Finite Group Databasebeta

← #6407840

Subgroups — #6407840

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36696C211yesmaximal, characteristic
2#23378C211yesmaximal, characteristic, complemented
3#23375C211yesmaximal, characteristic, complemented
4#23355C211yesmaximal, characteristic, complemented
5#23330C213yesmaximal, complemented
6#23328C213yesmaximal, complemented
7#23319C213yesmaximal, complemented
8#23310C213yesmaximal, complemented
9#21681C211yesmaximal, characteristic, complemented
10#21666C211yesmaximal, characteristic, complemented
11C8×#126C211yesmaximal, characteristic, complemented
12#21571C213yesmaximal, complemented
13#21496C213yesmaximal, complemented
14#21494C213yesmaximal, complemented
15#21445C213yesmaximal, complemented
16C22×#131C2211yescharacteristic
17C2×#436C2213yes—
18C2×#436C2213yes—
19C2×#436C2213yes—
20C2×#435C2213yes—
21C2×#405C2213yes—
22#2843C411yescharacteristic
23#2830C413yes—
24#2829C413yes—
25#2829C413yes—
26#2828C413yes—
27#2823C411yescharacteristic
28#2818C411yescharacteristic
29#2818C411yescharacteristic
30#2395C2213yes—
31#2392C2213yes—
32#2392C2213yes—
33#2388C2213yes—
34C4×#127C2211yescharacteristic
35C4×#126C2211yescharacteristic
36C4×#126C2211yescharacteristic
37#1955C2213yes—
38#1954C2213yes—
39#1954C2213yes—
40#1954C2213yes—
41#1940C2213yes—
42#1939C2213yes—
43#1924C2213yes—
44#1924C2213yes—
45#1924C2213yes—
46#1923C2213yes—
47#1923C2213yes—
48#1923C2213yes—
49#1923C2213yes—
50#1923C2213yes—
51#1917C2213yes—
52#1916C2213yes—
53#1916C2213yes—
54#1916C2213yes—
55#1870C2213yes—
56#1869C2213yes—
57#1869C2213yes—
58#1869C2213yes—
59#1869C2213yes—
60#1869C2213yes—
61#1869C2213yes—
62#1869C2213yes—
63#1851C2213yes—
64#1850C2213yes—
65#1850C2213yes—
66#1850C2213yes—
67#1842C2211yescharacteristic
68#1839C2211yescharacteristic
69#1839C2211yescharacteristic
70#1818C2213yes—
71#1798C2213yes—
72#1798C2213yes—
73#1780C2213yes—
74#1771C2213yes—
75C8×C4⋊C4C2213yes—
76#1753C2213yes—
77#1753C2213yes—
78#1753C2213yes—
79C8×C22⋊C4C2213yes—
80C42×C8C2211yescharacteristic
81C22×#41C2311yescharacteristic
82C2×#131—46no—
83C22×M16C2313yes—
84C22×M16C2311yescharacteristic
85C22×M16C2311yescharacteristic
86C22×M16C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2313yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×#127C2×C411yescharacteristic
92C2×#126C2×C411yescharacteristic
93C2×#126C2×C411yescharacteristic
94C2×C4×D4C2313yes—
95C2×C4×D4C2313yes—
96C2×C4×D4C2313yes—
97C2×C4×D4C2313yes—
98C2×#117C2313yes—
99#436—46no—
100#436—46no—
101#436—46no—
102#435—46no—
103C2×C4⋊C8C2313yes—
104C2×C4⋊C8C2313yes—
105C2×C4⋊C8C2313yes—
106C2×C4⋊C8C2313yes—
107C2×C4⋊C8C2313yes—
108C2×C4⋊C8C2313yes—
109C2×C22⋊C8C2313yes—
110C2×C22⋊C8C2313yes—
111C2×C22⋊C8C2313yes—
112C2×C22⋊C8C2313yes—
113C2×C22⋊C8C2313yes—
114C2×C22⋊C8C2313yes—
115C2×C22⋊C8C2313yes—
116C2×C22⋊C8C2313yes—
117C2×C22⋊C8C2313yes—
118C2×C22⋊C8C2313yes—
119C2×C22⋊C8C2313yes—
120C2×C22⋊C8C2313yes—
121#405—46no—
122C2×#97C2313yes—
123C2×#97C2313yes—
124C2×C4×C8C2313yes—
125C2×C4×C8C2313yes—
126C2×C4×C8C2313yes—
127C2×C4×C8C2313yes—
128#400C2×C413yes—
129#399C2×C413yes—
130#398C2×C413yes—
131#398C2×C413yes—
132#398C2×C413yes—
133#397C2×C413yes—
134#397C2×C413yes—
135#397C2×C413yes—
136#397C2×C413yes—
137#397C2×C413yes—
138#396C2×C413yes—
139#396C2×C413yes—
140#396C2×C413yes—
141#396C2×C413yes—
142#395C2×C413yes—
143#394C2×C413yes—
144#394C2×C413yes—
145#394C2×C413yes—
146#394C2×C413yes—
147#394C2×C413yes—
148#394C2×C413yes—
149#394C2×C413yes—
150#394C2×C413yes—
151#394C2×C413yes—
152#394C2×C413yes—
153#394C2×C413yes—
154#393C2×C413yes—
155#392C2×C413yes—
156#392C2×C413yes—
157#392C2×C413yes—
158#392C2×C413yes—
159#392C2×C413yes—
160#390C2×C413yes—
161#390C2×C413yes—
162#388C2×C413yes—
163#388C2×C413yes—
164#387C2×C413yes—
165#387C2×C413yes—
166#384C2×C411yescharacteristic
167#383C2×C411yescharacteristic
168#383C2×C411yescharacteristic
169C4×C4⋊C4C2313yes—
170C4×C4⋊C4C2313yes—
171C4×C22⋊C4C2313yes—
172C4×C22⋊C4C2313yes—
173C4×C22⋊C4C2313yes—
174C4×C22⋊C4C2313yes—
175C43C2311yescharacteristic
176#336C2313yes—
177#336C2313yes—
178#336C2313yes—
179#336C2313yes—
180#336C2313yes—
181#336C2313yes—
182#336C2313yes—
183#336C2313yes—
184#336C2313yes—
185#336C2313yes—
186#336C2313yes—
187#336C2313yes—
188C2×#41—43no—
189C2×#41—43no—
190C22×Q8C22×C411yescharacteristic
191C22×D4C22×C413yes—
192C22×D4C22×C413yes—
193C22×D4C22×C413yes—
194C22×D4C22×C413yes—
195C22×D4C22×C411yescharacteristic
196C22×D4C22×C411yescharacteristic
197C22×D4C22×C411yescharacteristic
198C23×C4C2413yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202#131—416no—
203C2×M16—26no—
204C2×M16—26no—
205C2×M16—26no—
206C2×M16—26no—
207C2×M16—26no—
208C2×M16—43no—
209C22×C8—26no—
210C22×C8—43no—
211C22×C8—43no—
212C22×C8—43no—
213C22×C8—43no—
214C22×C8—43no—
215C22×C8C2411yescharacteristic
216C22×C8C2411yescharacteristic
217C22×C8C2411yescharacteristic
218C22×C8C2413yes—
219C22×C8C2413yes—
220C22×C8C2413yes—
221C22×C8C2413yes—
222C22×C8C2411yescharacteristic
223#127—24no—
224#126—24no—
225#126—24no—
226C4×D4—43no—
227C4×D4—43no—
228C4×D4—43no—
229C4×D4—43no—
230C4×D4—43no—
231C4×D4—43no—
232C4×D4—43no—
233C4×D4—43no—
234#117—43no—
235#117—43no—
236C2×C4⋊C4C22×C413yes—
237C2×C4⋊C4C22×C413yes—
238C2×C4⋊C4C22×C413yes—
239C2×C4⋊C4C22×C413yes—
240C2×C4⋊C4C22×C413yes—
241C2×C4⋊C4C22×C413yes—
242C2×C4⋊C4C22×C413yes—
243C2×C4⋊C4C22×C413yes—
244C2×C4⋊C4C22×C413yes—
245C2×C4⋊C4C22×C413yes—
246C2×C22⋊C4C22×C413yes—
247C2×C22⋊C4C22×C413yes—
248C2×C22⋊C4C22×C413yes—
249C2×C22⋊C4C22×C413yes—
250C2×C22⋊C4C22×C413yes—
251C2×C22⋊C4C22×C413yes—
252C2×C22⋊C4C22×C413yes—
253C2×C22⋊C4C22×C413yes—
254C2×C22⋊C4C22×C413yes—
255C2×C22⋊C4C22×C413yes—
256C2×C22⋊C4C22×C413yes—
257C2×C22⋊C4C22×C413yes—
258C2×C22⋊C4C22×C413yes—
259C2×C22⋊C4C22×C413yes—
260C2×C22⋊C4C22×C413yes—
261C2×C22⋊C4C22×C413yes—
262C2×C22⋊C4C22×C413yes—
263C2×C22⋊C4C22×C413yes—
264C2×C42C22×C413yes—
265C2×C42C22×C411yescharacteristic
266C2×C42C2413yes—
267C2×C42C2413yes—
268C2×C42C2413yes—
269C4⋊C8—43no—
270C4⋊C8—43no—
271C4⋊C8—43no—
272C4⋊C8—43no—
273C4⋊C8—43no—
274C4⋊C8—43no—
275C22⋊C8—43no—
276C22⋊C8—43no—
277C22⋊C8—43no—
278C22⋊C8—43no—
279C22⋊C8—43no—
280C22⋊C8—43no—
281C22⋊C8—43no—
282C22⋊C8—43no—
283C22⋊C8—43no—
284C22⋊C8—43no—
285C22⋊C8—43no—
286C22⋊C8—43no—
287#97—26no—
288#97—43no—
289C4×C8—26no—
290C4×C8—26no—
291C4×C8—43no—
292C4×C8—43no—
293#95C22×C413yes—
294#95C22×C413yes—
295#95C22×C413yes—
296#95C22×C413yes—
297#95C22×C413yes—
298#95C22×C413yes—
299#95C22×C413yes—
300#95C22×C413yes—
301#95C22×C413yes—
302#95C22×C413yes—
303#95C22×C413yes—
304#95C22×C413yes—
305C24C23×C413yes—
306C24C23×C411yescharacteristic
307C24C23×C411yescharacteristic
308C24C23×C411yescharacteristic
309#41—44no—
310#41—412no—
311C2×Q8—43no—
312C2×D4—26no—
313C2×D4—26no—
314C2×D4—43no—
315C2×D4—43no—
316C2×D4—43no—
317C2×D4—43no—
318C2×D4—43no—
319C2×D4—43no—
320C2×D4—43no—
321C2×D4—43no—
322C2×D4—43no—
323C2×D4—43no—
324C2×D4—43no—
325C2×D4—43no—
326C2×D4—43no—
327C22×C4—23no—
328C22×C4—23no—
329C22×C4—23no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—23no—
339C22×C4C23×C413yes—
340C22×C4C23×C413yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C23×C411yescharacteristic
343C22×C4C23×C411yescharacteristic
344C22×C4C23×C413yes—
345C22×C4C23×C413yes—
346C22×C4C23×C413yes—
347C22×C4C23×C413yes—
348C22×C4C23×C413yes—
349C22×C4C2511yescharacteristic
350M16—44no—
351M16—412no—
352M16—44no—
353M16—44no—
354C2×C8—23no—
355C2×C8—23no—
356C2×C8—23no—
357C2×C8—23no—
358C2×C8—23no—
359C2×C8—23no—
360C2×C8—23no—
361C2×C8—23no—
362C2×C8—23no—
363C2×C8—23no—
364C2×C8—23no—
365C2×C8—23no—
366C2×C8—23no—
367C2×C8—23no—
368C2×C8—23no—
369C2×C8—412no—
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—44no—
373C2×C8H(5,𝔽2)16yes—
374C4⋊C4—26no—
375C4⋊C4—26no—
376C4⋊C4—43no—
377C4⋊C4—43no—
378C4⋊C4—43no—
379C4⋊C4—43no—
380C4⋊C4—43no—
381C4⋊C4—43no—
382C4⋊C4—43no—
383C4⋊C4—43no—
384C22⋊C4—26no—
385C22⋊C4—26no—
386C22⋊C4—43no—
387C22⋊C4—43no—
388C22⋊C4—43no—
389C22⋊C4—43no—
390C22⋊C4—43no—
391C22⋊C4—43no—
392C22⋊C4—43no—
393C22⋊C4—43no—
394C22⋊C4—43no—
395C22⋊C4—43no—
396C22⋊C4—43no—
397C22⋊C4—43no—
398C22⋊C4—43no—
399C22⋊C4—43no—
400C22⋊C4—43no—
401C22⋊C4—43no—
402C42—23no—
403C42—23no—
404C42—23no—
405C42—23no—
406C42—23no—
407C42—23no—
408C42—43no—
409C42#13114yescomplemented
410C23—23no—
411C23—23no—
412C23—23no—
413C23—23no—
414C23—23no—
415C23—23no—
416C23—21no—
417C23—21no—
418C23—23no—
419C23—21no—
420C23—23no—
421C23—23no—
422C23—23no—
423C23—23no—
424C23—23no—
425C23—23no—
426C23C24×C411yescharacteristic
427Q8—44no—
428D4—44no—
429D4—44no—
430D4—44no—
431D4—86no—
432D4—86no—
433D4—86no—
434D4—86no—
435C2×C4—23no—
436C2×C4—23no—
437C2×C4—23no—
438C2×C4—23no—
439C2×C4—23no—
440C2×C4—23no—
441C2×C4—23no—
442C2×C4—23no—
443C2×C4—23no—
444C2×C4—23no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—23no—
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—23no—
457C2×C4—23no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—43no—
462C2×C4—43no—
463C2×C4—43no—
464C2×C4—43no—
465C2×C4—43no—
466C2×C4—43no—
467C2×C4—43no—
468C2×C4C2×#13116yes—
469C2×C4C2×H(5,𝔽2)13yes—
470C2×C4#58513yes—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—43no—
475C8—43no—
476C8—43no—
477C8—43no—
478C8#56014yes—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—43no—
483C22—43no—
484C22—43no—
485C22—43no—
486C22—43no—
487C22—43no—
488C22—43no—
489C22—43no—
490C22—43no—
491C22—43no—
492C22—43no—
493C22—43no—
494C22—43no—
495C22C4×H(5,𝔽2)13yes—
496C22C22×#13111yescharacteristic
497C22#357313yes—
498C4—26no—
499C4—43no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—43no—
504C4—43no—
505C4—43no—
506C4—43no—
507C4—43no—
508C4C2×#56011yescharacteristic
509C4#352913yes—
510C2—81no—
511C2—81no—
512C2—81no—
513C2—83no—
514C2C4×#56011yescharacteristic
515C2#6032113yes—
516C2#6032713yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radical#36696C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #23378C2semidirect
3. #23375C2semidirect
4. #23355C2semidirect
5. #23330C2semidirect
6. #23328C2semidirect
7. #23319C2semidirect
8. #23310C2semidirect
9. #21681C2semidirect
10. #21666C2semidirect
11. C8×#126C2semidirect
12. #21571C2semidirect
13. #21496C2semidirect
14. #21494C2semidirect
15. #21445C2semidirect
409. C42#131semidirect

Maximal subgroups

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

subgroupindex
#366962
#233782
#233752
#233552
#233302
#233282
#233192
#233102
#216812
#216662
C8×#1262
#215712
#214962
#214942
#214452