Finite Group Databasebeta

← #4462772

Subgroups — #4462772

Order 512.

2518 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
1C2×#2658C211yesmaximal, characteristic, complemented
2C2×#2401C211yesmaximal, characteristic, complemented
3#17226C212yesmaximal, complemented
4#17204C212yesmaximal, complemented
5#17161C216yesmaximal, complemented
6#17156C216yesmaximal, complemented
7#15217C213yesmaximal, complemented
8#15215C213yesmaximal
9#15180C213yesmaximal, complemented
10#15178C213yesmaximal, complemented
11#14350C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2213yes—
13C22×#124C2213yes—
14C22×C22⋊Q8C2213yes—
15C22×#121C2211yescharacteristic
16#2658—28no—
17#2401—28no—
18C2×#401C2211yescharacteristic
19C2×#400C2213yes—
20C2×#399C2216yes—
21C2×#399C2212yes—
22C2×#397C2213yes—
23C2×#396C2211yescharacteristic
24C2×#391C2213yes—
25C2×#391C2213yes—
26C2×#391C2213yes—
27C2×#390C2213yes—
28C2×C4×C4⋊C4C2213yes—
29C2×C4×C22⋊C4C2213yes—
30C2×#376C2211yescharacteristic
31#1444C2216yes—
32#1444C2216yes—
33#1443C2216yes—
34#1443C2212yes—
35#1442C2216yes—
36#1442C2216yes—
37#1441C2216yes—
38#1441C2216yes—
39#1440C2216yes—
40#1440C2216yes—
41#1439C2216yes—
42#1439C2216yes—
43#1439C2216yes—
44#1439C2216yes—
45#1439C2216yes—
46#1439C2216yes—
47#1438C2216yes—
48#1438C2216yes—
49C4⋊#95C2213yes—
50C4⋊#95C2213yes—
51C23×Q8C2313yes—
52C23×D4C2311yescharacteristic
53C23×D4C2312yes—
54C24×C4C2311yescharacteristic
55C2×C4⋊Q8—212no—
56C2×C4⋊Q8—46no—
57C2×C4⋊Q8—46no—
58C2×#124—212no—
59C2×#124—46no—
60C2×#124—46no—
61C2×C22⋊Q8—46no—
62C2×C22⋊Q8—46no—
63C2×C22⋊Q8—46no—
64C2×#121—46no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2313yes—
67C22×C4⋊C4C2313yes—
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C2316yes—
70C22×C4⋊C4C2316yes—
71C22×C4⋊C4C2316yes—
72C22×C22⋊C4C2313yes—
73C22×C22⋊C4C2313yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2316yes—
80C22×C42C2313yes—
81C22×C42C2313yes—
82#401—24no—
83#400—212no—
84#399—28no—
85#399—224no—
86#397—212no—
87#396—24no—
88#391—46no—
89#391—46no—
90#391—46no—
91#390—46no—
92C4×C4⋊C4—212no—
93C4×C22⋊C4—212no—
94#376D418yescomplemented
95C2×#95C2313yes—
96C2×#95C2313yes—
97C2×#95C2313yes—
98C2×#95C2313yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2311yescharacteristic
110C2×#95C2313yes—
111C25C2412yes—
112C25C2411yescharacteristic
113C22×Q8—26no—
114C22×Q8—26no—
115C22×Q8—26no—
116C22×Q8—26no—
117C22×Q8—26no—
118C22×Q8—26no—
119C22×Q8—26no—
120C22×D4—24no—
121C22×D4—22no—
122C22×D4—26no—
123C22×D4—26no—
124C22×D4—46no—
125C22×D4—46no—
126C23×C4—22no—
127C23×C4—26no—
128C23×C4—26no—
129C23×C4C2416yes—
130C23×C4C2416yes—
131C23×C4C2413yes—
132C23×C4C2411yescharacteristic
133C23×C4C2412yes—
134C23×C4C2413yes—
135C23×C4C2413yes—
136C23×C4C2413yes—
137C23×C4C2411yescharacteristic
138C4⋊Q8—448no—
139#124—448no—
140C22⋊Q8—448no—
141#121—416no—
142C2×C4⋊C4—212no—
143C2×C4⋊C4—212no—
144C2×C4⋊C4—26no—
145C2×C4⋊C4—26no—
146C2×C4⋊C4—26no—
147C2×C4⋊C4—26no—
148C2×C4⋊C4—26no—
149C2×C4⋊C4—26no—
150C2×C4⋊C4—26no—
151C2×C4⋊C4—26no—
152C2×C4⋊C4—43no—
153C2×C4⋊C4—43no—
154C2×C4⋊C4—46no—
155C2×C4⋊C4—46no—
156C2×C4⋊C4—46no—
157C2×C4⋊C4—46no—
158C2×C4⋊C4—46no—
159C2×C4⋊C4—46no—
160C2×C4⋊C4—46no—
161C2×C22⋊C4—26no—
162C2×C22⋊C4—212no—
163C2×C22⋊C4—212no—
164C2×C22⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—212no—
168C2×C22⋊C4—43no—
169C2×C22⋊C4—43no—
170C2×C22⋊C4—46no—
171C2×C22⋊C4—46no—
172C2×C22⋊C4—46no—
173C2×C22⋊C4—46no—
174C2×C22⋊C4—46no—
175C2×C22⋊C4—46no—
176C2×C22⋊C4—46no—
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×C42—26no—
184C2×C42—26no—
185C2×C42—26no—
186C2×C42—26no—
187C2×C42—26no—
188C2×C42C2×D4112yes—
189#95—26no—
190#95—26no—
191#95—26no—
192#95—26no—
193#95—46no—
194#95—46no—
195#95—46no—
196#95—46no—
197#95—46no—
198#95—46no—
199#95—46no—
200#95—46no—
201#95—46no—
202#95—46no—
203#95C2×D414yes—
204#95C2×D4112yes—
205C24—21no—
206C24—23no—
207C24—23no—
208C24—23no—
209C24—21no—
210C24—23no—
211C24—22no—
212C24—26no—
213C24—26no—
214C24—26no—
215C24—22no—
216C24—22no—
217C24—26no—
218C242−1+412yes—
219C24C2511yescharacteristic
220C2×Q8—412no—
221C2×Q8—412no—
222C2×Q8—412no—
223C2×Q8—412no—
224C2×Q8—412no—
225C2×Q8—412no—
226C2×Q8—412no—
227C2×D4—48no—
228C2×D4—412no—
229C2×D4—44no—
230C2×D4—412no—
231C2×D4—812no—
232C2×D4—812no—
233C22×C4—23no—
234C22×C4—23no—
235C22×C4—23no—
236C22×C4—23no—
237C22×C4—23no—
238C22×C4—23no—
239C22×C4—23no—
240C22×C4—23no—
241C22×C4—23no—
242C22×C4—23no—
243C22×C4—23no—
244C22×C4—23no—
245C22×C4—23no—
246C22×C4—23no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—21no—
252C22×C4—23no—
253C22×C4—26no—
254C22×C4—23no—
255C22×C4—23no—
256C22×C4—26no—
257C22×C4—26no—
258C22×C4—26no—
259C22×C4—26no—
260C22×C4—26no—
261C22×C4—26no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—23no—
265C22×C4—26no—
266C22×C4—26no—
267C22×C4—23no—
268C22×C4—23no—
269C22×C4—26no—
270C22×C4—23no—
271C22×C4—22no—
272C22×C4—26no—
273C22×C4—26no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—23no—
277C22×C4—23no—
278C22×C4—412no—
279C22×C4—44no—
280C22×C4—412no—
281C22×C4C22×D416yes—
282C22×C4C22×D416yes—
283C22×C4C22×D412yes—
284C22×C4H(5,𝔽2)16yes—
285C22×C4H(5,𝔽2)16yes—
286C22×C42−1+416yes—
287C4⋊C4—412no—
288C4⋊C4—412no—
289C4⋊C4—424no—
290C4⋊C4—424no—
291C4⋊C4—424no—
292C4⋊C4—44no—
293C4⋊C4—412no—
294C22⋊C4—412no—
295C22⋊C4—412no—
296C22⋊C4—424no—
297C22⋊C4—424no—
298C22⋊C4—48no—
299C22⋊C4—424no—
300C22⋊C4—812no—
301C22⋊C4—812no—
302C42—224no—
303C42—412no—
304C23—22no—
305C23—26no—
306C23—26no—
307C23—43no—
308C23—43no—
309C23—43no—
310C23—41no—
311C23—43no—
312C23—43no—
313C23—43no—
314C23—43no—
315C23—43no—
316C23—43no—
317C23—46no—
318C23—46no—
319C23—42no—
320C23—42no—
321C23—46no—
322C23—46no—
323C23—46no—
324C23—46no—
325C23—46no—
326C23—46no—
327C23—46no—
328C23—46no—
329C23—46no—
330C23C23×D411yescharacteristic
331C23C2×H(5,𝔽2)13yes—
332C23C2×H(5,𝔽2)13yes—
333C23C2×2−1+413yes—
334C23C2×2−1+411yescharacteristic
335C23#58511yescharacteristic
336C23#58513yes—
337Q8—824no—
338D4—816no—
339D4—88no—
340C2×C4—26no—
341C2×C4—26no—
342C2×C4—26no—
343C2×C4—26no—
344C2×C4—26no—
345C2×C4—26no—
346C2×C4—26no—
347C2×C4—26no—
348C2×C4—26no—
349C2×C4—43no—
350C2×C4—43no—
351C2×C4—43no—
352C2×C4—43no—
353C2×C4—43no—
354C2×C4—43no—
355C2×C4—43no—
356C2×C4—43no—
357C2×C4—43no—
358C2×C4—43no—
359C2×C4—43no—
360C2×C4—43no—
361C2×C4—43no—
362C2×C4—46no—
363C2×C4—46no—
364C2×C4—46no—
365C2×C4—43no—
366C2×C4—43no—
367C2×C4—43no—
368C2×C4—41no—
369C2×C4—46no—
370C2×C4—42no—
371C2×C4—46no—
372C2×C4—43no—
373C2×C4—46no—
374C2×C4—46no—
375C2×C4—46no—
376C2×C4—43no—
377C2×C4—43no—
378C2×C4—46no—
379C2×C4—46no—
380C2×C4—46no—
381C2×C4—43no—
382C2×C4—46no—
383C2×C4—43no—
384C2×C4—46no—
385C2×C4—46no—
386C2×C4—46no—
387C2×C4—46no—
388C2×C4—88no—
389C2×C4#535112yes—
390C2×C4#560112yes—
391C2×C4#56014yes—
392C22—46no—
393C22—46no—
394C22—42no—
395C22—83no—
396C22—81no—
397C22—81no—
398C22—83no—
399C22—86no—
400C22—86no—
401C22—86no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—86no—
406C22C2×#53513yes—
407C22#348913yes—
408C22#348911yescharacteristic
409C22C2×#56013yes—
410C22C2×#56011yescharacteristic
411C22#355213yes—
412C22#355213yes—
413C22#355413yes—
414C22#355413yes—
415C22#356113yes—
416C22#356113yes—
417C22#356213yes—
418C22#356213yes—
419C4—46no—
420C4—46no—
421C4—46no—
422C4—83no—
423C4—82no—
424C4—81no—
425C4—86no—
426C4—86no—
427C4#281818yes—
428C2—82no—
429C2—162no—
430C2C2×#281811yescharacteristic
431C2#3737913yes—
432C2#3737913yes—
433C2#5304113yes—
434C2#5304311yescharacteristic
435C2#5308511yescharacteristic
436C2#5308713yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2658C2semidirect
2. C2×#2401C2semidirect
3. #17226C2semidirect
4. #17204C2semidirect
5. #17161C2semidirect
6. #17156C2semidirect
7. #15217C2semidirect
9. #15180C2semidirect
10. #15178C2semidirect
11. #14350C2semidirect
94. #376D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26582
C2×#24012
#172262
#172042
#171612
#171562
#152172
#152152
#151802
#151782
#143502