Finite Group Databasebeta

← #406677

Subgroups — #406677

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic
2C2×#1935C212yesmaximal
3#14703C211yesmaximal, characteristic
4#11293C212yesmaximal
5#11208C211yesmaximal, characteristic
6#11208C211yesmaximal, characteristic
7#11067C212yesmaximal
8#11052C211yesmaximal, characteristic
9#11052C211yesmaximal, characteristic
10#10881C212yesmaximal
11#10693C211yesmaximal, characteristic
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#97C2212yes—
15#2864—28no—
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#387—22no—
23C2×#387—22no—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C43C2211yescharacteristic
27#1935—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1436—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40#1434—22no—
41C23×Q8C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C42D412yes—
53C22×C42D412yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#103—24no—
58C2×#103—24no—
59C2×#103—24no—
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—42no—
63C2×#97—24no—
64C2×#97—24no—
65C2×#97—42no—
66#391—22no—
67#391—22no—
68#391—24no—
69#391—22no—
70#391—22no—
71#389—24no—
72#389—24no—
73#389—24no—
74#389—22no—
75#389—22no—
76#389D418yes—
77#387—216no—
78#387—216no—
79#382—24no—
80#382—24no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C43—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#336—24no—
102#336—22no—
103#336—22no—
104C22×Q8—21no—
105C22×Q8—22no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8#4112yes—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2411yescharacteristic
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×C8—22no—
136C22×C8—22no—
137C22×C8#4114yes—
138C22×C8#4114yes—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4#4112yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42C2×D414yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191#103—216no—
192#103—48no—
193#97—48no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—28no—
199#95—28no—
200#95—28no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—42no—
205#95—42no—
206#95—42no—
207C24C22×D411yescharacteristic
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
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—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
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—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4#12014yes—
281C22×C4#12412yes—
282C22×C4#12412yes—
283C22×C4#13714yes—
284C22×C4C22×D412yes—
285C22×C4C2×#4111yescharacteristic
286C22×C4C2×#4112yes—
287C22×C4C2×#4112yes—
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4H(5,𝔽2)11yescharacteristic
292C22×C42−1+412yes—
293C22×C42−1+411yescharacteristic
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—416no—
315C4⋊C4—416no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—84no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—48no—
332C42—48no—
333C42—42no—
334C42—42no—
335C23C2×#12011yescharacteristic
336C23#52511yescharacteristic
337C23C2×#12411yescharacteristic
338C23#53211yescharacteristic
339C23#53211yescharacteristic
340C23#53611yescharacteristic
341C23#53711yescharacteristic
342C23C2×#13712yes—
343C23#57512yes—
344C23#57512yes—
345C23#57512yes—
346Q8—44no—
347Q8—44no—
348Q8—82no—
349Q8—82no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4#52512yes—
411C2×C4#52912yes—
412C2×C4#52912yes—
413C2×C4#53212yes—
414C2×C4#53912yes—
415C2×C4#54112yes—
416C2×C4#54612yes—
417C2×C4#54712yes—
418C2×C4#54812yes—
419C2×C4#54812yes—
420C2×C4#55212yes—
421C2×C4#55211yescharacteristic
422C2×C4#55211yescharacteristic
423C2×C4#55212yes—
424C2×C4#55311yescharacteristic
425C2×C4#55311yescharacteristic
426C8—44no—
427C8—44no—
428C22#238811yescharacteristic
429C22#243011yescharacteristic
430C22#243611yescharacteristic
431C22#256211yescharacteristic
432C22#261111yescharacteristic
433C22#266411yescharacteristic
434C22#286111yescharacteristic
435C22#300512yes—
436C22#300712yes—
437C22#311112yes—
438C22#311712yes—
439C22#313511yescharacteristic
440C22#313511yescharacteristic
441C22#313911yescharacteristic
442C22#314011yescharacteristic
443C22#317212yes—
444C22#318112yes—
445C22#318312yes—
446C22#318312yes—
447C22#323712yes—
448C22#324012yes—
449C22#325212yes—
450C22#325612yes—
451C4—44no—
452C4—42no—
453C4—42no—
454C4—44no—
455C4—84no—
456C4—82no—
457C4—84no—
458C4—82no—
459C4—161no—
460C4—161no—
461C4#246012yes—
462C4#247512yes—
463C4#251912yes—
464C4#255812yes—
465C2#1493311yescharacteristic
466C2#2357812yes—
467C2#2550512yes—
468C2#2590911yescharacteristic
469C2#2591011yescharacteristic
470C2#2912811yescharacteristic
471C2#2912911yescharacteristic
472C2#2958312yes—
473C2#3038712yes—
474C2#3328111yescharacteristic
475C2#3328811yescharacteristic
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×#28642
C2×#19352
#147032
#112932
#112082
#110672
#110522
#108812
#106932