Finite Group Databasebeta

← #6404367

Subgroups — #6404367

Order 512.

2232 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3358C211yesmaximal, characteristic
2#34313C211yesmaximal, characteristic, complemented
3#34311C211yesmaximal, characteristic, complemented
4#23216C213yesmaximal, complemented
5#23093C213yesmaximal, complemented
6#22962C213yesmaximal, complemented
7#22766C213yesmaximal, complemented
8#22740C211yesmaximal, characteristic, complemented
9#22540C211yesmaximal, characteristic, complemented
10Q8×#102C211yesmaximal, characteristic, complemented
11#22145C213yesmaximal, complemented
12#22110C213yesmaximal, complemented
13C4×#486C213yesmaximal, complemented
14#21275C213yesmaximal, complemented
15C4×C8×Q8C211yesmaximal, characteristic, complemented
16C2×#550C2211yescharacteristic
17C2×D4×Q8C2211yescharacteristic
18C2×#520—21no—
19C2×#519—21no—
20#3358C4116yescomplemented
21C2×#486C2213yes—
22C2×#474C2213yes—
23C2×#459C2213yes—
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C4×D8C2213yes—
27#2366C2213yes—
28#2365C2213yes—
29#2362—23no—
30#2361—23no—
31#2354C2213yes—
32Q8×C4⋊C4C2211yescharacteristic
33Q8×C4⋊C4C2211yescharacteristic
34D4×C4⋊C4C2213yes—
35#2341—21no—
36Q8×C22⋊C4—21no—
37#2324—23no—
38#2323—23no—
39C4×C4⋊Q8C2213yes—
40C4×#127C2213yes—
41#2293C2213yes—
42#2293C2213yes—
43C42×Q8C2211yescharacteristic
44#1990C2213yes—
45#1988C2213yes—
46#1982C2213yes—
47#1970C2213yes—
48#1970C2213yes—
49#1957C2213yes—
50#1947C2213yes—
51#1947C2213yes—
52#1942C2213yes—
53#1942C2213yes—
54#1934C2211yescharacteristic
55#1934C2213yes—
56#1934C2213yes—
57#1934C2213yes—
58#1934C2216yes—
59#1925C2216yes—
60#1918C2216yes—
61#1879C2216yes—
62#1862C2216yes—
63#1834C2213yes—
64#1802C2213yes—
65#1802C2213yes—
66C4×#107C2213yes—
67C8×C4⋊C4C2213yes—
68C8×C4⋊C4C2213yes—
69C4×C4⋊C8C2213yes—
70C4×#102C2213yes—
71C4×#102C2213yes—
72C4×#102C2216yes—
73C4×#102C2216yes—
74C42×C8C2213yes—
75C22×#41—21no—
76C23×Q8—21no—
77C22×D8C2313yes—
78#550—24no—
79#550C2×C418yes—
80D4×Q8—24no—
81D4×Q8C2×C418yescomplemented
82C2×C4⋊Q8C2313yes—
83C2×#127C2313yes—
84C2×C22⋊Q8—23no—
85C2×#121—23no—
86#520—48no—
87#519—48no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—23no—
91C2×C4×Q8—23no—
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4—23no—
96C2×C4×D4—23no—
97C2×C4×D4—23no—
98C2×C4×D4C2313yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2316yes—
101C2×#117—23no—
102C2×#117—23no—
103C2×#117—23no—
104C22×C4⋊C4—23no—
105#486C2×C4124yescomplemented
106#474C2×C4124yescomplemented
107#459C2×C4124yescomplemented
108C8×Q8—24no—
109C8×Q8C2×C418yescomplemented
110C4×D8C2×C4124yescomplemented
111C2×#107C2313yes—
112C2×#107C2313yes—
113C2×#107C2313yes—
114C2×C4⋊C8C2313yes—
115C2×C4⋊C8C2313yes—
116C2×#102C2313yes—
117C2×#102C2311yescharacteristic
118C2×#102C2313yes—
119C2×#102C2316yes—
120C2×#102C2313yes—
121C2×#102C2313yes—
122C2×#102C2316yes—
123C2×#102C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2313yes—
126C2×C4×C8C2313yes—
127C2×C4×C8C2313yes—
128C2×C4×C8C2313yes—
129C2×C4×C8C2313yes—
130#391—23no—
131#391—23no—
132#390—23no—
133#390—23no—
134#390—23no—
135#390C2313yes—
136#390C2313yes—
137#390C2313yes—
138#390C2311yescharacteristic
139#389—23no—
140#389—23no—
141#389—23no—
142#389C2313yes—
143#389C2313yes—
144#389C2313yes—
145#389C2311yescharacteristic
146#388—23no—
147#388—23no—
148#388—23no—
149#388—23no—
150#388—26no—
151#387—26no—
152#386—26no—
153#385—26no—
154#384C2313yes—
155#384C2313yes—
156#380—23no—
157#380—23no—
158C4×C4⋊C4—23no—
159C4×C4⋊C4D413yes—
160C4×C4⋊C4D413yes—
161C4×C4⋊C4C2316yes—
162C4×C4⋊C4C2313yes—
163C4×C4⋊C4C2313yes—
164C4×C4⋊C4C2313yes—
165C4×C22⋊C4—23no—
166C4×C22⋊C4—23no—
167C43C2313yes—
168#340C2313yes—
169#340C2316yes—
170#340C2313yes—
171#340C2316yes—
172#340C2313yes—
173#340C2311yescharacteristic
174#340C2311yescharacteristic
175#340C2313yes—
176#336C2316yes—
177C2×#41—28no—
178C2×#41—44no—
179C2×#41—44no—
180C22×Q8—28no—
181C22×Q8—23no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8C2×D411yescharacteristic
185C22×D4—23no—
186C22×D4—23no—
187C22×D4C2413yes—
188C22×D4C2411yescharacteristic
189C23×C4—23no—
190C23×C4—23no—
191C23×C4—26no—
192C2×D8—43no—
193C2×D8—43no—
194C2×D8C22×C4112yes—
195C22×C8C2413yes—
196C22×C8C2413yes—
197C22×C8C2411yescharacteristic
198C22×C8C2411yescharacteristic
199C4⋊Q8—26no—
200C4⋊Q8C22×C4112yes—
201#127—26no—
202#127C22×C4112yes—
203C22⋊Q8—224no—
204#121—224no—
205C4×Q8—24no—
206C4×Q8—44no—
207C4×Q8—44no—
208C4×Q8—412no—
209C4×Q8—412no—
210C4×Q8D818yescomplemented
211C4×Q8C22×C414yes—
212C4×Q8#4114yes—
213C4×D4—26no—
214C4×D4—26no—
215C4×D4—46no—
216C4×D4—46no—
217C4×D4—412no—
218C4×D4—412no—
219C4×D4—86no—
220C4×D4C22×C4112yes—
221C4×D4C22×C4112yes—
222#117—412no—
223#117—412no—
224#117—412no—
225C2×C4⋊C4—23no—
226C2×C4⋊C4—23no—
227C2×C4⋊C4—23no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—23no—
230C2×C4⋊C4—23no—
231C2×C4⋊C4—23no—
232C2×C4⋊C4—23no—
233C2×C4⋊C4—23no—
234C2×C4⋊C4—412no—
235C2×C4⋊C4C2×D413yes—
236C2×C4⋊C4C2×D413yes—
237C2×C4⋊C4C2×D413yes—
238C2×C4⋊C4C2413yes—
239C2×C4⋊C4C2413yes—
240C2×C4⋊C4C2411yescharacteristic
241C2×C4⋊C4C2411yescharacteristic
242C2×C4⋊C4C2411yescharacteristic
243C2×C4⋊C4C2413yes—
244C2×C22⋊C4—23no—
245C2×C22⋊C4—23no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—23no—
248C2×C22⋊C4—23no—
249C2×C22⋊C4—23no—
250C2×C22⋊C4—23no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—26no—
253C2×C22⋊C4—23no—
254C2×C22⋊C4—23no—
255C2×C42—23no—
256C2×C42—23no—
257C2×C42—26no—
258C2×C42C2×D413yes—
259C2×C42C2×D413yes—
260C2×C42C2×D413yes—
261C2×C42C2×D413yes—
262C2×C42C2413yes—
263C2×C42C2411yescharacteristic
264C2×C42C2413yes—
265#107—43no—
266#107—43no—
267#107C22×C4112yes—
268C4⋊C8—26no—
269C4⋊C8C22×C4112yes—
270#102—26no—
271#102—22no—
272#102—43no—
273#102—46no—
274#102—43no—
275#102C22×C414yes—
276#102C22×C4112yes—
277#102C22×C4124yes—
278C4×C8—22no—
279C4×C8—26no—
280C4×C8—43no—
281C4×C8—43no—
282C4×C8—43no—
283C4×C8C22×C4112yes—
284#95—21no—
285#95—23no—
286#95—23no—
287#95—23no—
288#95—26no—
289#95—23no—
290#95—23no—
291#95C2×D416yes—
292C24—21no—
293C24—23no—
294#41—432no—
295C2×Q8—212no—
296C2×Q8—432no—
297C2×Q8—46no—
298C2×Q8—46no—
299C2×Q8C4×D414yes—
300C2×Q8C2×D814yes—
301C2×Q8#13514yes—
302C2×D4—212no—
303C2×D4—212no—
304C2×D4—21no—
305C2×D4—23no—
306C2×D4—23no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—23no—
310C2×D4—23no—
311C2×D4—23no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—46no—
315C2×D4—46no—
316C2×D4—83no—
317C2×D4—83no—
318C2×D4C23×C416yes—
319C2×D4C23×C412yes—
320C22×C4—23no—
321C22×C4—21no—
322C22×C4—212no—
323C22×C4—212no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—23no—
327C22×C4—23no—
328C22×C4—46no—
329C22×C4—46no—
330C22×C4—46no—
331C22×C4—46no—
332C22×C4—412no—
333C22×C4—86no—
334C22×C4—86no—
335C22×C4C22×D413yes—
336C22×C4C22×D413yes—
337C22×C4C22×D413yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D413yes—
341C22×C4C2511yescharacteristic
342D8—412no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—23no—
348C2×C8—23no—
349C2×C8—23no—
350C2×C8—23no—
351C2×C8—23no—
352C2×C8C23×C412yes—
353C2×C8C23×C416yes—
354C2×C82−1+412yes—
355C4⋊C4—212no—
356C4⋊C4—26no—
357C4⋊C4—21no—
358C4⋊C4—23no—
359C4⋊C4—23no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—23no—
363C4⋊C4—23no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—23no—
367C4⋊C4—46no—
368C4⋊C4—46no—
369C4⋊C4—42no—
370C4⋊C4—46no—
371C4⋊C4—46no—
372C4⋊C4—46no—
373C4⋊C4—83no—
374C4⋊C4—83no—
375C4⋊C4C4×D4112yes—
376C4⋊C4C2×D8112yes—
377C4⋊C4C23×C416yes—
378C4⋊C4C23×C412yes—
379C22⋊C4—212no—
380C22⋊C4—212no—
381C22⋊C4—46no—
382C22⋊C4—42no—
383C22⋊C4—46no—
384C22⋊C4—46no—
385C22⋊C4—42no—
386C22⋊C4—46no—
387C22⋊C4—83no—
388C22⋊C4—83no—
389C22⋊C4—86no—
390C42—26no—
391C42—26no—
392C42—21no—
393C42—21no—
394C42—23no—
395C42—23no—
396C42—43no—
397C42—46no—
398C42—412no—
399C42—46no—
400C42C2×D8112yes—
401C42C23×C416yes—
402C42C2×#4116yes—
403C23—22no—
404C23—26no—
405C23—41no—
406C23—43no—
407C23—43no—
408C23—41no—
409C23—43no—
410C23—41no—
411C23—41no—
412C23—43no—
413C23—41no—
414C23—43no—
415C23—43no—
416C23—41no—
417C23C23×D411yescharacteristic
418Q8—424no—
419Q8C4×D8116yes—
420D4—22no—
421D4—26no—
422D4—22no—
423D4—26no—
424D4—424no—
425D4—41no—
426D4—43no—
427D4—41no—
428D4—43no—
429D4—812no—
430C2×C4—21no—
431C2×C4—23no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—23no—
435C2×C4—21no—
436C2×C4—23no—
437C2×C4—23no—
438C2×C4—23no—
439C2×C4—22no—
440C2×C4—26no—
441C2×C4—23no—
442C2×C4—23no—
443C2×C4—43no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—43no—
447C2×C4—424no—
448C2×C4—424no—
449C2×C4—41no—
450C2×C4—43no—
451C2×C4—43no—
452C2×C4—41no—
453C2×C4—43no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—43no—
458C2×C4—43no—
459C2×C4—43no—
460C2×C4—41no—
461C2×C4—1612no—
462C2×C4C2×C4×D416yes—
463C2×C4C2×C4×D416yes—
464C2×C4C22×D816yes—
465C2×C4C22×D812yes—
466C2×C4C22×D816yes—
467C2×C4C2×#13516yes—
468C2×C4#57712yes—
469C2×C4C24×C411yescharacteristic
470C2×C4C22×#4111yescharacteristic
471C2×C4C2×2−1+411yescharacteristic
472C2×C4#58511yescharacteristic
473C2×C4#58511yescharacteristic
474C2×C4#58511yescharacteristic
475C8—26no—
476C8—22no—
477C8—22no—
478C8—43no—
479C22—42no—
480C22—46no—
481C22—46no—
482C22—42no—
483C22—42no—
484C22—46no—
485C22—83no—
486C22—81no—
487C22—81no—
488C22—83no—
489C22—81no—
490C22—83no—
491C22—81no—
492C22—83no—
493C22C22×C4×D411yescharacteristic
494C22#349711yescharacteristic
495C22#350011yescharacteristic
496C22C23×D811yescharacteristic
497C22C22×#13511yescharacteristic
498C22C2×#57711yescharacteristic
499C22#358411yescharacteristic
500C4—26no—
501C4—26no—
502C4—22no—
503C4—22no—
504C4—43no—
505C4—42no—
506C4—46no—
507C4—81no—
508C4—83no—
509C4—81no—
510C4—83no—
511C4C2×C4×D8112yes—
512C4#342811yescharacteristic
513C4C4×2−1+411yescharacteristic
514C4#351611yescharacteristic
515C4#352411yescharacteristic
516C2—86no—
517C2—82no—
518C2#3431211yescharacteristic
519C2C22×C4×D811yescharacteristic
520C2C4×#57711yescharacteristic
521C2#6235911yescharacteristic
522C2#6236211yescharacteristic
523C2#6252911yescharacteristic
524C2#6253411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3358C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34313C2semidirect
3. #34311C2semidirect
4. #23216C2semidirect
5. #23093C2semidirect
6. #22962C2semidirect
7. #22766C2semidirect
8. #22740C2semidirect
9. #22540C2semidirect
10. Q8×#102C2semidirect
11. #22145C2semidirect
12. #22110C2semidirect
13. C4×#486C2semidirect
14. #21275C2semidirect
15. C4×C8×Q8C2semidirect
20. #3358C4semidirect
81. D4×Q8C2×C4semidirect
105. #486C2×C4semidirect
106. #474C2×C4semidirect
107. #459C2×C4semidirect
109. C8×Q8C2×C4semidirect
110. C4×D8C2×C4semidirect
210. C4×Q8D8semidirect

Maximal subgroups

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

subgroupindex
C2×#33582
#343132
#343112
#232162
#230932
#229622
#227662
#227402
#225402
Q8×#1022
#221452
#221102
C4×#4862
#212752
C4×C8×Q82