Finite Group Databasebeta

← #6798753

Subgroups — #6798753

Order 512.

1662 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31744C218yesmaximal, complemented
5#29844C212yesmaximal, complemented
6#29208C211yesmaximal, characteristic, complemented
7#28717C211yesmaximal, characteristic, complemented
8#27878C212yesmaximal, complemented
9#26026C214yesmaximal, complemented
10#25902C212yesmaximal, complemented
11#24971C214yesmaximal
12#21834C212yesmaximal, complemented
13C4×#487C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×D4×Q8—21no—
16C22×C4⋊Q8C2212yes—
17C2×#487C2211yescharacteristic
18C2×#487C2212yes—
19C2×#431C2212yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2571—21no—
25#2563—21no—
26#2551—21no—
27#2551C2212yes—
28#2546—21no—
29#2538C2212yes—
30#2497—22no—
31#2492C2214yes—
32#2440—22no—
33#2440C2214yes—
34#2439—22no—
35#2434—22no—
36#2369—21no—
37#2355C2212yes—
38#2354—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43#2285C2212yes—
44C42×D4C2211yescharacteristic
45#2076C2218yes—
46#2072C2214yes—
47#2068C2218yes—
48#2057C2214yes—
49#2036C2218yes—
50#2024C2218yes—
51#1990C2214yes—
52#1990C2212yes—
53#1990C2212yes—
54#1990C2212yes—
55#1990C2212yes—
56#1990C2211yescharacteristic
57#1987C2218yes—
58#1956C2211yescharacteristic
59#1954C2212yes—
60#1947C2214yes—
61#1939C2212yes—
62#1933C2214yes—
63#1923C2214yes—
64#1921C2218yes—
65#1872C2218yes—
66#1845C2212yes—
67#1834C2214yes—
68#1834C2211yescharacteristic
69#1824C2218yes—
70#1803C2214yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×#103C2214yes—
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C23×Q8—22no—
76D4×Q8—48no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—28no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—44no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8C2314yes—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×C4×D4—22no—
97C2×C4×D4D411yescharacteristic
98C2×C4×D4D411yescharacteristic
99C2×C4×D4C2311yescharacteristic
100C2×#117—22no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4C2314yes—
103C22×C42C2312yes—
104#487—24no—
105#487—44no—
106#431—44no—
107C2×#107C2318yes—
108C2×C4⋊C8C2312yes—
109C2×C4⋊C8C2314yes—
110C2×#103C2314yes—
111C2×#103C2318yes—
112C2×#103C2314yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×C4×C8C2312yes—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2312yes—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2312yes—
120#400—24no—
121#398—24no—
122#396—24no—
123#395—24no—
124#395—24no—
125#393—24no—
126#391—21no—
127#391—22no—
128#391—24no—
129#391C2312yes—
130#391C2314yes—
131#390—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389C2312yes—
142#389C2314yes—
143#388—24no—
144#387—24no—
145#385—24no—
146#385—24no—
147#384—21no—
148#384—21no—
149#384—21no—
150#384—21no—
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#384C2312yes—
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#382—22no—
158#382—22no—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380C2314yes—
164#380C2314yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C22⋊C4D412yes—
176C4×C22⋊C4D412yes—
177C4×C22⋊C4C2312yes—
178C43C2311yescharacteristic
179#340C2318yes—
180#340C2318yes—
181#340C2318yes—
182#340C2314yes—
183#340C2318yes—
184#340C2314yes—
185#336C2312yes—
186#336C2314yes—
187C22×Q8—21no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—48no—
192C22×Q8—48no—
193C22×Q8—84no—
194C22×Q8C2412yes—
195C22×D4C2×D411yescharacteristic
196C23×C4—24no—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2412yes—
200C22×C8C2412yes—
201C22×C8C2414yes—
202C22×C8C2412yes—
203C4⋊Q8—28no—
204C4⋊Q8—24no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—24no—
208C4⋊Q8—22no—
209C4⋊Q8—44no—
210C4⋊Q8—44no—
211C4⋊Q8—48no—
212C4⋊Q8—48no—
213C4⋊Q8—48no—
214C4⋊Q8—44no—
215C4⋊Q8—44no—
216C4⋊Q8—82no—
217C4⋊Q8—82no—
218C4⋊Q8#4114yes—
219C22⋊Q8—416no—
220C22⋊Q8—48no—
221C22⋊Q8—48no—
222C4×Q8—28no—
223C4×Q8—44no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—48no—
227C4×D4Q1618yes—
228#117—84no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4C2×D412yes—
267C2×C22⋊C4C2×D412yes—
268C2×C22⋊C4C2×D412yes—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2411yescharacteristic
281C2×C42C2412yes—
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284#107—48no—
285C4⋊C8—24no—
286C4⋊C8—44no—
287#103—28no—
288#103—44no—
289#103—48no—
290C22⋊C8—44no—
291C22⋊C8—44no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—24no—
295C4×C8—24no—
296C4×C8—42no—
297#95—24no—
298#95—24no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310C24C22×D412yes—
311C2×Q8—24no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—44no—
319C2×Q8—42no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—42no—
323C2×Q8—44no—
324C2×Q8—84no—
325C2×Q8—82no—
326C2×Q8—82no—
327C2×Q8—832no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4C2×Q1614yes—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—84no—
349C22×C4C2×Q1618yes—
350C22×C4C22×D412yes—
351C22×C4C22×D412yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2511yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—84no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4C2×Q1614yes—
401C4⋊C4C2×#4114yes—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—48no—
405C22⋊C4—84no—
406C22⋊C4—84no—
407C22⋊C4C2×Q1618yes—
408C42—24no—
409C42—24no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—44no—
416C42—44no—
417C42—84no—
418C42C2×Q1614yes—
419C42AGL(1,ℤ/8)14yes—
420C42C2×#4112yes—
421C42H(5,𝔽2)12yes—
422C422−1+414yes—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23C22×Q1614yes—
430C23C23×D411yescharacteristic
431Q8—48no—
432Q8—44no—
433Q8—44no—
434Q8—816no—
435Q8—88no—
436Q8—164no—
437D4—28no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—216no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—816no—
475C2×C4#53612yes—
476C2×C4#53612yes—
477C2×C4#54612yes—
478C2×C4#54612yes—
479C2×C4C22×Q1612yes—
480C2×C4C22×Q1614yes—
481C2×C4C22×Q1612yes—
482C2×C4C22×Q1612yes—
483C2×C4C2×AGL(1,ℤ/8)12yes—
484C2×C4C2×AGL(1,ℤ/8)12yes—
485C2×C4C2×AGL(1,ℤ/8)12yes—
486C2×C4#57712yes—
487C2×C4#57712yes—
488C2×C4C22×#4111yescharacteristic
489C2×C4C2×H(5,𝔽2)11yescharacteristic
490C2×C4C2×2−1+412yes—
491C2×C4#58512yes—
492C8—42no—
493C8—42no—
494C8—44no—
495C22—24no—
496C22—24no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22C2×#53611yescharacteristic
503C22C2×#54611yescharacteristic
504C22#350011yescharacteristic
505C22C23×Q1611yescharacteristic
506C22C22×AGL(1,ℤ/8)11yescharacteristic
507C22C2×#57711yescharacteristic
508C22C2×#57711yescharacteristic
509C4—22no—
510C4—22no—
511C4—24no—
512C4—22no—
513C4—42no—
514C4—42no—
515C4—44no—
516C4—82no—
517C4—82no—
518C4—84no—
519C4—82no—
520C4—84no—
521C4#323014yes—
522C4#329914yes—
523C4#332414yes—
524C4#351812yes—
525C4#354712yes—
526C2—44no—
527C2#3895811yescharacteristic
528C2C2×#323011yescharacteristic
529C2#6088111yescharacteristic
530C2C2×#329911yescharacteristic
531C2C2×#332411yescharacteristic
532C2#6253011yescharacteristic
533C2#6254111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33289C2semidirect
4. #31744C2semidirect
5. #29844C2semidirect
6. #29208C2semidirect
7. #28717C2semidirect
8. #27878C2semidirect
9. #26026C2semidirect
10. #25902C2semidirect
12. #21834C2semidirect
13. C4×#487C2semidirect

Maximal subgroups

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

subgroupindex
#361942
D4×C4⋊Q82
#332892
#317442
#298442
#292082
#287172
#278782
#260262
#259022
#249712
#218342
C4×#4872
#212642