Finite Group Databasebeta

← #465218

Subgroups — #465218

Order 512.

926 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2085C212yesmaximal
2C2×#1948C211yesmaximal, characteristic
3C2×#1918C211yesmaximal, characteristic
4#15242C211yesmaximal, characteristic
5#13918C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11316C212yesmaximal
8#11268C212yesmaximal
9#11255C211yesmaximal, characteristic
10#11227C211yesmaximal, characteristic
11#11220C212yesmaximal
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×#382—21no—
19C2×#382C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21#2085—28no—
22#1948—24no—
23#1918—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1445—22no—
36#1442—22no—
37#1441—21no—
38#1441—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43C4×#95C2211yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#107—22no—
61C2×#107—22no—
62C2×#107D414yes—
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—24no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—22no—
68#389—24no—
69#389—22no—
70#389—22no—
71#389—42no—
72#384—22no—
73#384—22no—
74#382—24no—
75#382—44no—
76#382—42no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4D414yes—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95D411yescharacteristic
91C2×#95D411yescharacteristic
92C2×#95D412yes—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—24no—
100#340—42no—
101#336—24no—
102#336—24no—
103#336—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8C2×D412yes—
131C22×C8#4114yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
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—44no—
160C2×C4⋊C4—44no—
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⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4#4112yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42C2×D412yes—
186C2×C42#4112yes—
187#107—28no—
188#106—48no—
189C4⋊C8—28no—
190#95—24no—
191#95—22no—
192#95—22no—
193#95—42no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—42no—
200#95—42no—
201#95—81no—
202#95—82no—
203#95—82no—
204C24C22×D411yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
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—22no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—41no—
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#12112yes—
264C22×C4#12112yes—
265C22×C4#13512yes—
266C22×C4AGL(1,ℤ/8)12yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4111yescharacteristic
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C42−1+412yes—
274C22×C42−1+412yes—
275C22×C42−1+412yes—
276C22×C42−1+411yescharacteristic
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
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—84no—
316C4⋊C4—84no—
317C4⋊C4—88no—
318C4⋊C4—84no—
319C4⋊C4#12114yes—
320C42—22no—
321C42—22no—
322C42—24no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—48no—
328C23C2×#12111yescharacteristic
329C23#53612yes—
330C23D4211yescharacteristic
331C23#54712yes—
332C23#54811yescharacteristic
333C23C2×#13511yescharacteristic
334C23C2×AGL(1,ℤ/8)11yescharacteristic
335C23#57512yes—
336C23#57712yes—
337C23#57711yescharacteristic
338C23#57811yescharacteristic
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
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—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—81no—
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—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4#46314yes—
401C2×C4C2×#12111yescharacteristic
402C2×C4#52511yescharacteristic
403C2×C4#53611yescharacteristic
404C2×C4#53611yescharacteristic
405C2×C4#53611yescharacteristic
406C2×C4#53612yes—
407C2×C4#53711yescharacteristic
408C2×C4#54112yes—
409C2×C4#54112yes—
410C2×C4#54112yes—
411C2×C4#54212yes—
412C2×C4#54711yescharacteristic
413C2×C4#54812yes—
414C2×C4#55212yes—
415C2×C4#55212yes—
416C2×C4#55311yescharacteristic
417C2×C4#56312yes—
418C2×C4#56412yes—
419C8—42no—
420C8—44no—
421C8—81no—
422C22#241311yescharacteristic
423C22#241611yescharacteristic
424C22#262111yescharacteristic
425C22#267412yes—
426C22#271912yes—
427C22C2×#46311yescharacteristic
428C22#304011yescharacteristic
429C22#304311yescharacteristic
430C22#304611yescharacteristic
431C22#323312yes—
432C22#323512yes—
433C22#324212yes—
434C22#324412yes—
435C22#327112yes—
436C22#327211yescharacteristic
437C22#328311yescharacteristic
438C22#333712yes—
439C22#334512yes—
440C22#334812yes—
441C22#335212yes—
442C22#336011yescharacteristic
443C22#336711yescharacteristic
444C22#337112yes—
445C4—24no—
446C4—42no—
447C4—44no—
448C4—82no—
449C4—82no—
450C4—82no—
451C4—82no—
452C4—161no—
453C4—162no—
454C4—162no—
455C4#253611yescharacteristic
456C4#262111yescharacteristic
457C4#263711yescharacteristic
458C4#271912yes—
459C4#276612yes—
460C4#282211yescharacteristic
461C2#1651711yescharacteristic
462C2#2458111yescharacteristic
463C2#2459511yescharacteristic
464C2#2472711yescharacteristic
465C2#2474711yescharacteristic
466C2#2986511yescharacteristic
467C2#2987311yescharacteristic
468C2#3078412yes—
469C2#3079412yes—
470C2#3232212yes—
471C2#3233612yes—
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×#20852
C2×#19482
C2×#19182
#152422
#139182
#113712
#113162
#112682
#112552
#112272
#112202