Finite Group Databasebeta

← #470093

Subgroups — #470093

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic
3#14697C212yesmaximal
4#11328C212yesmaximal, complemented
5#11315C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11220C212yesmaximal, complemented
8#11200C212yesmaximal
9#11095C211yesmaximal, characteristic
10C23×C42C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#385—22no—
18C2×#380C2212yes—
19C2×#380C2212yes—
20#2076—24no—
21#1817—24no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1436—22no—
33#1436—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C22⋊#95—22no—
37C24×C4D411yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C42—22no—
52C22×C42—22no—
53C22×C42—41no—
54C22×C42—41no—
55C22×C42—41no—
56C22×C42—41no—
57C22×C42D412yes—
58C22×C42C2312yes—
59C2×#107—22no—
60C2×#107—41no—
61C2×#107—41no—
62C2×#106—24no—
63C2×#106—41no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8D414yes—
70#389—42no—
71#389—42no—
72#389—42no—
73#389—42no—
74#385—48no—
75#380—24no—
76#380—24no—
77#380—24no—
78#380—24no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87#340—22no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—41no—
93#340—42no—
94#340—42no—
95#336—24no—
96#336—24no—
97C25C2×D411yescharacteristic
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—42no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4#4112yes—
125C23×C4C2411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
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⋊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—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4#4114yes—
172C2×C4⋊C4#4114yes—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—44no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42—44no—
192#107—44no—
193#106—44no—
194C22⋊C8—44no—
195C22⋊C8—44no—
196#95—28no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204C24—22no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24#12112yes—
212C24#13714yes—
213C24C22×D411yescharacteristic
214C22×C4—28no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—44no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
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—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4#12112yes—
287C22×C4#13514yes—
288C22×C4AGL(1,ℤ/8)14yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4H(5,𝔽2)11yescharacteristic
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C4H(5,𝔽2)11yescharacteristic
297C22×C42−1+412yes—
298C22×C42−1+412yes—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
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—42no—
332C4⋊C4—88no—
333C4⋊C4—88no—
334C22⋊C4—416no—
335C22⋊C4—88no—
336C42—48no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—24no—
345C23—24no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—22no—
350C23—24no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—42no—
356C23—41no—
357C23—42no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23#47014yes—
362C23C2×#12111yescharacteristic
363C23#52512yes—
364C23#53411yescharacteristic
365C23#53411yescharacteristic
366C23#53612yes—
367C23C2×#13512yes—
368C23C2×AGL(1,ℤ/8)12yes—
369C23C2×#13712yes—
370C23#57512yes—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—416no—
380C2×C4—416no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
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×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#53411yescharacteristic
427C2×C4#53511yescharacteristic
428C2×C4#53811yescharacteristic
429C2×C4#53811yescharacteristic
430C2×C4#53912yes—
431C2×C4#53912yes—
432C2×C4#54212yes—
433C2×C4#54212yes—
434C2×C4#54712yes—
435C2×C4#54712yes—
436C2×C4#55111yescharacteristic
437C2×C4#55212yes—
438C2×C4#55212yes—
439C2×C4#55312yes—
440C2×C4#55911yescharacteristic
441C2×C4#56311yescharacteristic
442C2×C4#56312yes—
443C2×C4#56311yescharacteristic
444C8—42no—
445C8—42no—
446C8—82no—
447C22—28no—
448C22—24no—
449C22—44no—
450C22—44no—
451C22—42no—
452C22—42no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22#241312yes—
457C22#260011yescharacteristic
458C22#261711yescharacteristic
459C22#263812yes—
460C22#281711yescharacteristic
461C22#305011yescharacteristic
462C22C2×#47011yescharacteristic
463C22#305512yes—
464C22#310212yes—
465C22#310612yes—
466C22#311012yes—
467C22#311712yes—
468C22#319012yes—
469C22#319012yes—
470C22#319211yescharacteristic
471C22#319412yes—
472C22#319711yescharacteristic
473C22#322812yes—
474C22#323312yes—
475C22#323612yes—
476C22#323912yes—
477C4—42no—
478C4—44no—
479C4—44no—
480C4—84no—
481C4—84no—
482C4—162no—
483C4—162no—
484C4#271711yescharacteristic
485C4#271812yes—
486C4#272411yescharacteristic
487C4#276411yescharacteristic
488C4#276512yes—
489C4#277111yescharacteristic
490C2—44no—
491C2#1667511yescharacteristic
492C2#2460512yes—
493C2#2460612yes—
494C2#2941412yes—
495C2#2973311yescharacteristic
496C2#2974211yescharacteristic
497C2#3017212yes—
498C2#3017412yes—
499C2#3278912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11328C2semidirect
5. #11315C2semidirect
6. #11292C2semidirect
7. #11220C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#18172
#146972
#113282
#113152
#112922
#112202
#112002
#110952