Finite Group Databasebeta

← #382766

Subgroups — #382766

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1798C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11373C211yesmaximal, characteristic, complemented
5#11371C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11358C211yesmaximal, characteristic, complemented
8#11200C212yesmaximal
9#11199C212yesmaximal
10#10850C212yesmaximal, complemented
11#10849C212yesmaximal, complemented
12C22×#117C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15#2314—24no—
16#2314C418yescomplemented
17C2×#388—22no—
18C2×#387—22no—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
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#1441—22no—
39#1439—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#117—24no—
46C2×#117—24no—
47C2×#117C2×C414yes—
48C2×#117D414yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4D412yes—
62C22×C42—22no—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—42no—
70C2×C22⋊C8—42no—
71#388—216no—
72#387—216no—
73#384—22no—
74#384—24no—
75#384C2×C414yes—
76#382—22no—
77#382—24no—
78#382C2×C414yes—
79#380—24no—
80#380—42no—
81#380—42no—
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—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#340—42no—
101#336—22no—
102#336—22no—
103C25C2×D411yescharacteristic
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C22×C412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138#117—216no—
139#117—216no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
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—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C22⋊C418yes—
175C2×C4⋊C4C22×C414yes—
176C2×C4⋊C4C22×C414yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4C22⋊C418yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—28no—
190C2×C42—82no—
191C2×C42—82no—
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42#4114yes—
197C2×C42#4114yes—
198C4⋊C8—48no—
199C22⋊C8—48no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—42no—
204#95—42no—
205#95—44no—
206#95—82no—
207#95—82no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24C2×C22⋊C412yes—
219C24C22×D411yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
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—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
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×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C414yes—
288C22×C4C2×C22⋊C414yes—
289C22×C4#12314yes—
290C22×C4#12314yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C4H(5,𝔽2)11yescharacteristic
297C22×C4H(5,𝔽2)12yes—
298C22×C42−1+411yescharacteristic
299C22×C42−1+411yescharacteristic
300C22×C42−1+412yes—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—216no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—88no—
333C4⋊C4—88no—
334C22⋊C4—216no—
335C22⋊C4—88no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C42—24no—
339C42—24no—
340C42—88no—
341C42#11718yes—
342C42#12318yes—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23C22×C22⋊C411yescharacteristic
367C23C2×#12311yescharacteristic
368C23C2×#12311yescharacteristic
369C23#53511yescharacteristic
370C23#53611yescharacteristic
371C23#53711yescharacteristic
372C23#53711yescharacteristic
373C23#57612yes—
374C23#57712yes—
375C23#57712yes—
376C23#57812yes—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
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—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4#38118yes—
436C2×C4C2×#11712yes—
437C2×C4#52012yes—
438C2×C4#52011yescharacteristic
439C2×C4#52011yescharacteristic
440C2×C4C2×#12312yes—
441C2×C4#53511yescharacteristic
442C2×C4#53611yescharacteristic
443C2×C4#53712yes—
444C2×C4#54012yes—
445C2×C4#54012yes—
446C2×C4#54112yes—
447C2×C4#54112yes—
448C2×C4#54212yes—
449C2×C4#54212yes—
450C2×C4#54212yes—
451C2×C4#54212yes—
452C8—82no—
453C8—82no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22—44no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—82no—
464C22—82no—
465C22C2×#38111yescharacteristic
466C22#231111yescharacteristic
467C22#231111yescharacteristic
468C22#262411yescharacteristic
469C22#262411yescharacteristic
470C22#262511yescharacteristic
471C22#262511yescharacteristic
472C22#289712yes—
473C22#289812yes—
474C22#309612yes—
475C22#309712yes—
476C22#309812yes—
477C22#309912yes—
478C22#322011yescharacteristic
479C22#322111yescharacteristic
480C22#322211yescharacteristic
481C22#322311yescharacteristic
482C22#324112yes—
483C22#324212yes—
484C22#325712yes—
485C22#325812yes—
486C22#326212yes—
487C22#326312yes—
488C4—28no—
489C4—42no—
490C4—44no—
491C4—84no—
492C4—84no—
493C4—161no—
494C4—161no—
495C4—161no—
496C4—161no—
497C4#231412yes—
498C4#231412yes—
499C4#262412yes—
500C4#262512yes—
501C2—82no—
502C2#1434311yescharacteristic
503C2#2070312yes—
504C2#2187211yescharacteristic
505C2#2187311yescharacteristic
506C2#2189612yes—
507C2#2993111yescharacteristic
508C2#2993211yescharacteristic
509C2#2993311yescharacteristic
510C2#2993411yescharacteristic
511C2#2994612yes—
512C2#2994712yes—
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
10. #10850C2semidirect
11. #10849C2semidirect
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×#17982
#143402
#113732
#113712
#113582
#112002
#111992
#108502
#108492