Finite Group Databasebeta

← #1393932

Subgroups — #1393932

Order 512.

5526 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#550C211yesmaximal, characteristic
2C2×#2452C213yesmaximal, complemented
3C2×#2616C212yesmaximal, complemented
4#16261C216yesmaximal, complemented
5#16253C216yesmaximal, complemented
6#15158C211yesmaximal, characteristic, complemented
7#15140C213yesmaximal, complemented
8#15127C213yesmaximal, complemented
9#14889C212yesmaximal, complemented
10#13920C213yesmaximal, complemented
11C4×#395C211yesmaximal, characteristic, complemented
12C23×#41C2212yes—
13C2×#550—224no—
14C22×C4⋊Q8C2211yescharacteristic
15C22×#127C2213yes—
16C22×C22⋊Q8C2216yes—
17C22×#121C2216yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2213yes—
20#2616—216no—
21#2452—224no—
22C2×#396C2216yes—
23C2×#395C2212yes—
24C2×#395C2211yescharacteristic
25C2×#394C2216yes—
26C2×#394C2216yes—
27C2×#394C2216yes—
28C2×#394C2216yes—
29C2×#393C2216yes—
30C2×#390C2213yes—
31C2×#390C2213yes—
32C2×#390C2216yes—
33C2×#389C2213yes—
34C2×#389C2213yes—
35C2×#385C2216yes—
36C2×#380C2216yes—
37C2×C4×C4⋊C4C2213yes—
38#1446C2211yescharacteristic
39#1444C2213yes—
40#1441C2213yes—
41#1441C2216yes—
42#1439C22112yes—
43#1438C2216yes—
44#1438C2216yes—
45#1438C22112yes—
46C4⋊#95C2213yes—
47C22⋊#95C2216yes—
48C4×#95C2213yes—
49C22×#41—28no—
50C22×#41—224no—
51C22×#41—412no—
52C23×Q8C2311yescharacteristic
53C23×Q8C2312yes—
54C23×D4C2313yes—
55C23×D4C2316yes—
56C23×D4C2316yes—
57C24×C4C2316yes—
58#550—2128no—
59C2×C4⋊Q8—212no—
60C2×#127—224no—
61C2×#127—212no—
62C2×C22⋊Q8—224no—
63C2×C22⋊Q8—224no—
64C2×C22⋊Q8—412no—
65C2×#121—224no—
66C2×#121—248no—
67C2×C4×Q8—212no—
68C2×C4×D4—224no—
69C2×C4×D4D4124yes—
70C22×C4⋊C4C2316yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2313yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2313yes—
75C22×C4⋊C4C2313yes—
76C22×C4⋊C4C2316yes—
77C22×C4⋊C4C2313yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4C2316yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2316yes—
82C22×C22⋊C4C2316yes—
83C22×C22⋊C4C2316yes—
84C22×C22⋊C4C23112yes—
85C22×C22⋊C4C23112yes—
86C22×C42C2311yescharacteristic
87C22×C42C2313yes—
88C22×C42C2313yes—
89#396—224no—
90#395—28no—
91#395D418yescomplemented
92#394—224no—
93#394—224no—
94#394—412no—
95#394—412no—
96#393—224no—
97#390—212no—
98#390—212no—
99#390—412no—
100#389—212no—
101#389—212no—
102#385—224no—
103#380—224no—
104C4×C4⋊C4—212no—
105C2×#95C2313yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2313yes—
109C2×#95C2316yes—
110C2×#95C2313yes—
111C2×#95C2316yes—
112C2×#95C2316yes—
113C2×#95C2313yes—
114C25C2416yes—
115C2×#41—264no—
116C2×#41—496no—
117C2×#41—496no—
118C22×Q8—26no—
119C22×Q8—24no—
120C22×Q8—212no—
121C22×Q8—26no—
122C22×Q8—46no—
123C22×Q8C2×D414yes—
124C22×D4—224no—
125C22×D4—224no—
126C22×D4—212no—
127C22×D4—212no—
128C22×D4—212no—
129C22×D4—212no—
130C22×D4—26no—
131C22×D4—212no—
132C22×D4—212no—
133C22×D4—26no—
134C22×D4—412no—
135C22×D4—412no—
136C22×D4—46no—
137C22×D4—46no—
138C22×D4C2×D4112yes—
139C23×C4—224no—
140C23×C4—212no—
141C23×C4—212no—
142C23×C4—412no—
143C23×C4C2×D4124yes—
144C23×C4C2412yes—
145C23×C4C2416yes—
146C23×C4C2413yes—
147C23×C4C2413yes—
148C23×C4C2411yescharacteristic
149C23×C4C2413yes—
150C23×C4C2411yescharacteristic
151C23×C4C2412yes—
152C23×C4C2413yes—
153C23×C4C2411yescharacteristic
154C4⋊Q8—416no—
155#127—296no—
156C22⋊Q8—496no—
157#121—2192no—
158C4×Q8—232no—
159C4×D4—296no—
160C2×C4⋊C4—212no—
161C2×C4⋊C4—26no—
162C2×C4⋊C4—26no—
163C2×C4⋊C4—26no—
164C2×C4⋊C4—212no—
165C2×C4⋊C4—212no—
166C2×C4⋊C4—212no—
167C2×C4⋊C4—212no—
168C2×C4⋊C4—212no—
169C2×C4⋊C4—26no—
170C2×C4⋊C4—26no—
171C2×C4⋊C4—26no—
172C2×C4⋊C4—212no—
173C2×C4⋊C4—26no—
174C2×C4⋊C4—26no—
175C2×C4⋊C4—26no—
176C2×C4⋊C4—43no—
177C2×C4⋊C4—46no—
178C2×C4⋊C4C2×D4112yes—
179C2×C4⋊C4C2×D4112yes—
180C2×C22⋊C4—224no—
181C2×C22⋊C4—212no—
182C2×C22⋊C4—212no—
183C2×C22⋊C4—212no—
184C2×C22⋊C4—212no—
185C2×C22⋊C4—212no—
186C2×C22⋊C4—212no—
187C2×C22⋊C4—212no—
188C2×C22⋊C4—224no—
189C2×C22⋊C4—46no—
190C2×C22⋊C4—46no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—412no—
193C2×C22⋊C4—412no—
194C2×C22⋊C4—412no—
195C2×C22⋊C4—412no—
196C2×C22⋊C4—412no—
197C2×C22⋊C4—412no—
198C2×C22⋊C4C2×D4124yes—
199C2×C42—212no—
200C2×C42—26no—
201C2×C42—26no—
202C2×C42—26no—
203C2×C42—26no—
204C2×C42C2×D4112yes—
205#95—26no—
206#95—26no—
207#95—212no—
208#95—212no—
209#95—212no—
210#95—26no—
211#95—46no—
212#95—46no—
213#95C2×D4112yes—
214C24—26no—
215C24—212no—
216C24—26no—
217C24—26no—
218C24—26no—
219C24—212no—
220C24—212no—
221C24—26no—
222C24—26no—
223C24—212no—
224C24C22×D4112yes—
225C24C2511yescharacteristic
226#41—4256no—
227C2×Q8—28no—
228C2×Q8—224no—
229C2×Q8—224no—
230C2×Q8—424no—
231C2×Q8—48no—
232C2×Q8—812no—
233C2×D4—248no—
234C2×D4—248no—
235C2×D4—224no—
236C2×D4—224no—
237C2×D4—224no—
238C2×D4—448no—
239C2×D4—448no—
240C2×D4—424no—
241C2×D4—424no—
242C2×D4—424no—
243C2×D4—424no—
244C2×D4—812no—
245C2×D4—812no—
246C2×D4—824no—
247C2×D4—824no—
248C22×C4—26no—
249C22×C4—22no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—23no—
253C22×C4—23no—
254C22×C4—23no—
255C22×C4—26no—
256C22×C4—23no—
257C22×C4—23no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—26no—
264C22×C4—23no—
265C22×C4—23no—
266C22×C4—248no—
267C22×C4—248no—
268C22×C4—23no—
269C22×C4—26no—
270C22×C4—26no—
271C22×C4—26no—
272C22×C4—22no—
273C22×C4—26no—
274C22×C4—26no—
275C22×C4—23no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—26no—
279C22×C4—23no—
280C22×C4—26no—
281C22×C4—26no—
282C22×C4—26no—
283C22×C4—424no—
284C22×C4—448no—
285C22×C4—448no—
286C22×C4C22×D416yes—
287C22×C4C22×D416yes—
288C22×C4C22×D412yes—
289C22×C4C22×D4112yes—
290C22×C4C22×D416yes—
291C22×C4C22×D416yes—
292C22×C4C22×D416yes—
293C22×C4H(5,𝔽2)12yes—
294C22×C4H(5,𝔽2)16yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C42−1+416yes—
297C4⋊C4—224no—
298C4⋊C4—224no—
299C4⋊C4—412no—
300C4⋊C4—424no—
301C4⋊C4—48no—
302C4⋊C4—44no—
303C4⋊C4—412no—
304C4⋊C4—424no—
305C4⋊C4#120116yes—
306C22⋊C4—248no—
307C22⋊C4—424no—
308C22⋊C4—424no—
309C22⋊C4—424no—
310C22⋊C4—824no—
311C22⋊C4—812no—
312C22⋊C4—824no—
313C42—224no—
314C42—44no—
315C42—412no—
316C23—212no—
317C23—212no—
318C23—212no—
319C23—212no—
320C23—212no—
321C23—212no—
322C23—212no—
323C23—412no—
324C23—412no—
325C23—412no—
326C23—412no—
327C23—412no—
328C23—46no—
329C23—412no—
330C23—46no—
331C23—412no—
332C23—412no—
333C23—46no—
334C23—46no—
335C23—412no—
336C23—412no—
337C23—412no—
338C23—412no—
339C23C23×D413yes—
340C23C23×D411yescharacteristic
341C23C2×H(5,𝔽2)11yescharacteristic
342C23C2×H(5,𝔽2)13yes—
343C23C2×H(5,𝔽2)11yescharacteristic
344C23C2×2−1+413yes—
345C23#58513yes—
346Q8—232no—
347Q8—816no—
348D4—496no—
349D4—448no—
350D4—848no—
351C2×C4—26no—
352C2×C4—212no—
353C2×C4—212no—
354C2×C4—212no—
355C2×C4—26no—
356C2×C4—212no—
357C2×C4—26no—
358C2×C4—26no—
359C2×C4—26no—
360C2×C4—26no—
361C2×C4—26no—
362C2×C4—26no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—496no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—43no—
369C2×C4—43no—
370C2×C4—43no—
371C2×C4—46no—
372C2×C4—43no—
373C2×C4—46no—
374C2×C4—43no—
375C2×C4—43no—
376C2×C4—42no—
377C2×C4—43no—
378C2×C4—46no—
379C2×C4—46no—
380C2×C4—46no—
381C2×C4—43no—
382C2×C4—46no—
383C2×C4—46no—
384C2×C4—46no—
385C2×C4—46no—
386C2×C4—43no—
387C2×C4—46no—
388C2×C4—46no—
389C2×C4—46no—
390C2×C4—46no—
391C2×C4C2×#12018yes—
392C2×C4C2×#12014yes—
393C2×C4#536112yes—
394C2×C4#537112yes—
395C2×C4D42112yes—
396C22—412no—
397C22—412no—
398C22—412no—
399C22—412no—
400C22—412no—
401C22—412no—
402C22—412no—
403C22—812no—
404C22—812no—
405C22—812no—
406C22—812no—
407C22C22×#12011yescharacteristic
408C22C2×#53613yes—
409C22C2×#53713yes—
410C22C2×D4213yes—
411C22#348413yes—
412C22#348813yes—
413C22#349713yes—
414C22#349811yescharacteristic
415C22#350013yes—
416C22#354613yes—
417C22#354813yes—
418C22#355416yes—
419C4—28no—
420C4—212no—
421C4—46no—
422C4—42no—
423C4—46no—
424C4—42no—
425C4—86no—
426C4—83no—
427C4—82no—
428C4#261618yescomplemented
429C2—812no—
430C2C2×#261611yescharacteristic
431C2#3608211yescharacteristic
432C2D4×#12011yescharacteristic
433C2#3934113yes—
434C2#3936313yes—
435C2#4242913yes—
436C2#4247913yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#550C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2452C2semidirect
3. C2×#2616C2semidirect
4. #16261C2semidirect
5. #16253C2semidirect
6. #15158C2semidirect
7. #15140C2semidirect
8. #15127C2semidirect
9. #14889C2semidirect
10. #13920C2semidirect
11. C4×#395C2semidirect
91. #395D4semidirect
428. C4#2616semidirect

Maximal subgroups

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

subgroupindex
C22×#5502
C2×#24522
C2×#26162
#162612
#162532
#151582
#151402
#151272
#148892
#139202
C4×#3952