Finite Group Databasebeta

← #6499337

Subgroups — #6499337

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C2×#2859C211yesmaximal, characteristic
3#33263C214yesmaximal, complemented
4#30305C212yesmaximal, complemented
5#30304C212yesmaximal, complemented
6#29320C214yesmaximal, complemented
7#29319C214yesmaximal, complemented
8#29302C214yesmaximal
9#29301C214yesmaximal
10#21982C212yesmaximal
11#21981C212yesmaximal
12#20463C211yesmaximal, characteristic
13C2×#533—21no—
14C22×#125—21no—
15C2×#431C2212yes—
16#2868C2211yescharacteristic
17#2868C2211yescharacteristic
18#2862—21no—
19#2859—21no—
20#2859—21no—
21#2859—21no—
22#2859—21no—
23#2859—21no—
24#2859C2212yes—
25#2859C2212yes—
26#2859C2212yes—
27#2859C2212yes—
28#2859C2212yes—
29#2857—21no—
30#2857—21no—
31#2857C2212yes—
32#2857C2212yes—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C2×#389—21no—
38C2×#389—21no—
39C2×#389C2212yes—
40C2×#389C2212yes—
41#2285—21no—
42#2285C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#382—21no—
46C2×C43C2211yescharacteristic
47#2084C2218yes—
48#2083C2218yes—
49#2058C2218yes—
50#2057C2218yes—
51#1948C2218yes—
52#1947C2218yes—
53#1944C22116yes—
54#1943C22116yes—
55#1853C2218yes—
56#1852C2218yes—
57#1845C2214yes—
58#1844C2214yes—
59#1834C2214yes—
60#1834C2214yes—
61#1767C2212yes—
62#1767C2212yes—
63#1749C2212yes—
64#1749C2212yes—
65#1744C2212yes—
66C4×#97C2212yes—
67#533—48no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8C2312yes—
73C2×C4⋊Q8C2312yes—
74C2×#125—22no—
75C2×#125—22no—
76C2×#125—22no—
77C2×#125—22no—
78C2×#125—22no—
79C2×#125—48no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×#117—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#431Q8116yescomplemented
101C2×#107C23116yes—
102C2×#106C23116yes—
103C2×C4⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C2×#97C2314yes—
106C2×#97C2314yes—
107C2×C4×C8C2314yes—
108#391—22no—
109#391—22no—
110#391—22no—
111#391—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#389C2314yes—
131#389C2314yes—
132#389C2314yes—
133#389C2314yes—
134#389C2314yes—
135#389C2314yes—
136#384—21no—
137#384—21no—
138#384—21no—
139#384—21no—
140#384—21no—
141#384—21no—
142#384—21no—
143#384—21no—
144#384C2312yes—
145#384C2312yes—
146#384C2312yes—
147#384C2312yes—
148#384C2312yes—
149#384C2312yes—
150#384C2312yes—
151#384C2312yes—
152#382—22no—
153#382—22no—
154#382—22no—
155#382—22no—
156#382—22no—
157#382—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⋊C4C2314yes—
166C2×#95—22no—
167C2×#95—22no—
168C43D411yescharacteristic
169C43D411yescharacteristic
170C43D411yescharacteristic
171C43D411yescharacteristic
172C43C2311yescharacteristic
173C43C2311yescharacteristic
174C43C2311yescharacteristic
175C43C2311yescharacteristic
176#340C23116yes—
177#340C23116yes—
178#336C2314yes—
179#336C2314yes—
180C22×Q8—22no—
181C22×Q8—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C22×C8C2418yes—
192C4⋊Q8—44no—
193C4⋊Q8—44no—
194C4⋊Q8—44no—
195C4⋊Q8—44no—
196C4⋊Q8—44no—
197C4⋊Q8—44no—
198#125—48no—
199#125—48no—
200#125—48no—
201#125—48no—
202#125—48no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205C4×Q8—48no—
206C4×Q8—48no—
207#117—48no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
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—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2414yes—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2414yes—
246C2×C22⋊C4—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
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×Q818yes—
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283#107—416no—
284#106—416no—
285C4⋊C8C2×Q8116yes—
286C22⋊C8C2×Q8116yes—
287#97—44no—
288#97—44no—
289C4×C8C2×Q8116yes—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298C24C22×D411yescharacteristic
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C22×C4—22no—
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—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4C4⋊Q814yes—
332C22×C4C4⋊Q814yes—
333C22×C4AGL(1,ℤ/8)14yes—
334C22×C4#13714yes—
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×D411yescharacteristic
348C22×C4C22×Q814yes—
349C22×C4C2511yescharacteristic
350C2×C8—28no—
351C2×C8—28no—
352C2×C8C22×Q8116yes—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C22⋊C4—48no—
384C42—22no—
385C42—22no—
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—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—48no—
407C42C4⋊Q814yes—
408C42C4⋊Q814yes—
409C42AGL(1,ℤ/8)14yes—
410C42#13714yes—
411C42C22×Q814yes—
412C42C22×Q814yes—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23C2×C4⋊Q812yes—
418C23C2×AGL(1,ℤ/8)12yes—
419C23C2×#13712yes—
420C23C23×D411yescharacteristic
421Q8—88no—
422Q8—88no—
423C2×C4—21no—
424C2×C4—21no—
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—28no—
436C2×C4—28no—
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—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—44no—
450C2×C4—44no—
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—88no—
466C2×C4—88no—
467C2×C4#501116yes—
468C2×C4C2×C4⋊Q812yes—
469C2×C4C2×C4⋊Q812yes—
470C2×C4C2×C4⋊Q812yes—
471C2×C4C2×C4⋊Q812yes—
472C2×C4C2×C4⋊Q812yes—
473C2×C4C2×AGL(1,ℤ/8)12yes—
474C2×C4C2×AGL(1,ℤ/8)12yes—
475C2×C4C2×AGL(1,ℤ/8)12yes—
476C2×C4C2×AGL(1,ℤ/8)12yes—
477C2×C4C2×AGL(1,ℤ/8)12yes—
478C2×C4C2×#13712yes—
479C2×C4C2×#13712yes—
480C2×C4C2×#13712yes—
481C2×C4C2×#13712yes—
482C2×C4C2×#13712yes—
483C2×C4C23×Q812yes—
484C2×C4#58512yes—
485C2×C4#58512yes—
486C8—216no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22C2×#50114yes—
494C22C22×C4⋊Q811yescharacteristic
495C22#344911yescharacteristic
496C22#344911yescharacteristic
497C22C22×AGL(1,ℤ/8)11yescharacteristic
498C22C22×AGL(1,ℤ/8)11yescharacteristic
499C22C22×#13711yescharacteristic
500C22C22×#13711yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—84no—
512C4—84no—
513C4—84no—
514C4—84no—
515C4C2×#50114yes—
516C4C2×#50114yes—
517C4#316514yes—
518C4#350414yes—
519C2—24no—
520C2#3533511yescharacteristic
521C2C22×#50111yescharacteristic
522C2C2×#316511yescharacteristic
523C2#6096811yescharacteristic
524C2#6096911yescharacteristic
525C2#6096911yescharacteristic
526C2#6097111yescharacteristic
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. #33263C2semidirect
4. #30305C2semidirect
5. #30304C2semidirect
6. #29320C2semidirect
7. #29319C2semidirect
100. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×#28592
#332632
#303052
#303042
#293202
#293192
#293022
#293012
#219822
#219812
#204632