Finite Group Databasebeta

← #4028495

Subgroups — #4028495

Order 512.

3864 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2680C213yesmaximal, complemented
2C2×#2406C213yesmaximal
3#17072C211yesmaximal, characteristic, complemented
4#16522C213yesmaximal, complemented
5#15298C216yesmaximal, complemented
6#15269C216yesmaximal, complemented
7#14720C213yesmaximal, complemented
8#14224C213yesmaximal, complemented
9#13880C213yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×#123C2216yes—
12C22×C22⋊Q8C2213yes—
13C23×C4⋊C4C2213yes—
14#2680—224no—
15#2406—224no—
16C2×#398C2216yes—
17C2×#397C2216yes—
18C2×#395C2213yes—
19C2×#395C2213yes—
20C2×#395C2213yes—
21C2×#395C2211yescharacteristic
22C2×#394C2213yes—
23C2×#394C2213yes—
24C2×#388C2216yes—
25C2×#386C2216yes—
26C2×#385C2216yes—
27C2×#385C2216yes—
28C2×#385C2216yes—
29C2×#384C2213yes—
30C2×#384C2211yescharacteristic
31C2×#381C2213yes—
32C2×#380C2213yes—
33C2×C4×C22⋊C4C2213yes—
34#1444C2216yes—
35#1444C2216yes—
36#1444C2216yes—
37#1443C2212yes—
38#1442C2216yes—
39#1441C2213yes—
40#1441C2213yes—
41#1441C2216yes—
42#1440C2216yes—
43#1439C2216yes—
44C4⋊#95C2213yes—
45C4⋊#95C2213yes—
46#1434C2216yes—
47C22⋊#95C2216yes—
48C22⋊#95C2213yes—
49C22×#41—212no—
50C22×#41—42no—
51C22×#41—46no—
52C23×Q8C2311yescharacteristic
53C23×Q8C2311yescharacteristic
54C23×D4C2313yes—
55C24×C4C2313yes—
56C24×C4C2313yes—
57C24×C4C2313yes—
58C2×#123—224no—
59C2×#123—224no—
60C2×#123—412no—
61C2×C22⋊Q8—224no—
62C2×C22⋊Q8—46no—
63C22×C4⋊C4—212no—
64C22×C4⋊C4—46no—
65C22×C4⋊C4—46no—
66C22×C4⋊C4C2316yes—
67C22×C4⋊C4C2316yes—
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C2316yes—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2313yes—
72C22×C4⋊C4C2313yes—
73C22×C4⋊C4C2313yes—
74C22×C4⋊C4C2313yes—
75C22×C4⋊C4C2313yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2316yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2316yes—
80C22×C22⋊C4C2316yes—
81C22×C42C2313yes—
82C22×C42C2311yescharacteristic
83#398—224no—
84#397—224no—
85#395—212no—
86#395—212no—
87#395—212no—
88#395—42no—
89#394—212no—
90#394D4124yes—
91#388—224no—
92#386—224no—
93#385—224no—
94#385—224no—
95#385—224no—
96#384—212no—
97#384—42no—
98#381—212no—
99#380—46no—
100C4×C22⋊C4—212no—
101C2×#95C2313yes—
102C2×#95C2313yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2313yes—
112C2×#95C2316yes—
113C2×#95C2312yes—
114C2×#95C2313yes—
115C25C2413yes—
116C2×#41—448no—
117C2×#41—448no—
118C2×#41—416no—
119C22×Q8—26no—
120C22×Q8—26no—
121C22×Q8—41no—
122C22×Q8—43no—
123C22×Q8—43no—
124C22×Q8—41no—
125C22×D4—212no—
126C22×D4—212no—
127C22×D4—43no—
128C22×D4—43no—
129C22×D4C2×D4112yes—
130C23×C4—26no—
131C23×C4—26no—
132C23×C4—26no—
133C23×C4—26no—
134C23×C4—26no—
135C23×C4—26no—
136C23×C4—212no—
137C23×C4—212no—
138C23×C4—26no—
139C23×C4—26no—
140C23×C4—26no—
141C23×C4—26no—
142C23×C4—43no—
143C23×C4—43no—
144C23×C4—46no—
145C23×C4—43no—
146C23×C4—46no—
147C23×C4C2416yes—
148C23×C4C2413yes—
149C23×C4C2416yes—
150C23×C4C2413yes—
151C23×C4C2413yes—
152C23×C4C2411yescharacteristic
153C23×C4C2413yes—
154C23×C4C2413yes—
155#123—496no—
156C22⋊Q8—448no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—26no—
160C2×C4⋊C4—212no—
161C2×C4⋊C4—212no—
162C2×C4⋊C4—212no—
163C2×C4⋊C4—212no—
164C2×C4⋊C4—212no—
165C2×C4⋊C4—212no—
166C2×C4⋊C4—212no—
167C2×C4⋊C4—26no—
168C2×C4⋊C4—26no—
169C2×C4⋊C4—26no—
170C2×C4⋊C4—26no—
171C2×C4⋊C4—43no—
172C2×C4⋊C4—43no—
173C2×C4⋊C4—43no—
174C2×C4⋊C4—43no—
175C2×C4⋊C4—43no—
176C2×C4⋊C4—43no—
177C2×C4⋊C4—43no—
178C2×C4⋊C4—46no—
179C2×C4⋊C4—46no—
180C2×C4⋊C4—46no—
181C2×C4⋊C4—448no—
182C2×C4⋊C4—43no—
183C2×C4⋊C4—43no—
184C2×C4⋊C4—43no—
185C2×C4⋊C4—43no—
186C2×C4⋊C4C2×D4112yes—
187C2×C22⋊C4—212no—
188C2×C22⋊C4—212no—
189C2×C22⋊C4—212no—
190C2×C22⋊C4—212no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—212no—
193C2×C22⋊C4—212no—
194C2×C22⋊C4—212no—
195C2×C22⋊C4—212no—
196C2×C22⋊C4—212no—
197C2×C22⋊C4—212no—
198C2×C22⋊C4—46no—
199C2×C22⋊C4—46no—
200C2×C22⋊C4C2×D4124yes—
201C2×C22⋊C4C2×D4124yes—
202C2×C42—26no—
203C2×C42—26no—
204C2×C42—26no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42#4114yescomplemented
208#95—212no—
209#95—26no—
210#95—212no—
211#95—212no—
212#95—212no—
213#95—212no—
214#95—212no—
215#95—26no—
216#95—212no—
217#95—212no—
218#95—42no—
219#95—46no—
220#95—43no—
221#95C2×D4112yes—
222C24—23no—
223C24—23no—
224C24—26no—
225C24—23no—
226C24—26no—
227C24—26no—
228C24—23no—
229C24—23no—
230C24—23no—
231C24C22×D4112yes—
232C24H(5,𝔽2)16yes—
233C24C2511yescharacteristic
234#41—4128no—
235C2×Q8—412no—
236C2×Q8—412no—
237C2×Q8—412no—
238C2×Q8—44no—
239C2×Q8—86no—
240C2×Q8—82no—
241C2×D4—248no—
242C2×D4—412no—
243C2×D4—412no—
244C2×D4—424no—
245C2×D4—412no—
246C22×C4—23no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—224no—
251C22×C4—224no—
252C22×C4—23no—
253C22×C4—23no—
254C22×C4—26no—
255C22×C4—26no—
256C22×C4—26no—
257C22×C4—23no—
258C22×C4—26no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—23no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—23no—
265C22×C4—23no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—26no—
269C22×C4—26no—
270C22×C4—26no—
271C22×C4—26no—
272C22×C4—23no—
273C22×C4—23no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—23no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—23no—
281C22×C4—23no—
282C22×C4—26no—
283C22×C4—26no—
284C22×C4—26no—
285C22×C4—26no—
286C22×C4—412no—
287C22×C4—412no—
288C22×C4—412no—
289C22×C4—412no—
290C22×C4—424no—
291C22×C4—424no—
292C22×C4—412no—
293C22×C4—412no—
294C22×C4—424no—
295C22×C4—412no—
296C22×C4—412no—
297C22×C4—412no—
298C22×C4—412no—
299C22×C4—812no—
300C22×C4—86no—
301C22×C4C22×D4112yes—
302C22×C4C22×D416yes—
303C22×C4C22×D416yes—
304C22×C4C22×D416yes—
305C22×C4C2×#4116yes—
306C22×C4H(5,𝔽2)12yes—
307C22×C42−1+416yes—
308C22×C42−1+416yes—
309C4⋊C4—412no—
310C4⋊C4—412no—
311C4⋊C4—412no—
312C4⋊C4—412no—
313C4⋊C4—424no—
314C4⋊C4—424no—
315C4⋊C4—424no—
316C4⋊C4—412no—
317C4⋊C4—86no—
318C4⋊C4—86no—
319C22⋊C4—248no—
320C22⋊C4—248no—
321C22⋊C4—424no—
322C22⋊C4—424no—
323C22⋊C4—424no—
324C42—44no—
325C42—412no—
326C23—212no—
327C23—212no—
328C23—26no—
329C23—212no—
330C23—212no—
331C23—26no—
332C23—212no—
333C23—212no—
334C23—26no—
335C23—26no—
336C23—26no—
337C23—26no—
338C23—43no—
339C23—43no—
340C23—43no—
341C23—46no—
342C23—46no—
343C23—46no—
344C23—43no—
345C23—46no—
346C23—46no—
347C23—46no—
348C23D42112yes—
349C23C23×D413yes—
350C23C22×#4111yescharacteristic
351C23C2×H(5,𝔽2)11yescharacteristic
352C23C2×H(5,𝔽2)13yes—
353C23C2×2−1+413yes—
354C23C2×2−1+413yes—
355C23#58511yescharacteristic
356Q8—416no—
357Q8—88no—
358D4—448no—
359C2×C4—26no—
360C2×C4—26no—
361C2×C4—26no—
362C2×C4—26no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—212no—
366C2×C4—212no—
367C2×C4—212no—
368C2×C4—212no—
369C2×C4—26no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—212no—
375C2×C4—26no—
376C2×C4—448no—
377C2×C4—448no—
378C2×C4—43no—
379C2×C4—43no—
380C2×C4—43no—
381C2×C4—43no—
382C2×C4—46no—
383C2×C4—43no—
384C2×C4—43no—
385C2×C4—46no—
386C2×C4—46no—
387C2×C4—46no—
388C2×C4—46no—
389C2×C4—41no—
390C2×C4—43no—
391C2×C4—43no—
392C2×C4—43no—
393C2×C4—46no—
394C2×C4—46no—
395C2×C4—46no—
396C2×C4—43no—
397C2×C4—43no—
398C2×C4—46no—
399C2×C4—46no—
400C2×C4—46no—
401C2×C4—46no—
402C2×C4—46no—
403C2×C4—43no—
404C2×C4—46no—
405C2×C4—46no—
406C2×C4—824no—
407C2×C4#547112yes—
408C22—212no—
409C22—212no—
410C22—212no—
411C22—46no—
412C22—412no—
413C22—46no—
414C22—46no—
415C22—412no—
416C22—412no—
417C22—46no—
418C22—412no—
419C22#345911yescharacteristic
420C22C2×D4213yes—
421C22C2×#54713yes—
422C22#348413yes—
423C22#348913yes—
424C22#349213yes—
425C22#349916yes—
426C22#353211yescharacteristic
427C22#353513yes—
428C22#353813yes—
429C22#354413yes—
430C22#354413yes—
431C4—212no—
432C4—42no—
433C4—46no—
434C4—46no—
435C4—46no—
436C4—412no—
437C4—83no—
438C4—86no—
439C2—412no—
440C2#3776013yes—
441C2#4475813yes—
442C2#4780211yescharacteristic
443C2#5154213yes—
444C2#5875011yescharacteristic
445C2#5899011yescharacteristic
446C2#5931313yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2680C2semidirect
3. #17072C2semidirect
4. #16522C2semidirect
5. #15298C2semidirect
6. #15269C2semidirect
7. #14720C2semidirect
8. #14224C2semidirect
9. #13880C2semidirect
207. C2×C42#41semidirect

Maximal subgroups

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

subgroupindex
C2×#26802
C2×#24062
#170722
#165222
#152982
#152692
#147202
#142242
#138802