Finite Group Databasebeta

← #473967

Subgroups — #473967

Order 512.

892 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14282C211yesmaximal, characteristic
2#14277C211yesmaximal, characteristic
3#11384C212yesmaximal
4#11372C212yesmaximal
5#11358C212yesmaximal
6#11293C212yesmaximal
7#11292C212yesmaximal
8#11267C212yesmaximal
9#10796C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#389C2212yes—
12C2×#384C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×#376C2211yescharacteristic
16C2×#340C2212yes—
17C2×#340C2212yes—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28#1446—21no—
29#1446—21no—
30#1445—21no—
31#1445—21no—
32#1444—21no—
33#1444—21no—
34#1442—21no—
35#1442—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C23×Q8—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#389—42no—
58#389—42no—
59#384—41no—
60#384—41no—
61#382—22no—
62#382—24no—
63#382Q814yescomplemented
64#376—41no—
65#376—41no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—22no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95D411yescharacteristic
78C2×#95D411yescharacteristic
79C2×#95D411yescharacteristic
80C2×#95D411yescharacteristic
81#340—24no—
82#340—24no—
83#340—42no—
84#340—42no—
85#340—42no—
86#340—42no—
87#340—42no—
88#340—42no—
89#336—24no—
90#336—24no—
91#336—41no—
92#336—41no—
93C22×Q8—24no—
94C22×Q8—42no—
95C22×Q8—42no—
96C22×Q8—82no—
97C22×Q8—82no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
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×C8—22no—
122C22×C8—22no—
123C4⋊Q8—48no—
124C4⋊Q8—48no—
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—24no—
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—42no—
149C2×C4⋊C4—42no—
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×Q814yes—
158C2×C4⋊C4#4114yes—
159C2×C42—22no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42C2×Q812yes—
171C2×C42#4112yes—
172C2×C42#4114yes—
173#95—24no—
174#95—24no—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—44no—
180#95—42no—
181#95—42no—
182#95—82no—
183#95—82no—
184#95—82no—
185#95—82no—
186#95—82no—
187#95—82no—
188C24C22×D411yescharacteristic
189C2×Q8—48no—
190C2×Q8—88no—
191C2×Q8—88no—
192C2×Q8—84no—
193C2×Q8—84no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
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—24no—
211C22×C4—24no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
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—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×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C22⋊Q812yes—
249C22×C4C22⋊Q812yes—
250C22×C4#12412yes—
251C22×C4#12412yes—
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4112yes—
255C22×C4C2×#4112yes—
256C22×C4H(5,𝔽2)11yescharacteristic
257C22×C4H(5,𝔽2)11yescharacteristic
258C22×C4H(5,𝔽2)12yes—
259C22×C42−1+411yescharacteristic
260C22×C42−1+411yescharacteristic
261C22×C42−1+412yes—
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—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—48no—
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—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—88no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—24no—
301C42—24no—
302C42—42no—
303C42—42no—
304C23C2×C22⋊Q811yescharacteristic
305C23#52511yescharacteristic
306C23C2×#12411yescharacteristic
307C23#53411yescharacteristic
308C23#53511yescharacteristic
309C23#53611yescharacteristic
310C23#53611yescharacteristic
311C23#57612yes—
312C23#57712yes—
313C23#57712yes—
314C23#57812yes—
315Q8—816no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4#53911yescharacteristic
375C2×C4#53911yescharacteristic
376C2×C4#54112yes—
377C2×C4#54112yes—
378C2×C4#54212yes—
379C2×C4#54212yes—
380C2×C4#54411yescharacteristic
381C2×C4#54411yescharacteristic
382C2×C4#55112yes—
383C2×C4#55112yes—
384C2×C4#55412yes—
385C2×C4#55612yes—
386C2×C4#55712yes—
387C2×C4#55912yes—
388C2×C4#56212yes—
389C2×C4#56212yes—
390C8—82no—
391C8—82no—
392C22#256111yescharacteristic
393C22#256811yescharacteristic
394C22#258011yescharacteristic
395C22#262711yescharacteristic
396C22#263411yescharacteristic
397C22#264011yescharacteristic
398C22#281611yescharacteristic
399C22#308312yes—
400C22#308412yes—
401C22#311812yes—
402C22#312112yes—
403C22#313612yes—
404C22#313712yes—
405C22#319911yescharacteristic
406C22#320011yescharacteristic
407C22#320011yescharacteristic
408C22#320211yescharacteristic
409C22#322412yes—
410C22#322512yes—
411C22#324112yes—
412C22#324212yes—
413C22#324312yes—
414C22#324412yes—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—81no—
420C4—81no—
421C4—84no—
422C4—84no—
423C4—162no—
424C4—162no—
425C4#275512yes—
426C4#275512yes—
427C4#279812yes—
428C4#279812yes—
429C2#1689011yescharacteristic
430C2#2911611yescharacteristic
431C2#2911811yescharacteristic
432C2#2922812yes—
433C2#2926712yes—
434C2#2997611yescharacteristic
435C2#2997711yescharacteristic
436C2#3010012yes—
437C2#3021112yes—
438C2#3277911yescharacteristic
439C2#3278011yescharacteristic
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
63. #382Q8semidirect

Maximal subgroups

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

subgroupindex
#142822
#142772
#113842
#113722
#113582
#112932
#112922
#112672
#107962