Finite Group Databasebeta

← #392047

Subgroups — #392047

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1818C211yesmaximal, characteristic
2#14340C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11325C212yesmaximal, complemented
5#11324C211yesmaximal, characteristic, complemented
6#11323C211yesmaximal, characteristic, complemented
7#11268C211yesmaximal, characteristic, complemented
8#11268C211yesmaximal, characteristic, complemented
9#11267C212yesmaximal, complemented
10#11096C212yesmaximal
11#11095C212yesmaximal
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2211yescharacteristic
23#1818C418yescomplemented
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33#1441—22no—
34#1441—22no—
35#1439—22no—
36#1439—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#117—42no—
44C2×#117—42no—
45C2×#117C2×C414yes—
46C2×#117Q814yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D412yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4D412yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#107—42no—
63C2×#107—42no—
64C2×#107—42no—
65C2×#106—42no—
66C2×#106—42no—
67C2×#106—42no—
68C2×C22⋊C8—42no—
69C2×C22⋊C8—42no—
70C2×C22⋊C8C2×C418yes—
71#384—42no—
72#384—42no—
73#382—42no—
74#382—42no—
75#380—42no—
76#380—42no—
77#380—42no—
78#380—42no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#336C2×C418yes—
95#336C2×C414yes—
96#336C2×C414yes—
97C25C2×D411yescharacteristic
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4C22×C412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×Q812yes—
117C23×C4C2411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C22×C414yes—
131C22×C8C22×C414yes—
132#117—48no—
133#117—48no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C4⋊C418yes—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—82no—
177C2×C22⋊C4—82no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4C4⋊C418yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42C22×C412yes—
186C2×C42C22×C412yes—
187C2×C42C2×Q812yes—
188C2×C42C2×Q812yes—
189#107—48no—
190#106—48no—
191C22⋊C8—48no—
192#95—42no—
193#95—42no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—82no—
198#95—82no—
199#95—82no—
200#95—82no—
201C24—21no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24C2×C4⋊C412yes—
212C24C22×D411yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
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×C4C2×C4⋊C414yes—
282C22×C4C2×C4⋊C414yes—
283C22×C4C2×C4⋊C412yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4H(5,𝔽2)12yes—
287C22×C4H(5,𝔽2)12yes—
288C22×C4H(5,𝔽2)12yes—
289C22×C4H(5,𝔽2)11yescharacteristic
290C22×C4H(5,𝔽2)11yescharacteristic
291C22×C42−1+411yescharacteristic
292C22×C42−1+411yescharacteristic
293C22×C42−1+412yes—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C4⋊C4—48no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—88no—
326C4⋊C4—88no—
327C22⋊C4—48no—
328C22⋊C4—84no—
329C22⋊C4—84no—
330C22⋊C4—88no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—24no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—24no—
339C23—24no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—41no—
358C23C22×C4⋊C411yescharacteristic
359C23#53411yescharacteristic
360C23#53411yescharacteristic
361C23#53511yescharacteristic
362C23#53511yescharacteristic
363C23#53611yescharacteristic
364C23#53611yescharacteristic
365C23#57612yes—
366C23#57712yes—
367C23#57712yes—
368C23#57812yes—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
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—81no—
427C2×C4—81no—
428C2×C4#51812yes—
429C2×C4#52012yes—
430C2×C4#52011yescharacteristic
431C2×C4#52011yescharacteristic
432C2×C4#54312yes—
433C2×C4#54412yes—
434C2×C4#54411yescharacteristic
435C2×C4#54411yescharacteristic
436C2×C4#56012yes—
437C2×C4#56012yes—
438C2×C4#56212yes—
439C2×C4#56212yes—
440C2×C4#56212yes—
441C2×C4#56212yes—
442C2×C4#56312yes—
443C2×C4#56312yes—
444C8—44no—
445C8—44no—
446C22—42no—
447C22—44no—
448C22—42no—
449C22—42no—
450C22—44no—
451C22—82no—
452C22—81no—
453C22—81no—
454C22—82no—
455C22—81no—
456C22—81no—
457C22#231511yescharacteristic
458C22#231711yescharacteristic
459C22#231711yescharacteristic
460C22#281511yescharacteristic
461C22#281611yescharacteristic
462C22#281611yescharacteristic
463C22#281711yescharacteristic
464C22#291212yes—
465C22#291312yes—
466C22#319812yes—
467C22#319812yes—
468C22#320112yes—
469C22#320112yes—
470C22#322011yescharacteristic
471C22#322111yescharacteristic
472C22#322211yescharacteristic
473C22#322311yescharacteristic
474C22#322412yes—
475C22#322512yes—
476C22#324112yes—
477C22#324212yes—
478C22#324312yes—
479C22#324412yes—
480C4—42no—
481C4—44no—
482C4—44no—
483C4—162no—
484C4—161no—
485C4—161no—
486C4—162no—
487C4—161no—
488C4—161no—
489C4#237612yes—
490C4#237612yes—
491C4#284112yes—
492C4#284112yes—
493C2—82no—
494C2#1467411yescharacteristic
495C2#2194512yes—
496C2#2199011yescharacteristic
497C2#2199111yescharacteristic
498C2#2199212yes—
499C2#3276511yescharacteristic
500C2#3276711yescharacteristic
501C2#3278112yes—
502C2#3278211yescharacteristic
503C2#3278411yescharacteristic
504C2#3279312yes—
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. #11325C2semidirect
5. #11324C2semidirect
6. #11323C2semidirect
7. #11268C2semidirect
8. #11268C2semidirect
9. #11267C2semidirect
23. #1818C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18182
#143402
#113252
#113242
#113232
#112682
#112672
#110962
#110952