Finite Group Databasebeta

← #6806085

Subgroups — #6806085

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33225C211yesmaximal, characteristic, complemented
4#31639C218yesmaximal, complemented
5#30206C212yesmaximal, complemented
6#29218C211yesmaximal, characteristic, complemented
7#29144C212yesmaximal, complemented
8#28613C212yesmaximal
9#28438C211yesmaximal, characteristic
10#25934C214yesmaximal, complemented
11#25319C214yesmaximal, complemented
12#22268C212yesmaximal
13C4×#486C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic, complemented
15C2×#554C2212yes—
16C2×#492C2212yes—
17C2×#487C2214yes—
18C2×#486C2211yescharacteristic
19C2×#461C2214yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2862C2212yes—
23#2859C2212yes—
24#2582—21no—
25#2571—21no—
26#2570—22no—
27#2568C2212yes—
28#2566C2214yes—
29#2563C2212yes—
30#2538—21no—
31#2538—21no—
32#2511—22no—
33#2452—22no—
34#2452—22no—
35#2446C2214yes—
36#2365C2212yes—
37#2355—21no—
38#2355—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×#127C2211yescharacteristic
44C4×#127C2212yes—
45C42×Q8C2211yescharacteristic
46#2100C2214yes—
47#2074C2218yes—
48#2055C2218yes—
49#2026C2218yes—
50#2026C2218yes—
51#2013C2214yes—
52#1990C2212yes—
53#1990C2211yescharacteristic
54#1987C2214yes—
55#1984C2212yes—
56#1957C2211yescharacteristic
57#1956C2212yes—
58#1956C2214yes—
59#1956C2212yes—
60#1956C2212yes—
61#1939C2212yes—
62#1934C2214yes—
63#1916C2214yes—
64#1880C2218yes—
65#1877C2218yes—
66#1877C2218yes—
67#1834C2212yes—
68#1834C2211yescharacteristic
69#1805C2218yes—
70C4×C4⋊C8C2211yescharacteristic
71C4×C4⋊C8C2212yes—
72C4×#102C2214yes—
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75C22×SD16C2318yes—
76#554—48no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2312yes—
84C2×#127C2311yescharacteristic
85C2×#125C2314yes—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—24no—
88C2×C4×Q8—21no—
89C2×C4×Q8D411yescharacteristic
90C2×C4×Q8D411yescharacteristic
91C2×C4×Q8C2314yes—
92C2×C4×Q8C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4C2312yes—
98C2×C4×D4C2314yes—
99C22×C4⋊C4—22no—
100#492—28no—
101#487—48no—
102#486—24no—
103#461—48no—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2314yes—
106C2×#103C2318yes—
107C2×#103C2318yes—
108C2×#103C2318yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2312yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2312yes—
116#400—24no—
117#400—24no—
118#398—24no—
119#396—24no—
120#396—24no—
121#393—24no—
122#391—22no—
123#391—21no—
124#391—22no—
125#391—22no—
126#391C2314yes—
127#391C2314yes—
128#391C2314yes—
129#390—21no—
130#390—21no—
131#390—22no—
132#390—22no—
133#390C2314yes—
134#390C2312yes—
135#389—22no—
136#389—21no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389C2312yes—
144#387—24no—
145#387—24no—
146#387—24no—
147#385—24no—
148#385—24no—
149#384—21no—
150#384—21no—
151#384—21no—
152#384—21no—
153#384—21no—
154#384—21no—
155#384C2311yescharacteristic
156#384C2312yes—
157#384C2311yescharacteristic
158#384C2312yes—
159#382—22no—
160#382—22no—
161#382—22no—
162#382C2314yes—
163#380—22no—
164#380—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4D412yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C43C2312yes—
177C43C2311yescharacteristic
178#340C2318yes—
179#340C2314yes—
180#340C2318yes—
181#340C2314yes—
182#336C2314yes—
183#336C2312yes—
184#336C2314yes—
185#336C2314yes—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8C2×D411yescharacteristic
189C22×Q8C2414yes—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4C2412yes—
193C23×C4—22no—
194C23×C4—24no—
195C2×SD16—216no—
196C2×SD16—48no—
197C2×SD16—48no—
198C22×C8C2414yes—
199C22×C8C2412yes—
200C22×C8C2412yes—
201C4⋊Q8—22no—
202C4⋊Q8—24no—
203C4⋊Q8—28no—
204C4⋊Q8—28no—
205C4⋊Q8—22no—
206C4⋊Q8—24no—
207C4⋊Q8—44no—
208C4⋊Q8—82no—
209C4⋊Q8—82no—
210#127—22no—
211#127#4114yes—
212#125—48no—
213C22⋊Q8—416no—
214C22⋊Q8—48no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—48no—
219C4×Q8—44no—
220C4×D4—28no—
221C4×D4—48no—
222C4×D4—44no—
223C4×D4—44no—
224C4×D4—84no—
225C4×D4—84no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2412yes—
274C2×C42C2412yes—
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C4⋊C8—24no—
279C4⋊C8—44no—
280#103—48no—
281#103—48no—
282#103—48no—
283#102—28no—
284#102—44no—
285C4×C8—22no—
286C4×C8—24no—
287C4×C8—24no—
288C4×C8—22no—
289C4×C8—42no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—24no—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95C2×D412yes—
301#95C2×D412yes—
302#95C2×D412yes—
303C24—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—28no—
309C2×Q8—24no—
310C2×Q8—28no—
311C2×Q8—44no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4C2×#4114yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—48no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C2511yescharacteristic
349SD16—816no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—84no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C22⋊C4—48no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—22no—
404C42—24no—
405C42—44no—
406C42—84no—
407C42#13514yes—
408C42#13714yes—
409C42C2×#4112yes—
410C42H(5,𝔽2)14yes—
411C422−1+412yes—
412C23—24no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23C23×D411yescharacteristic
420Q8—44no—
421Q8—48no—
422Q8—48no—
423Q8—84no—
424Q8—164no—
425D4—28no—
426D4—48no—
427D4—48no—
428D4—44no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—88no—
463C2×C4—168no—
464C2×C4#53512yes—
465C2×C4#53512yes—
466C2×C4#54812yes—
467C2×C4#54812yes—
468C2×C4C2×#13512yes—
469C2×C4C2×#13512yes—
470C2×C4C2×#13512yes—
471C2×C4C2×#13712yes—
472C2×C4C2×#13712yes—
473C2×C4C2×#13712yes—
474C2×C4#57712yes—
475C2×C4#57712yes—
476C2×C4C22×#4111yescharacteristic
477C2×C4C2×H(5,𝔽2)12yes—
478C2×C4C2×2−1+411yescharacteristic
479C2×C4#58512yes—
480C8—42no—
481C8—44no—
482C8—42no—
483C22—44no—
484C22—44no—
485C22—44no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22C2×#53511yescharacteristic
491C22C2×#54811yescharacteristic
492C22#350011yescharacteristic
493C22C22×#13511yescharacteristic
494C22C22×#13711yescharacteristic
495C22C2×#57711yescharacteristic
496C22C2×#57711yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4—82no—
508C4—84no—
509C4#321014yes—
510C4#336014yes—
511C4#337214yes—
512C4#351412yes—
513C4#355412yes—
514C2—84no—
515C2#3917111yescharacteristic
516C2C2×#321011yescharacteristic
517C2#6081811yescharacteristic
518C2C2×#336011yescharacteristic
519C2C2×#337211yescharacteristic
520C2#6253211yescharacteristic
521C2#6254011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
3. #33225C2semidirect
4. #31639C2semidirect
5. #30206C2semidirect
6. #29218C2semidirect
7. #29144C2semidirect
10. #25934C2semidirect
11. #25319C2semidirect
14. #21339C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#365592
#332252
#316392
#302062
#292182
#291442
#286132
#284382
#259342
#253192
#222682
C4×#4862
#213392