Finite Group Databasebeta

← #412648

Subgroups — #412648

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2085C212yesmaximal
2C2×#1948C211yesmaximal, characteristic
3C2×#1776C211yesmaximal, characteristic
4#13954C211yesmaximal, characteristic
5#13911C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11220C212yesmaximal
8#11208C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10843C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#2085—28no—
26#1948—24no—
27#1776—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1444—22no—
38#1442—21no—
39#1442—21no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42—21no—
59C22×C42—21no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×#107—22no—
63C2×#107—22no—
64C2×#107D414yes—
65C2×#106—22no—
66C2×#106—22no—
67C2×#106—24no—
68C2×#106—24no—
69C2×#106—24no—
70C2×#106Q814yes—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8Q814yes—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97Q814yes—
77#389—24no—
78#389—24no—
79#389—42no—
80#389—42no—
81#384—22no—
82#384—22no—
83#382—24no—
84#382—22no—
85#382—22no—
86#382—42no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4D414yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105#340—24no—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—22no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—42no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8C2×D412yes—
142C22×C8C2×Q812yes—
143C22×C8C2×Q814yes—
144C22×C8C2×Q812yes—
145C22×C8C2×Q812yes—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×Q812yes—
192C2×C4⋊C4C2×Q812yes—
193C2×C4⋊C4#4112yes—
194C2×C4⋊C4#4112yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—44no—
203C2×C42—42no—
204C2×C42—82no—
205C2×C42C2×D412yes—
206C2×C42C2×Q812yes—
207C2×C42C2×Q812yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211#107—28no—
212#106—28no—
213#106—48no—
214C4⋊C8—28no—
215#97—28no—
216#95—24no—
217#95—22no—
218#95—22no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—42no—
224#95—42no—
225#95—42no—
226#95—82no—
227#95—82no—
228C24C22×D411yescharacteristic
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—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
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—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12312yes—
284C22×C4#12312yes—
285C22×C4C4⋊Q812yes—
286C22×C4C4⋊Q812yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×Q811yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C22×Q811yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C42−1+411yescharacteristic
300C22×C42−1+411yescharacteristic
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8C4⋊Q814yes—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—48no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C4⋊C4—88no—
348C4⋊C4—88no—
349C4⋊C4#12114yes—
350C4⋊C4C22⋊Q814yes—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—84no—
356C42—84no—
357C42C22⋊Q814yes—
358C42C22⋊Q814yes—
359C42#12514yes—
360C42#12514yes—
361C23C2×#12111yescharacteristic
362C23C2×#12311yescharacteristic
363C23C2×C4⋊Q811yescharacteristic
364C23D4211yescharacteristic
365C23#54611yescharacteristic
366C23#54811yescharacteristic
367C23D4×Q811yescharacteristic
368C23#57512yes—
369C23#57512yes—
370C23#57712yes—
371C23#57712yes—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#39814yes—
429C2×C4#39814yes—
430C2×C4C2×#12111yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×#12511yescharacteristic
435C2×C4C2×#12511yescharacteristic
436C2×C4C2×C4⋊Q811yescharacteristic
437C2×C4#53211yescharacteristic
438C2×C4#53311yescharacteristic
439C2×C4#53311yescharacteristic
440C2×C4#53711yescharacteristic
441C2×C4#53711yescharacteristic
442C2×C4#54011yescharacteristic
443C2×C4#54111yescharacteristic
444C2×C4#54411yescharacteristic
445C2×C4#54411yescharacteristic
446C2×C4#54711yescharacteristic
447C2×C4D4×Q811yescharacteristic
448C2×C4#55011yescharacteristic
449C2×C4#55211yescharacteristic
450C2×C4#55311yescharacteristic
451C2×C4#55411yescharacteristic
452C2×C4#55411yescharacteristic
453C2×C4#55611yescharacteristic
454C2×C4#55711yescharacteristic
455C2×C4#55711yescharacteristic
456C2×C4Q8211yescharacteristic
457C2×C4#55911yescharacteristic
458C8—24no—
459C8—42no—
460C8—44no—
461C22C2×#39811yescharacteristic
462C22#241611yescharacteristic
463C22#241711yescharacteristic
464C22#244511yescharacteristic
465C22#244611yescharacteristic
466C22#255011yescharacteristic
467C22#255111yescharacteristic
468C22#304011yescharacteristic
469C22#304011yescharacteristic
470C22#304211yescharacteristic
471C22#304311yescharacteristic
472C22#309312yes—
473C22#309912yes—
474C22#316312yes—
475C22#316612yes—
476C22#326712yes—
477C22#327112yes—
478C22#331112yes—
479C22#332112yes—
480C22#337112yes—
481C22#337612yes—
482C22#338812yes—
483C22#338812yes—
484C4—24no—
485C4—24no—
486C4—44no—
487C4—44no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—162no—
493C4—162no—
494C4#250611yescharacteristic
495C4#250711yescharacteristic
496C4#253811yescharacteristic
497C4#255211yescharacteristic
498C4#257011yescharacteristic
499C4#258211yescharacteristic
500C4#262911yescharacteristic
501C4#285711yescharacteristic
502C2#1513311yescharacteristic
503C2#2365911yescharacteristic
504C2#2366111yescharacteristic
505C2#2474511yescharacteristic
506C2#2474711yescharacteristic
507C2#2482811yescharacteristic
508C2#2483411yescharacteristic
509C2#2624512yes—
510C2#2628912yes—
511C2#2880812yes—
512C2#2882212yes—
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×#20852
C2×#19482
C2×#17762
#139542
#139112
#112202
#112082
#108502
#108432
#108352