Finite Group Databasebeta

← #6477760

Subgroups — #6477760

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C2×#3095C212yesmaximal
3#60124C211yesmaximal, characteristic
4#33126C214yesmaximal, complemented
5#29792C214yesmaximal, complemented
6#24621C218yesmaximal, complemented
7#24522C218yesmaximal, complemented
8#21949C211yesmaximal, characteristic
9#21753C212yesmaximal
10C22×C22⋊Q8C2212yes—
11C22×#117C2211yescharacteristic
12#3195—22no—
13#3195—22no—
14#3195—24no—
15#3095—24no—
16#3095—22no—
17#3095—22no—
18#3095C2218yes—
19#3095C2214yes—
20#3095C2214yes—
21C2×#484—22no—
22C2×#485—24no—
23C2×#483—22no—
24C2×#483C2214yes—
25C2×#470C2214yes—
26C2×#451C2214yes—
27#2924—24no—
28#2924C2218yes—
29#2921—22no—
30#2920—22no—
31C2×#428C2214yes—
32#2905C2212yes—
33C2×#419C2214yes—
34C2×#407C2212yes—
35C2×#407C2211yescharacteristic
36C2×#407C2211yescharacteristic
37#2873C2212yes—
38#2090C2218yescomplemented
39#2064C22116yescomplemented
40#2047C2218yescomplemented
41#1979C2218yescomplemented
42#1951C2218yescomplemented
43#1930C22116yescomplemented
44#1901C2218yes—
45#1901C2218yes—
46#1855C2218yes—
47#1821C2214yes—
48#1811C2214yes—
49#1784C2214yes—
50C23×Q8C2312yes—
51C24×C4C2311yescharacteristic
52C22×Q16C2314yes—
53C22×M16C2312yes—
54C22×M16C2314yes—
55C22×M16C2312yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8C2314yes—
67C2×C22⋊Q8C2314yes—
68C2×#117—21no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—21no—
72C2×#117D414yes—
73C2×#117D412yes—
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2314yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4D412yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83#485—28no—
84#485—28no—
85#485—28no—
86#485—28no—
87#484—28no—
88#484—24no—
89#484—24no—
90#483—28no—
91#483—24no—
92#483—24no—
93#483—28no—
94#483—28no—
95#470—28no—
96#470—28no—
97#451—28no—
98#451—28no—
99#433—216no—
100#432—28no—
101#431—28no—
102C2×#108C2318yes—
103#428—24no—
104#428—24no—
105#428C2318yes—
106#428C2318yes—
107C2×#107—24no—
108C2×#106—24no—
109C2×#106C2314yes—
110#424—24no—
111#424C2318yes—
112#423—24no—
113#419—24no—
114#419—24no—
115#419C2318yes—
116#419C2318yes—
117C2×#103—24no—
118C2×#103—24no—
119C2×#103C2318yes—
120C2×#101C23116yes—
121#407—22no—
122#407—22no—
123#407—21no—
124#407—21no—
125#407D412yes—
126#407D412yes—
127#407D414yes—
128#407C2314yes—
129#407C2314yes—
130#407C2314yes—
131C2×C22⋊C8—22no—
132C2×C22⋊C8—22no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8C2314yes—
135#405—24no—
136#405C2318yes—
137C4×M16—24no—
138#343C23116yes—
139#341C2318yes—
140C25#4111yescharacteristic
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—48no—
147C22×Q8C2414yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—44no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4#4112yes—
159C23×C4#4111yescharacteristic
160C23×C4C2412yes—
161C23×C4C2411yescharacteristic
162C2×Q16—28no—
163C2×Q16—48no—
164C2×M16—22no—
165C2×M16—24no—
166C2×M16—44no—
167C2×M16—44no—
168C2×M16—48no—
169C2×M16C2×D414yes—
170C2×M16C2×D414yes—
171C2×M16C2418yes—
172C2×M16C2414yes—
173C22×C8—22no—
174C22×C8—24no—
175C22×C8C2414yes—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192#117—24no—
193#117—22no—
194#117—22no—
195#117—24no—
196#117—21no—
197#117—21no—
198#117—24no—
199#117—21no—
200#117—21no—
201#117—24no—
202#117C2×D414yes—
203#117C2×D414yes—
204#117C2×D412yes—
205#117C2×D412yes—
206#117C2412yes—
207#117C2412yes—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2414yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—24no—
243#108—48no—
244#107—28no—
245#107—28no—
246#106—28no—
247#106—28no—
248#106—28no—
249C4⋊C8—216no—
250#103—216no—
251#103—28no—
252#103—28no—
253#103—28no—
254#103—28no—
255#101—416no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258C22⋊C8—24no—
259C22⋊C8—24no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8C2×D418yes—
263C22⋊C8C2×D418yes—
264#97—28no—
265C4×C8—28no—
266C24—42no—
267C24—44no—
268C24#12111yescharacteristic
269C24#12112yes—
270C24#12111yescharacteristic
271C24C2×#4112yes—
272C24C2×#4111yescharacteristic
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—216no—
276C2×Q8—48no—
277C2×Q8—48no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—84no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4#12112yes—
307C22×C4#12111yescharacteristic
308C22×C4#12112yes—
309C22×C4#12111yescharacteristic
310C22×C4#12112yes—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D412yes—
316C22×C4C22×D412yes—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2511yescharacteristic
324Q16—88no—
325M16—28no—
326M16—216no—
327M16—44no—
328M16—44no—
329M16—48no—
330M16—44no—
331C2×C8—28no—
332C2×C8—24no—
333C2×C8—28no—
334C2×C8—48no—
335C2×C8—44no—
336C2×C8C22×D418yes—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4C22×D414yes—
350C4⋊C4C22×D414yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C42—22no—
362C42—22no—
363C42—24no—
364C42—24no—
365C42—22no—
366C42—22no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—22no—
371C23—24no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—88no—
380C23C2×#12111yescharacteristic
381C23C2×#12112yes—
382C23C2×#12111yescharacteristic
383C23C2×#12112yes—
384C23C22×#4111yescharacteristic
385Q8—48no—
386Q8—48no—
387Q8—84no—
388Q8—84no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—48no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—84no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4C2×#12111yescharacteristic
421C2×C4C2×#12112yes—
422C2×C4C2×#12111yescharacteristic
423C2×C4C2×#12112yes—
424C2×C4C23×D411yescharacteristic
425C2×C4C23×D411yescharacteristic
426C2×C4#58514yes—
427C8—28no—
428C8—48no—
429C8—44no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—48no—
441C22—44no—
442C22—82no—
443C22—84no—
444C22—82no—
445C22#306914yes—
446C22C22×#12111yescharacteristic
447C22#349112yes—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—22no—
452C4—44no—
453C4—44no—
454C4—84no—
455C4—84no—
456C4#344912yes—
457C4#348512yes—
458C2—22no—
459C2—24no—
460C2—42no—
461C2—41no—
462C2—41no—
463C2#3508911yescharacteristic
464C2C2×#306912yes—
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33126C2semidirect
5. #29792C2semidirect
6. #24621C2semidirect
7. #24522C2semidirect
38. #2090C22semidirect
39. #2064C22semidirect
40. #2047C22semidirect
41. #1979C22semidirect
42. #1951C22semidirect
43. #1930C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C2×#30952
#601242
#331262
#297922
#246212
#245222
#219492
#217532