Finite Group Databasebeta

← #6461490

Subgroups — #6461490

Order 512.

2018 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#29378C218yesmaximal, complemented
4#29369C214yesmaximal, complemented
5#29368C214yesmaximal, complemented
6#23744C211yesmaximal, characteristic, complemented
7#23731C211yesmaximal, characteristic
8#23507C211yesmaximal, characteristic, complemented
9C2×#2013C211yesmaximal, characteristic
10#21064C212yesmaximal, complemented
11#20936C212yesmaximal
12#20933C212yesmaximal
13#20877C212yesmaximal, complemented
14#20774C211yesmaximal, characteristic, complemented
15C23×SD16C2212yes—
16C22×C4⋊Q8C2211yescharacteristic
17C2×#519C2212yes—
18C2×#518C2211yescharacteristic
19C2×#428C2212yes—
20C2×#419C2212yes—
21C2×#419C2212yes—
22C2×#418C2212yes—
23C22×#103C2212yes—
24#2365—22no—
25#2365—22no—
26#2357—22no—
27#2354—22no—
28#2344—24no—
29#2342—24no—
30#2327C2214yescomplemented
31#2319C2214yes—
32#2317C2211yescharacteristic
33#2310C2212yes—
34#2309C2211yescharacteristic
35C4×C4⋊Q8C2212yescomplemented
36C4×C4⋊Q8C2212yescomplemented
37#2296C2211yescharacteristic
38#2293C2214yes—
39C2×#390C2211yescharacteristic
40#2285C2211yescharacteristic
41#2274C2211yescharacteristic
42C4×#117C2211yescharacteristic
43#2101C2214yescomplemented
44#2100C2212yescomplemented
45#2099C2212yescomplemented
46#2094C2212yes—
47#2093C2212yes—
48#2092C2214yes—
49#2043C2214yescomplemented
50#2043C2214yescomplemented
51#2013C2214yes—
52#2013C2214yes—
53#1928C2218yescomplemented
54#1927C2218yescomplemented
55#1880C2218yes—
56#1880C2218yes—
57#1877C2218yes—
58#1877C2218yes—
59#1865C2218yescomplemented
60#1863C2218yescomplemented
61#1840C2214yescomplemented
62#1839C2214yescomplemented
63#1735C2212yes—
64#1735C2212yes—
65#1729C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C22×SD16—28no—
72C22×SD16—28no—
73C22×SD16—44no—
74C22×SD16C2318yes—
75C22×M16C2312yes—
76C23×C8C2312yes—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—22no—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81#519—44no—
82#519—44no—
83#518—24no—
84#518—24no—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8C2314yes—
88C2×C4×Q8C2314yes—
89C2×C4×D4—22no—
90C2×C4×D4C2314yes—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117C2311yescharacteristic
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C22×C4⋊C4—22no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—22no—
101C22×C22⋊C4—22no—
102C22×C42C2311yescharacteristic
103#428—44no—
104C2×#107C2314yes—
105C2×#106C2314yes—
106#419—28no—
107#419—28no—
108#418—44no—
109C2×#103—44no—
110C2×#103C2318yes—
111C2×#103C2318yes—
112C2×#103C2318yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115#391—24no—
116#391C2314yes—
117#391C2314yes—
118#390—21no—
119#390—21no—
120#390—22no—
121#390C2314yes—
122#390C2312yes—
123#390C2312yes—
124#389—24no—
125#389—24no—
126#389C2314yes—
127#389C2312yes—
128#389C2312yes—
129#388—24no—
130#388—24no—
131#388—24no—
132#388—24no—
133#387—28no—
134#387—28no—
135#385—22no—
136#385—22no—
137#384—21no—
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#382—21no—
141#382C2312yes—
142#381—24no—
143#381—24no—
144#380—22no—
145#380—24no—
146#379—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4D412yes—
153C4×C4⋊C4D412yes—
154C4×C4⋊C4C2314yes—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—24no—
157C4×C22⋊C4D414yes—
158#376C2312yes—
159C43C2312yes—
160#340C2314yes—
161#340C2314yes—
162#340C2314yes—
163#340C2314yes—
164#336C2312yes—
165#336C2312yes—
166#336C2312yes—
167#336C2318yes—
168#336C2318yes—
169#336C2312yes—
170C25—21no—
171C22×Q8—22no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8#4118yes—
178C22×Q8C2414yes—
179C22×D4—21no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4#4114yes—
184C22×D4C2412yes—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—21no—
188C23×C4C2×D412yes—
189C23×C4C2411yescharacteristic
190C2×SD16—216no—
191C2×SD16—216no—
192C2×SD16—48no—
193C2×SD16—432no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—24no—
197C22×C8—24no—
198C22×C8—24no—
199C22×C8—24no—
200C22×C8C2414yes—
201C22×C8C2414yes—
202C4⋊Q8—216no—
203C4⋊Q8—28no—
204C4⋊Q8—28no—
205C4×Q8—44no—
206C4×Q8—44no—
207C4×Q8—44no—
208C4×Q8—44no—
209C4×Q8—48no—
210C4×Q8—48no—
211C4×D4—216no—
212C4×D4—28no—
213C4×D4—28no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—24no—
218#117—24no—
219#117—24no—
220#117—24no—
221#117—416no—
222#117—88no—
223#117#4114yes—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2412yes—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—84no—
254C2×C22⋊C4C2×D412yes—
255C2×C22⋊C4C2×D412yes—
256C2×C42—21no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—24no—
263C2×C42C2×D414yes—
264C2×C42C2×D412yes—
265C2×C42C2×D414yes—
266C2×C42C2412yes—
267C2×C42C2412yes—
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270#107—44no—
271#106—44no—
272#103—216no—
273#103—216no—
274#103—48no—
275#102—44no—
276#102—44no—
277#95—22no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—22no—
282#95—28no—
283#95—28no—
284#95C2×D414yes—
285#95C2×D414yes—
286C24—22no—
287C24—24no—
288C24—42no—
289C24—42no—
290C24—82no—
291C24—82no—
292C24C22×D411yescharacteristic
293C2×Q8—28no—
294C2×Q8—24no—
295C2×Q8—216no—
296C2×Q8—216no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8C2×#4118yes—
306C2×D4—24no—
307C2×D4—216no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—48no—
314C2×D4C2×#4114yes—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—216no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—82no—
332C22×C4C22×D414yes—
333C22×C4C22×D412yes—
334C22×C4C22×D412yes—
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4114yes—
340C22×C4C2×#4112yes—
341C22×C4H(5,𝔽2)14yes—
342C22×C4C2511yescharacteristic
343SD16—432no—
344M16—48no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—48no—
352C4⋊C4—216no—
353C4⋊C4—216no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—44no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4C2×#4114yes—
373C22⋊C4—216no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—48no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—24no—
384C42—24no—
385C42—28no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—24no—
391C42—44no—
392C42—48no—
393C42—48no—
394C42—42no—
395C42—44no—
396C42—84no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—48no—
405C23—88no—
406C23—162no—
407C23—162no—
408C23—164no—
409C23#52512yes—
410C23#52512yes—
411C23#53512yes—
412C23C23×D411yescharacteristic
413Q8—28no—
414Q8—28no—
415Q8—416no—
416Q8—44no—
417Q8—44no—
418D4—28no—
419D4—48no—
420D4—44no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—48no—
436C2×C4—42no—
437C2×C4—416no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—816no—
451C2×C4—162no—
452C2×C4#52512yes—
453C2×C4#52512yes—
454C2×C4#53512yes—
455C2×C4#55118yes—
456C2×C4#55318yes—
457C2×C4#57714yes—
458C2×C4C22×#4111yescharacteristic
459C2×C4C22×#4112yes—
460C2×C4C22×#4111yescharacteristic
461C2×C4C2×H(5,𝔽2)12yes—
462C8—44no—
463C8—44no—
464C22—22no—
465C22—22no—
466C22—24no—
467C22—24no—
468C22—44no—
469C22—48no—
470C22—82no—
471C22—82no—
472C22—84no—
473C22—168no—
474C22#259814yes—
475C22C2×#52511yescharacteristic
476C22C2×#52511yescharacteristic
477C22C2×#53511yescharacteristic
478C22C2×#57712yes—
479C22#358612yes—
480C4—24no—
481C4—48no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—44no—
486C4—44no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4#259814yes—
492C4C2×#55112yes—
493C4C2×#55312yes—
494C2—24no—
495C2—84no—
496C2C2×#259811yescharacteristic
497C2#6058212yes—
498C2#6059112yes—
499C2#6082011yescharacteristic
500C2#6082811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34209C2semidirect
3. #29378C2semidirect
4. #29369C2semidirect
5. #29368C2semidirect
6. #23744C2semidirect
8. #23507C2semidirect
10. #21064C2semidirect
13. #20877C2semidirect
14. #20774C2semidirect
30. #2327C22semidirect
35. C4×C4⋊Q8C22semidirect
36. C4×C4⋊Q8C22semidirect
43. #2101C22semidirect
44. #2100C22semidirect
45. #2099C22semidirect
49. #2043C22semidirect
50. #2043C22semidirect
53. #1928C22semidirect
54. #1927C22semidirect
59. #1865C22semidirect
60. #1863C22semidirect
61. #1840C22semidirect
62. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#342552
#342092
#293782
#293692
#293682
#237442
#237312
#235072
C2×#20132
#210642
#209362
#209332
#208772
#207742