Finite Group Databasebeta

← #375053

Subgroups — #375053

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal
2C2×#1832C211yesmaximal, characteristic
3#13779C211yesmaximal, characteristic
4#13779C211yesmaximal, characteristic
5#11324C211yesmaximal, characteristic
6#11323C211yesmaximal, characteristic
7#11268C211yesmaximal, characteristic
8#11268C211yesmaximal, characteristic
9#10908C212yesmaximal
10#10907C212yesmaximal
11#10849C212yesmaximal
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C2×#384C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1949—28no—
24#1832C418yescomplemented
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1436—22no—
35#1436—22no—
36#1436—22no—
37#1436—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4C2311yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42—22no—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#107—24no—
58C2×#107—24no—
59C2×#107—42no—
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—42no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—24no—
65C2×C4⋊C8C2×C418yes—
66#384—24no—
67#384—22no—
68#384Q814yes—
69#382—24no—
70#382—22no—
71#382D414yes—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—42no—
75C4×C4⋊C4—42no—
76#376C2×C414yes—
77#376Q814yes—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D412yes—
88C2×#95D412yes—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—42no—
94#340—42no—
95#336C2×C414yes—
96#336C2×C414yes—
97#336C2×C418yes—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2411yescharacteristic
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8C22×C414yes—
122C22×C8C22×C414yes—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
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—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×Q812yes—
154C2×C4⋊C4C2×Q812yes—
155C2×C4⋊C4#4114yes—
156C2×C4⋊C4#4114yes—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—44no—
164C2×C42—82no—
165C2×C42—82no—
166C2×C42—82no—
167C2×C42—82no—
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C2×D412yes—
172C2×C42C2×Q812yes—
173C2×C42C2×Q812yes—
174C2×C42C2×Q812yes—
175C2×C42C2×Q812yes—
176C2×C42#4112yes—
177C2×C42#4112yes—
178#107—48no—
179#106—48no—
180C4⋊C8—216no—
181#95—28no—
182#95—28no—
183#95—42no—
184#95—42no—
185#95—44no—
186#95—82no—
187#95—82no—
188#95—82no—
189#95—82no—
190#95C4⋊C418yes—
191#95C4⋊C418yes—
192C24C22×D411yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
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×C4C2×C4⋊C412yes—
246C22×C4C2×C4⋊C412yes—
247C22×C4C2×C4⋊C414yes—
248C22×C4C2×C4⋊C414yes—
249C22×C4#12114yes—
250C22×C4C22⋊Q814yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×Q811yescharacteristic
254C22×C4C22×Q811yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4112yes—
258C22×C4C2×#4112yes—
259C22×C4H(5,𝔽2)12yes—
260C22×C42−1+412yes—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—88no—
292C4⋊C4—88no—
293C42—24no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—88no—
298C42—84no—
299C42—84no—
300C42C4×D414yes—
301C42C4×Q814yes—
302C42#12514yes—
303C42C4⋊Q814yes—
304C23C22×C4⋊C411yescharacteristic
305C23C2×#12111yescharacteristic
306C23C2×C22⋊Q811yescharacteristic
307C23#52511yescharacteristic
308C23#52511yescharacteristic
309C23#53511yescharacteristic
310C23#53611yescharacteristic
311C23#57612yes—
312C23#57712yes—
313C23#57712yes—
314C23#57812yes—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4#38918yes—
373C2×C4C2×C4×D411yescharacteristic
374C2×C4C2×C4×Q811yescharacteristic
375C2×C4C4×#4112yes—
376C2×C4#52012yes—
377C2×C4C2×#12511yescharacteristic
378C2×C4C2×C4⋊Q811yescharacteristic
379C2×C4#53312yes—
380C2×C4#54412yes—
381C2×C4#54612yes—
382C2×C4#54812yes—
383C2×C4#55212yes—
384C2×C4#55312yes—
385C2×C4#55412yes—
386C2×C4#55612yes—
387C2×C4#55712yes—
388C2×C4#55912yes—
389C8—44no—
390C8—44no—
391C22C2×#38911yescharacteristic
392C22#229611yescharacteristic
393C22#231711yescharacteristic
394C22#253511yescharacteristic
395C22#253611yescharacteristic
396C22#258011yescharacteristic
397C22#258111yescharacteristic
398C22#291212yes—
399C22#291312yes—
400C22#304512yes—
401C22#304612yes—
402C22#308112yes—
403C22#308212yes—
404C22#311812yes—
405C22#311912yes—
406C22#312012yes—
407C22#312112yes—
408C22#322011yescharacteristic
409C22#322111yescharacteristic
410C22#322211yescharacteristic
411C22#322311yescharacteristic
412C22#324312yes—
413C22#324412yes—
414C4—28no—
415C4—44no—
416C4—42no—
417C4—42no—
418C4—84no—
419C4—82no—
420C4—82no—
421C4—84no—
422C4—82no—
423C4—82no—
424C4#236712yes—
425C4#237012yes—
426C4#255012yes—
427C4#257012yes—
428C2#1415011yescharacteristic
429C2#2117512yes—
430C2#2136712yes—
431C2#2199011yescharacteristic
432C2#2199111yescharacteristic
433C2#2817111yescharacteristic
434C2#2817311yescharacteristic
435C2#2821512yes—
436C2#2926511yescharacteristic
437C2#2926611yescharacteristic
438C2#2928112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
24. #1832C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19492
C2×#18322
#137792
#113242
#113232
#112682
#109082
#109072
#108492