Finite Group Databasebeta

← #6594178

Subgroups — #6594178

Order 512.

1642 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
1#36188C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#33049C212yesmaximal, complemented
5#29219C211yesmaximal, characteristic, complemented
6#29141C211yesmaximal, characteristic, complemented
7#27972C212yesmaximal, complemented
8#27962C212yesmaximal, complemented
9#27737C214yesmaximal, complemented
10#27722C218yesmaximal, complemented
11#22504C214yesmaximal
12C4×#487C211yesmaximal, characteristic, complemented
13#20764C212yesmaximal, complemented
14C4×#436C211yesmaximal, characteristic
15C2×#556C2212yes—
16C2×#554—21no—
17C2×#487C2211yescharacteristic
18C2×#436C2212yes—
19C2×C4×M16C2212yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2568C2212yes—
25#2567—21no—
26#2552—21no—
27#2552C2212yes—
28#2547—21no—
29#2547—21no—
30#2502—22no—
31#2499C2214yes—
32#2496—22no—
33#2496—22no—
34#2440—22no—
35#2440C2214yes—
36#2365—21no—
37#2365—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42#2285C2212yes—
43C2×#384C2212yes—
44C42×D4C2211yescharacteristic
45#2101C2218yes—
46#2099C2214yes—
47#2072C2218yes—
48#2024C2214yes—
49#1990C2214yes—
50#1990C2214yes—
51#1990C2212yes—
52#1990C2211yescharacteristic
53#1988C2214yes—
54#1956C2212yes—
55#1956C2211yescharacteristic
56#1935C2218yes—
57#1933C2214yes—
58#1894C2218yes—
59#1872C2218yes—
60#1836C2218yes—
61#1834C2212yes—
62#1834C2212yes—
63#1834C2211yescharacteristic
64#1834C2212yes—
65#1834C2212yes—
66#1825C2218yes—
67#1803C2214yes—
68#1770C2212yes—
69C4×C4⋊C8C2211yescharacteristic
70C4×#103C2214yes—
71#1753C2214yes—
72C4×C22⋊C8C2212yes—
73#1736C2218yes—
74C42×C8C2211yescharacteristic
75C22×M16C2314yes—
76#556—216no—
77#554—48no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#126—24no—
87C2×#125—22no—
88C2×#124—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8C2314yes—
92C2×C4×Q8—22no—
93C2×C4×Q8—21no—
94C2×C4×Q8C2312yes—
95C2×C4×D4—22no—
96C2×C4×D4D411yescharacteristic
97C2×C4×D4D411yescharacteristic
98C2×C4×D4C2311yescharacteristic
99C2×#117C2314yes—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4C2314yes—
103C22×C42C2312yes—
104#487—24no—
105#436—28no—
106C2×C4⋊C8C2312yes—
107C2×#103C2314yes—
108C2×#103C2318yes—
109C2×#103C2314yes—
110C2×C22⋊C8C2314yes—
111C4×M16—28no—
112C2×#97C2314yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2312yes—
116C2×C4×C8C2312yes—
117C2×C4×C8C2311yescharacteristic
118#400—24no—
119#398—24no—
120#398—24no—
121#397—24no—
122#397—24no—
123#395—24no—
124#391—22no—
125#391—22no—
126#391—21no—
127#391—22no—
128#391C2312yes—
129#391C2314yes—
130#389—22no—
131#389—22no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389C2312yes—
139#389C2314yes—
140#387—24no—
141#387—24no—
142#385—24no—
143#385—24no—
144#384—22no—
145#384—21no—
146#384—21no—
147#384—21no—
148#384—21no—
149#384—22no—
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#384C2312yes—
153#384C2314yes—
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#382C2314yes—
158#380—22no—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380C2314yes—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2311yescharacteristic
173C4×C22⋊C4D412yes—
174C4×C22⋊C4D412yes—
175C4×C22⋊C4C2312yes—
176C43C2311yescharacteristic
177#340C2318yes—
178#340C2318yes—
179#340C2318yes—
180#340C2314yes—
181#340C2318yes—
182#340C2314yes—
183#340C2318yes—
184#336C2314yes—
185#336C2312yes—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8C2412yes—
190C22×D4C2×D411yescharacteristic
191C23×C4—24no—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2412yes—
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—44no—
198C22×C8C2414yes—
199C22×C8C2412yes—
200C22×C8C2412yes—
201C4⋊Q8—28no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—44no—
209C4⋊Q8—44no—
210C4⋊Q8—82no—
211#126—416no—
212#125—48no—
213#124—48no—
214C22⋊Q8—216no—
215C22⋊Q8—416no—
216C22⋊Q8—48no—
217C4×Q8—28no—
218C4×Q8—48no—
219C4×Q8—44no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—48no—
223C4×D4#4118yes—
224#117—216no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—84no—
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2412yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4C2×D412yes—
264C2×C22⋊C4C2×D412yes—
265C2×C22⋊C4C2×D412yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2412yes—
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281C4⋊C8—24no—
282#103—28no—
283#103—44no—
284#103—48no—
285C22⋊C8—28no—
286#97—28no—
287C4×C8—24no—
288C4×C8—24no—
289C4×C8—22no—
290C4×C8—42no—
291C4×C8#4114yes—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95C2×D412yes—
303#95C2×D412yes—
304#95C2×D412yes—
305C24C22×D412yes—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—44no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—42no—
317C2×Q8—82no—
318C2×Q8—82no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4#12414yes—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—48no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4118yes—
350C22×C4C2511yescharacteristic
351M16—416no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8C2×#4114yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—28no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—48no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4#12414yes—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4#12418yes—
400C42—24no—
401C42—24no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—48no—
408C42—44no—
409C42—44no—
410C42AGL(1,ℤ/8)14yes—
411C42#13714yes—
412C42C2×#4114yes—
413C42C2×#4112yes—
414C422−1+412yes—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23C2×#12414yes—
422C23C23×D411yescharacteristic
423Q8—44no—
424Q8—44no—
425Q8—84no—
426Q8—88no—
427Q8—164no—
428D4—28no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—168no—
466C2×C4C2×#12414yes—
467C2×C4C2×#12412yes—
468C2×C4C2×#12412yes—
469C2×C4#54712yes—
470C2×C4#54712yes—
471C2×C4C2×AGL(1,ℤ/8)12yes—
472C2×C4C2×AGL(1,ℤ/8)12yes—
473C2×C4C2×AGL(1,ℤ/8)12yes—
474C2×C4C2×#13712yes—
475C2×C4C2×#13712yes—
476C2×C4C2×#13712yes—
477C2×C4#57712yes—
478C2×C4#57812yes—
479C2×C4C22×#4111yescharacteristic
480C2×C4C22×#4112yes—
481C2×C4C2×2−1+411yescharacteristic
482C2×C4#58512yes—
483C8—24no—
484C8—44no—
485C8—42no—
486C22—24no—
487C22—24no—
488C22—24no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22C22×#12411yescharacteristic
494C22C2×#54711yescharacteristic
495C22#350011yescharacteristic
496C22C22×AGL(1,ℤ/8)11yescharacteristic
497C22C22×#13711yescharacteristic
498C22C2×#57711yescharacteristic
499C22C2×#57811yescharacteristic
500C4—24no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4—84no—
510C4—84no—
511C4—82no—
512C4#314114yes—
513C4#335314yes—
514C4#335614yes—
515C4#347512yes—
516C4#351512yes—
517C2—44no—
518C2#3616411yescharacteristic
519C2C2×#314111yescharacteristic
520C2#6062311yescharacteristic
521C2C2×#335311yescharacteristic
522C2C2×#335611yescharacteristic
523C2#6253611yescharacteristic
524C2#6253911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33289C2semidirect
4. #33049C2semidirect
5. #29219C2semidirect
6. #29141C2semidirect
7. #27972C2semidirect
8. #27962C2semidirect
9. #27737C2semidirect
10. #27722C2semidirect
12. C4×#487C2semidirect
13. #20764C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#361802
#332892
#330492
#292192
#291412
#279722
#279622
#277372
#277222
#225042
C4×#4872
#207642
C4×#4362