Finite Group Databasebeta

← #4462745

Subgroups — #4462745

Order 512.

2834 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2645C211yesmaximal, characteristic, complemented
2C2×#2397C211yesmaximal, characteristic
3#17226C212yesmaximal, complemented
4#17161C216yesmaximal, complemented
5#17149C212yesmaximal, complemented
6#17095C216yesmaximal, complemented
7#15181C213yesmaximal, complemented
8#15177C213yesmaximal, complemented
9#15177C213yesmaximal, complemented
10#15129C213yesmaximal, complemented
11#14350C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2213yes—
14C22×C4×D4C2213yes—
15C22×C4×D4C2213yes—
16#2645—28no—
17C2×#401C2212yes—
18#2397—28no—
19C2×#400C2216yes—
20C2×#399C2213yes—
21C2×#399C2211yescharacteristic
22C2×#394C2216yes—
23C2×#392C2213yes—
24C2×#392C2211yescharacteristic
25C2×#391C2213yes—
26C2×#389C2213yes—
27C2×#380C2213yes—
28C2×#380C2213yes—
29C2×#380C2213yes—
30C2×#380C2213yes—
31C2×#376C2211yescharacteristic
32#1444C2216yes—
33#1444C2216yes—
34#1443C2216yes—
35#1443C2212yes—
36#1442C2216yes—
37#1441C2216yes—
38#1440C2216yes—
39#1440C2216yes—
40#1439C2216yes—
41#1439C2216yes—
42#1439C2216yes—
43#1439C2216yes—
44#1439C2216yes—
45#1439C2216yes—
46#1438C2216yes—
47#1438C2216yes—
48#1438C2216yes—
49C4⋊#95C2213yes—
50C4⋊#95C2213yes—
51C23×Q8C2311yescharacteristic
52C23×D4C2313yes—
53C23×D4C2313yes—
54C24×C4C2311yescharacteristic
55C24×C4C2313yes—
56C2×C22⋊Q8—46no—
57C2×C22⋊Q8—46no—
58C2×C22⋊Q8—46no—
59C2×C22⋊Q8—46no—
60C2×C4×D4—212no—
61C2×C4×D4—212no—
62C2×C4×D4—46no—
63C2×C4×D4—46no—
64C2×C4×D4—46no—
65C2×C4×D4—46no—
66C22×C4⋊C4C2313yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C2316yes—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2313yes—
72C22×C4⋊C4C2316yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4C2313yes—
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2316yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2316yes—
83C22×C22⋊C4C2316yes—
84C22×C42C2313yes—
85#401—28no—
86#400—224no—
87#399—212no—
88#399—24no—
89#394—412no—
90#392—212no—
91#392—24no—
92#391—212no—
93#389—212no—
94#380—46no—
95#380—46no—
96#380—46no—
97#380—46no—
98#376D418yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2313yes—
111C2×#95C2311yescharacteristic
112C25C2411yescharacteristic
113C25C2413yes—
114C22×Q8—22no—
115C22×Q8—26no—
116C22×Q8—26no—
117C22×D4—26no—
118C22×D4—26no—
119C22×D4—26no—
120C22×D4—26no—
121C22×D4—26no—
122C22×D4—26no—
123C22×D4—43no—
124C22×D4—43no—
125C22×D4—43no—
126C22×D4—43no—
127C22×D4—43no—
128C22×D4—43no—
129C22×D4—43no—
130C22×D4—43no—
131C23×C4—22no—
132C23×C4—26no—
133C23×C4—26no—
134C23×C4—26no—
135C23×C4—26no—
136C23×C4—26no—
137C23×C4—26no—
138C23×C4—26no—
139C23×C4—26no—
140C23×C4—26no—
141C23×C4C2412yes—
142C23×C4C2416yes—
143C23×C4C2416yes—
144C23×C4C2412yes—
145C23×C4C2413yes—
146C23×C4C2411yescharacteristic
147C23×C4C2413yes—
148C23×C4C2413yes—
149C23×C4C2411yescharacteristic
150C22⋊Q8—416no—
151C22⋊Q8—448no—
152C4×D4—448no—
153C4×D4—448no—
154C2×C4⋊C4—26no—
155C2×C4⋊C4—26no—
156C2×C4⋊C4—26no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—26no—
160C2×C4⋊C4—26no—
161C2×C4⋊C4—26no—
162C2×C4⋊C4—26no—
163C2×C4⋊C4—212no—
164C2×C4⋊C4—43no—
165C2×C4⋊C4—43no—
166C2×C4⋊C4—43no—
167C2×C4⋊C4—43no—
168C2×C4⋊C4—46no—
169C2×C4⋊C4—46no—
170C2×C4⋊C4—46no—
171C2×C4⋊C4—46no—
172C2×C4⋊C4—46no—
173C2×C4⋊C4—46no—
174C2×C22⋊C4—26no—
175C2×C22⋊C4—26no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—26no—
178C2×C22⋊C4—212no—
179C2×C22⋊C4—212no—
180C2×C22⋊C4—43no—
181C2×C22⋊C4—43no—
182C2×C22⋊C4—43no—
183C2×C22⋊C4—43no—
184C2×C22⋊C4—43no—
185C2×C22⋊C4—43no—
186C2×C22⋊C4—43no—
187C2×C22⋊C4—43no—
188C2×C22⋊C4—46no—
189C2×C22⋊C4—46no—
190C2×C22⋊C4—46no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4—46no—
196C2×C22⋊C4—46no—
197C2×C22⋊C4—46no—
198C2×C22⋊C4—46no—
199C2×C22⋊C4—46no—
200C2×C22⋊C4—46no—
201C2×C22⋊C4—46no—
202C2×C42—26no—
203C2×C42—26no—
204C2×C42C2×D4112yes—
205#95—212no—
206#95—212no—
207#95—212no—
208#95—46no—
209#95—46no—
210#95—46no—
211#95—46no—
212#95—46no—
213#95—46no—
214#95—46no—
215#95—46no—
216#95C2×D4112yes—
217#95C2×D414yes—
218C24—21no—
219C24—23no—
220C24—23no—
221C24—23no—
222C24—23no—
223C24—23no—
224C24—23no—
225C24—23no—
226C24—23no—
227C24—23no—
228C24—23no—
229C24—23no—
230C24—23no—
231C24—23no—
232C24—23no—
233C24—23no—
234C24—23no—
235C24—23no—
236C24—21no—
237C24—23no—
238C24H(5,𝔽2)12yes—
239C242−1+416yes—
240C24C2511yescharacteristic
241C2×Q8—412no—
242C2×Q8—44no—
243C2×Q8—412no—
244C2×D4—412no—
245C2×D4—412no—
246C2×D4—412no—
247C2×D4—412no—
248C2×D4—412no—
249C2×D4—412no—
250C2×D4—412no—
251C2×D4—412no—
252C2×D4—412no—
253C2×D4—412no—
254C2×D4—412no—
255C2×D4—412no—
256C2×D4—412no—
257C2×D4—412no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—23no—
264C22×C4—23no—
265C22×C4—23no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—23no—
269C22×C4—23no—
270C22×C4—22no—
271C22×C4—26no—
272C22×C4—21no—
273C22×C4—23no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—23no—
278C22×C4—23no—
279C22×C4—23no—
280C22×C4—23no—
281C22×C4—26no—
282C22×C4—26no—
283C22×C4—26no—
284C22×C4—26no—
285C22×C4—26no—
286C22×C4—26no—
287C22×C4—26no—
288C22×C4—23no—
289C22×C4—23no—
290C22×C4—23no—
291C22×C4—23no—
292C22×C4—26no—
293C22×C4—26no—
294C22×C4—26no—
295C22×C4—26no—
296C22×C4—26no—
297C22×C4—26no—
298C22×C4—22no—
299C22×C4—26no—
300C22×C4—412no—
301C22×C4—412no—
302C22×C4—412no—
303C22×C4—412no—
304C22×C4—44no—
305C22×C4—412no—
306C22×C4—412no—
307C22×C4—412no—
308C22×C4—412no—
309C22×C4—412no—
310C22×C4C22×D416yes—
311C22×C4C22×D416yes—
312C22×C4C22×D412yes—
313C22×C4H(5,𝔽2)16yes—
314C22×C4H(5,𝔽2)16yes—
315C4⋊C4—412no—
316C4⋊C4—412no—
317C4⋊C4—48no—
318C4⋊C4—424no—
319C4⋊C4—412no—
320C4⋊C4—44no—
321C4⋊C4—412no—
322C4⋊C4—812no—
323C22⋊C4—412no—
324C22⋊C4—412no—
325C22⋊C4—412no—
326C22⋊C4—412no—
327C22⋊C4—48no—
328C22⋊C4—424no—
329C22⋊C4—812no—
330C22⋊C4—812no—
331C22⋊C4—812no—
332C22⋊C4—812no—
333C42—224no—
334C23—22no—
335C23—26no—
336C23—26no—
337C23—26no—
338C23—26no—
339C23—26no—
340C23—26no—
341C23—26no—
342C23—26no—
343C23—26no—
344C23—43no—
345C23—43no—
346C23—43no—
347C23—43no—
348C23—43no—
349C23—43no—
350C23—43no—
351C23—43no—
352C23—41no—
353C23—43no—
354C23—43no—
355C23—43no—
356C23—43no—
357C23—43no—
358C23—43no—
359C23—43no—
360C23—43no—
361C23—43no—
362C23—43no—
363C23—43no—
364C23—43no—
365C23—43no—
366C23—43no—
367C23—43no—
368C23—43no—
369C23—43no—
370C23—43no—
371C23—43no—
372C23—43no—
373C23—43no—
374C23—43no—
375C23—43no—
376C23—43no—
377C23—43no—
378C23—43no—
379C23—43no—
380C23—43no—
381C23—43no—
382C23C23×D411yescharacteristic
383C23C2×H(5,𝔽2)13yes—
384C23C2×H(5,𝔽2)13yes—
385C23C2×H(5,𝔽2)11yescharacteristic
386C23C2×2−1+413yes—
387C23#58511yescharacteristic
388C23#58513yes—
389Q8—88no—
390D4—448no—
391D4—448no—
392C2×C4—26no—
393C2×C4—26no—
394C2×C4—26no—
395C2×C4—26no—
396C2×C4—26no—
397C2×C4—26no—
398C2×C4—43no—
399C2×C4—43no—
400C2×C4—43no—
401C2×C4—43no—
402C2×C4—43no—
403C2×C4—43no—
404C2×C4—43no—
405C2×C4—43no—
406C2×C4—43no—
407C2×C4—43no—
408C2×C4—43no—
409C2×C4—43no—
410C2×C4—46no—
411C2×C4—46no—
412C2×C4—46no—
413C2×C4—46no—
414C2×C4—42no—
415C2×C4—46no—
416C2×C4—42no—
417C2×C4—46no—
418C2×C4—41no—
419C2×C4—43no—
420C2×C4—43no—
421C2×C4—43no—
422C2×C4—43no—
423C2×C4—43no—
424C2×C4—46no—
425C2×C4—46no—
426C2×C4—46no—
427C2×C4—46no—
428C2×C4—46no—
429C2×C4—46no—
430C2×C4—46no—
431C2×C4—46no—
432C2×C4—46no—
433C2×C4—46no—
434C2×C4—46no—
435C2×C4—46no—
436C2×C4—88no—
437C2×C4—824no—
438C2×C4#535112yes—
439C2×C4#56014yes—
440C2×C4#560112yes—
441C22—46no—
442C22—42no—
443C22—46no—
444C22—46no—
445C22—46no—
446C22—46no—
447C22—46no—
448C22—46no—
449C22—46no—
450C22—46no—
451C22—81no—
452C22—83no—
453C22—81no—
454C22—83no—
455C22—83no—
456C22—83no—
457C22—83no—
458C22—83no—
459C22—83no—
460C22—83no—
461C22—83no—
462C22—83no—
463C22C2×#53513yes—
464C22#348811yescharacteristic
465C22#348913yes—
466C22C2×#56011yescharacteristic
467C22C2×#56013yes—
468C22#353613yes—
469C22#353813yes—
470C22#353913yes—
471C22#354013yes—
472C22#354113yes—
473C22#354113yes—
474C22#354113yes—
475C22#354213yes—
476C4—46no—
477C4—46no—
478C4—81no—
479C4—83no—
480C4—82no—
481C4—82no—
482C4—86no—
483C4—86no—
484C4#281818yes—
485C2—82no—
486C2—86no—
487C2C2×#281811yescharacteristic
488C2#3727113yes—
489C2#3737513yes—
490C2#5299311yescharacteristic
491C2#5307811yescharacteristic
492C2#5309813yes—
493C2#5313713yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2397C2
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×#2645C2semidirect
3. #17226C2semidirect
4. #17161C2semidirect
5. #17149C2semidirect
6. #17095C2semidirect
7. #15181C2semidirect
8. #15177C2semidirect
9. #15177C2semidirect
10. #15129C2semidirect
11. #14350C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26452
C2×#23972
#172262
#171612
#171492
#170952
#151812
#151772
#151292
#143502