Finite Group Databasebeta

← #464506

Subgroups — #464506

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#499C211yesmaximal, characteristic
2C2×#2087C211yesmaximal, characteristic
3C2×#2086C211yesmaximal, characteristic
4C2×#1916C211yesmaximal, characteristic
5#15218C212yesmaximal
6#11385C212yesmaximal
7#11379C211yesmaximal, characteristic
8#11377C211yesmaximal, characteristic
9#11327C212yesmaximal
10#11257C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×#125C2212yes—
13C2×#499—24no—
14C2×#499—24no—
15C2×#499—24no—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#2087—24no—
25#2086—24no—
26#1916—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1444—22no—
39#1444—22no—
40#1441—22no—
41#1441—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#125—24no—
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—42no—
54C2×#125—42no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#499—232no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107D414yes—
74C2×#107D414yes—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—24no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8D414yes—
84#389—24no—
85#389—24no—
86#389—42no—
87#389—42no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D411yescharacteristic
99C2×#95D411yescharacteristic
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340Q818yes—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C22×C8C2×Q814yes—
142#125—216no—
143#125—48no—
144#125—48no—
145#125—48no—
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—22no—
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—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—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×Q814yes—
186C2×C4⋊C4C2×Q814yes—
187C2×C42—21no—
188C2×C42—21no—
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×C42C2×D412yes—
202C2×C42#4112yes—
203#107—28no—
204#107—28no—
205#106—216no—
206C4⋊C8—44no—
207C4×C8—28no—
208#95—22no—
209#95—22no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—82no—
215#95—82no—
216#95—82no—
217#95—82no—
218C24C22×D411yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—42no—
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—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#12112yes—
278C22×C4#12112yes—
279C22×C4#13512yes—
280C22×C4#13512yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×Q812yes—
285C22×C4C2×#4111yescharacteristic
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C42−1+412yes—
289C22×C42−1+412yes—
290C22×C42−1+412yes—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—22no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8#12014yes—
310C2×C8C4⋊Q814yes—
311C4⋊C4—28no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—44no—
321C4⋊C4—416no—
322C4⋊C4—416no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—88no—
332C4⋊C4—88no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—42no—
342C42—41no—
343C42—41no—
344C23C2×#12111yescharacteristic
345C23#53612yes—
346C23D4211yescharacteristic
347C23D4211yescharacteristic
348C23D4×Q812yes—
349C23C2×#13511yescharacteristic
350C23C2×#13511yescharacteristic
351C23#57512yes—
352C23#57611yescharacteristic
353C23#57712yes—
354C23#57811yescharacteristic
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4C2×#12011yescharacteristic
418C2×C4C2×C4⋊Q811yescharacteristic
419C2×C4#53511yescharacteristic
420C2×C4#53611yescharacteristic
421C2×C4#53611yescharacteristic
422C2×C4#53611yescharacteristic
423C2×C4#53612yes—
424C2×C4#53612yes—
425C2×C4#54112yes—
426C2×C4#54412yes—
427C2×C4#54412yes—
428C2×C4#54611yescharacteristic
429C2×C4#54611yescharacteristic
430C2×C4D4×Q812yes—
431C2×C4D4×Q812yes—
432C2×C4#55712yes—
433C2×C4#56312yes—
434C2×C4#56312yes—
435C8—24no—
436C8—24no—
437C8—81no—
438C8—81no—
439C22#240811yescharacteristic
440C22#254611yescharacteristic
441C22#262111yescharacteristic
442C22#267712yes—
443C22#267712yes—
444C22#305011yescharacteristic
445C22#305411yescharacteristic
446C22#306011yescharacteristic
447C22#306111yescharacteristic
448C22#323212yes—
449C22#323312yes—
450C22#324112yes—
451C22#324212yes—
452C22#326712yes—
453C22#327112yes—
454C22#328011yescharacteristic
455C22#328311yescharacteristic
456C22#328511yescharacteristic
457C22#328711yescharacteristic
458C22#338112yes—
459C22#338112yes—
460C22#338912yes—
461C22#339012yes—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—81no—
467C4—84no—
468C4—81no—
469C4—84no—
470C4—84no—
471C4—84no—
472C4#243411yescharacteristic
473C4#261411yescharacteristic
474C4#266011yescharacteristic
475C4#271312yes—
476C4#271312yes—
477C4#282111yescharacteristic
478C2#1650711yescharacteristic
479C2#2426611yescharacteristic
480C2#2427111yescharacteristic
481C2#2867811yescharacteristic
482C2#2868011yescharacteristic
483C2#2987911yescharacteristic
484C2#2988211yescharacteristic
485C2#3095012yes—
486C2#3095212yes—
487C2#3095912yes—
488C2#3096112yes—
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
C22×#4992
C2×#20872
C2×#20862
C2×#19162
#152182
#113852
#113792
#113772
#113272
#112572
#108202