Finite Group Databasebeta

← #465328

Subgroups — #465328

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2056C211yesmaximal, characteristic
2C2×#1969C211yesmaximal, characteristic
3C2×#1916C211yesmaximal, characteristic
4C2×#1835C211yesmaximal, characteristic
5#15237C212yesmaximal
6#11328C211yesmaximal, characteristic
7#11232C211yesmaximal, characteristic
8#11190C212yesmaximal
9#11067C212yesmaximal
10#11052C212yesmaximal
11#11024C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#382C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#2056—24no—
26#1969—24no—
27#1916—24no—
28#1835—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1443—22no—
39#1443—22no—
40#1441—22no—
41#1441—22no—
42C4⋊#95C2212yes—
43C23×Q8C2312yes—
44C22×Q16—24no—
45C22×Q16—22no—
46C22×Q16—22no—
47C22×Q16—22no—
48C22×Q16—42no—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C22×C4⋊C4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#107—22no—
63C2×#107—42no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×C4×C8—22no—
77C2×C4×C8—22no—
78C2×C4×C8D414yes—
79#391—24no—
80#391—24no—
81#391—24no—
82#391—42no—
83#391—42no—
84#391—42no—
85#389—42no—
86#389—42no—
87#382—24no—
88#382—42no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104#340—22no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340D414yes—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8#4114yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C2×Q16—28no—
139C2×Q16—48no—
140C2×Q16—48no—
141C2×Q16—44no—
142C2×Q16—44no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8#4112yes—
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—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4C2×D414yes—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C2×D412yes—
204C2×C42#4112yes—
205#107—44no—
206C4⋊C8—44no—
207#103—48no—
208#103—48no—
209#103—48no—
210C4×C8—28no—
211#95—22no—
212#95—22no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224C24C22×D411yescharacteristic
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#13512yes—
314C22×C4#13512yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4112yes—
319C22×C4C2×#4111yescharacteristic
320C22×C4H(5,𝔽2)12yes—
321C22×C4H(5,𝔽2)12yes—
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C42−1+412yes—
324C22×C42−1+411yescharacteristic
325Q16—416no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8#12114yes—
346C4⋊C4—28no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—88no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—42no—
370C42—41no—
371C42—41no—
372C23C2×#12111yescharacteristic
373C23#53512yes—
374C23D4211yescharacteristic
375C23#54711yescharacteristic
376C23#54812yes—
377C23C2×#13511yescharacteristic
378C23C2×#13511yescharacteristic
379C23#57512yes—
380C23#57811yescharacteristic
381C23#57811yescharacteristic
382C23#57812yes—
383Q8—44no—
384Q8—44no—
385Q8—44no—
386Q8—44no—
387Q8—82no—
388Q8—82no—
389Q8—82no—
390Q8—82no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4C2×#12111yescharacteristic
454C2×C4#52511yescharacteristic
455C2×C4#53511yescharacteristic
456C2×C4#53511yescharacteristic
457C2×C4#53611yescharacteristic
458C2×C4#53711yescharacteristic
459C2×C4#53712yes—
460C2×C4#53812yes—
461C2×C4#53912yes—
462C2×C4#54012yes—
463C2×C4#54212yes—
464C2×C4#54612yes—
465C2×C4#54811yescharacteristic
466C2×C4#55112yes—
467C2×C4#55212yes—
468C2×C4#55311yescharacteristic
469C2×C4#56212yes—
470C2×C4#56312yes—
471C8—24no—
472C8—42no—
473C8—81no—
474C8—81no—
475C22#241611yescharacteristic
476C22#253611yescharacteristic
477C22#261911yescharacteristic
478C22#267612yes—
479C22#272312yes—
480C22#305011yescharacteristic
481C22#305411yescharacteristic
482C22#305911yescharacteristic
483C22#305911yescharacteristic
484C22#321112yes—
485C22#321212yes—
486C22#322312yes—
487C22#322312yes—
488C22#326912yes—
489C22#328311yescharacteristic
490C22#328511yescharacteristic
491C22#333611yescharacteristic
492C22#334211yescharacteristic
493C22#334912yes—
494C22#336412yes—
495C22#336412yes—
496C22#337712yes—
497C22#337712yes—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—81no—
503C4—84no—
504C4—84no—
505C4—81no—
506C4—162no—
507C4—162no—
508C4#241111yescharacteristic
509C4#262011yescharacteristic
510C4#263611yescharacteristic
511C4#270712yes—
512C4#276112yes—
513C4#282011yescharacteristic
514C2#1651911yescharacteristic
515C2#2477911yescharacteristic
516C2#2478511yescharacteristic
517C2#2819011yescharacteristic
518C2#2819311yescharacteristic
519C2#2980911yescharacteristic
520C2#2981211yescharacteristic
521C2#3088512yes—
522C2#3088812yes—
523C2#3243412yes—
524C2#3243912yes—
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×#20562
C2×#19692
C2×#19162
C2×#18352
#152372
#113282
#112322
#111902
#110672
#110522
#110242