Finite Group Databasebeta

← #6477819

Subgroups — #6477819

Order 512.

1520 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3#33159C214yesmaximal, complemented
4#29975C214yesmaximal, complemented
5#26010C214yesmaximal, complemented
6#26008C214yesmaximal, complemented
7#26006C214yesmaximal, complemented
8#26004C214yesmaximal, complemented
9#21943C211yesmaximal, characteristic
10#21943C211yesmaximal, characteristic
11#21935C211yesmaximal, characteristic
12#20700C211yesmaximal, characteristic
13#20700C211yesmaximal, characteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#407C2211yescharacteristic
16C2×#407C2211yescharacteristic
17#2842—21no—
18#2840—21no—
19#2665—22no—
20#2665—22no—
21#2628C2214yes—
22#2607—22no—
23#2439C2214yes—
24#2439C2214yes—
25#2345—22no—
26#2345C2212yes—
27#2345C2212yes—
28#2344—22no—
29#2344C2212yes—
30#2344C2212yes—
31#2316C2212yes—
32#2313—22no—
33#2280—21no—
34#2280—21no—
35#2280C2211yescharacteristic
36#2280C2211yescharacteristic
37#2280C2211yescharacteristic
38#2280C2211yescharacteristic
39#2274C2212yes—
40#2101C2214yescomplemented
41#2101C2214yescomplemented
42#2088C2214yescomplemented
43#2088C2214yescomplemented
44#2087C2214yescomplemented
45#2086C2214yescomplemented
46#2058C2214yescomplemented
47#2057C2214yescomplemented
48#1988C2214yescomplemented
49#1987C2214yescomplemented
50#1949C2214yescomplemented
51#1949C2214yescomplemented
52#1921C2214yescomplemented
53#1920C2214yescomplemented
54#1919C2214yescomplemented
55#1918C2214yescomplemented
56#1854C2214yes—
57#1854C2214yes—
58#1854C2214yes—
59#1854C2214yes—
60#1851C2214yes—
61#1851C2214yes—
62#1824C2212yes—
63#1824C2212yes—
64#1824C2212yes—
65#1824C2212yes—
66#1823C2212yes—
67#1823C2212yes—
68#1823C2212yes—
69#1823C2212yes—
70#1809C2214yes—
71#1809C2214yes—
72#1798C2214yes—
73#1736C2212yes—
74#1736C2212yes—
75#1736C2212yes—
76#1736C2212yes—
77C24×C4—21no—
78C24×C4C2311yescharacteristic
79C22×M16C2312yes—
80C22×M16C2312yes—
81C2×C4⋊Q8C2314yes—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117C2312yes—
85C2×#117C2312yes—
86C2×#117C2312yes—
87C2×#117C2312yes—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4Q818yescomplemented
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C2×#107C2314yes—
100C2×#107C2314yes—
101C2×#106C2314yes—
102C2×#106C2314yes—
103C2×C4⋊C8C2314yes—
104C2×C4⋊C8C2314yes—
105#407—24no—
106#407—24no—
107C2×C22⋊C8C2312yes—
108C2×C22⋊C8C2312yes—
109C2×C22⋊C8C2312yes—
110C2×C22⋊C8C2312yes—
111#400—24no—
112#399—24no—
113#396—24no—
114#396—24no—
115#395—24no—
116#393—24no—
117#393—24no—
118#393—24no—
119#391—24no—
120#391—24no—
121#389—24no—
122#389C2314yes—
123#389C2314yes—
124#389C2314yes—
125#389C2314yes—
126#387—24no—
127#387—24no—
128#387—24no—
129#387—24no—
130#387—24no—
131#387—24no—
132#382C2314yes—
133#381—21no—
134#381—21no—
135#381—21no—
136#381—22no—
137#381—22no—
138#381D412yes—
139#380—22no—
140#380—22no—
141#380—22no—
142#380—22no—
143#380D412yes—
144#380D412yes—
145#380C2312yes—
146#380C2312yes—
147#380C2312yes—
148#380C2312yes—
149#380C2312yes—
150#380C2312yes—
151#379—21no—
152#379—21no—
153#379—21no—
154#379D411yescharacteristic
155#379D411yescharacteristic
156C4×C4⋊C4—24no—
157C4×C4⋊C4C2314yes—
158C4×C4⋊C4C2314yes—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—22no—
166C4×C22⋊C4—22no—
167C4×C22⋊C4—22no—
168C4×C22⋊C4—22no—
169C4×C22⋊C4C2312yes—
170C4×C22⋊C4C2312yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2314yes—
185C25C2×D411yescharacteristic
186C22×Q8—24no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—28no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—84no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×Q814yes—
201C23×C4#4114yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C2×M16—24no—
206C2×M16—24no—
207C2×M16—42no—
208C2×M16—42no—
209C2×M16—42no—
210C2×M16—42no—
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—44no—
214C4⋊Q8—44no—
215#117—42no—
216#117—42no—
217#117—42no—
218#117—42no—
219#117—42no—
220#117—42no—
221#117—42no—
222#117—42no—
223#117—84no—
224#117—84no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×Q814yes—
247C2×C4⋊C4C2×Q814yes—
248C2×C4⋊C4C2×Q814yes—
249C2×C4⋊C4C2×Q814yes—
250C2×C4⋊C4C2×Q814yes—
251C2×C4⋊C4C2×Q814yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2411yescharacteristic
256C2×C4⋊C4C2411yescharacteristic
257C2×C4⋊C4C2411yescharacteristic
258C2×C4⋊C4C2411yescharacteristic
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4C2×D412yes—
269C2×C22⋊C4C2×D412yes—
270C2×C22⋊C4C2×D412yes—
271C2×C22⋊C4C2×D412yes—
272C2×C22⋊C4C2×D412yes—
273C2×C22⋊C4C2×D412yes—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42C2414yes—
280#107—44no—
281#107—44no—
282#106—44no—
283#106—44no—
284C4⋊C8—44no—
285C4⋊C8—44no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95C2×D414yes—
305#95C2×D414yes—
306C24—42no—
307C24—42no—
308C24—42no—
309C24C22⋊Q814yes—
310C24C22×D411yescharacteristic
311C24C22×D411yescharacteristic
312C24C22×D411yescharacteristic
313C2×Q8—84no—
314C2×Q8—84no—
315C2×Q8—84no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—48no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—816no—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q814yes—
350C22×C4C22⋊Q814yes—
351C22×C4C22×D414yes—
352C22×C4C22×D414yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q812yes—
357C22×C4C22×Q812yes—
358C22×C4C22×Q812yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4C2×#4112yes—
362C22×C4C2511yescharacteristic
363M16—84no—
364M16—84no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—44no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4C22×Q812yes—
394C4⋊C4C22×Q812yes—
395C4⋊C4C22×Q812yes—
396C4⋊C4C22×Q812yes—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C42—24no—
413C42—24no—
414C42—44no—
415C42—82no—
416C42—82no—
417C42—84no—
418C42—84no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—416no—
432C23C2×C22⋊Q812yes—
433C23C2×C22⋊Q812yes—
434C23C2×C22⋊Q812yes—
435C23C23×D411yescharacteristic
436Q8—168no—
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—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4C2×C22⋊Q812yes—
482C2×C4C2×C22⋊Q812yes—
483C2×C4C2×C22⋊Q812yes—
484C2×C4C23×Q811yescharacteristic
485C2×C4C22×#4111yescharacteristic
486C2×C4#58512yes—
487C2×C4#58512yes—
488C8—44no—
489C8—44no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—416no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22C22×C22⋊Q811yescharacteristic
504C22#348512yes—
505C22#358512yes—
506C22#358612yes—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—44no—
515C4—44no—
516C4—84no—
517C4—84no—
518C4—84no—
519C4#345811yescharacteristic
520C4#345811yescharacteristic
521C4#349112yes—
522C2—42no—
523C2—42no—
524C2—42no—
525C2#3509011yescharacteristic
526C2#6049011yescharacteristic
527C2#6049111yescharacteristic
528C2#6177514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33159C2semidirect
4. #29975C2semidirect
5. #26010C2semidirect
6. #26008C2semidirect
7. #26006C2semidirect
8. #26004C2semidirect
40. #2101C22semidirect
41. #2101C22semidirect
42. #2088C22semidirect
43. #2088C22semidirect
44. #2087C22semidirect
45. #2086C22semidirect
46. #2058C22semidirect
47. #2057C22semidirect
48. #1988C22semidirect
49. #1987C22semidirect
50. #1949C22semidirect
51. #1949C22semidirect
52. #1921C22semidirect
53. #1920C22semidirect
54. #1919C22semidirect
55. #1918C22semidirect
92. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#374802
#341782
#331592
#299752
#260102
#260082
#260062
#260042
#219432
#219352
#207002