Finite Group Databasebeta

← #497856

Subgroups — #497856

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2171C211yesmaximal, characteristic
2#33411C211yesmaximal, characteristic, complemented
3#33410C211yesmaximal, characteristic, complemented
4#22692C211yesmaximal, characteristic
5#22691C211yesmaximal, characteristic
6#13181C211yesmaximal, characteristic, complemented
7#13180C211yesmaximal, characteristic, complemented
8#12926C211yesmaximal, characteristic, complemented
9#12926C211yesmaximal, characteristic, complemented
10#12925C211yesmaximal, characteristic, complemented
11#12925C211yesmaximal, characteristic, complemented
12#12908C211yesmaximal, characteristic, complemented
13#12843C211yesmaximal, characteristic, complemented
14#12842C211yesmaximal, characteristic, complemented
15#12671C211yesmaximal, characteristic, complemented
16C2×#504C2211yescharacteristic
17C2×#504C2211yescharacteristic
18C2×#443C2211yescharacteristic
19#2952—21no—
20#2951—21no—
21#2913C2211yescharacteristic
22#2913C2211yescharacteristic
23#2913C2211yescharacteristic
24#2913C2211yescharacteristic
25#2171C418yescomplemented
26C2×#368C2211yescharacteristic
27#2159—21no—
28#2159—21no—
29#2159—21no—
30#2159—21no—
31#2159C2211yescharacteristic
32#2159C2211yescharacteristic
33#2159C2211yescharacteristic
34#2159C2211yescharacteristic
35#2158—21no—
36#2158—21no—
37#2158C2211yescharacteristic
38C2×#367—21no—
39C2×#367C2211yescharacteristic
40C2×#367C2211yescharacteristic
41C2×#366—21no—
42C2×#366C2211yescharacteristic
43C2×#366C2211yescharacteristic
44C2×#364C2211yescharacteristic
45C2×#350C2211yescharacteristic
46C2×#350C2211yescharacteristic
47C2×#346C2211yescharacteristic
48#1952—21no—
49#1951—21no—
50#1860—21no—
51#1860—21no—
52#1859—21no—
53#1859—21no—
54#1846C2211yescharacteristic
55#1845C2211yescharacteristic
56#1816C2211yescharacteristic
57#1816C2211yescharacteristic
58#1816C2211yescharacteristic
59#1816C2211yescharacteristic
60#1772C2211yescharacteristic
61#1772C2211yescharacteristic
62#1387C2211yescharacteristic
63#1387C2211yescharacteristic
64#1387C2211yescharacteristic
65#1387C2211yescharacteristic
66#1382C2211yescharacteristic
67#1382C2211yescharacteristic
68C2×#135D411yescharacteristic
69C2×#131C2311yescharacteristic
70C2×#131C2311yescharacteristic
71#520—21no—
72#520—21no—
73#520—21no—
74#520—21no—
75#504—22no—
76#504—22no—
77#504C2×C414yescomplemented
78#504C2×C414yescomplemented
79C2×M32—21no—
80C2×M32—21no—
81C2×M32C2311yescharacteristic
82C2×M32C2311yescharacteristic
83C22×C16—21no—
84C22×C16—21no—
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#443C2×C414yes—
88#443Q814yes—
89#442—41no—
90#441—41no—
91#440—41no—
92#440—41no—
93C4×Q16—41no—
94C4×SD16—41no—
95C4×SD16—41no—
96C4×D8—41no—
97C2×#108D411yescharacteristic
98C2×#108C2311yescharacteristic
99C2×#108C2311yescharacteristic
100#428—21no—
101#428—21no—
102#428—21no—
103#428—21no—
104#428—41no—
105#428—41no—
106#428—41no—
107#428—41no—
108#427—21no—
109#427—21no—
110#427—21no—
111#427—21no—
112#427C2311yescharacteristic
113#427C2311yescharacteristic
114C2×#107—21no—
115C2×#107—21no—
116C2×#107—21no—
117C2×#107—21no—
118C2×#107—21no—
119C2×#107C2311yescharacteristic
120C2×#107C2311yescharacteristic
121C2×#107C2311yescharacteristic
122C2×#107C2311yescharacteristic
123C2×#106—21no—
124C2×#104D411yescharacteristic
125C2×#104D411yescharacteristic
126#417—21no—
127#417—21no—
128#417—21no—
129#417—21no—
130#405—21no—
131#405—21no—
132C2×C4×C8C2311yescharacteristic
133#384—21no—
134#382—21no—
135#368—22no—
136#367—21no—
137#367—21no—
138#367—21no—
139#367—21no—
140#367—21no—
141#367—21no—
142#367—41no—
143#367D411yescharacteristic
144#367D411yescharacteristic
145#366—21no—
146#366—21no—
147#366—21no—
148#366—21no—
149#366—21no—
150#366—21no—
151#366—41no—
152#366D411yescharacteristic
153#366D411yescharacteristic
154#364C2×C414yescomplemented
155#350C2×C414yescomplemented
156#350C2×C414yescomplemented
157#346C2×C414yescomplemented
158#343—21no—
159#343—21no—
160#343—21no—
161#343—21no—
162#339—21no—
163#339—21no—
164C2×#41C2×D411yescharacteristic
165C2×#41C2×D411yescharacteristic
166#135C4⋊C414yes—
167#135Q1614yes—
168C2×Q16D811yescharacteristic
169C2×SD16D811yescharacteristic
170C2×SD16D811yescharacteristic
171C2×D8D811yescharacteristic
172#131—21no—
173#131—21no—
174#131—21no—
175#131—21no—
176#131C22×C412yes—
177#131C22×C412yes—
178#131C2×Q812yes—
179#131C2×Q812yes—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16C2×D411yescharacteristic
185C2×M16C2×D411yescharacteristic
186C22×C8—21no—
187C22×C8—21no—
188C22×C8C2×D411yescharacteristic
189C22×C8C2411yescharacteristic
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—81no—
199C4×D4—81no—
200C4×D4—81no—
201C4×D4—81no—
202#117—21no—
203#117—21no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—81no—
217C2×C42C2×D411yescharacteristic
218M32—21no—
219M32—21no—
220M32—24no—
221M32—24no—
222M32C22×C412yes—
223M32C22×C412yes—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—24no—
229C2×C16—24no—
230C2×C16C22×C412yes—
231C2×C16C22×C412yes—
232C2×C16C2×D411yescharacteristic
233C2×C16C2×D411yescharacteristic
234C2×C16#4112yes—
235#108—21no—
236#108—21no—
237#108—21no—
238#108—21no—
239#108C4⋊C414yes—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—41no—
249#107—41no—
250#107—41no—
251#107—41no—
252#107—41no—
253#107—41no—
254#107C2×D411yescharacteristic
255#107C2×D411yescharacteristic
256#107C2×D411yescharacteristic
257#107C2×D411yescharacteristic
258#106—21no—
259#106—21no—
260#106—41no—
261#106—41no—
262#106—41no—
263#106—41no—
264#106—41no—
265#106—41no—
266#104C4⋊C414yes—
267#104C4⋊C414yes—
268#103—41no—
269#103—41no—
270#103—41no—
271#103—41no—
272#102—41no—
273#102—41no—
274#102—41no—
275#102—41no—
276#97—41no—
277#97—41no—
278C4×C8—41no—
279C4×C8—41no—
280C4×C8C22×C412yes—
281C4×C8C2×Q812yes—
282#95—81no—
283#41—21no—
284#41—21no—
285#41—21no—
286#41—21no—
287#41C2×C4⋊C412yes—
288#41C2×C4⋊C412yes—
289#41C2×Q1612yes—
290#41C2×Q1612yes—
291C2×Q8C2×D811yescharacteristic
292C2×Q8C2×D811yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4C2×D811yescharacteristic
296C2×D4C2×D811yescharacteristic
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4C22×D411yescharacteristic
309Q16#10714yes—
310SD16#10714yes—
311SD16#10714yes—
312D8#10714yes—
313M16—21no—
314M16—21no—
315M16—22no—
316M16—22no—
317M16—22no—
318M16—22no—
319M16—41no—
320M16—41no—
321M16—41no—
322M16—41no—
323M16C2×C4⋊C412yes—
324M16C2×C4⋊C412yes—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8C2×C4⋊C412yes—
334C2×C8C2×D811yescharacteristic
335C2×C8C2×D811yescharacteristic
336C2×C8C2×Q1612yes—
337C2×C8C23×C411yescharacteristic
338C2×C8C22×D411yescharacteristic
339C2×C8C22×Q811yescharacteristic
340C2×C8C2×#4111yescharacteristic
341C2×C8H(5,𝔽2)11yescharacteristic
342C2×C82−1+411yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—162no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—81no—
363C22⋊C4—81no—
364C22⋊C4—81no—
365C22⋊C4—81no—
366C42—21no—
367C42—41no—
368C42—41no—
369C42—81no—
370C42—81no—
371C42—81no—
372C42—81no—
373C42C2×C4⋊C412yes—
374C16—22no—
375C16—22no—
376C16—22no—
377C16C4×D412yes—
378C16C4×D412yes—
379C23—21no—
380C23—21no—
381C23#57611yescharacteristic
382Q8—21no—
383Q8—21no—
384Q8C2×#10712yes—
385Q8C2×#10712yes—
386D4—22no—
387D4—22no—
388D4—22no—
389D4—22no—
390D4—21no—
391D4—21no—
392D4C2×#10712yes—
393D4C2×#10712yes—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—161no—
412C2×C4—161no—
413C2×C4—161no—
414C2×C4—161no—
415C2×C4—321no—
416C2×C4—321no—
417C2×C4—321no—
418C2×C4—321no—
419C2×C4C22×C4⋊C411yescharacteristic
420C2×C4D4211yescharacteristic
421C2×C4#54711yescharacteristic
422C2×C4C22×D811yescharacteristic
423C2×C4C22×Q1611yescharacteristic
424C2×C4#57811yescharacteristic
425C8—22no—
426C8—22no—
427C8—81no—
428C8—81no—
429C8C2×#10712yes—
430C8C2×#10712yes—
431C8C2×C4×D411yescharacteristic
432C8#52011yescharacteristic
433C8D4×Q811yescharacteristic
434C8#55411yescharacteristic
435C22—21no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22#291311yescharacteristic
444C22D4×D811yescharacteristic
445C22#333811yescharacteristic
446C4—22no—
447C4—22no—
448C4—42no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4D4×C4⋊C411yescharacteristic
458C4C22×#10711yescharacteristic
459C4D4×Q1611yescharacteristic
460C4#333411yescharacteristic
461C2—21no—
462C2—21no—
463C2—42no—
464C2—42no—
465C2D4×#10711yescharacteristic
466C2#3385411yescharacteristic
467C2#3385511yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#135D4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33411C2semidirect
3. #33410C2semidirect
6. #13181C2semidirect
7. #13180C2semidirect
8. #12926C2semidirect
9. #12926C2semidirect
10. #12925C2semidirect
11. #12925C2semidirect
12. #12908C2semidirect
13. #12843C2semidirect
14. #12842C2semidirect
15. #12671C2semidirect
25. #2171C4semidirect
77. #504C2×C4semidirect
78. #504C2×C4semidirect
154. #364C2×C4semidirect
155. #350C2×C4semidirect
156. #350C2×C4semidirect
157. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21712
#334112
#334102
#226922
#226912
#131812
#131802
#129262
#129252
#129082
#128432
#128422
#126712