Finite Group Databasebeta

← #6495019

Subgroups — #6495019

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2#36645C211yesmaximal, characteristic
3#33314C212yesmaximal, complemented
4#33293C212yesmaximal, complemented
5#30312C214yesmaximal, complemented
6#29212C214yesmaximal, complemented
7#28689C212yesmaximal, complemented
8#28687C212yesmaximal, complemented
9#27955C214yesmaximal
10#25812C214yesmaximal
11#21988C214yesmaximal
12C4×#432C211yesmaximal, characteristic
13C2×#533—21no—
14C2×#533C2212yes—
15C2×#500C2214yes—
16C2×#498C2214yes—
17C2×#432C2212yes—
18#2868C2211yescharacteristic
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2862—21no—
23#2862C2212yes—
24#2859—21no—
25#2859C2212yes—
26#2859C2212yes—
27#2859C2212yes—
28#2857—21no—
29#2857—21no—
30#2857—21no—
31#2857—21no—
32#2857—21no—
33#2857—21no—
34#2857C2212yes—
35#2857C2212yes—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8—21no—
38C4×C4⋊Q8—21no—
39C4×C4⋊Q8—21no—
40C4×C4⋊Q8—21no—
41C4×C4⋊Q8—21no—
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44#2285—21no—
45#2285C2212yes—
46#2285C2212yes—
47#2285C2212yes—
48C2×C43C2211yescharacteristic
49#2084C2218yes—
50#2083C2218yes—
51#2059C22116yes—
52#1993C2218yes—
53#1990C2214yes—
54#1990C2214yes—
55#1948C2218yes—
56#1947C2218yes—
57#1854C22116yes—
58#1847C2218yes—
59#1847C2218yes—
60#1836C2218yes—
61#1767C2212yes—
62C4×C4⋊C8C2212yes—
63#1749C2212yes—
64C4×C22⋊C8C2212yes—
65#1745C2212yes—
66C4×#97C2212yes—
67#533—48no—
68#533—48no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×#125—22no—
82C2×#125C2314yes—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C2×C4×Q8—22no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×#117—22no—
92C2×#117C2314yes—
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#500—48no—
101#498—48no—
102#432Q8116yescomplemented
103C2×#107C23116yes—
104C2×#106C23116yes—
105C2×C4⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107C2×#97C2314yes—
108C2×#97C2314yes—
109C2×C4×C8C2314yes—
110#391—22no—
111#391—22no—
112#391—22no—
113#391—22no—
114#391—22no—
115#391—22no—
116#391—22no—
117#391—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389C2314yes—
133#389C2314yes—
134#384—21no—
135#384—21no—
136#384—21no—
137#384—21no—
138#384—21no—
139#384—21no—
140#384—21no—
141#384—21no—
142#384C2312yes—
143#384C2312yes—
144#384C2312yes—
145#384C2312yes—
146#384C2312yes—
147#384C2312yes—
148#384C2312yes—
149#384C2312yes—
150#382—22no—
151#382—22no—
152#382C2314yes—
153#382C2314yes—
154#380—22no—
155#380—22no—
156#380—22no—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380C2314yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4C2314yes—
170C43D411yescharacteristic
171C43D411yescharacteristic
172C43D411yescharacteristic
173C43D411yescharacteristic
174C43C2311yescharacteristic
175C43C2311yescharacteristic
176C43C2311yescharacteristic
177C43C2311yescharacteristic
178#340C23116yes—
179#340C23116yes—
180#336C2314yes—
181#336C2314yes—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2411yescharacteristic
193C22×C8C2418yes—
194C4⋊Q8—28no—
195C4⋊Q8—28no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4⋊Q8—28no—
199C4⋊Q8—44no—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C4⋊Q8—44no—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207#125—48no—
208#125—48no—
209C22⋊Q8—48no—
210C22⋊Q8—48no—
211C22⋊Q8—48no—
212C22⋊Q8—48no—
213C4×Q8—48no—
214C4×Q8—48no—
215C4×Q8—48no—
216C4×Q8—48no—
217#117—48no—
218#117—48no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2414yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×Q818yes—
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286#107—416no—
287#106—416no—
288C4⋊C8C2×Q8116yes—
289C22⋊C8C2×Q8116yes—
290#97—28no—
291#97C2×Q8116yes—
292C4×C8—28no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301C24C22×D411yescharacteristic
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—44no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4C4⋊Q814yes—
337C22×C4C4⋊Q814yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×Q814yes—
352C22×C4C2511yescharacteristic
353C2×C8—28no—
354C2×C8—28no—
355C2×C8C22×Q8116yes—
356C4⋊C4—24no—
357C4⋊C4—28no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—28no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—48no—
408C42—48no—
409C42C4⋊Q814yes—
410C42C4⋊Q814yes—
411C42AGL(1,ℤ/8)14yes—
412C42#13714yes—
413C42C22×Q814yes—
414C42C22×Q814yes—
415C42H(5,𝔽2)14yes—
416C422−1+414yes—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23C2×C4⋊Q812yes—
424C23C23×D411yescharacteristic
425Q8—88no—
426Q8—88no—
427Q8—88no—
428Q8—88no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4C2×C4⋊Q812yes—
471C2×C4C2×C4⋊Q812yes—
472C2×C4C2×C4⋊Q812yes—
473C2×C4C2×C4⋊Q812yes—
474C2×C4C2×C4⋊Q812yes—
475C2×C4#53512yes—
476C2×C4#53512yes—
477C2×C4#53612yes—
478C2×C4#53612yes—
479C2×C4C2×AGL(1,ℤ/8)12yes—
480C2×C4C2×AGL(1,ℤ/8)12yes—
481C2×C4C2×AGL(1,ℤ/8)12yes—
482C2×C4C2×#13712yes—
483C2×C4C2×#13712yes—
484C2×C4C2×#13712yes—
485C2×C4#57512yes—
486C2×C4#57512yes—
487C2×C4C23×Q812yes—
488C2×C4C2×H(5,𝔽2)12yes—
489C2×C4C2×2−1+412yes—
490C8—216no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22C22×C4⋊Q811yescharacteristic
499C22C2×#53511yescharacteristic
500C22C2×#53611yescharacteristic
501C22C22×AGL(1,ℤ/8)11yescharacteristic
502C22C22×#13711yescharacteristic
503C22C2×#57511yescharacteristic
504C22C2×#57511yescharacteristic
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—41no—
515C4—84no—
516C4—84no—
517C4—84no—
518C4—84no—
519C4#316514yes—
520C4#321414yes—
521C4#321614yes—
522C4#350314yes—
523C2—42no—
524C2#3528111yescharacteristic
525C2C2×#316511yescharacteristic
526C2#6066311yescharacteristic
527C2C2×#321411yescharacteristic
528C2C2×#321611yescharacteristic
529C2#6086411yescharacteristic
530C2#6086511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33314C2semidirect
4. #33293C2semidirect
5. #30312C2semidirect
6. #29212C2semidirect
7. #28689C2semidirect
8. #28687C2semidirect
102. #432Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
#333142
#332932
#303122
#292122
#286892
#286872
#279552
#258122
#219882
C4×#4322