Finite Group Databasebeta

← #6495025

Subgroups — #6495025

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C2×#2859C211yesmaximal, characteristic
3#33316C212yesmaximal, complemented
4#33294C212yesmaximal, complemented
5#30312C214yesmaximal, complemented
6#29207C214yesmaximal, complemented
7#28692C214yesmaximal, complemented
8#27950C212yesmaximal
9#27948C212yesmaximal
10#25819C214yesmaximal
11#21986C214yesmaximal
12C4×#432C211yesmaximal, characteristic
13C2×#533—21no—
14C22×#125C2212yes—
15C2×#501C2218yes—
16C2×#432C2212yes—
17#2868C2211yescharacteristic
18#2868C2211yescharacteristic
19#2862—21no—
20#2859—21no—
21#2859—21no—
22#2859—21no—
23#2859—21no—
24#2859—21no—
25#2859C2212yes—
26#2859C2212yes—
27#2859C2212yes—
28#2859C2212yes—
29#2859C2212yes—
30#2857—21no—
31#2857—21no—
32#2857C2212yes—
33#2857C2212yes—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C2×#389—21no—
39C2×#389—21no—
40C2×#389—21no—
41C2×#389—21no—
42#2285—21no—
43#2285C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382C2212yes—
47C2×C43C2211yescharacteristic
48#2084C2218yes—
49#2083C2218yes—
50#2059C22116yes—
51#1992C2218yes—
52#1990C2214yes—
53#1990C2214yes—
54#1949C22116yes—
55#1853C2218yes—
56#1852C2218yes—
57#1847C2218yes—
58#1845C2214yes—
59#1844C2214yes—
60#1836C2218yes—
61#1767C2212yes—
62C4×C4⋊C8C2212yes—
63#1749C2212yes—
64C4×C22⋊C8C2212yes—
65#1745C2212yes—
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—48no—
78C2×#125C2314yes—
79C2×#125C2314yes—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×#117—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
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#501—416no—
101#432Q8116yescomplemented
102C2×#107C23116yes—
103C2×#106C23116yes—
104C2×C4⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106C2×#97C2314yes—
107C2×#97C2314yes—
108C2×C4×C8C2314yes—
109#391—22no—
110#391—22no—
111#391—22no—
112#391—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#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389C2314yes—
136#389C2314yes—
137#384—21no—
138#384—21no—
139#384—21no—
140#384—21no—
141#384—21no—
142#384—21no—
143#384—21no—
144#384—21no—
145#384C2312yes—
146#384C2312yes—
147#384C2312yes—
148#384C2312yes—
149#384C2312yes—
150#384C2312yes—
151#384C2312yes—
152#384C2312yes—
153#382—22no—
154#382—22no—
155#382—22no—
156#382—22no—
157#382C2314yes—
158#382C2314yes—
159#380—22no—
160#380—22no—
161#380—22no—
162#380C2314yes—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4C2314yes—
167C2×#95—22no—
168C2×#95—22no—
169C43D411yescharacteristic
170C43D411yescharacteristic
171C43D411yescharacteristic
172C43D411yescharacteristic
173C43C2311yescharacteristic
174C43C2311yescharacteristic
175C43C2311yescharacteristic
176C43C2311yescharacteristic
177#340C23116yes—
178#340C23116yes—
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
192C22×C8C2418yes—
193C4⋊Q8—28no—
194C4⋊Q8—28no—
195C4⋊Q8—44no—
196C4⋊Q8—44no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199#125—216no—
200#125—48no—
201#125—48no—
202#125—48no—
203#125—48no—
204C22⋊Q8—48no—
205C22⋊Q8—48no—
206C4×Q8—48no—
207C4×Q8—48no—
208#117—48no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
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—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
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—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2414yes—
247C2×C22⋊C4—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—42no—
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×D411yescharacteristic
277C2×C42C2×Q818yes—
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284#107—416no—
285#106—416no—
286C4⋊C8C2×Q8116yes—
287C22⋊C8C2×Q8116yes—
288#97—28no—
289#97C2×Q8116yes—
290C4×C8—28no—
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×C4AGL(1,ℤ/8)14yes—
335C22×C4#13714yes—
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×D411yescharacteristic
349C22×C4C22×Q814yes—
350C22×C4C2511yescharacteristic
351C2×C8—28no—
352C2×C8—28no—
353C2×C8C22×Q8116yes—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—48no—
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—48no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C22⋊C4—48no—
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—22no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—48no—
408C42C4⋊Q814yes—
409C42C4⋊Q814yes—
410C42C22×Q814yes—
411C42C22×Q814yes—
412C42H(5,𝔽2)14yes—
413C422−1+414yes—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23C2×C4⋊Q812yes—
419C23C2×AGL(1,ℤ/8)12yes—
420C23C2×#13712yes—
421C23C23×D411yescharacteristic
422Q8—88no—
423Q8—88no—
424C2×C4—28no—
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—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
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#53512yes—
476C2×C4#53612yes—
477C2×C4#53612yes—
478C2×C4C2×AGL(1,ℤ/8)12yes—
479C2×C4C2×AGL(1,ℤ/8)12yes—
480C2×C4C2×#13712yes—
481C2×C4C2×#13712yes—
482C2×C4#57512yes—
483C2×C4#57512yes—
484C2×C4C23×Q812yes—
485C2×C4C2×H(5,𝔽2)12yes—
486C2×C4C2×2−1+412yes—
487C8—216no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22#316514yes—
495C22C22×C4⋊Q811yescharacteristic
496C22C2×#53511yescharacteristic
497C22C2×#53611yescharacteristic
498C22C22×AGL(1,ℤ/8)11yescharacteristic
499C22C22×#13711yescharacteristic
500C22C2×#57511yescharacteristic
501C22C2×#57511yescharacteristic
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—41no—
512C4—41no—
513C4—84no—
514C4—84no—
515C4—84no—
516C4—84no—
517C4#323414yes—
518C4#323614yes—
519C4#350314yes—
520C2—24no—
521C2#3528111yescharacteristic
522C2C2×#316511yescharacteristic
523C2#6066311yescharacteristic
524C2#6080311yescharacteristic
525C2#6080411yescharacteristic
526C2C2×#323411yescharacteristic
527C2C2×#323611yescharacteristic
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. #33316C2semidirect
4. #33294C2semidirect
5. #30312C2semidirect
6. #29207C2semidirect
7. #28692C2semidirect
101. #432Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×#28592
#333162
#332942
#303122
#292072
#286922
#279502
#279482
#258192
#219862
C4×#4322