Finite Group Databasebeta

← #409309

Subgroups — #409309

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2096C211yesmaximal, characteristic
2C2×#2074C211yesmaximal, characteristic
3#14348C211yesmaximal, characteristic
4#13786C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic
6#11358C211yesmaximal, characteristic
7#11257C212yesmaximal
8#11190C212yesmaximal
9#10908C212yesmaximal
10#10796C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×#376C2211yescharacteristic
22#2096—24no—
23#2074—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1442—22no—
34#1442—22no—
35#1436—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40#1434—22no—
41C23×Q8C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8—24no—
47C2×C4×Q8—42no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42—21no—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—42no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—42no—
69#391—22no—
70#391—22no—
71#391—22no—
72#391D414yes—
73#384—22no—
74#384—22no—
75#382—22no—
76#382—22no—
77C4×C4⋊C4—42no—
78#376—24no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D412yes—
90C2×#95D412yes—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#336—24no—
97#336—22no—
98#336—22no—
99#336—24no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8C2×D412yes—
108C22×Q8#4112yes—
109C22×Q8#4112yes—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8—22no—
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#4114yes—
135C4×Q8—48no—
136C4×Q8—48no—
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—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
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—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4#4112yes—
166C2×C4⋊C4#4112yes—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—82no—
181C2×C42C2×D412yes—
182C2×C42#4112yes—
183C2×C42#4114yes—
184#106—48no—
185#103—48no—
186#103—48no—
187#95—28no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—42no—
192#95—44no—
193#95—82no—
194#95—82no—
195#95—82no—
196#95—82no—
197#95—82no—
198#95—82no—
199C24C22×D411yescharacteristic
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8#12414yes—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
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—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
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—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#12114yes—
274C22×C4#12312yes—
275C22×C4#12312yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4C2×#4112yes—
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4H(5,𝔽2)12yes—
285C22×C42−1+412yes—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—88no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—48no—
321C42—42no—
322C42—42no—
323C42—84no—
324C23C2×#12111yescharacteristic
325C23C2×#12311yescharacteristic
326C23#52511yescharacteristic
327C23#52511yescharacteristic
328C23#53211yescharacteristic
329C23#53511yescharacteristic
330C23#53611yescharacteristic
331C23#57712yes—
332C23#57712yes—
333C23#57712yes—
334C23#57812yes—
335Q8—28no—
336Q8—44no—
337Q8—44no—
338Q8—44no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4C2×#12411yescharacteristic
400C2×C4#52911yescharacteristic
401C2×C4#52912yes—
402C2×C4#52912yes—
403C2×C4#52911yescharacteristic
404C2×C4#53211yescharacteristic
405C2×C4#53912yes—
406C2×C4#53912yes—
407C2×C4#54612yes—
408C2×C4#54712yes—
409C2×C4#55112yes—
410C2×C4#55112yes—
411C2×C4#55212yes—
412C2×C4#55212yes—
413C2×C4#55212yes—
414C2×C4#55912yes—
415C8—44no—
416C8—82no—
417C22#239511yescharacteristic
418C22#248211yescharacteristic
419C22#248611yescharacteristic
420C22#254411yescharacteristic
421C22#261111yescharacteristic
422C22#263311yescharacteristic
423C22#285011yescharacteristic
424C22#304312yes—
425C22#304612yes—
426C22#309712yes—
427C22#309912yes—
428C22#312112yes—
429C22#312112yes—
430C22#312512yes—
431C22#312512yes—
432C22#317412yes—
433C22#317812yes—
434C22#322111yescharacteristic
435C22#322111yescharacteristic
436C22#322211yescharacteristic
437C22#322311yescharacteristic
438C22#324212yes—
439C22#324412yes—
440C4—44no—
441C4—44no—
442C4—42no—
443C4—42no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—162no—
449C4—162no—
450C4#248812yes—
451C4#251712yes—
452C4#255312yes—
453C4#257512yes—
454C2#1501611yescharacteristic
455C2#2377212yes—
456C2#2759611yescharacteristic
457C2#2759711yescharacteristic
458C2#2765712yes—
459C2#2862311yescharacteristic
460C2#2862911yescharacteristic
461C2#2958912yes—
462C2#3008611yescharacteristic
463C2#3009011yescharacteristic
464C2#3303712yes—
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

Maximal subgroups

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

subgroupindex
C2×#20962
C2×#20742
#143482
#137862
#113712
#113582
#112572
#111902
#109082
#107962
#107892