Finite Group Databasebeta

← #410653

Subgroups — #410653

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2092C212yesmaximal, complemented
2C2×#1970C211yesmaximal, characteristic
3C2×#1926C212yesmaximal, complemented
4C2×C4×#107C211yesmaximal, characteristic, complemented
5#14002C212yesmaximal, complemented
6#11325C212yesmaximal, complemented
7#11123C212yesmaximal, complemented
8#11022C212yesmaximal
9#10797C211yesmaximal, characteristic, complemented
10C23×D8C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#390C2212yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#382C2212yes—
23C2×C4×C4⋊C4C2212yes—
24#2092—28no—
25#1970—24no—
26#1926—28no—
27C4×#107—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1440—24no—
34#1437—24no—
35C4⋊#95C2212yes—
36C4×#95C2211yescharacteristic
37C23×D4C2312yes—
38C22×D8—24no—
39C22×D8—22no—
40C22×D8—24no—
41C22×D8—41no—
42C22×D8—41no—
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—24no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—24no—
52C22×C22⋊C4—22no—
53C22×C42—22no—
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—24no—
60C2×#107—22no—
61C2×#107—41no—
62C2×#107—41no—
63C2×#107D414yes—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—42no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—42no—
76C2×C4×C8—24no—
77C2×C4×C8D414yes—
78#390—24no—
79#390—24no—
80#390—24no—
81#390—24no—
82#390—24no—
83#390—42no—
84#382—24no—
85#382—42no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C2×#95—24no—
89C2×#95—22no—
90C2×#95—24no—
91C2×#95D412yes—
92C2×#95D412yes—
93#340—24no—
94#340—24no—
95#336—24no—
96#336—24no—
97#336D414yes—
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×D414yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C2×D8—28no—
121C2×D8—48no—
122C2×D8—48no—
123C2×D8—44no—
124C2×D8—44no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8C2×D414yes—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8#4112yes—
141C22×C8#4114yes—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
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⋊C4C2×D414yes—
159C2×C4⋊C4#4114yes—
160C2×C22⋊C4—216no—
161C2×C22⋊C4—48no—
162C2×C22⋊C4—48no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—48no—
166C2×C22⋊C4—84no—
167C2×C22⋊C4—84no—
168C2×C22⋊C4—84no—
169C2×C22⋊C4—84no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—44no—
179C2×C42—84no—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182#107—28no—
183#107—28no—
184#107—44no—
185#106—48no—
186#102—216no—
187#102—48no—
188#102—48no—
189C4×C8—28no—
190#95—24no—
191#95—42no—
192#95—48no—
193#95—44no—
194#95—84no—
195C24—24no—
196C24—42no—
197C24—42no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24—44no—
203C24—44no—
204C24—44no—
205C24—44no—
206C24C22×D411yescharacteristic
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
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—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4#12712yes—
263C22×C4#12712yes—
264C22×C4#13514yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C4H(5,𝔽2)11yescharacteristic
274C22×C42−1+412yes—
275D8—416no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8#12114yes—
292C2×C8#12314yes—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C22⋊C4—88no—
306C22⋊C4—816no—
307C22⋊C4—816no—
308C22⋊C4—88no—
309C42—22no—
310C42—22no—
311C42—24no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—88no—
317C23—48no—
318C23—48no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—84no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23—84no—
328C23—84no—
329C23—82no—
330C23—84no—
331C23—82no—
332C23—82no—
333C23—82no—
334C23—82no—
335C23—82no—
336C23—82no—
337C23—84no—
338C23—84no—
339C23—82no—
340C23#52512yes—
341C23C2×#12711yescharacteristic
342C23D4211yescharacteristic
343C23#54712yes—
344C23#55011yescharacteristic
345C23C2×#13512yes—
346C23#57512yes—
347C23#57612yes—
348C23#57612yes—
349D4—44no—
350D4—44no—
351D4—44no—
352D4—44no—
353D4—82no—
354D4—82no—
355D4—82no—
356D4—82no—
357C2×C4—28no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—84no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4—82no—
386C2×C4#49514yes—
387C2×C4C2×#12111yescharacteristic
388C2×C4C2×#12311yescharacteristic
389C2×C4#52511yescharacteristic
390C2×C4#52512yes—
391C2×C4#52511yescharacteristic
392C2×C4#52912yes—
393C2×C4#53212yes—
394C2×C4#53511yescharacteristic
395C2×C4#53611yescharacteristic
396C2×C4#54212yes—
397C2×C4#54212yes—
398C2×C4D4211yescharacteristic
399C2×C4#54712yes—
400C2×C4#54812yes—
401C2×C4#55011yescharacteristic
402C2×C4#55112yes—
403C2×C4#55112yes—
404C2×C4#55212yes—
405C8—24no—
406C8—24no—
407C8—44no—
408C22—84no—
409C22—88no—
410C22—88no—
411C22—84no—
412C22—84no—
413C22—162no—
414C22—162no—
415C22—162no—
416C22—162no—
417C22—164no—
418C22—164no—
419C22#239711yescharacteristic
420C22#246112yes—
421C22#247912yes—
422C22#250111yescharacteristic
423C22#254111yescharacteristic
424C22#310214yes—
425C22#312214yes—
426C22C2×#49511yescharacteristic
427C22#315011yescharacteristic
428C22#315511yescharacteristic
429C22#315511yescharacteristic
430C22#327312yes—
431C22#328712yes—
432C22#333914yes—
433C22#334714yes—
434C22#339612yes—
435C22#340312yes—
436C4—24no—
437C4—42no—
438C4—84no—
439C4—84no—
440C4—84no—
441C4—162no—
442C4#246112yes—
443C4#247912yes—
444C4#248211yescharacteristic
445C4#253611yescharacteristic
446C4#263211yescharacteristic
447C4#285411yescharacteristic
448C2—164no—
449C2#1508411yescharacteristic
450C2#2380011yescharacteristic
451C2#2381011yescharacteristic
452C2#2679814yes—
453C2#2750014yes—
454C2#2793211yescharacteristic
455C2#2793511yescharacteristic
456C2#2846311yescharacteristic
457C2#2846711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2092C2semidirect
3. C2×#1926C2semidirect
4. C2×C4×#107C2semidirect
5. #14002C2semidirect
6. #11325C2semidirect
7. #11123C2semidirect
9. #10797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20922
C2×#19702
C2×#19262
C2×C4×#1072
#140022
#113252
#111232
#110222
#107972