Finite Group Databasebeta

← #362874

Subgroups — #362874

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1798C211yesmaximal, characteristic
3#13777C211yesmaximal, characteristic
4#11373C211yesmaximal, characteristic, complemented
5#11372C211yesmaximal, characteristic, complemented
6#11372C211yesmaximal, characteristic, complemented
7#11371C211yesmaximal, characteristic, complemented
8#10850C212yesmaximal, complemented
9#10849C212yesmaximal, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17#2317—24no—
18#2317C418yescomplemented
19C2×#389—22no—
20C2×#385—22no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C4⋊C4C2212yes—
27#1798—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1437—22no—
39#1436—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#117—22no—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—42no—
50C2×#117—42no—
51C2×#117C2×C414yes—
52C2×#117C2×C414yes—
53C2×#117D414yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—42no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4C2×C414yes—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D412yes—
71C22×C42—21no—
72C22×C42—22no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—24no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81#389—216no—
82#385—216no—
83#384—22no—
84#384—22no—
85#384C2×C414yes—
86#384C2×C414yes—
87#382—22no—
88#382—22no—
89#382C2×C414yes—
90#382C2×C414yes—
91#380—24no—
92#380Q818yes—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101#340—42no—
102#340—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—22no—
111C25C2×D411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—24no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C22×C412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×Q814yes—
133C23×C4#4114yes—
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149#117—28no—
150#117—28no—
151#117—216no—
152#117—48no—
153#117—48no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
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—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C22⋊C418yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C2×Q814yes—
195C2×C4⋊C4C2×Q814yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4C22⋊C418yes—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—44no—
211C2×C42—82no—
212C2×C42—82no—
213C2×C42C22×C412yes—
214C2×C42C22×C412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42#4114yes—
218C2×C42#4114yes—
219C4⋊C8—48no—
220C22⋊C8—48no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—44no—
225#95—82no—
226#95—82no—
227C24—22no—
228C24—21no—
229C24—21no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24C2×C22⋊C412yes—
235C24C22⋊Q814yes—
236C24C22×D411yescharacteristic
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
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—24no—
255C22×C4—24no—
256C22×C4—24no—
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—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
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×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—88no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—88no—
302C22×C4C2×C22⋊C414yes—
303C22×C4C2×C22⋊C414yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C4×Q818yes—
306C22×C4C22⋊Q814yes—
307C22×C4#12314yes—
308C22×C4#12314yes—
309C22×C4#12414yes—
310C22×C4C4⋊Q814yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×Q812yes—
314C22×C4C2×#4112yes—
315C22×C4C2×#4112yes—
316C22×C4C2×#4112yes—
317C22×C4H(5,𝔽2)12yes—
318C22×C4H(5,𝔽2)12yes—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—216no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—48no—
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—216no—
354C22⋊C4—48no—
355C22⋊C4—88no—
356C42—22no—
357C42—22no—
358C42—24no—
359C42—48no—
360C42—88no—
361C42#11714yes—
362C42#11714yes—
363C42#12318yes—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—22no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23#39114yes—
385C23C22×C22⋊C411yescharacteristic
386C23C2×C22⋊Q812yes—
387C23C2×#12312yes—
388C23#53511yescharacteristic
389C23#53511yescharacteristic
390C23#57612yes—
391C23#57712yes—
392C23#57712yes—
393C23#57812yes—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
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—816no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4#38118yes—
451C2×C4#39114yes—
452C2×C4C2×#11711yescharacteristic
453C2×C4C2×#11711yescharacteristic
454C2×C4C2×C4×Q812yes—
455C2×C4#51812yes—
456C2×C4C2×#12312yes—
457C2×C4C2×#12411yescharacteristic
458C2×C4C2×C4⋊Q811yescharacteristic
459C2×C4#53512yes—
460C2×C4#55312yes—
461C2×C4#55312yes—
462C2×C4#55312yes—
463C2×C4#55312yes—
464C2×C4#55412yes—
465C2×C4#55412yes—
466C2×C4#55412yes—
467C2×C4#55412yes—
468C8—82no—
469C8—82no—
470C22—28no—
471C22—24no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—42no—
477C22—42no—
478C22—44no—
479C22C2×#38111yescharacteristic
480C22C2×#39111yescharacteristic
481C22#230911yescharacteristic
482C22#248712yes—
483C22#248712yes—
484C22#289712yes—
485C22#289812yes—
486C22#308112yes—
487C22#308212yes—
488C22#308312yes—
489C22#308412yes—
490C22#309612yes—
491C22#309712yes—
492C22#309812yes—
493C22#309912yes—
494C22#322011yescharacteristic
495C22#322111yescharacteristic
496C22#322211yescharacteristic
497C22#322311yescharacteristic
498C22#322412yes—
499C22#322512yes—
500C4—24no—
501C4—28no—
502C4—44no—
503C4—84no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4#234512yes—
510C4#234512yes—
511C4#250212yes—
512C4#250412yes—
513C2—44no—
514C2#1392411yescharacteristic
515C2#2070312yes—
516C2#2131512yes—
517C2#2181311yescharacteristic
518C2#2181411yescharacteristic
519C2#2768712yes—
520C2#2768812yes—
521C2#2768912yes—
522C2#2769012yes—
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. #11372C2semidirect
6. #11372C2semidirect
7. #11371C2semidirect
8. #10850C2semidirect
9. #10849C2semidirect
18. #2317C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#17982
#137772
#113732
#113722
#113712
#108502
#108492
#107312
#107302