Finite Group Databasebeta

← #1409134

Subgroups — #1409134

Order 512.

2733 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C213yesmaximal, complemented
2C2×#2521C213yesmaximal
3#17064C213yesmaximal, complemented
4#17059C213yesmaximal, complemented
5#16813C213yesmaximal, complemented
6#16807C213yesmaximal
7#15199C211yesmaximal, characteristic, complemented
8#15082C213yesmaximal, complemented
9#15079C213yesmaximal, complemented
10#14910C211yesmaximal, characteristic, complemented
11#14836C211yesmaximal, characteristic
12#14353C211yesmaximal, characteristic, complemented
13#14322C211yesmaximal, characteristic, complemented
14#13856C211yesmaximal, characteristic, complemented
15#13809C211yesmaximal, characteristic, complemented
16C22×#126C2213yes—
17C22×#124C2211yescharacteristic
18#2521—224no—
19#2496—224no—
20C2×#400C2216yes—
21C2×#399C2211yescharacteristic
22C2×#399C2213yes—
23C2×#397C2213yes—
24C2×#397C2216yes—
25C2×#395C2213yes—
26C2×#394C2213yes—
27C2×#391C2213yes—
28C2×#389C2213yes—
29C2×#389C2213yes—
30C2×#388C2213yes—
31C2×#388C2213yes—
32C2×#388C2213yes—
33C2×#387C2213yes—
34C2×#387C2216yes—
35C2×#386C2213yes—
36C2×#384C2213yes—
37C2×#380C2213yes—
38C2×#380C2213yes—
39C2×#380C2213yes—
40C2×C4×C4⋊C4C2213yes—
41C2×C4×C4⋊C4C2213yes—
42C2×C4×C22⋊C4C2213yes—
43C2×#376C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1445C2213yes—
46#1445C2216yes—
47#1445C2213yes—
48#1445C2216yes—
49#1445C2213yes—
50#1444C2211yescharacteristic
51#1444C2211yescharacteristic
52#1444C2213yes—
53#1444C2213yes—
54#1444C2211yescharacteristic
55#1444C2213yes—
56#1444C2213yes—
57#1444C2213yes—
58#1444C2213yes—
59#1444C2216yes—
60#1439C2213yes—
61#1439C2216yes—
62#1439C2213yes—
63#1439C2213yes—
64#1434C2213yes—
65#1434C2213yes—
66C22⋊#95C2213yes—
67C4×#95C2211yescharacteristic
68C23×Q8C2311yescharacteristic
69C23×D4C2311yescharacteristic
70C24×C4C2313yes—
71C2×#126—224no—
72C2×#126—212no—
73C2×#124—212no—
74C22×C4⋊C4C2313yes—
75C22×C4⋊C4C2313yes—
76C22×C4⋊C4C2313yes—
77C22×C4⋊C4C2313yes—
78C22×C4⋊C4C2313yes—
79C22×C4⋊C4C2313yes—
80C22×C4⋊C4C2313yes—
81C22×C4⋊C4C2316yes—
82C22×C4⋊C4C2313yes—
83C22×C4⋊C4C2313yes—
84C22×C4⋊C4C2313yes—
85C22×C4⋊C4C2313yes—
86C22×C22⋊C4C2313yes—
87C22×C22⋊C4C2311yescharacteristic
88C22×C22⋊C4C2313yes—
89C22×C22⋊C4C2311yescharacteristic
90C22×C22⋊C4C2313yes—
91C22×C22⋊C4C2311yescharacteristic
92C22×C22⋊C4C2313yes—
93C22×C22⋊C4C2313yes—
94C22×C22⋊C4C2313yes—
95C22×C22⋊C4C2313yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2313yes—
99C22×C42C2313yes—
100C22×C42C2313yes—
101#400—224no—
102#399—212no—
103#399—24no—
104#397—224no—
105#397—212no—
106#395—212no—
107#394—212no—
108#391—212no—
109#389—212no—
110#389—212no—
111#388—212no—
112#388—212no—
113#388—212no—
114#387—224no—
115#387—212no—
116#386—212no—
117#384—212no—
118#380—212no—
119#380—212no—
120#380—212no—
121C4×C4⋊C4—212no—
122C4×C4⋊C4—212no—
123C4×C22⋊C4D4124yes—
124#376—24no—
125C2×#95—24no—
126C2×#95C2313yes—
127C2×#95C2313yes—
128C2×#95C2313yes—
129C2×#95C2316yes—
130C2×#95C2316yes—
131C2×#95C2316yes—
132C2×#95C2313yes—
133C2×#95C2313yes—
134C2×#95C2316yes—
135C2×#95C2316yes—
136C2×#95C2313yes—
137C2×#95C2313yes—
138C2×#95C2313yes—
139C2×#95C2313yes—
140C2×#95C2313yes—
141C2×#95C2313yes—
142C2×#95C2316yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2311yescharacteristic
146C25C2411yescharacteristic
147C25C2411yescharacteristic
148C22×Q8—26no—
149C22×Q8—22no—
150C22×Q8—43no—
151C22×D4—26no—
152C22×D4—22no—
153C22×D4—43no—
154C23×C4—26no—
155C23×C4—26no—
156C23×C4—46no—
157C23×C4—46no—
158C23×C4C2×D4112yes—
159C23×C4C2413yes—
160C23×C4C2411yescharacteristic
161C23×C4C2413yes—
162C23×C4C2413yes—
163C23×C4C2413yes—
164C23×C4C2411yescharacteristic
165C23×C4C2413yes—
166C23×C4C2413yes—
167C23×C4C2413yes—
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2411yescharacteristic
171C23×C4C2413yes—
172#126—448no—
173#124—416no—
174C2×C4⋊C4—212no—
175C2×C4⋊C4—26no—
176C2×C4⋊C4—212no—
177C2×C4⋊C4—26no—
178C2×C4⋊C4—212no—
179C2×C4⋊C4—26no—
180C2×C4⋊C4—26no—
181C2×C4⋊C4—26no—
182C2×C4⋊C4—26no—
183C2×C4⋊C4—26no—
184C2×C4⋊C4—26no—
185C2×C4⋊C4—26no—
186C2×C4⋊C4—26no—
187C2×C4⋊C4—212no—
188C2×C4⋊C4—26no—
189C2×C4⋊C4—26no—
190C2×C4⋊C4—26no—
191C2×C4⋊C4—26no—
192C2×C4⋊C4—43no—
193C2×C4⋊C4—46no—
194C2×C4⋊C4—46no—
195C2×C4⋊C4—46no—
196C2×C4⋊C4—43no—
197C2×C4⋊C4—43no—
198C2×C4⋊C4—46no—
199C2×C4⋊C4—46no—
200C2×C4⋊C4—46no—
201C2×C4⋊C4—46no—
202C2×C22⋊C4—26no—
203C2×C22⋊C4—26no—
204C2×C22⋊C4—212no—
205C2×C22⋊C4—212no—
206C2×C22⋊C4—26no—
207C2×C22⋊C4—26no—
208C2×C22⋊C4—26no—
209C2×C22⋊C4—26no—
210C2×C22⋊C4—26no—
211C2×C22⋊C4—26no—
212C2×C22⋊C4—26no—
213C2×C22⋊C4—26no—
214C2×C22⋊C4—26no—
215C2×C22⋊C4—43no—
216C2×C22⋊C4—43no—
217C2×C22⋊C4—43no—
218C2×C22⋊C4—43no—
219C2×C22⋊C4—46no—
220C2×C22⋊C4—46no—
221C2×C22⋊C4C2×D4112yes—
222C2×C22⋊C4C2×D4112yes—
223C2×C42—26no—
224C2×C42—26no—
225C2×C42—26no—
226C2×C42—212no—
227C2×C42—26no—
228C2×C42—46no—
229C2×C42C2×D4112yes—
230C2×C42C2×D4112yes—
231#95—26no—
232#95—26no—
233#95—212no—
234#95—212no—
235#95—212no—
236#95—26no—
237#95—26no—
238#95—26no—
239#95—212no—
240#95—26no—
241#95—26no—
242#95—26no—
243#95—212no—
244#95—26no—
245#95—26no—
246#95—26no—
247#95—46no—
248#95C2×D4112yes—
249#95C2×D4112yes—
250#95#4114yes—
251C24—21no—
252C24—23no—
253C24—21no—
254C24—21no—
255C24—23no—
256C24—23no—
257C24—23no—
258C24—21no—
259C24—23no—
260C24—21no—
261C24—21no—
262C24—23no—
263C24—23no—
264C24C22×D416yes—
265C24C2511yescharacteristic
266C2×Q8—412no—
267C2×Q8—44no—
268C2×Q8—86no—
269C2×D4—412no—
270C2×D4—44no—
271C2×D4—86no—
272C22×C4—21no—
273C22×C4—23no—
274C22×C4—21no—
275C22×C4—23no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—23no—
279C22×C4—23no—
280C22×C4—23no—
281C22×C4—26no—
282C22×C4—23no—
283C22×C4—23no—
284C22×C4—23no—
285C22×C4—23no—
286C22×C4—26no—
287C22×C4—26no—
288C22×C4—23no—
289C22×C4—23no—
290C22×C4—23no—
291C22×C4—23no—
292C22×C4—26no—
293C22×C4—23no—
294C22×C4—224no—
295C22×C4—26no—
296C22×C4—23no—
297C22×C4—23no—
298C22×C4—26no—
299C22×C4—23no—
300C22×C4—23no—
301C22×C4—26no—
302C22×C4—23no—
303C22×C4—26no—
304C22×C4—23no—
305C22×C4—23no—
306C22×C4—23no—
307C22×C4—26no—
308C22×C4—23no—
309C22×C4—21no—
310C22×C4—23no—
311C22×C4—23no—
312C22×C4—26no—
313C22×C4—26no—
314C22×C4—21no—
315C22×C4—23no—
316C22×C4—26no—
317C22×C4—23no—
318C22×C4—23no—
319C22×C4—23no—
320C22×C4—23no—
321C22×C4—412no—
322C22×C4—412no—
323C22×C4—812no—
324C22×C4—812no—
325C22×C4C22×D416yes—
326C22×C4C22×D416yes—
327C22×C4C22×D416yes—
328C22×C4C22×D416yes—
329C22×C4C22×D416yes—
330C22×C4C22×D416yes—
331C22×C4C2×#4116yes—
332C22×C42−1+412yes—
333C22×C42−1+416yes—
334C4⋊C4—412no—
335C4⋊C4—412no—
336C4⋊C4—412no—
337C4⋊C4—412no—
338C4⋊C4—412no—
339C4⋊C4—424no—
340C4⋊C4—412no—
341C4⋊C4—412no—
342C4⋊C4—412no—
343C4⋊C4—412no—
344C4⋊C4—412no—
345C4⋊C4—86no—
346C22⋊C4—412no—
347C22⋊C4—412no—
348C22⋊C4—412no—
349C22⋊C4—44no—
350C22⋊C4—412no—
351C22⋊C4—412no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—412no—
355C22⋊C4—86no—
356C22⋊C4#127116yes—
357C42—224no—
358C42—412no—
359C42—44no—
360C42—412no—
361C23—26no—
362C23—26no—
363C23—26no—
364C23—26no—
365C23—26no—
366C23—26no—
367C23—41no—
368C23—43no—
369C23—43no—
370C23—46no—
371C23—43no—
372C23—41no—
373C23—43no—
374C23—43no—
375C23—43no—
376C23—43no—
377C23—43no—
378C23—43no—
379C23—41no—
380C23—43no—
381C23—43no—
382C23—46no—
383C23—43no—
384C23C2×#12714yes—
385C23C23×D413yes—
386C23C22×#4111yescharacteristic
387C23C2×2−1+413yes—
388C23C2×2−1+411yescharacteristic
389C23#58511yescharacteristic
390C23#58513yes—
391C23#58513yes—
392Q8—88no—
393D4—88no—
394C2×C4—26no—
395C2×C4—26no—
396C2×C4—26no—
397C2×C4—26no—
398C2×C4—26no—
399C2×C4—26no—
400C2×C4—212no—
401C2×C4—212no—
402C2×C4—26no—
403C2×C4—26no—
404C2×C4—26no—
405C2×C4—26no—
406C2×C4—26no—
407C2×C4—26no—
408C2×C4—26no—
409C2×C4—26no—
410C2×C4—26no—
411C2×C4—26no—
412C2×C4—41no—
413C2×C4—43no—
414C2×C4—43no—
415C2×C4—43no—
416C2×C4—46no—
417C2×C4—43no—
418C2×C4—46no—
419C2×C4—46no—
420C2×C4—46no—
421C2×C4—43no—
422C2×C4—41no—
423C2×C4—43no—
424C2×C4—43no—
425C2×C4—41no—
426C2×C4—43no—
427C2×C4—43no—
428C2×C4—43no—
429C2×C4—43no—
430C2×C4—43no—
431C2×C4—43no—
432C2×C4—46no—
433C2×C4—46no—
434C2×C4—43no—
435C2×C4—46no—
436C2×C4—43no—
437C2×C4—46no—
438C2×C4—43no—
439C2×C4—46no—
440C2×C4—43no—
441C2×C4—46no—
442C2×C4—43no—
443C2×C4—43no—
444C2×C4—46no—
445C2×C4—43no—
446C2×C4—43no—
447C2×C4—46no—
448C2×C4—824no—
449C2×C4C2×#12714yes—
450C2×C4C2×#12714yes—
451C2×C4#547112yes—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—46no—
456C22—46no—
457C22—46no—
458C22—46no—
459C22—83no—
460C22—83no—
461C22—81no—
462C22—81no—
463C22—83no—
464C22—83no—
465C22—81no—
466C22C22×#12711yescharacteristic
467C22#344913yes—
468C22#344913yes—
469C22C2×#54713yes—
470C22#348513yes—
471C22#348913yes—
472C22#349913yes—
473C22#351811yescharacteristic
474C22#352113yes—
475C22#355513yes—
476C22#356416yes—
477C22#356513yes—
478C4—24no—
479C4—212no—
480C4—24no—
481C4—46no—
482C4—46no—
483C4—46no—
484C4—42no—
485C4—81no—
486C4—83no—
487C4—83no—
488C4—83no—
489C4—83no—
490C4—81no—
491C2—44no—
492C2—161no—
493C2#3464111yescharacteristic
494C2#3606011yescharacteristic
495C2#3617911yescharacteristic
496C2#3804613yes—
497C2#3805113yes—
498C2#4231013yes—
499C2#4231513yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2496C2semidirect
3. #17064C2semidirect
4. #17059C2semidirect
5. #16813C2semidirect
7. #15199C2semidirect
8. #15082C2semidirect
9. #15079C2semidirect
10. #14910C2semidirect
12. #14353C2semidirect
13. #14322C2semidirect
14. #13856C2semidirect
15. #13809C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#25212
#170642
#170592
#168132
#168072
#151992
#150822
#150792
#149102
#148362
#143532
#143222
#138562
#138092