Finite Group Databasebeta

← #465219

Subgroups — #465219

Order 512.

912 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2085C212yesmaximal
2C2×#1947C211yesmaximal, characteristic
3C2×#1919C211yesmaximal, characteristic
4#15242C211yesmaximal, characteristic
5#13933C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11268C212yesmaximal
9#11255C211yesmaximal, characteristic
10#11227C211yesmaximal, characteristic
11#11219C212yesmaximal
12C22×#107C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×#382C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20#2085—28no—
21#1947—24no—
22#1919—24no—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1445—22no—
35#1442—22no—
36#1441—21no—
37#1441—21no—
38C4⋊#95—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—24no—
61C2×#107—24no—
62C2×#107—24no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106D414yes—
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—42no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4D414yes—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95D411yescharacteristic
90C2×#95D411yescharacteristic
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—24no—
99#340—42no—
100#336—24no—
101#336—24no—
102#336—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2411yescharacteristic
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8C2×D412yes—
130C22×C8#4114yes—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
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—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4#4112yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—82no—
183C2×C42C2×D412yes—
184C2×C42#4112yes—
185#107—48no—
186#106—28no—
187C4⋊C8—28no—
188#95—24no—
189#95—22no—
190#95—22no—
191#95—42no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—42no—
196#95—42no—
197#95—81no—
198#95—82no—
199#95—82no—
200#95—82no—
201#95—82no—
202C24C22×D411yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
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—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—42no—
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—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4#12112yes—
260C22×C4#12112yes—
261C22×C4#13512yes—
262C22×C4#13712yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4112yes—
267C22×C4C2×#4111yescharacteristic
268C22×C4H(5,𝔽2)11yescharacteristic
269C22×C42−1+412yes—
270C22×C42−1+412yes—
271C22×C42−1+412yes—
272C22×C42−1+411yescharacteristic
273C2×C8—22no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—88no—
314C4⋊C4—84no—
315C4⋊C4#12114yes—
316C42—22no—
317C42—22no—
318C42—24no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—84no—
324C23C2×#12111yescharacteristic
325C23#53612yes—
326C23D4211yescharacteristic
327C23#54712yes—
328C23#54811yescharacteristic
329C23C2×#13511yescharacteristic
330C23C2×#13711yescharacteristic
331C23#57512yes—
332C23#57612yes—
333C23#57711yescharacteristic
334C23#57811yescharacteristic
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
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—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4#46414yes—
395C2×C4C2×#12111yescharacteristic
396C2×C4#52511yescharacteristic
397C2×C4#53611yescharacteristic
398C2×C4#53611yescharacteristic
399C2×C4#53611yescharacteristic
400C2×C4#53612yes—
401C2×C4#53711yescharacteristic
402C2×C4#54112yes—
403C2×C4#54112yes—
404C2×C4#54112yes—
405C2×C4#54212yes—
406C2×C4#54711yescharacteristic
407C2×C4#54812yes—
408C2×C4#55212yes—
409C2×C4#55212yes—
410C2×C4#55311yescharacteristic
411C2×C4#56312yes—
412C2×C4#56412yes—
413C8—42no—
414C8—44no—
415C8—81no—
416C22#241311yescharacteristic
417C22#241611yescharacteristic
418C22#262111yescharacteristic
419C22#267412yes—
420C22#271912yes—
421C22C2×#46411yescharacteristic
422C22#304011yescharacteristic
423C22#304111yescharacteristic
424C22#304611yescharacteristic
425C22#323312yes—
426C22#323712yes—
427C22#324112yes—
428C22#324312yes—
429C22#326812yes—
430C22#327611yescharacteristic
431C22#328411yescharacteristic
432C22#333912yes—
433C22#334612yes—
434C22#334912yes—
435C22#335112yes—
436C22#336411yescharacteristic
437C22#336911yescharacteristic
438C22#337012yes—
439C4—24no—
440C4—42no—
441C4—44no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—161no—
447C4—162no—
448C4—162no—
449C4#253611yescharacteristic
450C4#262111yescharacteristic
451C4#263711yescharacteristic
452C4#271912yes—
453C4#276612yes—
454C4#282211yescharacteristic
455C2#1651711yescharacteristic
456C2#2458311yescharacteristic
457C2#2459311yescharacteristic
458C2#2472911yescharacteristic
459C2#2474911yescharacteristic
460C2#2986711yescharacteristic
461C2#2987211yescharacteristic
462C2#3078812yes—
463C2#3079012yes—
464C2#3232412yes—
465C2#3233412yes—
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×#19472
C2×#19192
#152422
#139332
#113712
#113152
#112682
#112552
#112272
#112192