Finite Group Databasebeta

← #375490

Subgroups — #375490

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#1729C211yesmaximal, characteristic
3#13777C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic, complemented
5#11385C211yesmaximal, characteristic, complemented
6#11373C211yesmaximal, characteristic, complemented
7#11371C211yesmaximal, characteristic, complemented
8#10885C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10722C212yesmaximal
11#10721C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C22×#117C2211yescharacteristic
16#2629—24no—
17#2629C418yescomplemented
18C2×#398—22no—
19C2×#396—22no—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×C4×C4⋊C4C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33#1729—24no—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1437—22no—
39#1436—22no—
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×#125—22no—
58C2×#125—42no—
59C2×#125—42no—
60C2×#125C2×C414yes—
61C2×#117—42no—
62C2×#117—42no—
63C2×#117C2×C414yes—
64C2×#117D414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C42—22no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#398—216no—
80#396—216no—
81#389—24no—
82#389C2×C418yes—
83#384—22no—
84#384C2×C414yes—
85#380—24no—
86#380—24no—
87#380C2×C418yes—
88#380Q818yes—
89C4×C4⋊C4—42no—
90C4×C4⋊C4—42no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#340—42no—
104#340—42no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109C25C2×D411yescharacteristic
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C22×C412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×Q814yes—
126C23×C4#4114yes—
127C23×C4C2411yescharacteristic
128C2×M16—28no—
129C2×M16—48no—
130C2×M16—48no—
131C2×M16—44no—
132C2×M16—44no—
133C22×C8—28no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—48no—
151C22×C8—48no—
152C22×C8—44no—
153C22×C8—44no—
154C22×C8#4112yes—
155C22×C8#4112yes—
156C22×C8#4112yes—
157C22×C8#4112yes—
158#125—48no—
159#125—44no—
160#125—44no—
161#117—48no—
162#117—44no—
163#117—44no—
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—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C22⋊C418yes—
196C2×C4⋊C4C22×C414yes—
197C2×C4⋊C4C22×C414yes—
198C2×C4⋊C4C2×Q814yes—
199C2×C4⋊C4C2×Q814yes—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4C22⋊C418yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—44no—
214C2×C42—82no—
215C2×C42—82no—
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—44no—
224#95—82no—
225#95—82no—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24C2×C22⋊C412yes—
234C24C22⋊Q814yes—
235C24C22×D411yescharacteristic
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
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×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4C2×C22⋊C414yes—
295C22×C4C2×C22⋊C414yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C4×Q818yes—
298C22×C4C22⋊Q814yes—
299C22×C4#12414yes—
300C22×C4C4⋊Q814yes—
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×Q812yes—
304C22×C4C2×#4112yes—
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4H(5,𝔽2)12yes—
310C22×C4H(5,𝔽2)12yes—
311M16—416no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—416no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—48no—
342C4⋊C4—416no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—88no—
352C4⋊C4—88no—
353C22⋊C4—48no—
354C22⋊C4—88no—
355C22⋊C4—88no—
356C42—24no—
357C42—24no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—88no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23#39114yes—
384C23C22×C22⋊C411yescharacteristic
385C23C2×C22⋊Q812yes—
386C23#53211yescharacteristic
387C23#53211yescharacteristic
388C23#53412yes—
389C23#57612yes—
390C23#57712yes—
391C23#57712yes—
392C23#57812yes—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#39114yes—
446C2×C4C2×C4×Q812yes—
447C2×C4C4×#4111yescharacteristic
448C2×C4C4×#4111yescharacteristic
449C2×C4#51812yes—
450C2×C4#52511yescharacteristic
451C2×C4#52511yescharacteristic
452C2×C4C2×#12411yescharacteristic
453C2×C4C2×C4⋊Q811yescharacteristic
454C2×C4#53411yescharacteristic
455C2×C4#53411yescharacteristic
456C2×C4#55112yes—
457C2×C4#55112yes—
458C2×C4#55112yes—
459C2×C4#55112yes—
460C2×C4#55412yes—
461C2×C4#55412yes—
462C2×C4#55412yes—
463C2×C4#55412yes—
464C8—42no—
465C8—42no—
466C8—42no—
467C8—42no—
468C22—28no—
469C22—24no—
470C22—44no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22C2×#39111yescharacteristic
478C22#229711yescharacteristic
479C22#230811yescharacteristic
480C22#256412yes—
481C22#256412yes—
482C22#289712yes—
483C22#289812yes—
484C22#308112yes—
485C22#308212yes—
486C22#308312yes—
487C22#308412yes—
488C22#317411yescharacteristic
489C22#317411yescharacteristic
490C22#317611yescharacteristic
491C22#317711yescharacteristic
492C22#317812yes—
493C22#317912yes—
494C22#319812yes—
495C22#320112yes—
496C22#320312yes—
497C22#320412yes—
498C4—24no—
499C4—44no—
500C4—44no—
501C4—84no—
502C4—84no—
503C4—84no—
504C4—84no—
505C4#236812yes—
506C4#236812yes—
507C4#256111yescharacteristic
508C4#256111yescharacteristic
509C4#258311yescharacteristic
510C4#258311yescharacteristic
511C2—44no—
512C2#1415911yescharacteristic
513C2#2131512yes—
514C2#2141311yescharacteristic
515C2#2141411yescharacteristic
516C2#2177212yes—
517C2#2916812yes—
518C2#2916912yes—
519C2#2917212yes—
520C2#2917312yes—
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. #11385C2semidirect
5. #11385C2semidirect
6. #11373C2semidirect
7. #11371C2semidirect
8. #10885C2semidirect
9. #10884C2semidirect
17. #2629C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#17292
#137772
#113852
#113732
#113712
#108852
#108842
#107222
#107212