Finite Group Databasebeta

← #7616347

Subgroups — #7616347

Order 512.

1400 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic
2#59828C211yesmaximal, characteristic
3#32505C214yesmaximal
4#29287C211yesmaximal, characteristic
5#29287C211yesmaximal, characteristic
6#28829C212yesmaximal
7#28677C212yesmaximal
8#27990C214yesmaximal
9#26510C212yesmaximal
10#26508C212yesmaximal
11#26280C212yesmaximal
12#26278C212yesmaximal
13#23156C212yesmaximal
14Q8×#106C212yesmaximal
15C4×#498C211yesmaximal, characteristic
16C4×#498C211yesmaximal, characteristic
17C8×C4⋊Q8C211yesmaximal, characteristic
18C2×#557—21no—
19C2×#498C2211yescharacteristic
20C2×#498C2211yescharacteristic
21C2×C8×Q8C2212yes—
22#2806—22no—
23#2726—22no—
24#2584—22no—
25#2582—22no—
26#2582—21no—
27#2567C2212yes—
28#2552—21no—
29#2552C2212yes—
30#2552C2212yes—
31#2546C2212yes—
32#2507C2214yes—
33#2506C2214yes—
34#2453—22no—
35#2446—22no—
36#2369C2212yes—
37#2365—21no—
38#2365C2212yes—
39Q8×C4⋊C4C2212yes—
40Q8×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46#2100C2214yes—
47#2099C2214yes—
48#2088C2214yes—
49#2088C2214yes—
50#2088C2214yes—
51#2088C2214yes—
52#2084C2214yes—
53#2084C2214yes—
54#1990C2211yescharacteristic
55#1990C2211yescharacteristic
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1988C2212yes—
59#1988C2212yes—
60#1987C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1948C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1948C2214yes—
68#1948C2214yes—
69#1941C2214yes—
70#1941C2214yes—
71#1941C2212yes—
72#1941C2212yes—
73#1941C2212yes—
74#1941C2212yes—
75#1939C2212yes—
76#1939C2212yes—
77#1920C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1919C2214yes—
81#1844C2214yes—
82#1844C2211yescharacteristic
83#1844C2211yescharacteristic
84#1842C2211yescharacteristic
85C4×#106C2212yes—
86C4×#106C2214yes—
87C4×#106C2212yes—
88C4×#106C2212yes—
89C4×#106C2212yes—
90C8×C4⋊C4C2212yes—
91C8×C4⋊C4C2212yes—
92C42×C8C2211yescharacteristic
93#557—48no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8D411yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125—22no—
101C2×#125—24no—
102C2×#125—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2312yes—
111#498—24no—
112#498Q818yescomplemented
113C8×Q8—28no—
114C2×#106C2312yes—
115C2×#106C2312yes—
116C2×#106C2312yes—
117C2×#106C2312yes—
118C2×#106C2312yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2312yes—
122C2×#106C2312yes—
123C2×#106C2312yes—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4×C8C2314yes—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#400—24no—
136#398—24no—
137#398—24no—
138#398—24no—
139#398—24no—
140#398—24no—
141#398—24no—
142#395—24no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—24no—
151#391D412yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—24no—
156#389—22no—
157#389—22no—
158#389—24no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—24no—
164#389D412yes—
165#389D412yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#387—24no—
175#387—24no—
176#387—24no—
177#384—22no—
178#384D411yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2314yes—
184#384C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C43C2311yescharacteristic
199#340C2312yes—
200#340C2312yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2312yes—
206#340C2312yes—
207#340C2314yes—
208#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#336C2314yes—
214#336C2314yes—
215C22×Q8—21no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8C2×D412yes—
220C22×C8C2411yescharacteristic
221C22×C8C2414yes—
222C22×C8C2411yescharacteristic
223C22×C8C2411yescharacteristic
224C22×C8C2411yescharacteristic
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—48no—
232C4⋊Q8SD1618yes—
233C4⋊Q8C2×Q814yes—
234C4⋊Q8C2×Q814yes—
235#125—416no—
236#125—48no—
237#125—84no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—48no—
241C4×Q8—48no—
242C4×Q8—44no—
243C4×Q8—44no—
244C4×Q8—82no—
245C4×Q8—82no—
246C4×Q8—84no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2414yes—
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299#106—24no—
300#106—24no—
301#106—24no—
302#106—24no—
303#106—42no—
304#106—42no—
305#106—44no—
306#106—44no—
307#106C2×Q818yes—
308#106C2×Q818yes—
309C4⋊C8—28no—
310C4⋊C8—24no—
311C4⋊C8—24no—
312C4⋊C8—42no—
313C4×C8—28no—
314C4×C8—22no—
315C4×C8—22no—
316C4×C8—22no—
317C4×C8—41no—
318C4×C8C2×Q814yes—
319C4×C8#4114yes—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95C2×D414yes—
335#95C2×D414yes—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×Q8C2×SD1618yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D414yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D414yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2511yescharacteristic
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—24no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8C22×Q812yes—
379C2×C8C22×Q812yes—
380C2×C8C2×#4112yes—
381C2×C8C2×#4112yes—
382C2×C82−1+412yes—
383C4⋊C4—28no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4—81no—
419C4⋊C4—81no—
420C4⋊C4—84no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4C2×SD1618yes—
426C4⋊C4C2×SD1618yes—
427C4⋊C4C22×Q814yes—
428C4⋊C4C22×Q814yes—
429C42—22no—
430C42—22no—
431C42—24no—
432C42—21no—
433C42—24no—
434C42—21no—
435C42—41no—
436C42—48no—
437C42—44no—
438C42—44no—
439C42—44no—
440C42—44no—
441C42—84no—
442C42C2×SD1614yes—
443C42C22×Q812yes—
444C42C2×#4112yes—
445C422−1+412yes—
446C422−1+412yes—
447C23C23×D411yescharacteristic
448Q8—216no—
449Q8—48no—
450Q8—84no—
451Q8—84no—
452Q8—84no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4#54712yes—
492C2×C4#54712yes—
493C2×C4D4×Q812yes—
494C2×C4D4×Q812yes—
495C2×C4C22×SD1618yes—
496C2×C4C22×SD1612yes—
497C2×C4C22×SD1612yes—
498C2×C4C22×SD1612yes—
499C2×C4#57712yes—
500C2×C4#57712yes—
501C2×C4C23×Q811yescharacteristic
502C2×C4C22×#4111yescharacteristic
503C2×C4C2×2−1+411yescharacteristic
504C2×C4C2×2−1+411yescharacteristic
505C2×C4C2×2−1+411yescharacteristic
506C2×C4#58511yescharacteristic
507C8—22no—
508C8—22no—
509C8—44no—
510C8—41no—
511C8#55714yescomplemented
512C22C2×#54711yescharacteristic
513C22C2×D4×Q811yescharacteristic
514C22#349711yescharacteristic
515C22C23×SD1611yescharacteristic
516C22C2×#57711yescharacteristic
517C22C2×#57711yescharacteristic
518C22#358411yescharacteristic
519C4—28no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—44no—
524C4—41no—
525C4—44no—
526C4—44no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4#333514yes—
534C4Q8×SD1614yes—
535C4C2×#55711yescharacteristic
536C4#350211yescharacteristic
537C4#350711yescharacteristic
538C4#355611yescharacteristic
539C2#5982811yescharacteristic
540C2C2×#333511yescharacteristic
541C2#6128511yescharacteristic
542C2C2×Q8×SD1611yescharacteristic
543C2#6137111yescharacteristic
544C2#6235711yescharacteristic
545C2#6237111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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
112. #498Q8semidirect
511. C8#557semidirect

Maximal subgroups

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

subgroupindex
#598712
#598282
#325052
#292872
#288292
#286772
#279902
#265102
#265082
#262802
#262782
#231562
Q8×#1062
C4×#4982
C8×C4⋊Q82