Finite Group Databasebeta

← #6482127

Subgroups — #6482127

Order 512.

2154 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
1C2×#3234C212yesmaximal, complemented
2#35166C211yesmaximal, characteristic
3C2×#2370C211yesmaximal, characteristic, complemented
4#33107C212yesmaximal, complemented
5#33106C212yesmaximal, complemented
6#29755C212yesmaximal, complemented
7#29754C212yesmaximal, complemented
8#25465C218yesmaximal, complemented
9#25288C218yesmaximal, complemented
10#21955C211yesmaximal, characteristic, complemented
11#21395C212yesmaximal, complemented
12C2×#537—21no—
13C2×#537—21no—
14C2×#532C2212yes—
15C22×#125C2211yescharacteristic
16#3234—216no—
17C2×#501C2214yes—
18C2×#490C2214yes—
19C2×#481C2214yes—
20C2×#480C2214yes—
21C2×#478C2218yes—
22C2×#432C2212yes—
23#2856C2212yes—
24#2856C2212yes—
25#2656—21no—
26#2656—21no—
27#2542C2214yes—
28#2541C2214yes—
29#2540—24no—
30#2370—24no—
31#2370C2218yes—
32C2×#389—21no—
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#387—21no—
36C2×#387—21no—
37C2×C4×C4⋊C4—21no—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42#2084C2214yes—
43#2083C2214yes—
44#2067C2218yes—
45#2065C2218yes—
46#2041C2214yes—
47#2041C2214yes—
48#1987C2214yes—
49#1982C2214yes—
50#1982C2214yes—
51#1982C2214yes—
52#1948C2214yes—
53#1947C2214yes—
54#1940C2218yes—
55#1926C2218yes—
56#1925C2218yes—
57#1874C2214yes—
58#1874C2214yes—
59#1797C2212yes—
60#1797C2212yes—
61#1762C2218yes—
62C22×#41—22no—
63#537—48no—
64#537—48no—
65#532—28no—
66#532—28no—
67C2×C4⋊Q8C2312yes—
68C2×#127C2312yes—
69C2×#127C2312yes—
70C2×#127C2312yes—
71C2×#125—22no—
72C2×#125—24no—
73C2×#125D418yescomplemented
74C2×#125C2314yes—
75C2×#124—22no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—24no—
84C2×#121—24no—
85C2×#121C2314yes—
86C2×C4×D4—22no—
87C2×C4×D4C2314yes—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4D411yescharacteristic
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C42—21no—
99C22×C42—21no—
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#501—216no—
105#490—216no—
106#481—216no—
107#480—216no—
108#478—232no—
109#432—28no—
110C2×#107C2318yes—
111C2×#106C2318yes—
112C2×C4⋊C8C2314yes—
113C2×C4⋊C8C2314yes—
114C2×C4⋊C8C2314yes—
115C2×#102C2318yes—
116C2×#102C2318yes—
117C2×#102C2318yes—
118C2×#102C2318yes—
119C2×C22⋊C8C2314yes—
120C2×#97C2314yes—
121#399—24no—
122#399—24no—
123#392—24no—
124#392—24no—
125#390—22no—
126#390—22no—
127#390—22no—
128#390—22no—
129#390C2314yes—
130#390C2314yes—
131#389—28no—
132#389—22no—
133#389—22no—
134#389C2314yes—
135#389C2314yes—
136#387—28no—
137#387—28no—
138#384C2312yes—
139#384C2312yes—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4D414yes—
145C4×C4⋊C4C2314yes—
146C4×C4⋊C4C2314yes—
147C4×C4⋊C4C2314yes—
148C4×C4⋊C4C2314yes—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95—21no—
152C2×#95—21no—
153C2×#95—21no—
154C2×#95D411yescharacteristic
155C2×#95D411yescharacteristic
156#340C2318yes—
157#340C2318yes—
158#336C2314yes—
159#336C2314yes—
160C2×#41—48no—
161C2×#41—48no—
162C2×#41—48no—
163C22×Q8—22no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4C2414yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—24no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8C2418yes—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187#127—24no—
188#127—24no—
189#127—44no—
190#127—44no—
191#125—216no—
192#125—28no—
193#125C2×D4116yes—
194#124—48no—
195C22⋊Q8—416no—
196C22⋊Q8—416no—
197#121—28no—
198#121—28no—
199#121—416no—
200#121—416no—
201#121—48no—
202#121—48no—
203#121—48no—
204#121—48no—
205#121—48no—
206C4×D4—28no—
207C4×D4—28no—
208C4×D4—48no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
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—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2414yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—44no—
253C2×C42—82no—
254C2×C42—82no—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D414yes—
259C2×C42#4114yes—
260C2×C42C2412yes—
261C2×C42C2412yes—
262C2×C42C2412yes—
263#107—216no—
264#106—216no—
265C4⋊C8—28no—
266C4⋊C8—44no—
267C4⋊C8—44no—
268#102—216no—
269#102—216no—
270#102—216no—
271#102—48no—
272C22⋊C8—28no—
273#97—28no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95C2×D414yes—
280#95C2×D414yes—
281C24—24no—
282C24C22×D411yescharacteristic
283#41—832no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—28no—
290C2×D4—28no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4—82no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—48no—
348C22×C4#12114yes—
349C22×C4#12114yes—
350C22×C4AGL(1,ℤ/8)14yes—
351C22×C4AGL(1,ℤ/8)14yes—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C2×#4112yes—
361C22×C4C2511yescharacteristic
362C2×C8—28no—
363C2×C8—28no—
364C2×C8—28no—
365C4⋊C4—24no—
366C4⋊C4—216no—
367C4⋊C4—216no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—28no—
371C4⋊C4—28no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—84no—
386C4⋊C4C22×D414yes—
387C4⋊C4C22×D414yes—
388C4⋊C42−1+414yes—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C42—24no—
396C42—24no—
397C42—42no—
398C42—42no—
399C42—48no—
400C42—84no—
401C42—84no—
402C42#12118yes—
403C42C22×D414yes—
404C42C2×#4114yes—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23C2×#12112yes—
417C23C2×AGL(1,ℤ/8)12yes—
418C23C2×AGL(1,ℤ/8)12yes—
419C23C23×D411yescharacteristic
420Q8—88no—
421D4—48no—
422D4—48no—
423D4—88no—
424D4—88no—
425D4—816no—
426D4—164no—
427D4—164no—
428D4—164no—
429D4—164no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—48no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—816no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—84no—
474C2×C4C2×#12112yes—
475C2×C4C2×#12114yes—
476C2×C4C2×#12114yes—
477C2×C4C2×AGL(1,ℤ/8)14yes—
478C2×C4C2×AGL(1,ℤ/8)14yes—
479C2×C4C23×D411yescharacteristic
480C2×C4C22×#4111yescharacteristic
481C2×C4C2×2−1+412yes—
482C2×C4#58512yes—
483C8—48no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—84no—
491C22—84no—
492C22—84no—
493C22—84no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22#303814yes—
498C22C22×#12111yescharacteristic
499C22#348511yescharacteristic
500C22#348611yescharacteristic
501C22C22×AGL(1,ℤ/8)11yescharacteristic
502C22C22×AGL(1,ℤ/8)11yescharacteristic
503C22#358511yescharacteristic
504C22#358611yescharacteristic
505C4—24no—
506C4—24no—
507C4—24no—
508C4—22no—
509C4—24no—
510C4—42no—
511C4—44no—
512C4—42no—
513C4—44no—
514C4—44no—
515C4—82no—
516C4—84no—
517C4#346112yes—
518C4#349712yes—
519C2—24no—
520C2—164no—
521C2#3515311yescharacteristic
522C2C2×#303811yescharacteristic
523C2#6042511yescharacteristic
524C2#6170411yescharacteristic
525C2#6170711yescharacteristic
526C2#6182511yescharacteristic
527C2#6182811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35166C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3234C2semidirect
3. C2×#2370C2semidirect
4. #33107C2semidirect
5. #33106C2semidirect
6. #29755C2semidirect
7. #29754C2semidirect
8. #25465C2semidirect
9. #25288C2semidirect
10. #21955C2semidirect
11. #21395C2semidirect
73. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32342
#351662
C2×#23702
#331072
#331062
#297552
#297542
#254652
#252882
#219552
#213952