Finite Group Databasebeta

← #382746

Subgroups — #382746

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic
3#14343C211yesmaximal, characteristic
4#11373C211yesmaximal, characteristic, complemented
5#11372C211yesmaximal, characteristic, complemented
6#11372C211yesmaximal, characteristic, complemented
7#11371C211yesmaximal, characteristic, complemented
8#11358C212yesmaximal, complemented
9#11357C212yesmaximal, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16#2317—24no—
17#2317C418yescomplemented
18C2×#389—22no—
19C2×#385—22no—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#380C2212yes—
25#1818—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1445—22no—
38#1440—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—22no—
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⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D412yes—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77#389—216no—
78#385—216no—
79#384—22no—
80#384—22no—
81#384C2×C414yes—
82#384C2×C414yes—
83#382—22no—
84#382—22no—
85#382C2×C414yes—
86#382C2×C414yes—
87#380—24no—
88#380—24no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—42no—
102#340—42no—
103#340—42no—
104#336—24no—
105#336—22no—
106#336—22no—
107C25C2×D411yescharacteristic
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—82no—
122C23×C4—82no—
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4#4114yes—
129C23×C4#4114yes—
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
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—
145#117—28no—
146#117—28no—
147#117—216no—
148#117—48no—
149#117—48no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
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—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C22⋊C418yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4C22⋊C418yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C22⋊C8—48no—
213#95—28no—
214#95—28no—
215#95—44no—
216#95—82no—
217#95—82no—
218#95—82no—
219#95—82no—
220C24—21no—
221C24—21no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24C2×C22⋊C412yes—
228C24#12314yes—
229C24C22×D411yescharacteristic
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
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—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—48no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
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—88no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—88no—
297C22×C4C2×C22⋊C414yes—
298C22×C4C2×C22⋊C414yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4#11714yes—
301C22×C4#11714yes—
302C22×C4#12318yes—
303C22×C4#12314yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4112yes—
307C22×C4C2×#4112yes—
308C22×C4H(5,𝔽2)12yes—
309C22×C4H(5,𝔽2)12yes—
310C22×C42−1+412yes—
311C22×C42−1+412yes—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—216no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—48no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—88no—
345C4⋊C4—88no—
346C22⋊C4—216no—
347C22⋊C4—48no—
348C22⋊C4—88no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—24no—
354C42—24no—
355C42—48no—
356C23—22no—
357C23—22no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23#38114yes—
377C23C22×C22⋊C411yescharacteristic
378C23C2×#12312yes—
379C23#53511yescharacteristic
380C23#53511yescharacteristic
381C23#53612yes—
382C23#57612yes—
383C23#57712yes—
384C23#57712yes—
385C23#57812yes—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—816no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
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#38114yes—
444C2×C4C2×#11711yescharacteristic
445C2×C4C2×#11711yescharacteristic
446C2×C4#51812yes—
447C2×C4#51912yes—
448C2×C4C2×#12312yes—
449C2×C4#53512yes—
450C2×C4#53611yescharacteristic
451C2×C4#53611yescharacteristic
452C2×C4#53912yes—
453C2×C4#53912yes—
454C2×C4#53912yes—
455C2×C4#53912yes—
456C2×C4#53912yes—
457C2×C4#53912yes—
458C2×C4#53912yes—
459C2×C4#53912yes—
460C8—82no—
461C8—82no—
462C22—24no—
463C22—28no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—44no—
471C22C2×#38111yescharacteristic
472C22#230911yescharacteristic
473C22#231011yescharacteristic
474C22#262312yes—
475C22#262312yes—
476C22#289712yes—
477C22#289812yes—
478C22#309612yes—
479C22#309712yes—
480C22#309812yes—
481C22#309912yes—
482C22#322011yescharacteristic
483C22#322111yescharacteristic
484C22#322211yescharacteristic
485C22#322311yescharacteristic
486C22#322412yes—
487C22#322512yes—
488C22#324112yes—
489C22#324212yes—
490C22#324312yes—
491C22#324412yes—
492C4—24no—
493C4—44no—
494C4—44no—
495C4—82no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4—162no—
500C4—162no—
501C4#231312yes—
502C4#231312yes—
503C4#262312yes—
504C4#262312yes—
505C2—44no—
506C2#1434211yescharacteristic
507C2#2070312yes—
508C2#2181311yescharacteristic
509C2#2181411yescharacteristic
510C2#2184312yes—
511C2#2991612yes—
512C2#2991712yes—
513C2#2991812yes—
514C2#2991912yes—
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. #11358C2semidirect
9. #11357C2semidirect
17. #2317C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#18182
#143432
#113732
#113722
#113712
#113582
#113572
#107312
#107302