Finite Group Databasebeta

← #244074

Subgroups — #244074

Order 512.

836 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21409C211yesmaximal, characteristic
2#21409C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic
5#20800C211yesmaximal, characteristic
6#20764C211yesmaximal, characteristic
7#10686C211yesmaximal, characteristic, complemented
8#10684C211yesmaximal, characteristic, complemented
9#10681C211yesmaximal, characteristic, complemented
10#10680C211yesmaximal, characteristic, complemented
11#10680C211yesmaximal, characteristic, complemented
12#10679C211yesmaximal, characteristic, complemented
13#10676C211yesmaximal, characteristic, complemented
14#10675C211yesmaximal, characteristic, complemented
15#7852C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#428—22no—
18C2×C4×M16C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#1992—22no—
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1894—22no—
38#1894—22no—
39#1847—22no—
40#1845C2211yescharacteristic
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1844C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1836—22no—
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1809—22no—
54#1776—22no—
55#1763—22no—
56#1763—22no—
57#1736—22no—
58#1736—22no—
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×M16D412yes—
64C22×M16D412yes—
65C22×M16D412yes—
66#533—22no—
67#533—22no—
68#533—22no—
69#533Q814yes—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×#125—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C42C2311yescharacteristic
84#428—48no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C4×M16—22no—
96C4×M16—22no—
97C4×M16—22no—
98C4×M16—22no—
99C4×M16—22no—
100C4×M16—22no—
101C2×#97D412yes—
102C2×#97D412yes—
103C2×#97D412yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#384—21no—
111#384—21no—
112#384—21no—
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#382—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143C22×Q8—21no—
144C22×Q8—21no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151C2×M16—24no—
152C2×M16—24no—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16—42no—
156C2×M16—42no—
157C2×M16—42no—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C2×Q812yes—
171C4⋊Q8C2×Q812yes—
172#125—24no—
173#125—42no—
174C22⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—42no—
181C4×Q8—42no—
182#117—24no—
183#117—42no—
184#117—44no—
185#117—44no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×Q812yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230#107—44no—
231#107—44no—
232#107—44no—
233#106—44no—
234#106—44no—
235#106—44no—
236#103—44no—
237#103—44no—
238#103—44no—
239#103—44no—
240#97—24no—
241#97—24no—
242#97—24no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24#12711yescharacteristic
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4C22⋊Q812yes—
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4#12312yes—
289C22×C4#12312yes—
290C22×C4#12312yes—
291C22×C4#12711yescharacteristic
292C22×C4#12711yescharacteristic
293C22×C4#12711yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297M16—216no—
298M16—48no—
299M16—48no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8AGL(1,ℤ/8)14yes—
316C2×C8#13714yes—
317C2×C8#13714yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C22⋊C4—44no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C42—44no—
353C42—82no—
354C42—82no—
355C42C22⋊Q812yes—
356C42C22⋊Q812yes—
357C42C22⋊Q812yes—
358C42#12312yes—
359C42#12312yes—
360C42#12312yes—
361C42C22×Q811yescharacteristic
362C42C2×#4111yescharacteristic
363C42C2×#4111yescharacteristic
364C42C2×#4111yescharacteristic
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23#39912yes—
372C23C2×#12711yescharacteristic
373Q8—44no—
374Q8—44no—
375Q8—82no—
376Q8—82no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—164no—
414C2×C4#39912yes—
415C2×C4#39912yes—
416C2×C4#39912yes—
417C2×C4C2×C22⋊Q811yescharacteristic
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×C22⋊Q811yescharacteristic
420C2×C4C2×#12311yescharacteristic
421C2×C4C2×#12311yescharacteristic
422C2×C4C2×#12311yescharacteristic
423C2×C4C2×AGL(1,ℤ/8)12yes—
424C2×C4C2×#13712yes—
425C2×C4C2×#13712yes—
426C2×C4#57512yes—
427C2×C4#57512yes—
428C2×C4#57512yes—
429C8—24no—
430C8—24no—
431C8—24no—
432C8—42no—
433C8—42no—
434C8—42no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22C2×#39911yescharacteristic
443C22#315111yescharacteristic
444C22#315211yescharacteristic
445C22#315211yescharacteristic
446C22#315311yescharacteristic
447C22#315311yescharacteristic
448C22#315311yescharacteristic
449C4—22no—
450C4—22no—
451C4—22no—
452C4—82no—
453C4—82no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4#307812yes—
461C4#307912yes—
462C4#308012yes—
463C4#309212yes—
464C4#309312yes—
465C4#309512yes—
466C2—42no—
467C2#1974811yescharacteristic
468C2#1974911yescharacteristic
469C2#1975111yescharacteristic
470C2#1975211yescharacteristic
471C2#2373211yescharacteristic
472C2#2373311yescharacteristic
473C2#2373411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10686C2semidirect
8. #10684C2semidirect
9. #10681C2semidirect
10. #10680C2semidirect
11. #10680C2semidirect
12. #10679C2semidirect
13. #10676C2semidirect
14. #10675C2semidirect

Maximal subgroups

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

subgroupindex
#214092
#213222
#208242
#208002
#207642
#106862
#106842
#106812
#106802
#106792
#106762
#106752
#78522