Finite Group Databasebeta

← #474932

Subgroups — #474932

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2087C211yesmaximal, characteristic, complemented
2C2×#2054C211yesmaximal, characteristic
3C2×#1936C212yesmaximal, complemented
4#15134C212yesmaximal, complemented
5#11371C212yesmaximal, complemented
6#11123C212yesmaximal, complemented
7#11049C212yesmaximal, complemented
8#11022C212yesmaximal
9#10835C211yesmaximal, characteristic, complemented
10C23×D8C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#102C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#97C2211yescharacteristic
17C2×#396—24no—
18C2×#390C2212yes—
19C2×#390C2212yes—
20C2×#390C2212yes—
21C2×#389C2212yes—
22C2×#384C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#2087—24no—
25#2054—24no—
26#1936—28no—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1439—24no—
35C4⋊#95C2212yes—
36C23×D4C2312yes—
37C22×D8—24no—
38C22×D8—22no—
39C22×D8—42no—
40C22×D8—41no—
41C22×D8—41no—
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—24no—
54C22×C22⋊C4—22no—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57C2×#107—22no—
58C2×#107—41no—
59C2×#107—41no—
60C2×C4⋊C8—41no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8D414yes—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—42no—
70C2×#102—42no—
71C2×#102—42no—
72C2×#97—41no—
73C2×#97—41no—
74C2×#97D414yes—
75#396—416no—
76#390—24no—
77#390—24no—
78#390—24no—
79#390—42no—
80#390—42no—
81#390—42no—
82#389—24no—
83#389—24no—
84#384—24no—
85#384—42no—
86C4×C4⋊C4—41no—
87C4×C4⋊C4—41no—
88C2×#95—24no—
89C2×#95—22no—
90C2×#95D412yes—
91#340—22no—
92#340—22no—
93#340—24no—
94#340—24no—
95#340—24no—
96#336—42no—
97#336—42no—
98C25—22no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4#4114yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D414yes—
119C23×C4C2411yescharacteristic
120C2×D8—48no—
121C2×D8—44no—
122C2×D8—44no—
123C2×D8—48no—
124C2×D8—44no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8C2×D412yes—
140C22×C8C2×D414yes—
141C22×C8C2×D412yes—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—28no—
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—42no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4#4114yes—
167C2×C22⋊C4—48no—
168C2×C22⋊C4—48no—
169C2×C22⋊C4—48no—
170C2×C22⋊C4—48no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—48no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4—84no—
176C2×C22⋊C4—84no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189#107—44no—
190C4⋊C8—44no—
191#102—216no—
192#102—48no—
193#102—48no—
194#97—44no—
195#95—42no—
196#95—44no—
197#95—48no—
198C24—24no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24—44no—
206C24—44no—
207C24—44no—
208C24—44no—
209C24C22×D411yescharacteristic
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4#12712yes—
267C22×C4#12712yes—
268C22×C4AGL(1,ℤ/8)18yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4H(5,𝔽2)11yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C4H(5,𝔽2)11yescharacteristic
276C22×C4H(5,𝔽2)12yes—
277C22×C42−1+411yescharacteristic
278C22×C42−1+412yes—
279D8—416no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—48no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—88no—
311C4⋊C4—816no—
312C22⋊C4—816no—
313C22⋊C4—88no—
314C22⋊C4—816no—
315C22⋊C4—88no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—42no—
324C23—48no—
325C23—48no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23—84no—
330C23—84no—
331C23—82no—
332C23—82no—
333C23—82no—
334C23—82no—
335C23—82no—
336C23—84no—
337C23—84no—
338C23—82no—
339C23—82no—
340C23—84no—
341C23—84no—
342C23—82no—
343C23—84no—
344C23—84no—
345C23—84no—
346C23—84no—
347C23C2×#12711yescharacteristic
348C23#53411yescharacteristic
349C23#53711yescharacteristic
350C23#54612yes—
351C23#54812yes—
352C23C2×AGL(1,ℤ/8)14yes—
353C23#57612yes—
354C23#57812yes—
355D4—44no—
356D4—44no—
357D4—44no—
358D4—44no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—82no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—28no—
369C2×C4—28no—
370C2×C4—28no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—84no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4#52512yes—
395C2×C4#53411yescharacteristic
396C2×C4#53411yescharacteristic
397C2×C4#53711yescharacteristic
398C2×C4#53712yes—
399C2×C4#53711yescharacteristic
400C2×C4#53812yes—
401C2×C4#53912yes—
402C2×C4#54012yes—
403C2×C4#54212yes—
404C2×C4#54612yes—
405C2×C4#54812yes—
406C2×C4#55111yescharacteristic
407C2×C4#55212yes—
408C2×C4#55511yescharacteristic
409C2×C4#56011yescharacteristic
410C2×C4#56312yes—
411C2×C4#56311yescharacteristic
412C8—44no—
413C8—42no—
414C8—42no—
415C22—84no—
416C22—84no—
417C22—88no—
418C22—88no—
419C22—84no—
420C22—162no—
421C22—162no—
422C22—162no—
423C22—162no—
424C22—164no—
425C22—164no—
426C22#254012yes—
427C22#265611yescharacteristic
428C22#269911yescharacteristic
429C22#270212yes—
430C22#272811yescharacteristic
431C22#315112yes—
432C22#315511yescharacteristic
433C22#315611yescharacteristic
434C22#319212yes—
435C22#320312yes—
436C22#325312yes—
437C22#326212yes—
438C22#330814yes—
439C22#331814yes—
440C22#336614yes—
441C22#336714yes—
442C4—42no—
443C4—42no—
444C4—48no—
445C4—84no—
446C4—162no—
447C4—162no—
448C4#260911yescharacteristic
449C4#263012yes—
450C4#269711yescharacteristic
451C4#270712yes—
452C4#273111yescharacteristic
453C4#282311yescharacteristic
454C2—164no—
455C2#1701511yescharacteristic
456C2#2842112yes—
457C2#2842612yes—
458C2#3024711yescharacteristic
459C2#3025511yescharacteristic
460C2#3173612yes—
461C2#3181814yes—
462C2#3255912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2087C2semidirect
3. C2×#1936C2semidirect
4. #15134C2semidirect
5. #11371C2semidirect
6. #11123C2semidirect
7. #11049C2semidirect
9. #10835C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#20872
C2×#20542
C2×#19362
#151342
#113712
#111232
#110492
#110222
#108352