Finite Group Databasebeta

← #401487

Subgroups — #401487

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal
2C2×#1947C211yesmaximal, characteristic
3C2×#1919C211yesmaximal, characteristic
4#15242C211yesmaximal, characteristic
5#13818C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11322C211yesmaximal, characteristic
8#11227C211yesmaximal, characteristic
9#11227C212yesmaximal
10#11208C212yesmaximal
11#10907C212yesmaximal
12C22×#107C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C2×#389C2211yescharacteristic
17C2×#389C2212yes—
18C2×#384C2211yescharacteristic
19C2×#382C2212yes—
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22#1949—28no—
23#1947—24no—
24#1919—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1445—22no—
36#1441—21no—
37#1441—21no—
38#1436—22no—
39C4⋊#95—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4×#95—21no—
43C4×#95C2211yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—22no—
59C22×C42C2311yescharacteristic
60C22×C42C2312yes—
61C2×#107—24no—
62C2×#107—24no—
63C2×#107—24no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106D414yes—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—24no—
72#389—24no—
73#389—22no—
74#389—22no—
75#389Q818yes—
76#384—22no—
77#384—22no—
78#382—24no—
79#382—42no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—42no—
83C4×C4⋊C4D414yes—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D411yescharacteristic
95C2×#95D411yescharacteristic
96C2×#95D412yes—
97#340—24no—
98#340—24no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#340—42no—
105#336—22no—
106#336—24no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8C2×D412yes—
134C22×C8#4114yes—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×Q814yes—
175C2×C4⋊C4C2×Q814yes—
176C2×C4⋊C4#4112yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—24no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42—82no—
187C2×C42—82no—
188C2×C42C2×D412yes—
189C2×C42C2×Q814yes—
190C2×C42#4112yes—
191C2×C42#4114yes—
192#107—48no—
193#106—28no—
194C4⋊C8—28no—
195C4⋊C8—216no—
196#95—24no—
197#95—28no—
198#95—22no—
199#95—22no—
200#95—42no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—42no—
205#95—42no—
206#95—82no—
207#95—82no—
208#95—81no—
209C24C22×D411yescharacteristic
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
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—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4#12112yes—
265C22×C4#12112yes—
266C22×C4C22⋊Q814yes—
267C22×C4C22⋊Q814yes—
268C22×C4#13512yes—
269C22×C4#13712yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×Q812yes—
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4C2×#4112yes—
276C22×C4C2×#4111yescharacteristic
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C42−1+412yes—
279C22×C42−1+411yescharacteristic
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—88no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4#12114yes—
322C4⋊C4C4⋊Q814yes—
323C42—22no—
324C42—24no—
325C42—22no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—84no—
331C42—88no—
332C23C2×#12111yescharacteristic
333C23C2×C22⋊Q812yes—
334C23D4211yescharacteristic
335C23#54712yes—
336C23#54811yescharacteristic
337C23C2×#13511yescharacteristic
338C23C2×#13711yescharacteristic
339C23#57512yes—
340C23#57612yes—
341C23#57711yescharacteristic
342C23#57811yescharacteristic
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4#46414yes—
403C2×C4C2×#12111yescharacteristic
404C2×C4#52511yescharacteristic
405C2×C4#52512yes—
406C2×C4#52912yes—
407C2×C4C2×C4⋊Q811yescharacteristic
408C2×C4#53211yescharacteristic
409C2×C4#53312yes—
410C2×C4#53611yescharacteristic
411C2×C4#53711yescharacteristic
412C2×C4#53912yes—
413C2×C4#54112yes—
414C2×C4#54711yescharacteristic
415C2×C4D4×Q812yes—
416C2×C4#55212yes—
417C2×C4#55311yescharacteristic
418C2×C4#55412yes—
419C2×C4#55712yes—
420C2×C4#55712yes—
421C8—42no—
422C8—44no—
423C8—81no—
424C22#241311yescharacteristic
425C22#241611yescharacteristic
426C22#244912yes—
427C22#245312yes—
428C22#285711yescharacteristic
429C22C2×#46411yescharacteristic
430C22#304011yescharacteristic
431C22#304111yescharacteristic
432C22#304611yescharacteristic
433C22#307612yes—
434C22#307812yes—
435C22#308112yes—
436C22#308312yes—
437C22#326812yes—
438C22#327611yescharacteristic
439C22#328411yescharacteristic
440C22#333912yes—
441C22#334612yes—
442C22#334812yes—
443C22#335212yes—
444C22#336411yescharacteristic
445C22#336911yescharacteristic
446C22#337012yes—
447C4—24no—
448C4—42no—
449C4—44no—
450C4—44no—
451C4—44no—
452C4—44no—
453C4—82no—
454C4—84no—
455C4—161no—
456C4—162no—
457C4#239211yescharacteristic
458C4#246212yes—
459C4#254611yescharacteristic
460C4#258411yescharacteristic
461C4#258512yes—
462C4#266411yescharacteristic
463C2#1481811yescharacteristic
464C2#2458311yescharacteristic
465C2#2459511yescharacteristic
466C2#2472911yescharacteristic
467C2#2474911yescharacteristic
468C2#2634512yes—
469C2#2636212yes—
470C2#2651612yes—
471C2#2651912yes—
472C2#3320211yescharacteristic
473C2#3320311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#19492
C2×#19472
C2×#19192
#152422
#138182
#113712
#113222
#112272
#112082
#109072