Finite Group Databasebeta

← #6436005

Subgroups — #6436005

Order 512.

1710 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C2×#2859C211yesmaximal, characteristic
3#30312C214yesmaximal, complemented
4#30310C214yesmaximal, complemented
5#30307C214yesmaximal, complemented
6#30305C212yesmaximal, complemented
7#30304C212yesmaximal, complemented
8#23657C214yesmaximal
9#23554C214yesmaximal
10#20800C212yesmaximal
11#20764C212yesmaximal
12#20364C211yesmaximal, characteristic
13C2×#533C2212yes—
14C22×#125—21no—
15C2×#428C2218yes—
16C2×C4×M16C2212yes—
17#2868C2211yescharacteristic
18#2868C2211yescharacteristic
19#2862C2212yes—
20#2859—21no—
21#2859—21no—
22#2859—21no—
23#2859—21no—
24#2859C2212yes—
25#2859C2212yes—
26#2859C2212yes—
27#2859C2212yes—
28#2859C2212yes—
29#2859C2212yes—
30#2857—21no—
31#2857—21no—
32#2857—21no—
33#2857—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8—21no—
38C2×#389—21no—
39C2×#389—21no—
40C2×#389C2212yes—
41C2×#389C2212yes—
42#2285C2212yes—
43#2285C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382—21no—
47C2×C43C2211yescharacteristic
48#2085C22116yes—
49#2084C2218yes—
50#2083C2218yes—
51#2059C22116yes—
52#2058C2218yes—
53#2057C2218yes—
54#1847C2218yes—
55#1845C2214yes—
56#1844C2214yes—
57#1836C2218yes—
58#1834C2214yes—
59#1834C2214yes—
60#1745C2212yes—
61#1745C2212yes—
62#1744C2212yes—
63#1744C2212yes—
64#1736C2218yes—
65#1728C2212yes—
66#1728C2212yes—
67C22×M16C2314yes—
68#533—48no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8C2312yes—
74C2×C4⋊Q8C2312yes—
75C2×#125—22no—
76C2×#125—22no—
77C2×#125—22no—
78C2×#125—22no—
79C2×#125—48no—
80C2×#125C2314yes—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8—22no—
85C2×#117C2314yes—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42D411yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101#428—416no—
102C2×#107C23116yes—
103C2×#106C23116yes—
104C4×M16Q8116yescomplemented
105C2×#97C2314yes—
106C2×C4×C8C2314yes—
107#391—22no—
108#391—22no—
109#391—22no—
110#391—22no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—22no—
117#389—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#389C2314yes—
132#389C2314yes—
133#389C2314yes—
134#389C2314yes—
135#384—21no—
136#384—21no—
137#384—21no—
138#384—21no—
139#384—21no—
140#384—21no—
141#384—21no—
142#384—21no—
143#384C2312yes—
144#384C2312yes—
145#384C2312yes—
146#384C2312yes—
147#384C2312yes—
148#384C2312yes—
149#384C2312yes—
150#384C2312yes—
151#382—22no—
152#382—22no—
153#382—22no—
154#382—22no—
155#382C2314yes—
156#382C2314yes—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C2×#95—22no—
166C2×#95—22no—
167C43D411yescharacteristic
168C43D411yescharacteristic
169C43D411yescharacteristic
170C43D411yescharacteristic
171C43C2311yescharacteristic
172C43C2311yescharacteristic
173C43C2311yescharacteristic
174C43C2311yescharacteristic
175#340C23116yes—
176#340C23116yes—
177#336C2314yes—
178#336C2314yes—
179#336C2314yes—
180#336C2314yes—
181C22×Q8—22no—
182C22×Q8—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C2×M16—28no—
193C2×M16—28no—
194C2×M16C2×Q8116yes—
195C22×C8C2418yes—
196C4⋊Q8—44no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199C4⋊Q8—44no—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202#125—48no—
203#125—48no—
204#125—48no—
205#125—48no—
206#125—48no—
207C22⋊Q8—48no—
208C22⋊Q8—48no—
209C4×Q8—48no—
210C4×Q8—48no—
211#117—216no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
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—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4C2414yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2414yes—
250C2×C22⋊C4—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—42no—
257C2×C42—42no—
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×D411yescharacteristic
280C2×C42C2×Q818yes—
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287#107—416no—
288#106—416no—
289#97C2×Q8116yes—
290C4×C8C2×Q8116yes—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299C24C22×D411yescharacteristic
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4C4⋊Q814yes—
333C22×C4C4⋊Q814yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
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×Q814yes—
348C22×C4H(5,𝔽2)14yes—
349C22×C42−1+414yes—
350C22×C4C2511yescharacteristic
351M16—232no—
352C2×C8—28no—
353C2×C8—28no—
354C2×C8C22×Q8116yes—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C22⋊C4—48no—
386C42—22no—
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—22no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—48no—
409C42C4⋊Q814yes—
410C42C4⋊Q814yes—
411C42AGL(1,ℤ/8)14yes—
412C42#13714yes—
413C42C22×Q814yes—
414C42C22×Q814yes—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23C2×C4⋊Q812yes—
420C23#53512yes—
421C23#53612yes—
422C23C23×D411yescharacteristic
423Q8—88no—
424Q8—88no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
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—28no—
438C2×C4—28no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—44no—
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—88no—
468C2×C4—88no—
469C2×C4C2×C4⋊Q812yes—
470C2×C4C2×C4⋊Q812yes—
471C2×C4C2×C4⋊Q812yes—
472C2×C4C2×C4⋊Q812yes—
473C2×C4C2×C4⋊Q812yes—
474C2×C4#53512yes—
475C2×C4#53612yes—
476C2×C4#544116yes—
477C2×C4C2×AGL(1,ℤ/8)12yes—
478C2×C4C2×AGL(1,ℤ/8)12yes—
479C2×C4C2×AGL(1,ℤ/8)12yes—
480C2×C4C2×#13712yes—
481C2×C4C2×#13712yes—
482C2×C4C2×#13712yes—
483C2×C4#57512yes—
484C2×C4#57512yes—
485C2×C4C23×Q812yes—
486C2×C4C2×H(5,𝔽2)12yes—
487C2×C4C2×2−1+412yes—
488C8—216no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22#266014yes—
496C22C22×C4⋊Q811yescharacteristic
497C22C2×#53511yescharacteristic
498C22C2×#53611yescharacteristic
499C22C22×AGL(1,ℤ/8)11yescharacteristic
500C22C22×#13711yescharacteristic
501C22C2×#57511yescharacteristic
502C22C2×#57511yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—41no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—84no—
514C4—84no—
515C4—84no—
516C4—84no—
517C4#266014yes—
518C4#316514yes—
519C4#316614yes—
520C4C2×#54414yes—
521C2—24no—
522C2C2×#266011yescharacteristic
523C2C2×#316511yescharacteristic
524C2C2×#316611yescharacteristic
525C2#6080311yescharacteristic
526C2#6080411yescharacteristic
527C2#6086411yescharacteristic
528C2#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. #30312C2semidirect
4. #30310C2semidirect
5. #30307C2semidirect
6. #30305C2semidirect
7. #30304C2semidirect
104. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×#28592
#303122
#303102
#303072
#303052
#303042
#236572
#235542
#208002
#207642
#203642