Finite Group Databasebeta

← #414557

Subgroups — #414557

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic
2C2×#1935C212yesmaximal
3C2×#1933C211yesmaximal, characteristic
4#15201C211yesmaximal, characteristic
5#15158C211yesmaximal, characteristic
6#11228C212yesmaximal
7#11227C211yesmaximal, characteristic
8#11077C212yesmaximal
9#11077C211yesmaximal, characteristic
10#10821C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#446—28no—
14C2×#446—24no—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×#97C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#395—21no—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27#1935—28no—
28#1933—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36#1446—21no—
37#1442—22no—
38#1442—21no—
39#1441—22no—
40#1441—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4×Q8—24no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C42C2311yescharacteristic
64C22×C42C2312yes—
65#446—232no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—24no—
71C2×#103—22no—
72C2×#103—24no—
73C2×#103—22no—
74C2×#103—24no—
75C2×#103—22no—
76C2×#103—24no—
77C2×#103—22no—
78C2×#103—24no—
79C2×#97—24no—
80C2×#97—24no—
81C2×#97D418yes—
82C2×C4×C8—24no—
83C2×C4×C8D414yes—
84#395—44no—
85#391—22no—
86#391—24no—
87#391—22no—
88#391—24no—
89#389—22no—
90#389—24no—
91#389—22no—
92#389—24no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107#340—22no—
108#340—22no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114C22×Q8—24no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—42no—
128C22×Q8#4112yes—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8C2×D412yes—
154C22×C8C2×D414yes—
155C22×C8C2×D414yes—
156C22×C8C2×D412yes—
157C4×Q8—28no—
158C4×Q8—216no—
159C4×Q8—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4#4114yes—
196C2×C4⋊C4#4112yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C2×D412yes—
204C2×C42C2×D414yes—
205C2×C42#4114yes—
206C2×C42#4112yes—
207C4⋊C8—28no—
208C4⋊C8—216no—
209#103—28no—
210#103—216no—
211#103—44no—
212#97—216no—
213C4×C8—28no—
214#95—44no—
215#95—44no—
216#95—42no—
217#95—44no—
218#95—42no—
219#95—44no—
220#95—44no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—81no—
225#95—82no—
226#95—81no—
227#95—82no—
228C24C22×D411yescharacteristic
229C2×Q8—28no—
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8—44no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4#12112yes—
307C22×C4#12114yes—
308C22×C4#12114yes—
309C22×C4#12112yes—
310C22×C4AGL(1,ℤ/8)12yes—
311C22×C4#13712yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D412yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4111yescharacteristic
319C22×C42−1+411yescharacteristic
320C22×C42−1+412yes—
321C22×C42−1+411yescharacteristic
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8#12014yes—
339C4⋊C4—28no—
340C4⋊C4—216no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—88no—
360C4⋊C4—84no—
361C42—22no—
362C42—24no—
363C42—22no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—24no—
368C23C2×#12111yescharacteristic
369C23C2×#12112yes—
370C23#54812yes—
371C23#54811yescharacteristic
372C23#54811yescharacteristic
373C23C2×AGL(1,ℤ/8)11yescharacteristic
374C23C2×#13711yescharacteristic
375C23#57512yes—
376C23#57711yescharacteristic
377C23#57711yescharacteristic
378C23#57812yes—
379Q8—416no—
380Q8—42no—
381Q8—44no—
382Q8—42no—
383Q8—81no—
384Q8—82no—
385Q8—81no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4#46814yes—
446C2×C4C2×#12011yescharacteristic
447C2×C4#52511yescharacteristic
448C2×C4#52512yes—
449C2×C4#52511yescharacteristic
450C2×C4#52512yes—
451C2×C4#53211yescharacteristic
452C2×C4#53212yes—
453C2×C4#53611yescharacteristic
454C2×C4#53612yes—
455C2×C4#53611yescharacteristic
456C2×C4#53612yes—
457C2×C4#55211yescharacteristic
458C2×C4#55212yes—
459C2×C4#55211yescharacteristic
460C2×C4#55212yes—
461C2×C4#55612yes—
462C2×C4#55612yes—
463C2×C4#55612yes—
464C8—24no—
465C8—42no—
466C8—44no—
467C22#253611yescharacteristic
468C22#253612yes—
469C22#254511yescharacteristic
470C22#254512yes—
471C22#284711yescharacteristic
472C22C2×#46811yescharacteristic
473C22#305411yescharacteristic
474C22#305512yes—
475C22#305612yes—
476C22#305911yescharacteristic
477C22#306011yescharacteristic
478C22#306312yes—
479C22#306312yes—
480C22#336712yes—
481C22#336811yescharacteristic
482C22#336911yescharacteristic
483C22#337112yes—
484C22#337211yescharacteristic
485C22#337311yescharacteristic
486C22#337412yes—
487C22#337612yes—
488C22#337712yes—
489C22#337712yes—
490C4—24no—
491C4—44no—
492C4—42no—
493C4—44no—
494C4—82no—
495C4—84no—
496C4—82no—
497C4—161no—
498C4—161no—
499C4—162no—
500C4#238811yescharacteristic
501C4#259911yescharacteristic
502C4#259912yes—
503C4#260411yescharacteristic
504C4#260412yes—
505C4#261411yescharacteristic
506C2#1517611yescharacteristic
507C2#2819311yescharacteristic
508C2#2819511yescharacteristic
509C2#2819812yes—
510C2#2819912yes—
511C2#2864411yescharacteristic
512C2#2864611yescharacteristic
513C2#2864912yes—
514C2#2865212yes—
515C2#3293511yescharacteristic
516C2#3294011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ 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
C22×#4462
C2×#19352
C2×#19332
#152012
#151582
#112282
#112272
#110772
#108212
#108202