Finite Group Databasebeta

← #6433235

Subgroups — #6433235

Order 512.

1976 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60519C211yesmaximal, characteristic, complemented
2#60518C211yesmaximal, characteristic
3C2×#3057C211yesmaximal, characteristic
4#60100C211yesmaximal, characteristic, complemented
5#29793C218yesmaximal, complemented
6#29790C218yesmaximal, complemented
7#23674C212yesmaximal, complemented
8#23603C212yesmaximal, complemented
9#23602C212yesmaximal, complemented
10C2×#2046C212yesmaximal, complemented
11#20991C211yesmaximal, characteristic, complemented
12#20990C211yesmaximal, characteristic
13#20669C211yesmaximal, characteristic, complemented
14C23×SD16C2211yescharacteristic
15C2×#568C2211yescharacteristic
16C22×#131C2211yescharacteristic
17C2×#536C2211yescharacteristic
18C2×#535C2211yescharacteristic
19#3428C2211yescharacteristic
20#3097—28no—
21#3096—28no—
22#3095C2218yescomplemented
23#3094C2218yes—
24#3057—28no—
25C2×#468C2214yes—
26C2×#465C2214yes—
27C2×#430C2212yes—
28#2911C2212yes—
29#2911C2212yes—
30C2×#427C2211yescharacteristic
31#2906—28no—
32#2905C2212yes—
33#2905C2214yescomplemented
34C2×#424C2212yes—
35C2×#424C2214yescomplemented
36C2×#419C2212yes—
37C2×#419C2211yescharacteristic
38C2×#418C2212yes—
39C2×#418C2211yescharacteristic
40C22×#101C2212yes—
41C22×#100C2212yes—
42#2879C2211yescharacteristic
43#2879C2211yescharacteristic
44#2090C2218yescomplemented
45#2089C2218yescomplemented
46#2064C22116yescomplemented
47#2063C22116yes—
48#2046C2218yes—
49#2046C2218yes—
50#1901C2218yescomplemented
51#1900C2218yes—
52#1811C2214yescomplemented
53#1810C2214yescomplemented
54#1742C2212yes—
55#1742C2212yes—
56C2×#341C2212yes—
57C22×#41C2311yescharacteristic
58C23×Q8C2311yescharacteristic
59C23×D4C2311yescharacteristic
60C22×SD16—24no—
61C22×SD16—44no—
62C22×SD16C2312yes—
63C22×SD16C2312yes—
64#568—44no—
65C2×#131—24no—
66C2×#131—44no—
67C2×#131D418yes—
68C22×M16C2312yes—
69C22×M16C2312yes—
70C22×M16C2312yes—
71C22×M16C2314yes—
72C22×M16C2311yescharacteristic
73C22×M16C2312yes—
74C23×C8C2311yescharacteristic
75C23×C8C2312yes—
76#536—24no—
77#536—24no—
78#535—24no—
79#535—24no—
80C2×C4⋊Q8—21no—
81C2×#127—21no—
82C2×#124—21no—
83C2×#124—21no—
84C2×#123—22no—
85C2×C22⋊Q8C2314yes—
86C2×#121C2314yes—
87C2×#120—22no—
88#520D418yes—
89#519—21no—
90#518—22no—
91#518—21no—
92C4×#41—24no—
93C2×#117C2314yes—
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97#485—216no—
98#484—28no—
99#483—28no—
100#482—216no—
101#481—28no—
102#480—28no—
103#468—216no—
104#465—216no—
105#430—44no—
106C2×#108C2318yes—
107#428—24no—
108#428—24no—
109#428C2318yes—
110#428C2318yes—
111#427—24no—
112#427—24no—
113#427—24no—
114C2×#107C2312yes—
115C2×#106C2312yes—
116#424—24no—
117#424—28no—
118#424—24no—
119#424—28no—
120#424C2318yes—
121#424C2314yes—
122#424C2314yes—
123#423—24no—
124#423—28no—
125C2×C4⋊C8—24no—
126C2×C4⋊C8—28no—
127#419—24no—
128#419—24no—
129#419C2318yes—
130#418—24no—
131#418—24no—
132#418C2318yes—
133C2×#103—24no—
134C2×#103C2314yes—
135C2×#102—24no—
136C2×#102C2314yes—
137C2×#101—44no—
138C2×#101C2318yes—
139C2×#100—44no—
140C2×#100C2318yes—
141#408—24no—
142#408—24no—
143#407C2314yes—
144#407C2314yes—
145#343C2318yes—
146#343C2318yes—
147#341C2314yes—
148#341C2314yes—
149C25—21no—
150C2×#41—24no—
151C2×#41—24no—
152C2×#41C2×D414yes—
153C2×#41C2×D414yes—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8#4111yescharacteristic
160C22×Q8C2412yes—
161C22×D4—24no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4#4112yes—
168C22×D4#4111yescharacteristic
169C22×D4C2412yes—
170C23×C4—21no—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C2×SD16—24no—
174C2×SD16—24no—
175C2×SD16—44no—
176C2×SD16—44no—
177C2×SD16—416no—
178#131—416no—
179#131—416no—
180C2×M16—24no—
181C2×M16—24no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—44no—
192C2×M16—48no—
193C2×M16C2×D414yes—
194C2×M16C2×D418yes—
195C2×M16C2414yes—
196C2×M16C2414yes—
197C2×M16C2414yes—
198C22×C8—24no—
199C22×C8—24no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—44no—
204C22×C8C2×D414yes—
205C22×C8C2×D418yes—
206C22×C8C2411yescharacteristic
207C22×C8C2411yescharacteristic
208C22×C8C2412yes—
209C4⋊Q8—44no—
210#127—44no—
211#124—44no—
212#124—44no—
213#123—48no—
214C22⋊Q8—28no—
215C22⋊Q8—28no—
216#121—28no—
217#121—28no—
218#120—48no—
219C4×Q8—28no—
220C4×D4—216no—
221C4×D4—28no—
222#117—24no—
223#117—22no—
224#117—24no—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117C2×D414yes—
231#117C2×D418yes—
232#117C2414yes—
233C2×C4⋊C4—24no—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D418yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2412yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248#108—48no—
249#107—24no—
250#107—28no—
251#107—28no—
252#106—24no—
253#106—28no—
254#106—28no—
255C4⋊C8—28no—
256C4⋊C8—216no—
257C4⋊C8—28no—
258#103—28no—
259#103—216no—
260#102—28no—
261#102—216no—
262#101—48no—
263#100—48no—
264C22⋊C8—28no—
265C22⋊C8—28no—
266C24—22no—
267C24—42no—
268C24—44no—
269C24—162no—
270C24C2×#4112yes—
271C24C2×#4111yescharacteristic
272#41—28no—
273#41—28no—
274#41—28no—
275#41—28no—
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—42no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8#12114yes—
288C2×D4—28no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—48no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—164no—
302C2×D4#12118yes—
303C2×D4#12114yes—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—28no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4C22×D414yes—
314C22×C4C22×D414yes—
315C22×C4C22×D412yes—
316C22×C4C22×D412yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4111yescharacteristic
319C22×C4H(5,𝔽2)14yes—
320C22×C42−1+414yes—
321C22×C4C2511yescharacteristic
322SD16—48no—
323SD16—48no—
324M16—28no—
325M16—24no—
326M16—24no—
327M16—24no—
328M16—24no—
329M16—24no—
330M16—24no—
331M16—48no—
332M16—48no—
333M16—44no—
334M16—44no—
335M16—44no—
336C2×C8—28no—
337C2×C8—24no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—48no—
344C2×C8—48no—
345C2×C8—44no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8C22×D414yes—
349C2×C8C22×D412yes—
350C2×C8C22×D412yes—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—48no—
358C4⋊C4C22×D418yes—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C42—28no—
367C42—24no—
368C42—24no—
369C42—42no—
370C42—42no—
371C23—24no—
372C23—42no—
373C23—44no—
374C23—42no—
375C23—44no—
376C23—84no—
377C23—84no—
378C23—164no—
379C23—168no—
380C23C2×#12114yes—
381C23C2×#12112yes—
382C23#53914yes—
383C23C22×#4111yescharacteristic
384Q8—28no—
385Q8—42no—
386Q8—42no—
387Q8—44no—
388Q8—84no—
389D4—216no—
390D4—28no—
391D4—42no—
392D4—42no—
393D4—44no—
394D4—84no—
395D4—168no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—84no—
406C2×C4—88no—
407C2×C4—84no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4C2×#12114yes—
411C2×C4C2×#12112yes—
412C2×C4#53414yes—
413C2×C4#53614yes—
414C2×C4#53714yes—
415C2×C4#53714yes—
416C2×C4#53914yes—
417C2×C4C23×D411yescharacteristic
418C2×C4C23×D411yescharacteristic
419C2×C4C2×H(5,𝔽2)12yes—
420C2×C4C2×2−1+412yes—
421C8—22no—
422C8—22no—
423C8—24no—
424C8—44no—
425C8—44no—
426C8—44no—
427C22—22no—
428C22—24no—
429C22—28no—
430C22—24no—
431C22—44no—
432C22—84no—
433C22—84no—
434C22—82no—
435C22—164no—
436C22—168no—
437C22#261814yes—
438C22C22×#12111yescharacteristic
439C22C2×#53912yes—
440C4—24no—
441C4—48no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4#261814yes—
446C4C2×#53411yescharacteristic
447C4C2×#53611yescharacteristic
448C4C2×#53712yes—
449C2—22no—
450C2—24no—
451C2—44no—
452C2—162no—
453C2C2×#261811yescharacteristic
454C2#6047012yes—
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60519C2semidirect
4. #60100C2semidirect
5. #29793C2semidirect
6. #29790C2semidirect
7. #23674C2semidirect
8. #23603C2semidirect
9. #23602C2semidirect
10. C2×#2046C2semidirect
11. #20991C2semidirect
13. #20669C2semidirect
22. #3095C22semidirect
33. #2905C22semidirect
35. C2×#424C22semidirect
44. #2090C22semidirect
45. #2089C22semidirect
46. #2064C22semidirect
50. #1901C22semidirect
52. #1811C22semidirect
53. #1810C22semidirect

Maximal subgroups

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

subgroupindex
#605192
#605182
C2×#30572
#601002
#297932
#297902
#236742
#236032
#236022
C2×#20462
#209912
#209902
#206692