Finite Group Databasebeta

← #400855

Subgroups — #400855

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2074C211yesmaximal, characteristic
2C2×#1933C212yesmaximal
3C2×C4×C4⋊C8C211yesmaximal, characteristic
4#14707C211yesmaximal, characteristic
5#14703C211yesmaximal, characteristic
6#11358C211yesmaximal, characteristic
7#11268C211yesmaximal, characteristic
8#11227C212yesmaximal
9#11077C212yesmaximal
10#11052C211yesmaximal, characteristic
11#10880C212yesmaximal
12C22×C4⋊C8C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×C4×C8C2212yes—
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C43C2211yescharacteristic
24#2074—24no—
25#1933—28no—
26C4×C4⋊C8—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1436—22no—
36#1436—22no—
37#1436—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41#1434—22no—
42C23×Q8C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C42D412yes—
53C22×C42D412yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—24no—
58C2×C4⋊C8—24no—
59C2×C4⋊C8—24no—
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—42no—
66C2×C4×C8—24no—
67C2×C4×C8—24no—
68C2×C4×C8—42no—
69#391—22no—
70#391—22no—
71#391—22no—
72#391—24no—
73#391D414yes—
74#389—24no—
75#389—24no—
76#389—24no—
77#382—24no—
78#382—22no—
79#382—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C43—22no—
93C43—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—42no—
100#336—22no—
101#336D414yes—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8C2×D412yes—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—22no—
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×C8—22no—
134C22×C8—22no—
135C22×C8C2×D414yes—
136C22×C8#4114yes—
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—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4#4114yes—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—42no—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C2×C42#4114yes—
186C4⋊C8—216no—
187#103—216no—
188#103—48no—
189C4×C8—48no—
190#95—28no—
191#95—28no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—42no—
198#95—42no—
199#95—42no—
200#95—44no—
201#95—44no—
202C24C22×D411yescharacteristic
203C2×Q8—24no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—24no—
207C2×Q8—42no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4#12114yes—
273C22×C4#12114yes—
274C22×C4#12412yes—
275C22×C4#12412yes—
276C22×C4#13514yes—
277C22×C4#13514yes—
278C22×C4#13714yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4112yes—
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C42−1+412yes—
288C22×C42−1+411yescharacteristic
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
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—42no—
316C4⋊C4—42no—
317C4⋊C4—88no—
318C4⋊C4—84no—
319C42—216no—
320C42—24no—
321C42—24no—
322C42—22no—
323C42—22no—
324C42—24no—
325C42—22no—
326C42—24no—
327C42—22no—
328C42—48no—
329C23C2×#12111yescharacteristic
330C23C2×#12111yescharacteristic
331C23#52511yescharacteristic
332C23C2×#12411yescharacteristic
333C23#53211yescharacteristic
334C23#53611yescharacteristic
335C23#53711yescharacteristic
336C23C2×#13512yes—
337C23C2×#13512yes—
338C23C2×#13712yes—
339C23#57512yes—
340Q8—44no—
341Q8—44no—
342Q8—82no—
343Q8—82no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#46418yes—
403C2×C4#48814yes—
404C2×C4#52512yes—
405C2×C4#52512yes—
406C2×C4#52912yes—
407C2×C4#52912yes—
408C2×C4#53912yes—
409C2×C4#54112yes—
410C2×C4D4211yescharacteristic
411C2×C4#54712yes—
412C2×C4#54712yes—
413C2×C4#54711yescharacteristic
414C2×C4#54811yescharacteristic
415C2×C4#55212yes—
416C2×C4#55311yescharacteristic
417C2×C4#55612yes—
418C2×C4#55612yes—
419C2×C4#55912yes—
420C8—44no—
421C8—44no—
422C22#239211yescharacteristic
423C22#241311yescharacteristic
424C22#241611yescharacteristic
425C22#254711yescharacteristic
426C22#259211yescharacteristic
427C22#266411yescharacteristic
428C22#285011yescharacteristic
429C22C2×#46412yes—
430C22#303512yes—
431C22#305012yes—
432C22#305612yes—
433C22#310212yes—
434C22#310712yes—
435C22C2×#48811yescharacteristic
436C22#313111yescharacteristic
437C22#313911yescharacteristic
438C22#314211yescharacteristic
439C22#316812yes—
440C22#317212yes—
441C22#323112yes—
442C22#323312yes—
443C22#324812yes—
444C22#325012yes—
445C4—24no—
446C4—28no—
447C4—44no—
448C4—42no—
449C4—82no—
450C4—84no—
451C4—82no—
452C4—161no—
453C4—161no—
454C4—162no—
455C4#245512yes—
456C4#246912yes—
457C4#247812yes—
458C4#259612yes—
459C2#1480811yescharacteristic
460C2#2368112yes—
461C2#2460412yes—
462C2#2475512yes—
463C2#2871911yescharacteristic
464C2#2872111yescharacteristic
465C2#2934911yescharacteristic
466C2#2935111yescharacteristic
467C2#3038012yes—
468C2#3304411yescharacteristic
469C2#3305911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#20742
C2×#19332
C2×C4×C4⋊C82
#147072
#147032
#113582
#112682
#112272
#110772
#110522
#108802