×C42C2412yes—
289C2×C42C2412yes—
290#106—44no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293#103—28no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298#103—44no—
299#102—44no—
300#102—44no—
301#97—42no—
302C4×C8—42no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—21no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95C2×D414yes—
317#95C2×D414yes—
318C24—22no—
319C2×Q8—22no—
320C2×Q8—21no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—44no—
333C2×Q8—42no—
334C2×Q8—84no—
335C2×Q8—82no—
336C2×Q8#13714yes—
337C2×Q8H(5,𝔽2)12yes—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—82no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—44no—
354C22×C4—84no—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C2511yescharacteristic
364Q16—88no—
365SD16—88no—
366SD16—88no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C4⋊C4—24no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—82no—
395C4⋊C4—84no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4H(5,𝔽2)12yes—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—48no—
405C22⋊C4—48no—
406C22⋊C4—84no—
407C22⋊C4—84no—
408C42—24no—
409C42—22no—
410C42—24no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—21no—
416C42—21no—
417C42—24no—
418C42—41no—
419C42—44no—
420C42—48no—
421C42—42no—
422C42—82no—
423C42—82no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23C23×D411yescharacteristic
430Q8—24no—
431Q8—44no—
432Q8—42no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—82no—
438Q8—164no—
439Q8—162no—
440D4—44no—
441D4—42no—
442D4—42no—
443D4—84no—
444D4—162no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—164no—
479C2×C4C2×#13712yes—
480C2×C4C2×#13714yes—
481C2×C4#57714yes—
482C2×C4C2×H(5,𝔽2)11yescharacteristic
483C2×C4C2×H(5,𝔽2)11yescharacteristic
484C2×C4#58514yes—
485C8—44no—
486C8—44no—
487C22—82no—
488C22—82no—
489C22—84no—
490C22—84no—
491C22—82no—
492C22#348411yescharacteristic
493C22#348712yes—
494C22C22×#13711yescharacteristic
495C22C2×#57712yes—
496C22#358611yescharacteristic
497C4—22no—
498C4—24no—
499C4—24no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—41no—
505C4—42no—
506C4—42no—
507C4—84no—
508C4—82no—
509C4—84no—
510C4—82no—
511C4#354512yes—
512C4#355112yes—
513C2—162no—
514C2#5208311yescharacteristic
515C2#6157311yescharacteristic
516C2#6166211yescharacteristic
517C2#6190612yes—
518C2#6191612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59869C2semidirect
4. #32645C2semidirect
5. #30878C2semidirect
6. #30760C2semidirect
8. #29371C2semidirect
9. #27439C2semidirect
10. #27193C2semidirect
11. #25936C2semidirect
13. #23365C2semidirect

Maximal subgroups

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

subgroupindex
#598692
#391682
#326692
#326452
#308782
#307602
#300412
#293712
#274392
#271932
#259362
#256412
#233652