Finite Group Databasebeta

← #6477815

Subgroups — #6477815

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic
2#34179C211yesmaximal, characteristic
3#33159C214yesmaximal, complemented
4#29975C214yesmaximal, complemented
5#25995C214yesmaximal, complemented
6#25993C214yesmaximal, complemented
7#25991C214yesmaximal, complemented
8#25989C214yesmaximal, complemented
9#21939C211yesmaximal, characteristic
10#21939C211yesmaximal, characteristic
11#21931C211yesmaximal, characteristic
12#20699C211yesmaximal, characteristic
13#20699C211yesmaximal, characteristic
14C22×#117C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2842—21no—
18#2840—21no—
19#2665—22no—
20#2665—22no—
21#2628C2214yes—
22#2607—22no—
23#2439C2214yes—
24#2439C2214yes—
25#2347—22no—
26#2347C2214yes—
27#2346—22no—
28#2346C2214yes—
29#2316C2212yes—
30#2312—21no—
31#2312—21no—
32#2283—22no—
33#2280C2211yescharacteristic
34#2280C2211yescharacteristic
35#2280C2211yescharacteristic
36#2280C2211yescharacteristic
37#2274C2212yes—
38#2101C2214yescomplemented
39#2101C2214yescomplemented
40#2088C2214yescomplemented
41#2088C2214yescomplemented
42#2087C2214yescomplemented
43#2086C2214yescomplemented
44#2058C2214yescomplemented
45#2057C2214yescomplemented
46#1988C2214yescomplemented
47#1987C2214yescomplemented
48#1949C2214yescomplemented
49#1949C2214yescomplemented
50#1921C2214yescomplemented
51#1920C2214yescomplemented
52#1919C2214yescomplemented
53#1918C2214yescomplemented
54#1854C2214yes—
55#1854C2214yes—
56#1854C2214yes—
57#1854C2214yes—
58#1850C2212yes—
59#1850C2212yes—
60#1850C2212yes—
61#1850C2212yes—
62#1825C2214yes—
63#1825C2214yes—
64#1824C2212yes—
65#1824C2212yes—
66#1823C2212yes—
67#1823C2212yes—
68#1808C2212yes—
69#1808C2212yes—
70#1807C2212yes—
71#1807C2212yes—
72#1798C2214yes—
73#1735C2212yes—
74#1735C2212yes—
75#1735C2212yes—
76#1735C2212yes—
77C24×C4C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C2×C4⋊Q8C2314yes—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117—24no—
86C2×#117Q818yescomplemented
87C2×#117C2312yes—
88C2×#117C2312yes—
89C2×#117C2312yes—
90C2×#117C2312yes—
91C2×#117C2312yes—
92C2×#117C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4—21no—
96C22×C42—21no—
97C2×#107C2314yes—
98C2×#107C2314yes—
99C2×#106C2314yes—
100C2×#106C2314yes—
101C2×C4⋊C8C2314yes—
102C2×C4⋊C8C2314yes—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8C2312yes—
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2312yes—
108C2×C22⋊C8C2312yes—
109#400—24no—
110#399—24no—
111#396—24no—
112#396—24no—
113#395—24no—
114#393—24no—
115#393—24no—
116#393—24no—
117#391—24no—
118#391—24no—
119#389—24no—
120#389C2314yes—
121#389C2314yes—
122#389C2314yes—
123#389C2314yes—
124#388—24no—
125#388—24no—
126#388—24no—
127#388—24no—
128#387—24no—
129#387—24no—
130#383—24no—
131#382—24no—
132#382C2314yes—
133#381—21no—
134#381—21no—
135#381—21no—
136#381D412yes—
137#380—22no—
138#380—22no—
139#380D412yes—
140#380D412yes—
141#380C2312yes—
142#380C2312yes—
143#380C2312yes—
144#380C2312yes—
145#380C2312yes—
146#380C2312yes—
147#379—21no—
148#379—21no—
149#379—21no—
150#379D411yescharacteristic
151#379D411yescharacteristic
152C4×C4⋊C4—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4C2314yes—
155C4×C4⋊C4C2314yes—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—22no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4C2312yes—
165C4×C22⋊C4C2312yes—
166#376—24no—
167#340C2314yes—
168#340C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2314yes—
180#336C2314yes—
181C25C2×D411yescharacteristic
182C22×Q8—24no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×Q814yes—
191C23×C4#4114yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8—42no—
203C22×C8C2412yes—
204C22×C8C2412yes—
205C22×C8C2412yes—
206C22×C8C2412yes—
207C4⋊Q8—44no—
208C4⋊Q8—44no—
209#117—28no—
210#117—28no—
211#117—24no—
212#117—24no—
213#117—42no—
214#117—42no—
215#117—42no—
216#117—42no—
217#117—42no—
218#117—42no—
219#117—42no—
220#117—42no—
221#117C2×Q818yes—
222#117C2×Q818yes—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×Q814yes—
234C2×C4⋊C4C2×Q814yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2414yes—
237C2×C4⋊C4C2414yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4C2×D412yes—
253C2×C22⋊C4C2×D412yes—
254C2×C22⋊C4C2×D412yes—
255C2×C22⋊C4C2×D412yes—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C42—28no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42C2414yes—
267#107—44no—
268#107—44no—
269#106—44no—
270#106—44no—
271C4⋊C8—44no—
272C4⋊C8—44no—
273C22⋊C8—24no—
274C22⋊C8—24no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277#95—22no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—24no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95C2×D414yes—
292#95C2×D414yes—
293C24—42no—
294C24—42no—
295C24—42no—
296C24C22⋊Q814yes—
297C24C22×D411yescharacteristic
298C24C22×D411yescharacteristic
299C24C22×D411yescharacteristic
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—48no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4C22⋊Q814yes—
325C22×C4C22⋊Q814yes—
326C22×C4C22⋊Q814yes—
327C22×C4C22×D414yes—
328C22×C4C22×D414yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×Q812yes—
333C22×C4C22×Q812yes—
334C22×C4C22×Q812yes—
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4C2×#4112yes—
338C22×C4C2511yescharacteristic
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—82no—
348C2×C8—82no—
349C2×C8—82no—
350C2×C8—82no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—44no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4C22×Q814yes—
368C4⋊C4C22×Q814yes—
369C22⋊C4—28no—
370C22⋊C4—28no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C42—28no—
387C42—28no—
388C42—24no—
389C42—24no—
390C42—44no—
391C42—84no—
392C42—84no—
393C42—82no—
394C42—82no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—88no—
408C23C2×C22⋊Q812yes—
409C23C2×C22⋊Q812yes—
410C23C2×C22⋊Q812yes—
411C23C23×D411yescharacteristic
412Q8—168no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4C2×C22⋊Q812yes—
450C2×C4C2×C22⋊Q812yes—
451C2×C4C2×C22⋊Q812yes—
452C2×C4C23×Q811yescharacteristic
453C2×C4C22×#4111yescharacteristic
454C2×C4#58512yes—
455C2×C4#58512yes—
456C8—42no—
457C8—42no—
458C8—42no—
459C8—42no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—88no—
473C22C22×C22⋊Q811yescharacteristic
474C22#348512yes—
475C22#358412yes—
476C22#358412yes—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—44no—
481C4—44no—
482C4—44no—
483C4—44no—
484C4—84no—
485C4—84no—
486C4—84no—
487C4#345812yes—
488C4#349111yescharacteristic
489C4#349111yescharacteristic
490C2—42no—
491C2—42no—
492C2—42no—
493C2#3509011yescharacteristic
494C2#6048911yescharacteristic
495C2#6048911yescharacteristic
496C2#6176914yes—
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. #25995C2semidirect
6. #25993C2semidirect
7. #25991C2semidirect
8. #25989C2semidirect
38. #2101C22semidirect
39. #2101C22semidirect
40. #2088C22semidirect
41. #2088C22semidirect
42. #2087C22semidirect
43. #2086C22semidirect
44. #2058C22semidirect
45. #2057C22semidirect
46. #1988C22semidirect
47. #1987C22semidirect
48. #1949C22semidirect
49. #1949C22semidirect
50. #1921C22semidirect
51. #1920C22semidirect
52. #1919C22semidirect
53. #1918C22semidirect
86. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#374802
#341792
#331592
#299752
#259952
#259932
#259912
#259892
#219392
#219312
#206992