Finite Group Databasebeta

← #474681

Subgroups — #474681

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2058C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3#14282C211yesmaximal, characteristic
4#14276C211yesmaximal, characteristic
5#11325C212yesmaximal
6#11267C212yesmaximal
7#11257C212yesmaximal
8#11256C212yesmaximal
9#11208C211yesmaximal, characteristic
10#11208C211yesmaximal, characteristic
11#10796C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C2×#389C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#2058—24no—
22#2057—24no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1446—22no—
34#1445—22no—
35#1441—22no—
36#1441—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C23×Q8—21no—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—42no—
46C2×C4⋊Q8—42no—
47C2×C4⋊Q8D414yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#107—24no—
62C2×#107—42no—
63C2×#107—42no—
64C2×#106—24no—
65C2×#106—42no—
66C2×#106—42no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#384—22no—
72#384—22no—
73#382—42no—
74#382—42no—
75#376—42no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
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#336—22no—
96#336—22no—
97#336—42no—
98#336Q818yescomplemented
99C22×Q8—24no—
100C22×Q8—42no—
101C22×Q8—42no—
102C22×Q8—82no—
103C22×Q8—82no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2411yescharacteristic
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C2×Q814yes—
128C22×C8C2×Q814yes—
129C4⋊Q8—48no—
130C4⋊Q8—48no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4#4112yes—
172C2×C4⋊C4#4112yes—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42C2×D412yes—
183C2×C42C2×Q814yes—
184C2×C42#4112yes—
185#107—48no—
186#106—48no—
187#95—42no—
188#95—42no—
189#95—44no—
190#95—42no—
191#95—42no—
192#95—42no—
193#95—42no—
194#95—82no—
195#95—82no—
196#95—82no—
197#95—82no—
198#95—82no—
199#95—82no—
200C24C22×D411yescharacteristic
201C2×Q8—48no—
202C2×Q8—84no—
203C2×Q8—84no—
204C2×Q8—88no—
205C2×Q8—88no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
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—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
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#12112yes—
262C22×C4#12112yes—
263C22×C4C4⋊Q814yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×Q812yes—
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C22×C4H(5,𝔽2)12yes—
270C22×C4H(5,𝔽2)12yes—
271C22×C42−1+412yes—
272C22×C42−1+411yescharacteristic
273C22×C42−1+411yescharacteristic
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—48no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4—88no—
312C42—24no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—42no—
317C42—42no—
318C23C2×#12111yescharacteristic
319C23#52511yescharacteristic
320C23C2×C4⋊Q811yescharacteristic
321C23#53411yescharacteristic
322C23#53511yescharacteristic
323C23#53611yescharacteristic
324C23#53611yescharacteristic
325C23#57612yes—
326C23#57712yes—
327C23#57712yes—
328C23#57812yes—
329Q8—816no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4#53412yes—
389C2×C4#53812yes—
390C2×C4#54212yes—
391C2×C4#54212yes—
392C2×C4#54412yes—
393C2×C4#54412yes—
394C2×C4#54811yescharacteristic
395C2×C4#54811yescharacteristic
396C2×C4D4×Q812yes—
397C2×C4#55211yescharacteristic
398C2×C4#55211yescharacteristic
399C2×C4#55412yes—
400C2×C4#55412yes—
401C2×C4#55712yes—
402C2×C4#56312yes—
403C2×C4#56312yes—
404C8—44no—
405C8—44no—
406C22#253611yescharacteristic
407C22#254611yescharacteristic
408C22#258311yescharacteristic
409C22#261711yescharacteristic
410C22#263211yescharacteristic
411C22#266011yescharacteristic
412C22#281711yescharacteristic
413C22#304111yescharacteristic
414C22#304211yescharacteristic
415C22#304311yescharacteristic
416C22#304411yescharacteristic
417C22#311812yes—
418C22#312112yes—
419C22#316312yes—
420C22#316412yes—
421C22#319812yes—
422C22#320112yes—
423C22#322412yes—
424C22#322512yes—
425C22#324112yes—
426C22#324212yes—
427C22#324312yes—
428C22#324412yes—
429C4—42no—
430C4—42no—
431C4—44no—
432C4—82no—
433C4—84no—
434C4—82no—
435C4—82no—
436C4—161no—
437C4—161no—
438C4—162no—
439C4#272912yes—
440C4#272912yes—
441C4#277412yes—
442C4#277412yes—
443C2#1699311yescharacteristic
444C2#2821011yescharacteristic
445C2#2821311yescharacteristic
446C2#2869611yescharacteristic
447C2#2869711yescharacteristic
448C2#2930812yes—
449C2#2972711yescharacteristic
450C2#2972811yescharacteristic
451C2#3006812yes—
452C2#3031512yes—
453C2#3279312yes—
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
98. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
C2×#20572
#142822
#142762
#113252
#112672
#112572
#112562
#112082
#107962