Finite Group Databasebeta

← #379637

Subgroups — #379637

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic
2#13777C212yesmaximal
3#11200C212yesmaximal
4#11095C211yesmaximal, characteristic
5#10908C212yesmaximal, complemented
6#10907C212yesmaximal, complemented
7#10807C212yesmaximal, complemented
8#10806C212yesmaximal, complemented
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#380C2212yes—
16C2×#380C2212yes—
17C2×C4×C4⋊C4C2212yes—
18C2×C4×C4⋊C4C2212yes—
19#2075C418yescomplemented
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1437—22no—
31#1437—22no—
32#1436—22no—
33#1436—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C24×C4C2311yescharacteristic
37C23×C8C2312yes—
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40C2×#117—24no—
41C2×#117—24no—
42C2×#117—41no—
43C2×#117—41no—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D411yescharacteristic
52C22×C22⋊C4D411yescharacteristic
53C22×C42—22no—
54C22×C42—22no—
55C22×C42C2312yes—
56C2×#107—41no—
57C2×#107—41no—
58C2×#107C2×C414yes—
59C2×#106—24no—
60C2×#106—41no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—41no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8C2×C414yes—
67#380—24no—
68#380—24no—
69#380—24no—
70#380C2×C418yes—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—42no—
74C4×C4⋊C4—42no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—41no—
87#340—42no—
88#340—42no—
89#340C2×C414yes—
90#340C2×C418yes—
91#336—24no—
92#336—24no—
93C25C2×D411yescharacteristic
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4C22×C412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4#4112yes—
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C22×C412yes—
128C22×C8C22×C412yes—
129#117—44no—
130#117—48no—
131#117—44no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
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—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C22×C414yes—
156C2×C4⋊C4C22×C414yes—
157C2×C4⋊C4#4114yes—
158C2×C4⋊C4#4114yes—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—82no—
166C2×C22⋊C4—82no—
167C2×C22⋊C4—82no—
168C2×C22⋊C4—82no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—44no—
174C2×C42—44no—
175C2×C42—82no—
176C2×C42—82no—
177C2×C42—82no—
178C2×C42—82no—
179C2×C42#4114yes—
180C2×C42#4114yes—
181#107—44no—
182#106—44no—
183C22⋊C8—44no—
184C22⋊C8—44no—
185#95—28no—
186#95—28no—
187#95—44no—
188#95—44no—
189#95—82no—
190#95—82no—
191#95—82no—
192#95—82no—
193C24—22no—
194C24—22no—
195C24—21no—
196C24—21no—
197C24—22no—
198C24—22no—
199C24—22no—
200C24—22no—
201C24C4×D412yes—
202C24C22×D411yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
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—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
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×C4C4×D412yes—
266C22×C4#12314yes—
267C22×C4#12314yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4C2×#4111yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C4H(5,𝔽2)11yescharacteristic
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C42−1+411yescharacteristic
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—88no—
315C22⋊C4—88no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—88no—
321C42—88no—
322C23—22no—
323C23—24no—
324C23—22no—
325C23—22no—
326C23—24no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23C2×C4×D411yescharacteristic
344C23C2×#12312yes—
345C23#52512yes—
346C23#53411yescharacteristic
347C23#53711yescharacteristic
348C23#57612yes—
349C23#57712yes—
350C23#57712yes—
351C23#57812yes—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—82no—
390C2×C4—82no—
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×C4C4×#4112yes—
406C2×C4C4×#4112yes—
407C2×C4#51811yescharacteristic
408C2×C4#52011yescharacteristic
409C2×C4#52912yes—
410C2×C4#52912yes—
411C2×C4#53811yescharacteristic
412C2×C4#53911yescharacteristic
413C2×C4#55111yescharacteristic
414C2×C4#55111yescharacteristic
415C2×C4#55112yes—
416C2×C4#55111yescharacteristic
417C2×C4#55212yes—
418C2×C4#55312yes—
419C2×C4#55312yes—
420C2×C4#55512yes—
421C2×C4#55612yes—
422C2×C4#55911yescharacteristic
423C8—42no—
424C8—42no—
425C8—82no—
426C22—44no—
427C22—44no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22C4×#12312yes—
436C22#232511yescharacteristic
437C22#248311yescharacteristic
438C22#248512yes—
439C22#260011yescharacteristic
440C22#294712yes—
441C22#294811yescharacteristic
442C22#295211yescharacteristic
443C22#309612yes—
444C22#309712yes—
445C22#309812yes—
446C22#309912yes—
447C22#311912yes—
448C22#312112yes—
449C22#312312yes—
450C22#312412yes—
451C22#319812yes—
452C22#320112yes—
453C22#325712yes—
454C22#326111yescharacteristic
455C22#326311yescharacteristic
456C4—42no—
457C4—44no—
458C4—44no—
459C4—84no—
460C4—84no—
461C4—84no—
462C4—84no—
463C4#237211yescharacteristic
464C4#237312yes—
465C4#237411yescharacteristic
466C4#251711yescharacteristic
467C4#251812yes—
468C4#252311yescharacteristic
469C2—82no—
470C2#1424811yescharacteristic
471C2#2158612yes—
472C2#2159112yes—
473C2#2219311yescharacteristic
474C2#2219811yescharacteristic
475C2#2760612yes—
476C2#2763712yes—
477C2#2764312yes—
478C2#2941912yes—
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
5. #10908C2semidirect
6. #10907C2semidirect
7. #10807C2semidirect
8. #10806C2semidirect
19. #2075C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
#137772
#112002
#110952
#109082
#109072
#108072
#108062
#107132