Finite Group Databasebeta

← #468311

Subgroups — #468311

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic
2C2×#2085C212yesmaximal
3#14276C211yesmaximal, characteristic
4#11328C212yesmaximal
5#11327C212yesmaximal
6#11324C211yesmaximal, characteristic
7#11323C211yesmaximal, characteristic
8#11268C211yesmaximal, characteristic
9#11268C211yesmaximal, characteristic
10#11207C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14#2647—28no—
15C2×#400—22no—
16C2×#398—22no—
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#2085—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1441—22no—
38#1441—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#107—24no—
59C2×#107—42no—
60C2×#107—42no—
61C2×#106—24no—
62C2×#106—42no—
63C2×#106—42no—
64#400—216no—
65#398—216no—
66#389—24no—
67#389—24no—
68#389—24no—
69#389—42no—
70#389—42no—
71#389D418yes—
72#384—24no—
73#384—22no—
74#384—22no—
75#382—24no—
76#382—24no—
77#376—24no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D412yes—
87C2×#95D412yes—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—42no—
93#340—42no—
94#336—24no—
95#336—42no—
96#336Q814yes—
97#336Q814yes—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
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×C8C2×Q814yes—
122C22×C8C2×Q814yes—
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—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4#4112yes—
166C2×C4⋊C4#4112yes—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42C2×D414yes—
177C2×C42C2×Q812yes—
178C2×C42C2×Q812yes—
179#107—48no—
180#106—48no—
181#95—28no—
182#95—28no—
183#95—28no—
184#95—24no—
185#95—24no—
186#95—42no—
187#95—42no—
188#95—44no—
189#95—82no—
190#95—82no—
191C24C22×D411yescharacteristic
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—24no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
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—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4#12014yes—
249C22×C4C4⋊Q814yes—
250C22×C4C22×D412yes—
251C22×C4C22×Q811yescharacteristic
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4H(5,𝔽2)11yescharacteristic
256C22×C4H(5,𝔽2)12yes—
257C22×C42−1+411yescharacteristic
258C22×C42−1+412yes—
259C22×C42−1+412yes—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
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—
292C4⋊C4—88no—
293C4⋊C4—88no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C42—24no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—42no—
301C42—42no—
302C23C2×#12011yescharacteristic
303C23#52511yescharacteristic
304C23C2×C4⋊Q811yescharacteristic
305C23#53511yescharacteristic
306C23#53611yescharacteristic
307C23#53611yescharacteristic
308C23#53711yescharacteristic
309C23#57612yes—
310C23#57712yes—
311C23#57712yes—
312C23#57812yes—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4#53512yes—
372C2×C4#53712yes—
373C2×C4#53912yes—
374C2×C4#53912yes—
375C2×C4#54412yes—
376C2×C4#54412yes—
377C2×C4#54712yes—
378C2×C4#54712yes—
379C2×C4D4×Q812yes—
380C2×C4D4×Q812yes—
381C2×C4#55411yescharacteristic
382C2×C4#55411yescharacteristic
383C2×C4#55711yescharacteristic
384C2×C4#55711yescharacteristic
385C2×C4#56312yes—
386C2×C4#56412yes—
387C8—44no—
388C8—44no—
389C22#240611yescharacteristic
390C22#243411yescharacteristic
391C22#258411yescharacteristic
392C22#261511yescharacteristic
393C22#263311yescharacteristic
394C22#266011yescharacteristic
395C22#282211yescharacteristic
396C22#301212yes—
397C22#301312yes—
398C22#311812yes—
399C22#312112yes—
400C22#316312yes—
401C22#316412yes—
402C22#322011yescharacteristic
403C22#322111yescharacteristic
404C22#322211yescharacteristic
405C22#322311yescharacteristic
406C22#324112yes—
407C22#324212yes—
408C22#324312yes—
409C22#324412yes—
410C22#325712yes—
411C22#325812yes—
412C4—44no—
413C4—44no—
414C4—84no—
415C4—82no—
416C4—82no—
417C4—81no—
418C4—84no—
419C4—81no—
420C4—161no—
421C4—161no—
422C4#272012yes—
423C4#272012yes—
424C4#272612yes—
425C4#272612yes—
426C2#1660711yescharacteristic
427C2#2407512yes—
428C2#2582212yes—
429C2#2932512yes—
430C2#2967511yescharacteristic
431C2#2967611yescharacteristic
432C2#3008411yescharacteristic
433C2#3008611yescharacteristic
434C2#3031311yescharacteristic
435C2#3031411yescharacteristic
436C2#3285412yes—
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×#26472
C2×#20852
#142762
#113282
#113272
#113242
#113232
#112682
#112072
#107962