Finite Group Databasebeta

← #5084457

Subgroups — #5084457

Order 512.

2755 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2760C211yesmaximal, characteristic, complemented
2C2×#2656C211yesmaximal, characteristic
3#17226C212yesmaximal, complemented
4#17106C213yesmaximal, complemented
5#16270C216yesmaximal, complemented
6#15984C213yesmaximal, complemented
7#15916C213yesmaximal, complemented
8#15673C213yesmaximal, complemented
9#15312C216yesmaximal, complemented
10C22⋊#401C212yesmaximal, complemented
11#14651C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#121C2213yes—
14C22×#121C2213yes—
15#2760—28no—
16#2656—28no—
17C2×#401C2212yes—
18C2×#399C2213yes—
19C2×#399C2211yescharacteristic
20C2×#397C2216yes—
21C2×#395C2213yes—
22C2×#394C2216yes—
23C2×#393C2213yes—
24C2×#392C2213yes—
25C2×#392C2211yescharacteristic
26C2×#386C2216yes—
27C2×#386C2213yes—
28C2×#386C2213yes—
29C2×#381C2213yes—
30C22×#95C2211yescharacteristic
31#1444C2216yes—
32#1443C2216yes—
33#1443C2212yes—
34#1443C2216yes—
35#1442C2216yes—
36#1442C2216yes—
37#1440C2216yes—
38#1440C2213yes—
39#1440C2216yes—
40#1439C2216yes—
41#1439C2216yes—
42#1438C2213yes—
43#1438C2216yes—
44#1438C2216yes—
45#1438C2213yes—
46#1438C2213yes—
47#1437C2213yes—
48#1436C2213yes—
49C22⋊#95C2216yes—
50C22⋊#95C2216yes—
51C22⋊#95C2216yes—
52C23×Q8C2311yescharacteristic
53C23×D4C2313yes—
54C23×D4C2311yescharacteristic
55C23×D4C2313yes—
56C23×D4C2313yes—
57C24×C4C2313yes—
58C24×C4C2313yes—
59C2×#124—46no—
60C2×#121—212no—
61C2×#121—212no—
62C2×#121—46no—
63C2×#121—46no—
64C2×#121—46no—
65C2×#121—46no—
66C22×C4⋊C4C2313yes—
67C22×C4⋊C4C2313yes—
68C22×C4⋊C4C2316yes—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2316yes—
71C22×C22⋊C4C2313yes—
72C22×C22⋊C4C2316yes—
73C22×C22⋊C4C2316yes—
74C22×C22⋊C4C2313yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2313yes—
79C22×C22⋊C4C2316yes—
80C22×C22⋊C4C2313yes—
81C22×C42C2311yescharacteristic
82#401—28no—
83#399—212no—
84#399—24no—
85#397—412no—
86#395—46no—
87#394—412no—
88#393—212no—
89#392—212no—
90#392D418yes—
91#386—224no—
92#386—46no—
93#386—46no—
94#381—212no—
95C2×#95—24no—
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2313yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2313yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2311yescharacteristic
111C25C2411yescharacteristic
112C25C2413yes—
113C25C2411yescharacteristic
114C22×Q8—22no—
115C22×Q8—43no—
116C22×Q8—43no—
117C22×D4—26no—
118C22×D4—26no—
119C22×D4—26no—
120C22×D4—212no—
121C22×D4—26no—
122C22×D4—22no—
123C22×D4—26no—
124C22×D4—43no—
125C22×D4—43no—
126C22×D4—43no—
127C22×D4—43no—
128C22×D4—46no—
129C22×D4—43no—
130C22×D4—46no—
131C22×D4—46no—
132C22×D4—46no—
133C22×D4—46no—
134C22×D4C2×D4112yes—
135C23×C4—26no—
136C23×C4—26no—
137C23×C4—26no—
138C23×C4—26no—
139C23×C4—46no—
140C23×C4—46no—
141C23×C4—46no—
142C23×C4—46no—
143C23×C4—43no—
144C23×C4—43no—
145C23×C4C2412yes—
146C23×C4C2416yes—
147C23×C4C2412yes—
148C23×C4C2416yes—
149C23×C4C2413yes—
150C23×C4C2411yescharacteristic
151C23×C4C2413yes—
152C23×C4C2413yes—
153#124—416no—
154#121—448no—
155#121—448no—
156C2×C4⋊C4—26no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—43no—
160C2×C4⋊C4—43no—
161C2×C4⋊C4—43no—
162C2×C4⋊C4—46no—
163C2×C4⋊C4—46no—
164C2×C4⋊C4—46no—
165C2×C22⋊C4—212no—
166C2×C22⋊C4—212no—
167C2×C22⋊C4—26no—
168C2×C22⋊C4—212no—
169C2×C22⋊C4—26no—
170C2×C22⋊C4—212no—
171C2×C22⋊C4—212no—
172C2×C22⋊C4—43no—
173C2×C22⋊C4—43no—
174C2×C22⋊C4—46no—
175C2×C22⋊C4—43no—
176C2×C22⋊C4—43no—
177C2×C22⋊C4—46no—
178C2×C22⋊C4—46no—
179C2×C22⋊C4—46no—
180C2×C22⋊C4—46no—
181C2×C22⋊C4—46no—
182C2×C22⋊C4—46no—
183C2×C22⋊C4—46no—
184C2×C22⋊C4—46no—
185C2×C22⋊C4—46no—
186C2×C22⋊C4—46no—
187C2×C22⋊C4—43no—
188C2×C22⋊C4—43no—
189C2×C22⋊C4—46no—
190C2×C22⋊C4—46no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4C2×D4112yes—
194C2×C42—43no—
195#95—26no—
196#95—26no—
197#95—212no—
198#95—212no—
199#95—212no—
200#95—212no—
201#95—46no—
202#95—46no—
203#95—46no—
204#95—46no—
205#95—46no—
206#95—46no—
207#95—43no—
208#95—46no—
209#95C2×D414yes—
210C24—21no—
211C24—23no—
212C24—22no—
213C24—26no—
214C24—26no—
215C24—26no—
216C24—23no—
217C24—23no—
218C24—26no—
219C24—26no—
220C24—26no—
221C24—26no—
222C24—21no—
223C24—23no—
224C24—23no—
225C24—22no—
226C24—26no—
227C24C22×D416yes—
228C24H(5,𝔽2)16yes—
229C24C2511yescharacteristic
230C2×Q8—44no—
231C2×Q8—86no—
232C2×Q8—86no—
233C2×D4—412no—
234C2×D4—424no—
235C2×D4—412no—
236C2×D4—412no—
237C2×D4—412no—
238C2×D4—424no—
239C2×D4—412no—
240C2×D4—44no—
241C2×D4—412no—
242C2×D4—86no—
243C2×D4—812no—
244C2×D4—86no—
245C2×D4—812no—
246C2×D4—86no—
247C2×D4—86no—
248C2×D4—86no—
249C2×D4—812no—
250C2×D4—812no—
251C22×C4—23no—
252C22×C4—23no—
253C22×C4—23no—
254C22×C4—23no—
255C22×C4—23no—
256C22×C4—23no—
257C22×C4—26no—
258C22×C4—26no—
259C22×C4—26no—
260C22×C4—26no—
261C22×C4—224no—
262C22×C4—26no—
263C22×C4—22no—
264C22×C4—23no—
265C22×C4—26no—
266C22×C4—26no—
267C22×C4—26no—
268C22×C4—26no—
269C22×C4—26no—
270C22×C4—23no—
271C22×C4—26no—
272C22×C4—26no—
273C22×C4—26no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—23no—
277C22×C4—26no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—26no—
282C22×C4—26no—
283C22×C4—22no—
284C22×C4—26no—
285C22×C4—26no—
286C22×C4—26no—
287C22×C4—412no—
288C22×C4—812no—
289C22×C4—86no—
290C22×C4—86no—
291C22×C4—812no—
292C22×C4—86no—
293C22×C4—86no—
294C22×C4—812no—
295C22×C4—812no—
296C22×C4C22×D416yes—
297C22×C4C22×D412yes—
298C22×C4H(5,𝔽2)16yes—
299C4⋊C4—412no—
300C4⋊C4—412no—
301C4⋊C4—812no—
302C22⋊C4—412no—
303C22⋊C4—424no—
304C22⋊C4—412no—
305C22⋊C4—48no—
306C22⋊C4—48no—
307C22⋊C4—812no—
308C22⋊C4—86no—
309C22⋊C4—812no—
310C22⋊C4—86no—
311C22⋊C4—86no—
312C22⋊C4—812no—
313C22⋊C4—812no—
314C42—44no—
315C23—26no—
316C23—26no—
317C23—212no—
318C23—212no—
319C23—212no—
320C23—26no—
321C23—26no—
322C23—212no—
323C23—26no—
324C23—43no—
325C23—41no—
326C23—43no—
327C23—43no—
328C23—46no—
329C23—43no—
330C23—43no—
331C23—46no—
332C23—43no—
333C23—46no—
334C23—43no—
335C23—46no—
336C23—46no—
337C23—46no—
338C23—46no—
339C23—46no—
340C23—46no—
341C23—46no—
342C23—46no—
343C23—46no—
344C23—46no—
345C23—43no—
346C23—43no—
347C23—41no—
348C23—41no—
349C23—43no—
350C23—43no—
351C23—43no—
352C23—46no—
353C23—46no—
354C23—46no—
355C23#56014yes—
356C23C23×D411yescharacteristic
357C23C2×H(5,𝔽2)13yes—
358C23C2×H(5,𝔽2)13yes—
359C23#58516yes—
360C23#58512yes—
361Q8—88no—
362D4—88no—
363D4—824no—
364D4—824no—
365D4—824no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—212no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—46no—
373C2×C4—41no—
374C2×C4—43no—
375C2×C4—43no—
376C2×C4—43no—
377C2×C4—46no—
378C2×C4—46no—
379C2×C4—46no—
380C2×C4—43no—
381C2×C4—46no—
382C2×C4—46no—
383C2×C4—46no—
384C2×C4—46no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—43no—
388C2×C4—46no—
389C2×C4—46no—
390C2×C4—46no—
391C2×C4—46no—
392C2×C4—43no—
393C2×C4—46no—
394C2×C4—46no—
395C2×C4—46no—
396C2×C4—46no—
397C2×C4—46no—
398C2×C4—46no—
399C2×C4—46no—
400C2×C4—46no—
401C2×C4—46no—
402C2×C4—46no—
403C2×C4—46no—
404C2×C4—824no—
405C2×C4—824no—
406C22—28no—
407C22—24no—
408C22—46no—
409C22—46no—
410C22—412no—
411C22—46no—
412C22—412no—
413C22—412no—
414C22—46no—
415C22—46no—
416C22—86no—
417C22—86no—
418C22—86no—
419C22—86no—
420C22—82no—
421C22—83no—
422C22—83no—
423C22—86no—
424C22—81no—
425C22#348413yes—
426C22#348513yes—
427C22#350011yescharacteristic
428C22C2×#56011yescharacteristic
429C22#353313yes—
430C22#354313yes—
431C22#354513yes—
432C22#355116yes—
433C22#356016yes—
434C22#356016yes—
435C4—46no—
436C4—46no—
437C4—42no—
438C4—82no—
439C4—86no—
440C4—86no—
441C4—83no—
442C4—82no—
443C2—44no—
444C2—86no—
445C2—161no—
446C2#4632411yescharacteristic
447C2#4642713yes—
448C2#4877313yes—
449C2#5276512yes—
450C2#5628616yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2656C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2760C2semidirect
3. #17226C2semidirect
4. #17106C2semidirect
5. #16270C2semidirect
6. #15984C2semidirect
7. #15916C2semidirect
8. #15673C2semidirect
9. #15312C2semidirect
10. C22⋊#401C2semidirect
11. #14651C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27602
C2×#26562
#172262
#171062
#162702
#159842
#159162
#156732
#153122
C22⋊#4012
#146512