Finite Group Databasebeta

← #355599

Subgroups — #355599

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic
4C2×#1746C211yesmaximal, characteristic
5#13779C211yesmaximal, characteristic
6#10885C212yesmaximal
7#10884C212yesmaximal
8#10843C211yesmaximal, characteristic
9#10841C211yesmaximal, characteristic
10#10807C212yesmaximal
11#10806C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2330—24no—
16#2330C418yescomplemented
17C2×#389—22no—
18C2×#389C2212yes—
19C2×#387—22no—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1992C418yescomplemented
28#1991C418yescomplemented
29#1746—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1436—22no—
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125C2×C414yes—
50C2×#125C2×C414yes—
51C2×#125C2×C414yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C42—22no—
63C22×C42—21no—
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#97—22no—
69C2×#97—22no—
70C2×#97C2×C414yes—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8C2×C414yes—
74#389—24no—
75#389—216no—
76#389C2×C418yes—
77#387—216no—
78#384—22no—
79#384—22no—
80#384C2×C414yes—
81#384C2×C414yes—
82#382—22no—
83#382C2×C414yes—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—42no—
86C4×C4⋊C4—42no—
87C4×C4⋊C4C2×C418yes—
88#376C2×C414yes—
89#376D414yes—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—42no—
100#340—42no—
101#340—42no—
102#340—42no—
103#340C2×C418yes—
104#340C2×C418yes—
105#340C2×C418yes—
106#340C2×C418yes—
107#336—24no—
108#336—24no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2411yescharacteristic
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×C8—22no—
131C22×C8—22no—
132C22×C8C22×C414yes—
133C22×C8C22×C414yes—
134#125—28no—
135#125—28no—
136#125—28no—
137#125C42116yescomplemented
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
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—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4C22×C414yes—
176C2×C4⋊C4C22×C414yes—
177C2×C4⋊C4C22×C414yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—28no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—82no—
186C2×C42—82no—
187C2×C42—82no—
188C2×C42—82no—
189C2×C42C22×C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C412yes—
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199#97—28no—
200C4×C8—28no—
201#95—28no—
202#95—28no—
203#95—28no—
204#95—42no—
205#95—42no—
206#95—82no—
207#95—82no—
208#95C22⋊C418yes—
209#95C22⋊C418yes—
210C24C22×D411yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
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×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C414yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C414yes—
267C22×C4C4×D414yes—
268C22×C4C4×D414yes—
269C22×C4C23×C411yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4H(5,𝔽2)12yes—
276C22×C42−1+412yes—
277C22×C42−1+412yes—
278C22×C42−1+412yes—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—232no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—88no—
310C4⋊C4—88no—
311C4⋊C4—88no—
312C4⋊C4C2×C4218yes—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—88no—
318C42—84no—
319C42—84no—
320C42C2×C4214yes—
321C42#11714yes—
322C42C4×D414yes—
323C42C4×D414yes—
324C23C22×C22⋊C411yescharacteristic
325C23C2×C4×D411yescharacteristic
326C23C2×C4×D411yescharacteristic
327C23#53511yescharacteristic
328C23#53611yescharacteristic
329C23#53611yescharacteristic
330C23#53611yescharacteristic
331C23#57612yes—
332C23#57712yes—
333C23#57712yes—
334C23#57812yes—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—28no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4C4×C22⋊C418yes—
393C2×C4C22×C4211yescharacteristic
394C2×C4C2×#11711yescharacteristic
395C2×C4C2×C4×D411yescharacteristic
396C2×C4C2×C4×D411yescharacteristic
397C2×C4#51912yes—
398C2×C4#51912yes—
399C2×C4#51912yes—
400C2×C4#52012yes—
401C2×C4#52012yes—
402C2×C4#52012yes—
403C2×C4#53612yes—
404C2×C4#53712yes—
405C2×C4#54112yes—
406C2×C4#54112yes—
407C2×C4#54212yes—
408C2×C4#54212yes—
409C8—44no—
410C8—44no—
411C22C2×C4×C22⋊C411yescharacteristic
412C22#231011yescharacteristic
413C22#231111yescharacteristic
414C22#232611yescharacteristic
415C22#232611yescharacteristic
416C22#232711yescharacteristic
417C22#232711yescharacteristic
418C22#289712yes—
419C22#289812yes—
420C22#294612yes—
421C22#294712yes—
422C22#295112yes—
423C22#295212yes—
424C22#322011yescharacteristic
425C22#322111yescharacteristic
426C22#322211yescharacteristic
427C22#322311yescharacteristic
428C22#324112yes—
429C22#324112yes—
430C22#324212yes—
431C22#324212yes—
432C22#324312yes—
433C22#324412yes—
434C4—28no—
435C4—44no—
436C4—42no—
437C4—42no—
438C4—48no—
439C4—84no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4#227612yes—
445C4#232012yes—
446C4#232312yes—
447C4#232312yes—
448C2#1378611yescharacteristic
449C2#2041512yes—
450C2#2184312yes—
451C2#2187211yescharacteristic
452C2#2187311yescharacteristic
453C2#2222711yescharacteristic
454C2#2222811yescharacteristic
455C2#2223111yescharacteristic
456C2#2223211yescharacteristic
457C2#2225212yes—
458C2#2225512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2330C4semidirect
27. #1992C4semidirect
28. #1991C4semidirect
137. #125C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23302
C2×#19922
C2×#19912
C2×#17462
#137792
#108852
#108842
#108432
#108412
#108072
#108062