Finite Group Databasebeta

← #413566

Subgroups — #413566

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2084C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#1949C212yesmaximal
4#14277C211yesmaximal, characteristic
5#13779C211yesmaximal, characteristic
6#11257C212yesmaximal
7#11256C212yesmaximal
8#11208C211yesmaximal, characteristic
9#11208C211yesmaximal, characteristic
10#10849C212yesmaximal
11#10833C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C2×#389C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×#376C2211yescharacteristic
23#2084—24no—
24#2083—24no—
25#1949—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1444—22no—
36#1442—22no—
37#1436—22no—
38#1436—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×#107—24no—
60C2×#107—24no—
61C2×#107—42no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—42no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8Q818yes—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#384—22no—
73#384—24no—
74#384D414yes—
75#382—24no—
76#382—22no—
77#382D414yes—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—24no—
80#376—24no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D412yes—
92C2×#95D412yes—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#336—22no—
100#336—22no—
101#336—42no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2411yescharacteristic
113C22×C8—22no—
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×C8C2×Q814yes—
126C22×C8C2×Q814yes—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
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⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4#4114yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—44no—
177C2×C42—82no—
178C2×C42—82no—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×Q814yes—
182C2×C42#4112yes—
183C2×C42#4114yes—
184C2×C42#4112yes—
185#107—48no—
186#106—48no—
187C4⋊C8—216no—
188#95—24no—
189#95—24no—
190#95—28no—
191#95—24no—
192#95—24no—
193#95—44no—
194#95—42no—
195#95—42no—
196#95—82no—
197#95—82no—
198#95—82no—
199#95—82no—
200C24C22×D411yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
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—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4#12112yes—
255C22×C4#12112yes—
256C22×C4#12114yes—
257C22×C4#12314yes—
258C22×C4C4⋊Q814yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×Q812yes—
262C22×C4C2×#4112yes—
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4112yes—
266C22×C4H(5,𝔽2)11yescharacteristic
267C22×C42−1+411yescharacteristic
268C22×C42−1+412yes—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—88no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C42—24no—
309C42—24no—
310C42—42no—
311C42—42no—
312C42—88no—
313C42C22⋊Q818yes—
314C42#12518yes—
315C23C2×#12111yescharacteristic
316C23C2×#12111yescharacteristic
317C23C2×#12311yescharacteristic
318C23#52511yescharacteristic
319C23C2×C4⋊Q811yescharacteristic
320C23#53611yescharacteristic
321C23#53711yescharacteristic
322C23#57612yes—
323C23#57712yes—
324C23#57712yes—
325C23#57812yes—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4#39818yes—
384C2×C4C2×C22⋊Q812yes—
385C2×C4C2×#12512yes—
386C2×C4#52912yes—
387C2×C4#53212yes—
388C2×C4#54112yes—
389C2×C4#54212yes—
390C2×C4D4211yescharacteristic
391C2×C4#54711yescharacteristic
392C2×C4#54711yescharacteristic
393C2×C4#54812yes—
394C2×C4D4×Q812yes—
395C2×C4#55212yes—
396C2×C4#55412yes—
397C2×C4#55511yescharacteristic
398C2×C4#55712yes—
399C2×C4#55712yes—
400C8—44no—
401C8—44no—
402C22C2×#39811yescharacteristic
403C22#253611yescharacteristic
404C22#253711yescharacteristic
405C22#254611yescharacteristic
406C22#258411yescharacteristic
407C22#262511yescharacteristic
408C22#285411yescharacteristic
409C22#304111yescharacteristic
410C22#304211yescharacteristic
411C22#304311yescharacteristic
412C22#304411yescharacteristic
413C22#304512yes—
414C22#304612yes—
415C22#309612yes—
416C22#309712yes—
417C22#311912yes—
418C22#312012yes—
419C22#316312yes—
420C22#316412yes—
421C22#324312yes—
422C22#324412yes—
423C22#325712yes—
424C22#325812yes—
425C4—28no—
426C4—42no—
427C4—42no—
428C4—84no—
429C4—84no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4#245212yes—
436C4#245312yes—
437C4#250612yes—
438C4#250812yes—
439C2#1514911yescharacteristic
440C2#2366212yes—
441C2#2821512yes—
442C2#2825711yescharacteristic
443C2#2825811yescharacteristic
444C2#2869611yescharacteristic
445C2#2869711yescharacteristic
446C2#2932412yes—
447C2#2994512yes—
448C2#3311811yescharacteristic
449C2#3311911yescharacteristic
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×#20842
C2×#20832
C2×#19492
#142772
#137792
#112572
#112562
#112082
#108492
#108332