Finite Group Databasebeta

← #476064

Subgroups — #476064

Order 512.

920 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14282C211yesmaximal, characteristic
2#14276C211yesmaximal, characteristic
3#11384C212yesmaximal
4#11373C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic
6#11268C211yesmaximal, characteristic
7#11268C211yesmaximal, characteristic
8#11268C212yesmaximal
9#11228C212yesmaximal
10#11227C212yesmaximal
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C2×#389C2212yes—
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#376C2211yescharacteristic
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30#1446—22no—
31#1445—22no—
32#1441—22no—
33#1441—22no—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C23×Q8—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—42no—
44C2×C4⋊Q8—42no—
45C2×C4⋊Q8Q814yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—24no—
59C2×C4⋊C8—42no—
60C2×C4⋊C8—42no—
61#389—24no—
62#389—24no—
63#384—22no—
64#384Q814yescomplemented
65#382—42no—
66#382—42no—
67#376—42no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95D412yes—
79C2×#95D412yes—
80#340—42no—
81#340—42no—
82#340—42no—
83#340—42no—
84#340—42no—
85#340—42no—
86#340—42no—
87#340—42no—
88#336—24no—
89#336—22no—
90#336—22no—
91C22×Q8—24no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×Q8—82no—
95C22×Q8—82no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2411yescharacteristic
107C22×C8—22no—
108C22×C8—22no—
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—
121C4⋊Q8—48no—
122C4⋊Q8—48no—
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—24no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
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—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C2×Q812yes—
158C2×C4⋊C4C2×Q812yes—
159C2×C4⋊C4C2×Q814yes—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42C2×Q812yes—
168C2×C42C2×Q812yes—
169C2×C42#4112yes—
170C2×C42#4112yes—
171C2×C42#4114yes—
172C4⋊C8—48no—
173#95—42no—
174#95—42no—
175#95—44no—
176#95—42no—
177#95—42no—
178#95—44no—
179#95—82no—
180#95—82no—
181#95—82no—
182#95—82no—
183#95—82no—
184#95—82no—
185C24C22×D411yescharacteristic
186C2×Q8—48no—
187C2×Q8—88no—
188C2×Q8—88no—
189C2×Q8—84no—
190C2×Q8—84no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
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—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
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×C4C22⋊Q814yes—
246C22×C4C22⋊Q812yes—
247C22×C4C22⋊Q812yes—
248C22×C4#12414yes—
249C22×C4C22×Q811yescharacteristic
250C22×C4C22×Q811yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4112yes—
253C22×C4C2×#4111yescharacteristic
254C22×C4H(5,𝔽2)12yes—
255C22×C4H(5,𝔽2)11yescharacteristic
256C22×C4H(5,𝔽2)12yes—
257C22×C42−1+411yescharacteristic
258C22×C42−1+412yes—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—48no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—88no—
291C4⋊C4—88no—
292C42—24no—
293C42—24no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—24no—
298C23C2×C22⋊Q811yescharacteristic
299C23C2×C22⋊Q811yescharacteristic
300C23C2×#12411yescharacteristic
301C23#53411yescharacteristic
302C23#53511yescharacteristic
303C23#53611yescharacteristic
304C23#53711yescharacteristic
305C23#57612yes—
306C23#57712yes—
307C23#57712yes—
308C23#57812yes—
309Q8—816no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#53812yes—
369C2×C4#53812yes—
370C2×C4#53912yes—
371C2×C4#53912yes—
372C2×C4#54312yes—
373C2×C4#54412yes—
374C2×C4#55412yes—
375C2×C4#55512yes—
376C2×C4#55612yes—
377C2×C4#55711yescharacteristic
378C2×C4#55712yes—
379C2×C4#55711yescharacteristic
380C2×C4Q8211yescharacteristic
381C2×C4#55911yescharacteristic
382C2×C4#56212yes—
383C2×C4#56312yes—
384C8—82no—
385C8—82no—
386C22#256311yescharacteristic
387C22#256811yescharacteristic
388C22#258211yescharacteristic
389C22#262611yescharacteristic
390C22#262711yescharacteristic
391C22#263911yescharacteristic
392C22#282011yescharacteristic
393C22#308112yes—
394C22#308112yes—
395C22#308212yes—
396C22#308212yes—
397C22#313612yes—
398C22#313712yes—
399C22#319812yes—
400C22#320112yes—
401C22#322011yescharacteristic
402C22#322111yescharacteristic
403C22#322211yescharacteristic
404C22#322311yescharacteristic
405C22#324312yes—
406C22#324412yes—
407C22#326212yes—
408C22#326312yes—
409C4—42no—
410C4—42no—
411C4—44no—
412C4—44no—
413C4—82no—
414C4—82no—
415C4—84no—
416C4—161no—
417C4—161no—
418C4—162no—
419C4#280512yes—
420C4#280612yes—
421C4#280812yes—
422C4#280912yes—
423C2#1707911yescharacteristic
424C2#2915211yescharacteristic
425C2#2915311yescharacteristic
426C2#2922912yes—
427C2#2929312yes—
428C2#2996211yescharacteristic
429C2#2996311yescharacteristic
430C2#2997512yes—
431C2#3019512yes—
432C2#3282911yescharacteristic
433C2#3283011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
64. #384Q8semidirect

Maximal subgroups

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

subgroupindex
#142822
#142762
#113842
#113732
#113712
#112682
#112282
#112272
#108332