Finite Group Databasebeta

← #415152

Subgroups — #415152

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1992C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3C2×#1847C211yesmaximal, characteristic
4C2×#1768C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic
6#14277C211yesmaximal, characteristic
7#11232C212yesmaximal
8#11231C212yesmaximal
9#11220C212yesmaximal
10#11219C212yesmaximal
11#10841C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1992—24no—
25#1991—24no—
26#1847—24no—
27#1768—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1444—22no—
36#1442—22no—
37#1441—22no—
38#1441—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125Q814yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—24no—
66C2×#107—24no—
67C2×#107—42no—
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—42no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8Q818yes—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97D414yes—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80#389—42no—
81#389—42no—
82#384—22no—
83#384—22no—
84#384—22no—
85#384—22no—
86#382—22no—
87#382D414yes—
88#376—22no—
89#376—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100C2×#95D412yes—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#336—24no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2411yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8C2×D414yes—
130C22×C8C2×Q814yes—
131C22×C8C2×Q814yes—
132C22×C8#4114yes—
133#125—28no—
134#125—28no—
135#125—44no—
136#125—44no—
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—28no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
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—22no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×Q814yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×Q814yes—
188C2×C42C2×Q812yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4112yes—
192#107—48no—
193#106—48no—
194C4⋊C8—216no—
195#97—28no—
196C4×C8—44no—
197#95—28no—
198#95—24no—
199#95—24no—
200#95—42no—
201#95—42no—
202#95—82no—
203#95—82no—
204#95—82no—
205#95—82no—
206#95—82no—
207#95—82no—
208C24C22×D411yescharacteristic
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—41no—
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#12114yes—
264C22×C4#12114yes—
265C22×C4C22⋊Q814yes—
266C22×C4C4⋊Q814yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×Q811yescharacteristic
270C22×C4C22×Q812yes—
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4H(5,𝔽2)12yes—
276C22×C42−1+412yes—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8C22⋊Q818yes—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—416no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—88no—
309C4⋊C4—88no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C42—24no—
313C42—24no—
314C42—42no—
315C42—42no—
316C42#12114yes—
317C42C22⋊Q814yes—
318C42C22⋊Q814yes—
319C42#12614yes—
320C23C2×#12111yescharacteristic
321C23C2×#12111yescharacteristic
322C23C2×C22⋊Q811yescharacteristic
323C23C2×C4⋊Q811yescharacteristic
324C23#53211yescharacteristic
325C23#53511yescharacteristic
326C23#53611yescharacteristic
327C23#57612yes—
328C23#57712yes—
329C23#57712yes—
330C23#57812yes—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
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—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4#39618yes—
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4C2×C22⋊Q812yes—
392C2×C4C2×C22⋊Q811yescharacteristic
393C2×C4C2×#12611yescharacteristic
394C2×C4#53312yes—
395C2×C4#53712yes—
396C2×C4#54212yes—
397C2×C4#54612yes—
398C2×C4D4×Q812yes—
399C2×C4#55212yes—
400C2×C4#55412yes—
401C2×C4#55712yes—
402C2×C4#55712yes—
403C2×C4Q8212yes—
404C2×C4#55912yes—
405C8—28no—
406C8—44no—
407C22C2×#39611yescharacteristic
408C22#254411yescharacteristic
409C22#254611yescharacteristic
410C22#256511yescharacteristic
411C22#256711yescharacteristic
412C22#262011yescharacteristic
413C22#285711yescharacteristic
414C22#304512yes—
415C22#304612yes—
416C22#306212yes—
417C22#306312yes—
418C22#308312yes—
419C22#308412yes—
420C22#316312yes—
421C22#316412yes—
422C22#317812yes—
423C22#317912yes—
424C22#322011yescharacteristic
425C22#322111yescharacteristic
426C22#322211yescharacteristic
427C22#322311yescharacteristic
428C22#324112yes—
429C22#324212yes—
430C4—28no—
431C4—44no—
432C4—84no—
433C4—81no—
434C4—81no—
435C4—82no—
436C4—82no—
437C4—161no—
438C4—161no—
439C4—162no—
440C4#241712yes—
441C4#245112yes—
442C4#251112yes—
443C4#258212yes—
444C2#1518411yescharacteristic
445C2#2361612yes—
446C2#2862911yescharacteristic
447C2#2863011yescharacteristic
448C2#2869812yes—
449C2#2918812yes—
450C2#2921511yescharacteristic
451C2#2921611yescharacteristic
452C2#2985511yescharacteristic
453C2#2985611yescharacteristic
454C2#3321912yes—
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×#19922
C2×#19912
C2×#18472
C2×#17682
#142832
#142772
#112322
#112312
#112202
#112192
#108412