Finite Group Databasebeta

← #386233

Subgroups — #386233

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1808C211yesmaximal, characteristic
2#13974C212yesmaximal
3#11293C212yesmaximal, complemented
4#11219C212yesmaximal, complemented
5#11199C212yesmaximal
6#10907C212yesmaximal, complemented
7#10807C212yesmaximal, complemented
8#10721C211yesmaximal, characteristic
9#10714C211yesmaximal, characteristic
10C24×C8C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C2×#389C2212yes—
15C2×#388—22no—
16C2×#380C2212yes—
17C2×#380C2212yes—
18C2×C4×C4⋊C4C2212yes—
19C2×C4×C22⋊C4C2211yescharacteristic
20#1808C418yescomplemented
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1443—22no—
32#1440—22no—
33#1436—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C24×C4C2311yescharacteristic
37C23×C8—24no—
38C23×C8—42no—
39C23×C8—41no—
40C23×C8—41no—
41C23×C8C2×C414yes—
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4D412yes—
53C22×C42—22no—
54C22×C42C2312yes—
55C2×#107—41no—
56C2×#107—42no—
57C2×#107—41no—
58C2×#107C2×C414yes—
59C2×#107C2×C414yes—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—41no—
62C2×C22⋊C8—41no—
63#389—42no—
64#389—42no—
65#388—48no—
66#380—24no—
67#380—24no—
68#380—42no—
69#380C2×C418yes—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—42no—
72C4×C22⋊C4—41no—
73C4×C22⋊C4—41no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95D411yescharacteristic
83C2×#95D411yescharacteristic
84#340—24no—
85#340—24no—
86#340—41no—
87#340—42no—
88#340—42no—
89#340—42no—
90#340—41no—
91#340C2×C418yes—
92#336—24no—
93#336—42no—
94C25C2×D411yescharacteristic
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4C22×C412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4#4112yes—
112C23×C4C2411yescharacteristic
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—28no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—44no—
129C22×C8—44no—
130C22×C8—48no—
131C22×C8—48no—
132C22×C8C22×C412yes—
133C22×C8C22×C412yes—
134C22×C8#4114yes—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
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—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4C22×C414yes—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4#4114yes—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—82no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—44no—
178C2×C42—44no—
179C2×C42—82no—
180#107—44no—
181#107—44no—
182C22⋊C8—44no—
183#95—28no—
184#95—41no—
185#95—41no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—82no—
193C24—21no—
194C24—21no—
195C24—22no—
196C24—22no—
197C24—22no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—21no—
202C24C4×D412yes—
203C24AGL(1,ℤ/8)14yes—
204C24C22×D411yescharacteristic
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
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—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—42no—
267C22×C4—42no—
268C22×C4C4×D412yes—
269C22×C4#13514yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4C2×#4111yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C4H(5,𝔽2)11yescharacteristic
276C22×C4H(5,𝔽2)12yes—
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C42−1+412yes—
279C22×C42−1+411yescharacteristic
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—416no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8#11714yes—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—88no—
315C22⋊C4—48no—
316C22⋊C4—416no—
317C22⋊C4—88no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—88no—
323C23—24no—
324C23—24no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—24no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—24no—
333C23—24no—
334C23—24no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23#44214yes—
346C23C2×C4×D411yescharacteristic
347C23#53411yescharacteristic
348C23#53711yescharacteristic
349C23#54612yes—
350C23#54712yes—
351C23C2×#13512yes—
352C23C2×AGL(1,ℤ/8)12yes—
353C23#57611yescharacteristic
354C23#57811yescharacteristic
355C23#57811yescharacteristic
356C23#57811yescharacteristic
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
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—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4C2×#11711yescharacteristic
411C2×C4C4×#4111yescharacteristic
412C2×C4#51811yescharacteristic
413C2×C4#51811yescharacteristic
414C2×C4#51911yescharacteristic
415C2×C4#52011yescharacteristic
416C2×C4#53812yes—
417C2×C4#53812yes—
418C2×C4#53811yescharacteristic
419C2×C4#53912yes—
420C2×C4#53911yescharacteristic
421C2×C4#54212yes—
422C2×C4#55212yes—
423C2×C4#55212yes—
424C2×C4#55312yes—
425C2×C4#55512yes—
426C2×C4#56212yes—
427C2×C4#56312yes—
428C8—24no—
429C8—42no—
430C8—82no—
431C22—24no—
432C22—24no—
433C22—24no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—44no—
441C22—42no—
442C22#232511yescharacteristic
443C22#234411yescharacteristic
444C22#236411yescharacteristic
445C22#268612yes—
446C22#269512yes—
447C22C4×#13511yescharacteristic
448C22C2×#44211yescharacteristic
449C22#295011yescharacteristic
450C22#295111yescharacteristic
451C22#318812yes—
452C22#320211yescharacteristic
453C22#320311yescharacteristic
454C22#324712yes—
455C22#326111yescharacteristic
456C22#326211yescharacteristic
457C22#330012yes—
458C22#330212yes—
459C22#330812yes—
460C22#331812yes—
461C22#333912yes—
462C22#334212yes—
463C22#334312yes—
464C22#334312yes—
465C4—42no—
466C4—44no—
467C4—84no—
468C4—82no—
469C4—84no—
470C4—84no—
471C4—162no—
472C4#231211yescharacteristic
473C4#233211yescharacteristic
474C4#234611yescharacteristic
475C4#237811yescharacteristic
476C4#277012yes—
477C4#278112yes—
478C2—44no—
479C2#1442211yescharacteristic
480C2#2218411yescharacteristic
481C2#2219711yescharacteristic
482C2#2261411yescharacteristic
483C2#2261711yescharacteristic
484C2#2312911yescharacteristic
485C2#2313211yescharacteristic
486C2#3132812yes—
487C2#3133312yes—
488C2#3161212yes—
489C2#3161712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11293C2semidirect
4. #11219C2semidirect
6. #10907C2semidirect
7. #10807C2semidirect
20. #1808C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18082
#139742
#112932
#112192
#111992
#109072
#108072
#107212
#107142