Finite Group Databasebeta

← #381950

Subgroups — #381950

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2074C211yesmaximal, characteristic
2C2×#1863C212yesmaximal
3#14277C211yesmaximal, characteristic
4#13786C211yesmaximal, characteristic
5#11358C211yesmaximal, characteristic
6#11324C211yesmaximal, characteristic
7#11256C212yesmaximal
8#11190C212yesmaximal
9#11050C211yesmaximal, characteristic
10#10884C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#2074C418yescomplemented
24#1863—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34#1444—22no—
35#1442—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—22no—
41C23×Q8C2311yescharacteristic
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—22no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42—21no—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#107—24no—
62C2×#107—42no—
63C2×#107—42no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—42no—
68C2×#103—42no—
69C2×#103C2×C418yes—
70#391—22no—
71#391—22no—
72#391—22no—
73#391C2×C414yes—
74#389—24no—
75#389—24no—
76#384—42no—
77#382—22no—
78#382C2×C414yes—
79C4×C4⋊C4—42no—
80#376—42no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340C2×C418yes—
97#336—24no—
98#336—22no—
99#336—42no—
100#336C2×C414yes—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8C22×C412yes—
109C22×Q8C2×D412yes—
110C22×Q8#4112yes—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
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×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8C22×C414yes—
136C4×Q8—48no—
137C4×Q8—48no—
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—24no—
149C2×C4⋊C4—21no—
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—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C414yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4#4112yes—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—82no—
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188#107—48no—
189#103—216no—
190#103—48no—
191#95—28no—
192#95—28no—
193#95—28no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—44no—
198#95—82no—
199#95—82no—
200#95—82no—
201#95—82no—
202C24C22×D411yescharacteristic
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8C4×D414yes—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C4×D414yes—
277C22×C4#12112yes—
278C22×C4#12112yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4H(5,𝔽2)12yes—
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+412yes—
288C22×C42−1+412yes—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—48no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—88no—
319C4⋊C4—88no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—24no—
324C42—48no—
325C42—42no—
326C42—42no—
327C42—84no—
328C23C2×C4×D411yescharacteristic
329C23C2×#12111yescharacteristic
330C23#52511yescharacteristic
331C23#53411yescharacteristic
332C23#53511yescharacteristic
333C23#53611yescharacteristic
334C23#53611yescharacteristic
335C23#57612yes—
336C23#57712yes—
337C23#57812yes—
338C23#57812yes—
339Q8—28no—
340Q8—44no—
341Q8—44no—
342Q8—44no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4C2×C4×D411yescharacteristic
404C2×C4C4×#4111yescharacteristic
405C2×C4#52012yes—
406C2×C4#52012yes—
407C2×C4#52911yescharacteristic
408C2×C4#53211yescharacteristic
409C2×C4#53412yes—
410C2×C4#53612yes—
411C2×C4#53812yes—
412C2×C4#53912yes—
413C2×C4#53912yes—
414C2×C4#53912yes—
415C2×C4#53912yes—
416C2×C4#54112yes—
417C2×C4#54212yes—
418C2×C4#54212yes—
419C8—44no—
420C8—82no—
421C22C4×#12111yescharacteristic
422C22#232211yescharacteristic
423C22#232611yescharacteristic
424C22#261711yescharacteristic
425C22#262111yescharacteristic
426C22#263311yescharacteristic
427C22#263811yescharacteristic
428C22#294712yes—
429C22#295112yes—
430C22#304412yes—
431C22#304512yes—
432C22#312112yes—
433C22#312412yes—
434C22#319812yes—
435C22#320212yes—
436C22#322011yescharacteristic
437C22#322111yescharacteristic
438C22#322311yescharacteristic
439C22#322311yescharacteristic
440C22#324212yes—
441C22#324212yes—
442C22#324312yes—
443C22#324312yes—
444C4—42no—
445C4—42no—
446C4—44no—
447C4—44no—
448C4—84no—
449C4—82no—
450C4—82no—
451C4—82no—
452C4—82no—
453C4—162no—
454C4#232212yes—
455C4#233412yes—
456C4#265212yes—
457C4#266312yes—
458C2#1431311yescharacteristic
459C2#2151212yes—
460C2#2209412yes—
461C2#2222811yescharacteristic
462C2#2223111yescharacteristic
463C2#2972411yescharacteristic
464C2#2972511yescharacteristic
465C2#2987512yes—
466C2#3008611yescharacteristic
467C2#3008911yescharacteristic
468C2#3018312yes—
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
23. #2074C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20742
C2×#18632
#142772
#137862
#113582
#113242
#112562
#111902
#110502
#108842
#107962