Finite Group Databasebeta

← #1409133

Subgroups — #1409133

Order 512.

3311 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2491C213yesmaximal, complemented
2C2×#2521C213yesmaximal, complemented
3#16814C213yesmaximal, complemented
4#16807C213yesmaximal, complemented
5#16511C213yesmaximal, complemented
6#16363C213yesmaximal, complemented
7#15190C211yesmaximal, characteristic, complemented
8#15084C213yesmaximal, complemented
9#15079C213yesmaximal, complemented
10#14900C211yesmaximal, characteristic, complemented
11#14757C211yesmaximal, characteristic
12#14353C211yesmaximal, characteristic, complemented
13#14321C211yesmaximal, characteristic, complemented
14#13856C211yesmaximal, characteristic, complemented
15#13809C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2213yes—
17C22×#126C2211yescharacteristic
18C22×#124C2213yes—
19C22×#121C2213yes—
20#2521—224no—
21#2491—224no—
22C2×#399C2213yes—
23C2×#399C2216yes—
24C2×#397C2213yes—
25C2×#395C2213yes—
26C2×#394C2216yes—
27C2×#394C2213yes—
28C2×#393C2216yes—
29C2×#392C2211yescharacteristic
30C2×#391C2213yes—
31C2×#390C2213yes—
32C2×#390C2213yes—
33C2×#388C2213yes—
34C2×#388C2213yes—
35C2×#388C2213yes—
36C2×#388C2216yes—
37C2×#387C2213yes—
38C2×#386C2213yes—
39C2×#380C2213yes—
40C2×#380C2213yes—
41C2×C4×C4⋊C4C2213yes—
42C2×C4×C4⋊C4C2213yes—
43C2×C4×C22⋊C4C2213yes—
44C2×#376C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1445C2216yes—
47#1444C2211yescharacteristic
48#1444C2211yescharacteristic
49#1444C2213yes—
50#1444C2213yes—
51#1444C2211yescharacteristic
52#1444C2213yes—
53#1444C2213yes—
54#1444C2213yes—
55#1444C2213yes—
56#1439C2213yes—
57#1439C2213yes—
58#1439C2216yes—
59#1439C2213yes—
60#1439C2216yes—
61#1438C2213yes—
62#1438C2213yes—
63#1438C2213yes—
64#1434C2213yes—
65#1434C2213yes—
66C22⋊#95C2213yes—
67C4×#95C2211yescharacteristic
68C23×Q8C2313yes—
69C23×D4C2313yes—
70C23×D4C2313yes—
71C24×C4C2313yes—
72C2×C4⋊Q8—212no—
73C2×C4⋊Q8—412no—
74C2×#126—212no—
75C2×#124—224no—
76C2×#124—212no—
77C2×#121—212no—
78C2×#121—412no—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2313yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2313yes—
84C22×C4⋊C4C2316yes—
85C22×C4⋊C4C2313yes—
86C22×C4⋊C4C2313yes—
87C22×C4⋊C4C2313yes—
88C22×C4⋊C4C2313yes—
89C22×C4⋊C4C2313yes—
90C22×C22⋊C4C2313yes—
91C22×C22⋊C4C2313yes—
92C22×C22⋊C4C2311yescharacteristic
93C22×C22⋊C4C2313yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2313yes—
96C22×C22⋊C4C2316yes—
97C22×C22⋊C4C2316yes—
98C22×C22⋊C4C2313yes—
99C22×C22⋊C4C2313yes—
100C22×C22⋊C4C2316yes—
101C22×C22⋊C4C2313yes—
102C22×C22⋊C4C2313yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C42C2313yes—
105C22×C42C2311yescharacteristic
106C22×C42C2313yes—
107C22×C42C2313yes—
108#399—212no—
109#399—412no—
110#397—212no—
111#395—212no—
112#394—224no—
113#394—212no—
114#393—224no—
115#392—24no—
116#391—212no—
117#390—212no—
118#390—212no—
119#388—212no—
120#388—212no—
121#388—224no—
122#388—212no—
123#387—212no—
124#386—212no—
125#380—212no—
126#380—212no—
127C4×C4⋊C4—212no—
128C4×C4⋊C4—212no—
129C4×C22⋊C4D4124yescomplemented
130#376—24no—
131C2×#95—24no—
132C2×#95C2316yes—
133C2×#95C2316yes—
134C2×#95C2313yes—
135C2×#95C2313yes—
136C2×#95C2313yes—
137C2×#95C2313yes—
138C2×#95C2313yes—
139C2×#95C2313yes—
140C2×#95C2313yes—
141C2×#95C2313yes—
142C2×#95C2313yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2316yes—
146C2×#95C2313yes—
147C2×#95C2313yes—
148C2×#95C2311yescharacteristic
149C25C2413yes—
150C25C2411yescharacteristic
151C22×Q8—212no—
152C22×Q8—26no—
153C22×Q8—26no—
154C22×Q8—46no—
155C22×Q8—43no—
156C22×D4—212no—
157C22×D4—26no—
158C22×D4—26no—
159C22×D4—26no—
160C22×D4—26no—
161C22×D4—26no—
162C22×D4—46no—
163C22×D4—46no—
164C22×D4—46no—
165C22×D4—43no—
166C23×C4—26no—
167C23×C4—26no—
168C23×C4—46no—
169C23×C4—46no—
170C23×C4C2×D4112yes—
171C23×C4C2411yescharacteristic
172C23×C4C2413yes—
173C23×C4C2411yescharacteristic
174C23×C4C2413yes—
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C23×C4C2413yes—
178C23×C4C2413yes—
179C23×C4C2413yes—
180C23×C4C2413yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2413yes—
184C4⋊Q8—448no—
185#126—416no—
186#124—448no—
187#121—448no—
188C2×C4⋊C4—26no—
189C2×C4⋊C4—212no—
190C2×C4⋊C4—26no—
191C2×C4⋊C4—26no—
192C2×C4⋊C4—26no—
193C2×C4⋊C4—26no—
194C2×C4⋊C4—212no—
195C2×C4⋊C4—26no—
196C2×C4⋊C4—26no—
197C2×C4⋊C4—26no—
198C2×C4⋊C4—26no—
199C2×C4⋊C4—212no—
200C2×C4⋊C4—26no—
201C2×C4⋊C4—26no—
202C2×C4⋊C4—43no—
203C2×C4⋊C4—46no—
204C2×C4⋊C4—43no—
205C2×C4⋊C4—43no—
206C2×C4⋊C4—46no—
207C2×C4⋊C4—46no—
208C2×C4⋊C4—46no—
209C2×C4⋊C4—46no—
210C2×C22⋊C4—212no—
211C2×C22⋊C4—26no—
212C2×C22⋊C4—26no—
213C2×C22⋊C4—26no—
214C2×C22⋊C4—212no—
215C2×C22⋊C4—26no—
216C2×C22⋊C4—212no—
217C2×C22⋊C4—26no—
218C2×C22⋊C4—212no—
219C2×C22⋊C4—26no—
220C2×C22⋊C4—26no—
221C2×C22⋊C4—212no—
222C2×C22⋊C4—26no—
223C2×C22⋊C4—26no—
224C2×C22⋊C4—212no—
225C2×C22⋊C4—26no—
226C2×C22⋊C4—26no—
227C2×C22⋊C4—212no—
228C2×C22⋊C4—43no—
229C2×C22⋊C4—43no—
230C2×C22⋊C4—46no—
231C2×C22⋊C4—46no—
232C2×C22⋊C4—43no—
233C2×C22⋊C4—43no—
234C2×C22⋊C4—46no—
235C2×C22⋊C4—46no—
236C2×C22⋊C4—46no—
237C2×C22⋊C4—46no—
238C2×C22⋊C4—46no—
239C2×C22⋊C4C2×D4112yes—
240C2×C22⋊C4C2×D4112yes—
241C2×C42—26no—
242C2×C42—26no—
243C2×C42—26no—
244C2×C42—212no—
245C2×C42—26no—
246C2×C42—46no—
247C2×C42C2×D4112yes—
248C2×C42C2×D4112yes—
249#95—26no—
250#95—26no—
251#95—212no—
252#95—26no—
253#95—212no—
254#95—26no—
255#95—26no—
256#95—26no—
257#95—26no—
258#95—26no—
259#95—26no—
260#95—26no—
261#95—26no—
262#95—46no—
263#95C2×D4112yes—
264#95C2×D4112yes—
265#95#4114yes—
266C24—21no—
267C24—23no—
268C24—23no—
269C24—21no—
270C24—23no—
271C24—21no—
272C24—23no—
273C24—26no—
274C24—23no—
275C24—26no—
276C24—23no—
277C24—23no—
278C24—26no—
279C24—23no—
280C24—26no—
281C24—23no—
282C24—23no—
283C24C22×D416yes—
284C24C2511yescharacteristic
285C2×Q8—424no—
286C2×Q8—412no—
287C2×Q8—412no—
288C2×Q8—86no—
289C2×Q8—812no—
290C2×D4—424no—
291C2×D4—412no—
292C2×D4—412no—
293C2×D4—412no—
294C2×D4—412no—
295C2×D4—412no—
296C2×D4—812no—
297C2×D4—812no—
298C2×D4—86no—
299C2×D4—812no—
300C22×C4—21no—
301C22×C4—23no—
302C22×C4—26no—
303C22×C4—23no—
304C22×C4—26no—
305C22×C4—21no—
306C22×C4—23no—
307C22×C4—26no—
308C22×C4—23no—
309C22×C4—23no—
310C22×C4—23no—
311C22×C4—26no—
312C22×C4—26no—
313C22×C4—26no—
314C22×C4—26no—
315C22×C4—23no—
316C22×C4—23no—
317C22×C4—23no—
318C22×C4—23no—
319C22×C4—23no—
320C22×C4—23no—
321C22×C4—23no—
322C22×C4—26no—
323C22×C4—23no—
324C22×C4—224no—
325C22×C4—23no—
326C22×C4—23no—
327C22×C4—26no—
328C22×C4—23no—
329C22×C4—23no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—26no—
333C22×C4—21no—
334C22×C4—23no—
335C22×C4—26no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—21no—
339C22×C4—23no—
340C22×C4—23no—
341C22×C4—23no—
342C22×C4—23no—
343C22×C4—26no—
344C22×C4—23no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—412no—
348C22×C4—412no—
349C22×C4—812no—
350C22×C4—812no—
351C22×C4C22×D416yes—
352C22×C4C22×D416yes—
353C22×C4C22×D416yes—
354C22×C4C22×D416yes—
355C22×C4C22×D416yes—
356C22×C4C22×D416yes—
357C22×C4C2×#4116yes—
358C22×C4H(5,𝔽2)16yes—
359C22×C42−1+416yes—
360C4⋊C4—224no—
361C4⋊C4—412no—
362C4⋊C4—424no—
363C4⋊C4—412no—
364C4⋊C4—412no—
365C4⋊C4—412no—
366C4⋊C4—412no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—86no—
371C22⋊C4—412no—
372C22⋊C4—424no—
373C22⋊C4—424no—
374C22⋊C4—412no—
375C22⋊C4—412no—
376C22⋊C4—412no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—412no—
380C22⋊C4—412no—
381C22⋊C4—412no—
382C22⋊C4—86no—
383C22⋊C4—812no—
384C22⋊C4#127116yes—
385C42—224no—
386C42—412no—
387C42—412no—
388C42—44no—
389C23—26no—
390C23—26no—
391C23—26no—
392C23—26no—
393C23—26no—
394C23—26no—
395C23—43no—
396C23—41no—
397C23—43no—
398C23—46no—
399C23—43no—
400C23—43no—
401C23—43no—
402C23—46no—
403C23—43no—
404C23—46no—
405C23—43no—
406C23—46no—
407C23—46no—
408C23—43no—
409C23—46no—
410C23—43no—
411C23—43no—
412C23—46no—
413C23—43no—
414C23—46no—
415C23—43no—
416C23—46no—
417C23—43no—
418C23—46no—
419C23—46no—
420C23—46no—
421C23—43no—
422C23—46no—
423C23C2×#12714yes—
424C23C23×D413yes—
425C23C22×#4111yescharacteristic
426C23C2×H(5,𝔽2)13yes—
427C23C2×2−1+413yes—
428C23#58513yes—
429C23#58511yescharacteristic
430C23#58511yescharacteristic
431Q8—824no—
432D4—824no—
433D4—824no—
434C2×C4—26no—
435C2×C4—26no—
436C2×C4—26no—
437C2×C4—26no—
438C2×C4—26no—
439C2×C4—26no—
440C2×C4—212no—
441C2×C4—212no—
442C2×C4—26no—
443C2×C4—26no—
444C2×C4—26no—
445C2×C4—26no—
446C2×C4—26no—
447C2×C4—26no—
448C2×C4—26no—
449C2×C4—212no—
450C2×C4—26no—
451C2×C4—26no—
452C2×C4—26no—
453C2×C4—43no—
454C2×C4—43no—
455C2×C4—46no—
456C2×C4—43no—
457C2×C4—46no—
458C2×C4—43no—
459C2×C4—46no—
460C2×C4—43no—
461C2×C4—46no—
462C2×C4—46no—
463C2×C4—43no—
464C2×C4—43no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—43no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—46no—
471C2×C4—43no—
472C2×C4—43no—
473C2×C4—43no—
474C2×C4—43no—
475C2×C4—46no—
476C2×C4—43no—
477C2×C4—43no—
478C2×C4—43no—
479C2×C4—43no—
480C2×C4—46no—
481C2×C4—43no—
482C2×C4—46no—
483C2×C4—43no—
484C2×C4—43no—
485C2×C4—43no—
486C2×C4—824no—
487C2×C4C2×#12714yes—
488C2×C4C2×#12714yes—
489C2×C4#547112yes—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—46no—
494C22—46no—
495C22—46no—
496C22—46no—
497C22—86no—
498C22—83no—
499C22—83no—
500C22—86no—
501C22—83no—
502C22—83no—
503C22—83no—
504C22—86no—
505C22—83no—
506C22—83no—
507C22—86no—
508C22C22×#12711yescharacteristic
509C22#344913yes—
510C22#344913yes—
511C22C2×#54713yes—
512C22#348413yes—
513C22#348813yes—
514C22#349913yes—
515C22#351413yes—
516C22#352411yescharacteristic
517C22#354713yes—
518C22#356116yes—
519C22#356513yes—
520C4—24no—
521C4—24no—
522C4—212no—
523C4—46no—
524C4—46no—
525C4—46no—
526C4—46no—
527C4—83no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—83no—
532C4—81no—
533C2—44no—
534C2—163no—
535C2#3464111yescharacteristic
536C2#3592111yescharacteristic
537C2#3617911yescharacteristic
538C2#3804413yes—
539C2#3805013yes—
540C2#4231113yes—
541C2#4231713yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14757C2
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×#2491C2semidirect
2. C2×#2521C2semidirect
3. #16814C2semidirect
4. #16807C2semidirect
5. #16511C2semidirect
6. #16363C2semidirect
7. #15190C2semidirect
8. #15084C2semidirect
9. #15079C2semidirect
10. #14900C2semidirect
12. #14353C2semidirect
13. #14321C2semidirect
14. #13856C2semidirect
15. #13809C2semidirect
129. C4×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24912
C2×#25212
#168142
#168072
#165112
#163632
#151902
#150842
#150792
#149002
#147572
#143532
#143212
#138562
#138092