Finite Group Databasebeta

← #241367

Subgroups — #241367

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#10582C212yesmaximal, complemented
6#10581C212yesmaximal, complemented
7#10575C212yesmaximal, complemented
8#10575C212yesmaximal, complemented
9#8043C211yesmaximal, characteristic
10#7903C211yesmaximal, characteristic
11#7896C211yesmaximal, characteristic
12C2×#544—21no—
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2868—21no—
17#2859—21no—
18#2660—22no—
19#2552—21no—
20#2552C2212yes—
21#2551—21no—
22#2551C2212yes—
23#2440—22no—
24#2440—22no—
25#2317—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1948—21no—
31#1948C2211yescharacteristic
32#1948C2212yes—
33#1948C2211yescharacteristic
34#1947—21no—
35#1947C2211yescharacteristic
36#1947C2212yes—
37#1947C2211yescharacteristic
38#1854—21no—
39#1854—21no—
40#1836—22no—
41#1834C2212yes—
42#1834C2212yes—
43#1809—22no—
44C4×#107—21no—
45C4×#106—21no—
46#1736—22no—
47#1296C2212yes—
48#1296C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×M16—21no—
54C22×M16—21no—
55#544—44no—
56#544—44no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×#125—22no—
62C2×#125—22no—
63C2×C22⋊Q8—24no—
64C2×#117—21no—
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431Q814yescomplemented
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—22no—
77C2×#106—21no—
78C2×#106—21no—
79C2×#106—22no—
80#423—22no—
81#423Q814yescomplemented
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C22⋊C8—21no—
87C4×M16—22no—
88C4×M16Q814yescomplemented
89C2×#97—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2312yes—
92#398—24no—
93#398—24no—
94#395—24no—
95#395—24no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#385—24no—
103#384—22no—
104#384—22no—
105#384—22no—
106#384—21no—
107#384—21no—
108#384D411yescharacteristic
109#384D412yes—
110#384D411yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2312yes—
114#382—22no—
115#382—21no—
116#382—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380D412yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C43—21no—
133#340—22no—
134#340—22no—
135#340—24no—
136#340—24no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—24no—
142C23×C4—22no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C2×M16—81no—
151C2×M16—81no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C4⋊Q8—48no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C4⋊Q8—82no—
160C4⋊Q8—82no—
161C4⋊Q8—84no—
162#125—48no—
163#125—44no—
164#125—44no—
165C22⋊Q8—416no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—42no—
171#117—42no—
172#117—44no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D414yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—41no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—41no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×Q812yes—
221C2×C42C2×Q812yes—
222C2×C42C2×Q812yes—
223C2×C42#4112yes—
224C2×C42C2411yescharacteristic
225#107—42no—
226#107—42no—
227#107—82no—
228#106—42no—
229#106—42no—
230#106—82no—
231C4⋊C8—24no—
232C4⋊C8—22no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8C2×Q814yes—
236C4⋊C8C2×Q812yes—
237C4⋊C8C2×Q812yes—
238C22⋊C8—24no—
239#97—24no—
240C4×C8—24no—
241C4×C8—22no—
242C4×C8C2×Q814yes—
243#95—24no—
244#95—24no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—41no—
252#95—41no—
253C24#12011yescharacteristic
254C2×Q8—48no—
255C2×Q8—84no—
256C2×Q8—84no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—44no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4C22⋊Q812yes—
283C22×C4C22⋊Q812yes—
284C22×C4C22⋊Q812yes—
285C22×C4C4⋊Q812yes—
286C22×C4C4⋊Q812yes—
287C22×C4C4⋊Q812yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291M16—48no—
292M16—84no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—82no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4AGL(1,ℤ/8)12yes—
341C4⋊C4AGL(1,ℤ/8)12yes—
342C4⋊C4#13712yes—
343C4⋊C4#13712yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—44no—
347C22⋊C4—82no—
348C42—24no—
349C42—24no—
350C42—42no—
351C42—42no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—42no—
357C42—42no—
358C42—82no—
359C42C22⋊Q812yes—
360C42C22⋊Q812yes—
361C42C22⋊Q812yes—
362C42C4⋊Q812yes—
363C42C4⋊Q812yes—
364C42C4⋊Q812yes—
365C42C22×Q811yescharacteristic
366C42C22×Q811yescharacteristic
367C42C22×Q811yescharacteristic
368C42C2×#4111yescharacteristic
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23#39512yes—
376C23C2×#12011yescharacteristic
377Q8—168no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—164no—
422C2×C4#39512yes—
423C2×C4#39512yes—
424C2×C4#39512yes—
425C2×C4#45412yes—
426C2×C4#45512yes—
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×C22⋊Q811yescharacteristic
429C2×C4C2×C22⋊Q811yescharacteristic
430C2×C4C2×C4⋊Q811yescharacteristic
431C2×C4C2×C4⋊Q811yescharacteristic
432C2×C4C2×C4⋊Q811yescharacteristic
433C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
434C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
435C2×C4C2×#13711yescharacteristic
436C2×C4C2×#13711yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57811yescharacteristic
441C8—44no—
442C8—42no—
443C8—42no—
444C8—44no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22C2×#39511yescharacteristic
453C22#300311yescharacteristic
454C22#300411yescharacteristic
455C22#301211yescharacteristic
456C22#301311yescharacteristic
457C22C2×#45511yescharacteristic
458C22C2×#45411yescharacteristic
459C4—22no—
460C4—22no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—84no—
468C4—82no—
469C4—82no—
470C4—161no—
471C4—161no—
472C4#206114yes—
473C4#307911yescharacteristic
474C4#307911yescharacteristic
475C4#308111yescharacteristic
476C4#308211yescharacteristic
477C4#316311yescharacteristic
478C4#316411yescharacteristic
479C4#316511yescharacteristic
480C4#316511yescharacteristic
481C2—42no—
482C2#1857411yescharacteristic
483C2#1857511yescharacteristic
484C2#1858211yescharacteristic
485C2#1858311yescharacteristic
486C2#2355211yescharacteristic
487C2#2355711yescharacteristic
488C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
5. #10582C2semidirect
6. #10581C2semidirect
7. #10575C2semidirect
8. #10575C2semidirect
73. #431Q8semidirect
81. #423Q8semidirect
88. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#219642
#213622
#207642
#105822
#105812
#105752
#80432
#79032
#78962