Finite Group Databasebeta

← #206194

Subgroups — #206194

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2#21835C211yesmaximal, characteristic, complemented
3#21805C211yesmaximal, characteristic
4#21410C211yesmaximal, characteristic
5#21409C211yesmaximal, characteristic, complemented
6#20797C211yesmaximal, characteristic, complemented
7C2×#1451C211yesmaximal, characteristic, complemented
8#9842C212yesmaximal, complemented
9#9841C212yesmaximal, complemented
10#9827C212yesmaximal, complemented
11#9826C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3167—24no—
15#3167C418yescomplemented
16C2×#498C2211yescharacteristic
17C2×#492C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#486C2211yescharacteristic
20C2×#465—22no—
21C2×C4×SD16—22no—
22C2×#431C2211yescharacteristic
23C2×#427—22no—
24C2×C4×M16C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#1990C2212yes—
29#1990C2212yes—
30#1990C2212yes—
31#1990C2212yes—
32#1989—22no—
33#1983—22no—
34#1957C2212yes—
35#1957C2212yes—
36#1956C2212yes—
37#1956C2212yes—
38#1895—22no—
39#1894—22no—
40#1846—22no—
41#1845C2212yes—
42#1844C2212yes—
43#1825—22no—
44#1788—22no—
45#1763—22no—
46#1762—22no—
47#1451—24no—
48C2×#322C2212yes—
49C2×#322C2212yes—
50C22×#41—21no—
51C22×SD16—22no—
52C22×M16D412yes—
53C23×C8D412yes—
54#533—22no—
55#533—22no—
56#533—22no—
57#533C2×C414yes—
58#532—22no—
59#532—22no—
60#532—22no—
61#532C2×C414yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×#127C2311yescharacteristic
66C2×#125—21no—
67C2×#124—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×#117—21no—
77C2×#117—21no—
78C2×#117—21no—
79C22×C42C2311yescharacteristic
80#498—22no—
81#498C2×C414yes—
82#492—22no—
83#492C2×C414yes—
84#487—22no—
85#487C2×C414yes—
86#486—22no—
87#486C2×C414yes—
88#465—216no—
89C4×SD16—216no—
90#431—22no—
91#431—22no—
92#427—48no—
93C2×#107—24no—
94C2×#106—22no—
95C2×#106—24no—
96C2×C4⋊C8D412yes—
97C2×#103—22no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#102—22no—
101C2×#102—24no—
102C2×#102—24no—
103C2×C22⋊C8D412yes—
104C4×M16—22no—
105C4×M16—22no—
106C2×#97D412yes—
107C2×C4×C8C2×C414yes—
108C2×C4×C8D412yes—
109C2×C4×C8D414yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2312yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#384C2312yes—
115#384C2312yes—
116#382—21no—
117#382—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#322—24no—
131#322—24no—
132C2×#41—24no—
133C2×#41—42no—
134C2×#41—84no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—41no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×SD16—28no—
147C2×SD16—44no—
148C2×SD16—44no—
149C2×M16—24no—
150C2×M16—44no—
151C22×C8—24no—
152C22×C8—24no—
153C22×C8C22⋊C418yes—
154C22×C8C2×D414yes—
155C22×C8C2×D414yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C22×C412yes—
168C4⋊Q8C22×C412yes—
169C4⋊Q8C22×C412yes—
170#127—21no—
171#127—21no—
172#127—21no—
173#127C22×C412yes—
174#125—24no—
175#125—42no—
176#124—24no—
177#124—42no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182#121—24no—
183#121—24no—
184#121—42no—
185#121—42no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—42no—
193C4×D4—42no—
194#117—24no—
195#117—42no—
196#117—44no—
197#117—44no—
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—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42C22×C412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42C2411yescharacteristic
228#107—48no—
229#106—28no—
230#106—48no—
231C4⋊C8—24no—
232#103—28no—
233#103—48no—
234#103—84no—
235#102—28no—
236#102—48no—
237#102—84no—
238C22⋊C8—24no—
239#97—24no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8C22⋊C418yes—
245C4×C8C22×C412yes—
246C4×C8C22×C412yes—
247C4×C8C2×D412yes—
248C4×C8C2×D412yes—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24—41no—
254C24#12711yescharacteristic
255#41—88no—
256#41—88no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—84no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—84no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C4×D412yes—
292C22×C4C4×D412yes—
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4#12412yes—
296C22×C4#12711yescharacteristic
297C22×C4#12711yescharacteristic
298C22×C4#12711yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302SD16—416no—
303M16—48no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—216no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8C2×C22⋊C414yes—
314C2×C8C2×C22⋊C414yes—
315C2×C8AGL(1,ℤ/8)14yes—
316C2×C8#13714yes—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C42—44no—
343C42—82no—
344C42—82no—
345C42C2×C22⋊C412yes—
346C42C4×D412yes—
347C42C4×D412yes—
348C42#12112yes—
349C42#12112yes—
350C42#12412yes—
351C42C23×C411yescharacteristic
352C42C22×D411yescharacteristic
353C42C2×#4111yescharacteristic
354C42C2×#4111yescharacteristic
355C23—22no—
356C23—21no—
357C23—21no—
358C23—22no—
359C23—42no—
360C23—82no—
361C23—81no—
362C23—81no—
363C23—82no—
364C23#39012yes—
365C23C2×#12711yescharacteristic
366Q8—44no—
367Q8—44no—
368Q8—44no—
369Q8—82no—
370Q8—82no—
371Q8—82no—
372D4—44no—
373D4—44no—
374D4—44no—
375D4—88no—
376D4—82no—
377D4—82no—
378D4—82no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
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—42no—
392C2×C4—88no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4#39012yes—
404C2×C4#39012yes—
405C2×C4#39012yes—
406C2×C4C22×C22⋊C411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×#12111yescharacteristic
410C2×C4C2×#12111yescharacteristic
411C2×C4C2×#12411yescharacteristic
412C2×C4C2×AGL(1,ℤ/8)12yes—
413C2×C4C2×#13712yes—
414C2×C4#57514yes—
415C2×C4#57714yes—
416C8—24no—
417C8—44no—
418C8—44no—
419C8#41718yes—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—42no—
424C22—42no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—162no—
429C22—162no—
430C22C2×#39011yescharacteristic
431C22#315012yes—
432C22#315111yescharacteristic
433C22#315211yescharacteristic
434C22#315412yes—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4—162no—
442C4—162no—
443C4C2×#41712yes—
444C4#294312yes—
445C4#294912yes—
446C4#305412yes—
447C4#305712yes—
448C4#314112yes—
449C2—42no—
450C2—162no—
451C2#1969312yes—
452C2#1969412yes—
453C2#2120011yescharacteristic
454C2#2122811yescharacteristic
455C2#2123111yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ 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
2. #21835C2semidirect
5. #21409C2semidirect
6. #20797C2semidirect
7. C2×#1451C2semidirect
8. #9842C2semidirect
9. #9841C2semidirect
10. #9827C2semidirect
11. #9826C2semidirect
15. #3167C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
#218352
#218052
#214102
#214092
#207972
C2×#14512
#98422
#98412
#98272
#98262