Finite Group Databasebeta

← #415588

Subgroups — #415588

Order 512.

1192 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic
2C2×#1949C212yesmaximal
3#14276C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11328C212yesmaximal
7#11327C212yesmaximal
8#11324C211yesmaximal, characteristic
9#11323C211yesmaximal, characteristic
10#10849C212yesmaximal
11#10833C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16#2330—28no—
17C2×#389—22no—
18C2×#389C2212yes—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387—22no—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×#376C2211yescharacteristic
27#1949—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1441—22no—
38#1441—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#125—24no—
44C2×#125—24no—
45C2×#125—22no—
46C2×#125Q814yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42—22no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#107—24no—
64C2×#107—42no—
65C2×#107—42no—
66C2×#106—24no—
67C2×#106—42no—
68C2×#106—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8Q818yes—
72#389—216no—
73#389—24no—
74#389—24no—
75#389—42no—
76#389—42no—
77#387—216no—
78#384—24no—
79#384—22no—
80#384Q814yes—
81#382—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4D418yes—
84#376—24no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D412yes—
93C2×#95D412yes—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#336—22no—
101#336—22no—
102#336—42no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2411yescharacteristic
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—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8C2×Q814yes—
127C22×C8C2×Q814yes—
128#125—216no—
129#125—216no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—22no—
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—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×Q812yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C4⋊C4C2×Q814yes—
173C2×C4⋊C4#4114yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—28no—
181C2×C42—82no—
182C2×C42—82no—
183C2×C42C2×D414yes—
184C2×C42C2×Q812yes—
185C2×C42C2×Q814yes—
186C2×C42C2×Q812yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189#107—48no—
190#106—48no—
191C4⋊C8—216no—
192#95—28no—
193#95—28no—
194#95—24no—
195#95—24no—
196#95—42no—
197#95—42no—
198#95—44no—
199#95—82no—
200#95—82no—
201C24C22×D411yescharacteristic
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
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—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4#12014yes—
257C22×C4C22⋊Q814yes—
258C22×C4C22⋊Q814yes—
259C22×C4C4⋊Q814yes—
260C22×C4C22×D412yes—
261C22×C4C22×Q811yescharacteristic
262C22×C4C22×Q812yes—
263C22×C4C22×Q811yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4112yes—
266C22×C4C2×#4111yescharacteristic
267C22×C4H(5,𝔽2)11yescharacteristic
268C22×C4H(5,𝔽2)12yes—
269C22×C42−1+411yescharacteristic
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—232no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—88no—
301C4⋊C4—88no—
302C4⋊C4—88no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4C22⋊Q818yes—
306C42—24no—
307C42—24no—
308C42—42no—
309C42—42no—
310C42—88no—
311C42C22⋊Q818yes—
312C42C4⋊Q818yes—
313C23C2×#12011yescharacteristic
314C23C2×C22⋊Q811yescharacteristic
315C23C2×C22⋊Q811yescharacteristic
316C23#52511yescharacteristic
317C23C2×C4⋊Q811yescharacteristic
318C23#53511yescharacteristic
319C23#53711yescharacteristic
320C23#57612yes—
321C23#57712yes—
322C23#57712yes—
323C23#57812yes—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—28no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4#39518yes—
382C2×C4C2×C22⋊Q812yes—
383C2×C4C2×C22⋊Q812yes—
384C2×C4#52512yes—
385C2×C4C2×C4⋊Q812yes—
386C2×C4#53512yes—
387C2×C4#53712yes—
388C2×C4#55412yes—
389C2×C4#55412yes—
390C2×C4#55412yes—
391C2×C4#55711yescharacteristic
392C2×C4#55712yes—
393C2×C4#55711yescharacteristic
394C2×C4Q8211yescharacteristic
395C2×C4Q8212yes—
396C2×C4#55911yescharacteristic
397C2×C4#55912yes—
398C8—44no—
399C8—44no—
400C22C2×#39511yescharacteristic
401C22#256711yescharacteristic
402C22#258011yescharacteristic
403C22#258211yescharacteristic
404C22#258411yescharacteristic
405C22#261511yescharacteristic
406C22#284811yescharacteristic
407C22#301212yes—
408C22#301312yes—
409C22#308112yes—
410C22#308212yes—
411C22#308312yes—
412C22#308412yes—
413C22#311912yes—
414C22#312012yes—
415C22#316312yes—
416C22#316412yes—
417C22#322011yescharacteristic
418C22#322111yescharacteristic
419C22#322211yescharacteristic
420C22#322311yescharacteristic
421C22#325712yes—
422C22#325812yes—
423C4—28no—
424C4—44no—
425C4—48no—
426C4—82no—
427C4—82no—
428C4—81no—
429C4—84no—
430C4—81no—
431C4—161no—
432C4—161no—
433C4#256712yes—
434C4#258012yes—
435C4#258212yes—
436C4#258412yes—
437C2#1520211yescharacteristic
438C2#2355712yes—
439C2#2921511yescharacteristic
440C2#2921611yescharacteristic
441C2#2926511yescharacteristic
442C2#2926611yescharacteristic
443C2#2929212yes—
444C2#2932412yes—
445C2#2967511yescharacteristic
446C2#2967611yescharacteristic
447C2#3296812yes—
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×#23302
C2×#19492
#142762
#113852
#113282
#113272
#113242
#113232
#108492
#108332