Finite Group Databasebeta

← #416763

Subgroups — #416763

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1939C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic
3#13955C211yesmaximal, characteristic
4#13954C211yesmaximal, characteristic
5#11293C212yesmaximal
6#11292C212yesmaximal
7#11228C212yesmaximal
8#11227C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10850C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4×C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#382C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1939—24no—
23#1835—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1444—21no—
36#1444—21no—
37#1442—21no—
38#1442—21no—
39C4⋊#95—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1434—22no—
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42—21no—
57C22×C42—21no—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—24no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—42no—
65C2×C4⋊C8—42no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8D414yes—
69#389—24no—
70#389—24no—
71#389—42no—
72#389—42no—
73#382—22no—
74#382—22no—
75#382—22no—
76#382—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4Q814yes—
82C4×C4⋊C4Q814yes—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95D412yes—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#336—24no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
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—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4C2×Q814yes—
170C2×C4⋊C4C2×Q814yes—
171C2×C4⋊C4#4114yes—
172C2×C4⋊C4#4114yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—81no—
182C2×C42—81no—
183C2×C42—81no—
184C2×C42—81no—
185C2×C42C2×D412yes—
186C2×C42C2×Q812yes—
187C2×C42C2×Q812yes—
188C2×C42#4112yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C4⋊C8—48no—
192C4⋊C8—44no—
193C4×C8—28no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—42no—
201#95—42no—
202#95—42no—
203#95—42no—
204#95—44no—
205#95—82no—
206#95—82no—
207C24C22×D411yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
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—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4#12112yes—
262C22×C4#12112yes—
263C22×C4C22⋊Q812yes—
264C22×C4C22⋊Q812yes—
265C22×C4C22⋊Q812yes—
266C22×C4C22⋊Q812yes—
267C22×C4#13514yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×Q811yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4C2×#4112yes—
276C22×C4H(5,𝔽2)12yes—
277C22×C42−1+412yes—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—88no—
317C4⋊C4—88no—
318C4⋊C4#12518yes—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—84no—
324C42—84no—
325C42#12314yes—
326C42#12514yes—
327C42#12514yes—
328C42C4⋊Q814yes—
329C23C2×#12111yescharacteristic
330C23C2×C22⋊Q811yescharacteristic
331C23C2×C22⋊Q811yescharacteristic
332C23#54612yes—
333C23#54812yes—
334C23C2×#13512yes—
335C23#57512yes—
336C23#57611yescharacteristic
337C23#57711yescharacteristic
338C23#57711yescharacteristic
339C23#57811yescharacteristic
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4#39814yes—
399C2×C4#39814yes—
400C2×C4C2×#12311yescharacteristic
401C2×C4C2×#12511yescharacteristic
402C2×C4C2×#12511yescharacteristic
403C2×C4C2×#12512yes—
404C2×C4C2×C4⋊Q811yescharacteristic
405C2×C4#53211yescharacteristic
406C2×C4#53211yescharacteristic
407C2×C4#53312yes—
408C2×C4#53511yescharacteristic
409C2×C4#53611yescharacteristic
410C2×C4#54412yes—
411C2×C4#54412yes—
412C2×C4#55212yes—
413C2×C4#55312yes—
414C2×C4#55512yes—
415C2×C4#55612yes—
416C2×C4#55612yes—
417C2×C4#55912yes—
418C8—42no—
419C8—42no—
420C8—82no—
421C22C2×#39811yescharacteristic
422C22#243912yes—
423C22#244212yes—
424C22#254411yescharacteristic
425C22#254511yescharacteristic
426C22#305011yescharacteristic
427C22#305411yescharacteristic
428C22#306011yescharacteristic
429C22#306111yescharacteristic
430C22#307612yes—
431C22#307612yes—
432C22#308111yescharacteristic
433C22#308111yescharacteristic
434C22#308211yescharacteristic
435C22#308211yescharacteristic
436C22#329712yes—
437C22#330012yes—
438C22#332312yes—
439C22#332612yes—
440C22#336012yes—
441C22#336512yes—
442C22#337012yes—
443C22#337712yes—
444C4—24no—
445C4—24no—
446C4—48no—
447C4—44no—
448C4—84no—
449C4—82no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4#249911yescharacteristic
455C4#250011yescharacteristic
456C4#257012yes—
457C4#257112yes—
458C4#266011yescharacteristic
459C4#285911yescharacteristic
460C2#1523111yescharacteristic
461C2#2366611yescharacteristic
462C2#2366911yescharacteristic
463C2#2602812yes—
464C2#2603112yes—
465C2#2611412yes—
466C2#2611712yes—
467C2#2860111yescharacteristic
468C2#2860411yescharacteristic
469C2#2864211yescharacteristic
470C2#2864311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#19392
C2×#18352
#139552
#139542
#112932
#112922
#112282
#112272
#108502
C4×#3402