Finite Group Databasebeta

← #2747170

Subgroups — #2747170

Order 512.

3498 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2842C211yesmaximal, characteristic, complemented
2C2×#2397C211yesmaximal, characteristic, complemented
3#17126C216yesmaximal, complemented
4#17091C212yesmaximal, complemented
5#16338C213yesmaximal, complemented
6#16249C213yesmaximal, complemented
7#14899C212yesmaximal, complemented
8#14825C213yesmaximal, complemented
9#14732C216yesmaximal, complemented
10#14636C211yesmaximal, characteristic, complemented
11#13967C213yesmaximal, complemented
12C22×#120C2213yes—
13C22×C4×D4C2213yes—
14C22×C4×D4C2213yescomplemented
15C23×C22⋊C4C2211yescharacteristic
16#2842—28no—
17#2397—28no—
18C2×#400C2213yescomplemented
19C2×#399C2213yes—
20C2×#399C2211yescharacteristic
21C2×#399C2213yescomplemented
22C2×#396C2212yes—
23C2×#396C2216yes—
24C2×#394C2216yes—
25C2×#394C2216yescomplemented
26C2×#392C2213yes—
27C2×#392C2216yes—
28C2×#392C2211yescharacteristic, complemented
29C2×#386C2216yes—
30C2×#386C2213yes—
31C2×#380C2213yescomplemented
32C2×#379C2213yes—
33C2×#376C2211yescharacteristic, complemented
34#1443C2212yescomplemented
35#1442C2216yescomplemented
36#1441C2216yescomplemented
37#1441C2213yescomplemented
38#1440C2216yes—
39#1440C2216yescomplemented
40#1439C2213yes—
41#1439C2216yes—
42#1439C2216yescomplemented
43#1439C2216yescomplemented
44#1438C2216yes—
45#1438C2216yes—
46#1438C2216yes—
47#1437C2216yes—
48#1436C2216yescomplemented
49C22⋊#95C2213yes—
50C22⋊#95C2213yes—
51C22⋊#95C2213yescomplemented
52C26C2311yescharacteristic
53C23×D4C2313yes—
54C23×D4C2313yes—
55C23×D4C2316yes—
56C24×C4C2311yescharacteristic
57C24×C4C2313yes—
58C24×C4C2311yescharacteristic
59C2×#120—212no—
60C2×#120—46no—
61C2×#120—46no—
62C2×C4×D4—212no—
63C2×C4×D4—212no—
64C2×C4×D4—46no—
65C2×C4×D4—46no—
66C2×C4×D4—46no—
67C2×C4×D4—46no—
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C2316yes—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2313yes—
72C22×C4⋊C4C2313yes—
73C22×C22⋊C4—46no—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2316yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2316yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2313yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2316yes—
82C22×C22⋊C4C2313yes—
83C22×C22⋊C4C2313yes—
84C22×C22⋊C4C2316yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2313yes—
87C22×C42C2313yes—
88#400—46no—
89#399—212no—
90#399—212no—
91#399D418yescomplemented
92#396—224no—
93#396—28no—
94#394—412no—
95#394—412no—
96#392—212no—
97#392—24no—
98#392—412no—
99#386—412no—
100#386—46no—
101#380—46no—
102#379—212no—
103#376—24no—
104C2×#95C2313yes—
105C2×#95C2316yes—
106C2×#95C2313yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2313yes—
112C2×#95C2316yes—
113C2×#95C2316yes—
114C2×#95C2316yes—
115C2×#95C2311yescharacteristic
116C25—22no—
117C25—22no—
118C25—43no—
119C25—43no—
120C25—43no—
121C25—43no—
122C25#4114yes—
123C25C2412yes—
124C25C2413yes—
125C25C2411yescharacteristic
126C22×D4—26no—
127C22×D4—26no—
128C22×D4—26no—
129C22×D4—212no—
130C22×D4—26no—
131C22×D4—212no—
132C22×D4—212no—
133C22×D4—26no—
134C22×D4—26no—
135C22×D4—43no—
136C22×D4—43no—
137C22×D4—43no—
138C22×D4—43no—
139C22×D4—46no—
140C22×D4—46no—
141C22×D4—46no—
142C22×D4—46no—
143C22×D4—46no—
144C22×D4—46no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—26no—
148C23×C4—26no—
149C23×C4—26no—
150C23×C4—212no—
151C23×C4—43no—
152C23×C4—43no—
153C23×C4—46no—
154C23×C4—43no—
155C23×C4—43no—
156C23×C4C2416yes—
157C23×C4C2411yescharacteristic
158C23×C4C2412yes—
159C23×C4C2416yes—
160C23×C4C2413yes—
161C23×C4C2413yes—
162C23×C4C2413yes—
163C23×C4C2411yescharacteristic
164#120—448no—
165C4×D4—448no—
166C4×D4—448no—
167C2×C4⋊C4—26no—
168C2×C4⋊C4—26no—
169C2×C4⋊C4—212no—
170C2×C4⋊C4—26no—
171C2×C4⋊C4—43no—
172C2×C4⋊C4—43no—
173C2×C4⋊C4—43no—
174C2×C4⋊C4—46no—
175C2×C4⋊C4—43no—
176C2×C4⋊C4—43no—
177C2×C4⋊C4—46no—
178C2×C4⋊C4—46no—
179C2×C4⋊C4—43no—
180C2×C4⋊C4C2×D4112yes—
181C2×C22⋊C4—212no—
182C2×C22⋊C4—26no—
183C2×C22⋊C4—212no—
184C2×C22⋊C4—26no—
185C2×C22⋊C4—212no—
186C2×C22⋊C4—212no—
187C2×C22⋊C4—212no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—212no—
190C2×C22⋊C4—212no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—212no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—43no—
195C2×C22⋊C4—43no—
196C2×C22⋊C4—46no—
197C2×C22⋊C4—43no—
198C2×C22⋊C4—43no—
199C2×C22⋊C4—46no—
200C2×C22⋊C4—43no—
201C2×C22⋊C4—43no—
202C2×C22⋊C4—416no—
203C2×C22⋊C4—46no—
204C2×C22⋊C4—46no—
205C2×C22⋊C4—43no—
206C2×C22⋊C4—46no—
207C2×C22⋊C4—46no—
208C2×C22⋊C4—46no—
209C2×C22⋊C4—43no—
210C2×C22⋊C4—46no—
211C2×C22⋊C4—46no—
212C2×C22⋊C4—46no—
213C2×C22⋊C4—46no—
214C2×C22⋊C4—46no—
215C2×C22⋊C4—46no—
216C2×C22⋊C4C2×D4112yes—
217C2×C42—26no—
218C2×C42—26no—
219C2×C42—26no—
220#95—26no—
221#95—26no—
222#95—212no—
223#95—212no—
224#95—46no—
225#95—46no—
226#95—46no—
227#95—46no—
228#95—46no—
229#95—43no—
230#95—46no—
231#95C2×D414yes—
232C24—23no—
233C24—23no—
234C24—23no—
235C24—23no—
236C24—22no—
237C24—23no—
238C24—23no—
239C24—26no—
240C24—26no—
241C24—26no—
242C24—23no—
243C24—26no—
244C24—26no—
245C24—26no—
246C24—23no—
247C24—23no—
248C24—22no—
249C24—26no—
250C24—26no—
251C24—23no—
252C24—28no—
253C24—412no—
254C24—412no—
255C24—44no—
256C24—412no—
257C24—86no—
258C24—86no—
259C24—86no—
260C24—86no—
261C24—86no—
262C24—812no—
263C24—86no—
264C24C22×D412yes—
265C24C2×#4114yes—
266C24C2×#4112yes—
267C24H(5,𝔽2)16yes—
268C242−1+412yes—
269C24C2511yescharacteristic
270C2×D4—224no—
271C2×D4—412no—
272C2×D4—424no—
273C2×D4—412no—
274C2×D4—412no—
275C2×D4—412no—
276C2×D4—412no—
277C2×D4—412no—
278C2×D4—412no—
279C2×D4—424no—
280C2×D4—412no—
281C2×D4—412no—
282C2×D4—424no—
283C2×D4—424no—
284C2×D4—812no—
285C2×D4—812no—
286C2×D4—812no—
287C2×D4—812no—
288C2×D4—812no—
289C22×C4—21no—
290C22×C4—23no—
291C22×C4—23no—
292C22×C4—23no—
293C22×C4—23no—
294C22×C4—26no—
295C22×C4—26no—
296C22×C4—23no—
297C22×C4—23no—
298C22×C4—23no—
299C22×C4—23no—
300C22×C4—26no—
301C22×C4—23no—
302C22×C4—23no—
303C22×C4—23no—
304C22×C4—26no—
305C22×C4—26no—
306C22×C4—21no—
307C22×C4—26no—
308C22×C4—26no—
309C22×C4—26no—
310C22×C4—26no—
311C22×C4—26no—
312C22×C4—26no—
313C22×C4—26no—
314C22×C4—23no—
315C22×C4—26no—
316C22×C4—26no—
317C22×C4—26no—
318C22×C4—26no—
319C22×C4—26no—
320C22×C4—23no—
321C22×C4—22no—
322C22×C4—26no—
323C22×C4—26no—
324C22×C4—26no—
325C22×C4—424no—
326C22×C4—412no—
327C22×C4—412no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—412no—
331C22×C4—86no—
332C22×C4—812no—
333C22×C4—86no—
334C22×C4—86no—
335C22×C4—86no—
336C22×C4C22×D416yes—
337C22×C4C22×D416yes—
338C22×C4H(5,𝔽2)16yes—
339C22×C4H(5,𝔽2)16yes—
340C4⋊C4—412no—
341C4⋊C4—412no—
342C4⋊C4—86no—
343C4⋊C4—812no—
344C4⋊C4—86no—
345C22⋊C4—412no—
346C22⋊C4—424no—
347C22⋊C4—424no—
348C22⋊C4—412no—
349C22⋊C4—412no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—812no—
353C22⋊C4—812no—
354C22⋊C4—86no—
355C22⋊C4—812no—
356C22⋊C4—812no—
357C22⋊C4—812no—
358C42—224no—
359C23—26no—
360C23—26no—
361C23—26no—
362C23—26no—
363C23—26no—
364C23—26no—
365C23—212no—
366C23—24no—
367C23—26no—
368C23—212no—
369C23—26no—
370C23—26no—
371C23—212no—
372C23—212no—
373C23—41no—
374C23—43no—
375C23—43no—
376C23—46no—
377C23—43no—
378C23—416no—
379C23—43no—
380C23—46no—
381C23—43no—
382C23—43no—
383C23—46no—
384C23—46no—
385C23—46no—
386C23—46no—
387C23—46no—
388C23—42no—
389C23—43no—
390C23—46no—
391C23—43no—
392C23—46no—
393C23—43no—
394C23—46no—
395C23—46no—
396C23—46no—
397C23—46no—
398C23—43no—
399C23—46no—
400C23—46no—
401C23—46no—
402C23—46no—
403C23—46no—
404C23—46no—
405C23—43no—
406C23—46no—
407C23—824no—
408C23—824no—
409C23—824no—
410C23—824no—
411C23—88no—
412C23—88no—
413C23#54814yes—
414C23C23×D411yescharacteristic
415C23C22×#4111yescharacteristic
416C23C2×H(5,𝔽2)13yes—
417C23C2×H(5,𝔽2)13yes—
418C23C2×H(5,𝔽2)13yes—
419C23C2×2−1+411yescharacteristic
420C23#58513yes—
421D4—448no—
422D4—448no—
423D4—848no—
424C2×C4—26no—
425C2×C4—26no—
426C2×C4—26no—
427C2×C4—26no—
428C2×C4—212no—
429C2×C4—26no—
430C2×C4—212no—
431C2×C4—26no—
432C2×C4—26no—
433C2×C4—43no—
434C2×C4—43no—
435C2×C4—43no—
436C2×C4—43no—
437C2×C4—43no—
438C2×C4—43no—
439C2×C4—43no—
440C2×C4—46no—
441C2×C4—43no—
442C2×C4—46no—
443C2×C4—46no—
444C2×C4—43no—
445C2×C4—46no—
446C2×C4—46no—
447C2×C4—46no—
448C2×C4—46no—
449C2×C4—46no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—46no—
453C2×C4—46no—
454C2×C4—46no—
455C2×C4—46no—
456C2×C4—46no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—46no—
461C2×C4—46no—
462C2×C4—43no—
463C2×C4—824no—
464C2×C4—88no—
465C2×C4—88no—
466C2×C4#56014yes—
467C22—212no—
468C22—42no—
469C22—46no—
470C22—46no—
471C22—46no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—46no—
476C22—412no—
477C22—412no—
478C22—46no—
479C22—412no—
480C22—412no—
481C22—46no—
482C22—832no—
483C22—83no—
484C22—86no—
485C22—83no—
486C22—86no—
487C22—86no—
488C22—86no—
489C22—83no—
490C22—83no—
491C22—82no—
492C22—82no—
493C22#345913yes—
494C22C2×#54811yescharacteristic
495C22#348413yes—
496C22#348813yes—
497C22#349013yes—
498C22C2×#56011yescharacteristic
499C22#352913yes—
500C22#353513yes—
501C22#353613yes—
502C22#353713yes—
503C22#354216yes—
504C22#354313yes—
505C4—24no—
506C4—46no—
507C4—46no—
508C4—46no—
509C4—86no—
510C4—86no—
511C4—81no—
512C4—82no—
513C2—44no—
514C2—86no—
515C2—84no—
516C2#3667611yescharacteristic
517C2#3775213yes—
518C2#4092913yes—
519C2#4545213yes—
520C2#4632411yescharacteristic
521C2#5267313yes—
522C2#5298811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2842C2semidirect
2. C2×#2397C2semidirect
3. #17126C2semidirect
4. #17091C2semidirect
5. #16338C2semidirect
6. #16249C2semidirect
7. #14899C2semidirect
8. #14825C2semidirect
9. #14732C2semidirect
10. #14636C2semidirect
11. #13967C2semidirect
14. C22×C4×D4C22semidirect
18. C2×#400C22semidirect
21. C2×#399C22semidirect
25. C2×#394C22semidirect
28. C2×#392C22semidirect
31. C2×#380C22semidirect
33. C2×#376C22semidirect
34. #1443C22semidirect
35. #1442C22semidirect
36. #1441C22semidirect
37. #1441C22semidirect
39. #1440C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
48. #1436C22semidirect
51. C22⋊#95C22semidirect
91. #399D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28422
C2×#23972
#171262
#170912
#163382
#162492
#148992
#148252
#147322
#146362
#139672