Finite Group Databasebeta

← #364032

Subgroups — #364032

Order 512.

1386 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1729C211yesmaximal, characteristic
3#14343C211yesmaximal, characteristic
4#11373C211yesmaximal, characteristic, complemented
5#11371C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11358C211yesmaximal, characteristic, complemented
8#10807C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10722C212yesmaximal
11#10721C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15#2314—24no—
16#2314C418yescomplemented
17C2×#388—22no—
18C2×#387—22no—
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×C4×C4⋊C4C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32#1729—24no—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1445—22no—
38#1440—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—24no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—42no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—42no—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×#117—24no—
58C2×#117—24no—
59C2×#117C2×C414yes—
60C2×#117D414yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D412yes—
70C22×C42—22no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#388—216no—
74#387—216no—
75#384—22no—
76#384C2×C414yes—
77#382—22no—
78#382C2×C414yes—
79#380—24no—
80#380—24no—
81#380—24no—
82#380C2×C418yes—
83C4×C4⋊C4—24no—
84C4×C4⋊C4C2×C418yes—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—42no—
98#340—42no—
99#340—42no—
100#340—42no—
101#336—22no—
102#336—22no—
103#336—22no—
104#336—22no—
105C25C2×D411yescharacteristic
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C22×C412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4#4114yes—
122C23×C4#4114yes—
123C23×C4C2411yescharacteristic
124C2×M16—28no—
125C2×M16—48no—
126C2×M16—48no—
127C2×M16—44no—
128C2×M16—44no—
129C22×C8—28no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—48no—
147C22×C8—48no—
148C22×C8—44no—
149C22×C8—44no—
150C22×C8#4112yes—
151C22×C8#4112yes—
152C22×C8#4112yes—
153C22×C8#4112yes—
154#117—216no—
155#117—28no—
156#117—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C22⋊C418yes—
185C2×C4⋊C4C22×C414yes—
186C2×C4⋊C4C22×C414yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4C22⋊C418yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—28no—
201C2×C42—82no—
202C2×C42—82no—
203C2×C42C22×C412yes—
204C2×C42C22×C412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207#95—28no—
208#95—28no—
209#95—28no—
210#95—44no—
211#95—82no—
212#95—82no—
213#95—82no—
214#95—82no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24C2×C22⋊C412yes—
223C24#12314yes—
224C24C22×D411yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4C2×C22⋊C414yes—
284C22×C4C2×C22⋊C414yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4#11718yes—
287C22×C4#12314yes—
288C22×C4#12314yes—
289C22×C4#12314yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4112yes—
293C22×C4C2×#4112yes—
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4H(5,𝔽2)12yes—
299C22×C42−1+412yes—
300M16—416no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—416no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—216no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—42no—
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—88no—
340C22⋊C4—216no—
341C22⋊C4—88no—
342C22⋊C4—88no—
343C42—24no—
344C42—24no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—88no—
350C23—22no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—22no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23#38114yes—
371C23C22×C22⋊C411yescharacteristic
372C23C2×#12312yes—
373C23#53211yescharacteristic
374C23#53211yescharacteristic
375C23#53712yes—
376C23#57612yes—
377C23#57712yes—
378C23#57712yes—
379C23#57812yes—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—28no—
385C2×C4—28no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#38114yes—
433C2×C4C2×#11712yes—
434C2×C4C4×#4111yescharacteristic
435C2×C4C4×#4111yescharacteristic
436C2×C4#52012yes—
437C2×C4C2×#12311yescharacteristic
438C2×C4C2×#12311yescharacteristic
439C2×C4#52511yescharacteristic
440C2×C4#52511yescharacteristic
441C2×C4#53511yescharacteristic
442C2×C4#53611yescharacteristic
443C2×C4#55112yes—
444C2×C4#55112yes—
445C2×C4#55112yes—
446C2×C4#55112yes—
447C2×C4#55212yes—
448C2×C4#55212yes—
449C2×C4#55212yes—
450C2×C4#55212yes—
451C8—42no—
452C8—42no—
453C8—42no—
454C8—42no—
455C22—24no—
456C22—28no—
457C22—44no—
458C22—42no—
459C22—42no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22C2×#38111yescharacteristic
465C22#229711yescharacteristic
466C22#231111yescharacteristic
467C22#249812yes—
468C22#249812yes—
469C22#289712yes—
470C22#289812yes—
471C22#309612yes—
472C22#309712yes—
473C22#309812yes—
474C22#309912yes—
475C22#317411yescharacteristic
476C22#317411yescharacteristic
477C22#317611yescharacteristic
478C22#317711yescharacteristic
479C22#317812yes—
480C22#317912yes—
481C22#325712yes—
482C22#325812yes—
483C22#326212yes—
484C22#326312yes—
485C4—24no—
486C4—44no—
487C4—44no—
488C4—84no—
489C4—84no—
490C4—162no—
491C4—162no—
492C4#234312yes—
493C4#234312yes—
494C4#248211yescharacteristic
495C4#248211yescharacteristic
496C4#248411yescharacteristic
497C4#248411yescharacteristic
498C2—44no—
499C2#1395811yescharacteristic
500C2#2070312yes—
501C2#2141311yescharacteristic
502C2#2141411yescharacteristic
503C2#2189612yes—
504C2#2786712yes—
505C2#2786812yes—
506C2#2787112yes—
507C2#2787212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11373C2semidirect
5. #11371C2semidirect
6. #11358C2semidirect
7. #11358C2semidirect
8. #10807C2semidirect
9. #10806C2semidirect
16. #2314C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#17292
#143432
#113732
#113712
#113582
#108072
#108062
#107222
#107212