Finite Group Databasebeta

← #7194113

Subgroups — #7194113

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic, complemented
2#59823C211yesmaximal, characteristic
3#32157C214yesmaximal, complemented
4#31474C214yesmaximal, complemented
5#29277C211yesmaximal, characteristic, complemented
6#28202C211yesmaximal, characteristic
7#27018C214yesmaximal, complemented
8#26771C214yesmaximal
9#25839C212yesmaximal, complemented
10#25477C212yesmaximal, complemented
11#24990C212yesmaximal, complemented
12#23815C211yesmaximal, characteristic
13#22522C212yesmaximal, complemented
14#22248C211yesmaximal, characteristic, complemented
15#22242C211yesmaximal, characteristic, complemented
16C2×#554C2212yes—
17C2×#500C2211yescharacteristic
18C2×#476C2212yes—
19C2×#446C2212yes—
20#2806—22no—
21#2732—22no—
22#2584—22no—
23#2582—22no—
24#2567C2212yes—
25#2552—21no—
26#2552C2212yes—
27#2552C2212yes—
28#2542—21no—
29#2538—22no—
30#2538C2212yes—
31#2507C2214yes—
32#2454C2214yes—
33#2417—22no—
34#2369C2212yes—
35#2366C2212yes—
36#2365—21no—
37Q8×C4⋊C4C2212yes—
38D4×C4⋊C4—21no—
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44#2100C2212yes—
45#2099C2212yes—
46#2099C2214yes—
47#2088C2214yes—
48#2084C2212yes—
49#2071C2212yes—
50#2071C2214yes—
51#2067C2214yes—
52#2059C2214yes—
53#2059C2214yes—
54#2057C2212yes—
55#2041C2212yes—
56#2041C2212yes—
57#2023C2212yes—
58#1992C2214yes—
59#1990C2211yescharacteristic
60#1990C2211yescharacteristic
61#1989C2214yes—
62#1987C2212yes—
63#1984C2212yes—
64#1947C2212yes—
65#1943C2214yes—
66#1936C2214yes—
67#1926C2214yes—
68#1925C2214yes—
69#1921C2214yes—
70#1921C2214yes—
71#1921C2214yes—
72#1918C2214yes—
73#1917C2214yes—
74#1917C2214yes—
75#1862C2214yes—
76#1862C2214yes—
77#1834C2211yescharacteristic
78#1831C2211yescharacteristic
79#1802C2212yes—
80#1802C2214yes—
81#1762C2214yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85#554—28no—
86#554—44no—
87C2×C4⋊Q8D411yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#125—22no—
93C2×#125C2314yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C4×Q8—22no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8C2312yes—
102C2×C4×D4—22no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2314yes—
107C22×C4⋊C4—22no—
108#500—24no—
109#476—44no—
110#446—44no—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2314yes—
116C2×#102C2314yes—
117C2×#102C2312yes—
118C2×#102C2312yes—
119C2×#102C2312yes—
120C2×#102C2312yes—
121C2×#102C2314yes—
122C2×#97C2314yes—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#399—24no—
126#398—24no—
127#398—24no—
128#398—24no—
129#398—24no—
130#396—24no—
131#396—24no—
132#395—24no—
133#391—22no—
134#391—21no—
135#391—21no—
136#391—21no—
137#391—24no—
138#391D412yes—
139#390—22no—
140#390—21no—
141#390—21no—
142#390C2312yes—
143#390C2314yes—
144#389—22no—
145#389—24no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—24no—
153#389D412yes—
154#389D412yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2314yes—
159#389C2314yes—
160#389C2314yes—
161#387—24no—
162#386—24no—
163#385—24no—
164#384—22no—
165#384—22no—
166#384D411yescharacteristic
167#384C2311yescharacteristic
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2314yes—
172#380—22no—
173#380—24no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C43C2311yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2312yes—
190#340C2312yes—
191#340C2314yes—
192#340C2312yes—
193#340C2314yes—
194#340C2314yes—
195#340C2312yes—
196#336C2312yes—
197#336C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8C2×D412yes—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C2412yes—
209C23×C4—22no—
210C23×C4—24no—
211C22×C8C2412yes—
212C22×C8C2414yes—
213C22×C8C2412yes—
214C4⋊Q8—24no—
215C4⋊Q8—24no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8#4114yes—
220#127—24no—
221#125—28no—
222#125—44no—
223#125—84no—
224C22⋊Q8—48no—
225C22⋊Q8—416no—
226C22⋊Q8—84no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—82no—
230C4×Q8—82no—
231C4×Q8—82no—
232C4×Q8—84no—
233C4×D4—28no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—44no—
237C4×D4—44no—
238C4×D4—82no—
239C4×D4—84no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2414yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293#107—28no—
294#107—28no—
295#106—44no—
296C4⋊C8—24no—
297C4⋊C8—44no—
298#102—28no—
299#102—24no—
300#102—24no—
301#102—24no—
302#102—24no—
303#102—44no—
304#97—44no—
305C4×C8—22no—
306C4×C8—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95C2×D414yes—
318#95C2×D414yes—
319C24—22no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—42no—
326C2×Q8—81no—
327C2×Q8—81no—
328C2×Q8—81no—
329C2×Q8—81no—
330C2×Q8—81no—
331C2×Q8—81no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—81no—
342C2×D4—81no—
343C2×D42−1+414yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—48no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D414yes—
360C22×C4C22×D414yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C2511yescharacteristic
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—84no—
401C4⋊C4—81no—
402C4⋊C4—81no—
403C4⋊C4—81no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—81no—
407C4⋊C4—82no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4C2×#4114yes—
413C4⋊C42−1+414yes—
414C22⋊C4—48no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—84no—
418C22⋊C4—82no—
419C22⋊C4—84no—
420C42—22no—
421C42—24no—
422C42—24no—
423C42—22no—
424C42—22no—
425C42—41no—
426C42—41no—
427C42—44no—
428C42—44no—
429C42—44no—
430C42—84no—
431C42#13714yes—
432C42C2×#4112yes—
433C42H(5,𝔽2)12yes—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23C23×D411yescharacteristic
442Q8—48no—
443Q8—84no—
444Q8—84no—
445Q8—84no—
446D4—24no—
447D4—24no—
448D4—48no—
449D4—42no—
450D4—42no—
451D4—84no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—88no—
492C2×C4—168no—
493C2×C4#54612yes—
494C2×C4#54612yes—
495C2×C4C2×#13712yes—
496C2×C4C2×#13712yes—
497C2×C4C2×#13712yes—
498C2×C4#57612yes—
499C2×C4C22×#4111yescharacteristic
500C2×C4C2×H(5,𝔽2)11yescharacteristic
501C2×C4C2×2−1+412yes—
502C2×C4C2×2−1+412yes—
503C8—42no—
504C8—42no—
505C8—44no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22C2×#54611yescharacteristic
514C22#348611yescharacteristic
515C22#348611yescharacteristic
516C22C22×#13711yescharacteristic
517C22C2×#57611yescharacteristic
518C22#358511yescharacteristic
519C22#358511yescharacteristic
520C4—22no—
521C4—22no—
522C4—22no—
523C4—44no—
524C4—44no—
525C4—41no—
526C4—41no—
527C4—44no—
528C4—44no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4#333314yes—
535C4#352312yes—
536C4#355212yes—
537C2—84no—
538C2#4533611yescharacteristic
539C2C2×#333311yescharacteristic
540C2#6121811yescharacteristic
541C2#6182611yescharacteristic
542C2#6183011yescharacteristic
543C2#6184711yescharacteristic
544C2#6184711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59871C2semidirect
3. #32157C2semidirect
4. #31474C2semidirect
5. #29277C2semidirect
7. #27018C2semidirect
9. #25839C2semidirect
10. #25477C2semidirect
11. #24990C2semidirect
13. #22522C2semidirect
14. #22248C2semidirect
15. #22242C2semidirect

Maximal subgroups

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

subgroupindex
#598712
#598232
#321572
#314742
#292772
#282022
#270182
#267712
#258392
#254772
#249902
#238152
#225222
#222482
#222422