Finite Group Databasebeta

← #6433313

Subgroups — #6433313

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#29977C214yesmaximal, complemented
4#29976C214yesmaximal, complemented
5#29975C218yesmaximal, complemented
6#23746C212yesmaximal, complemented
7#23670C214yesmaximal, complemented
8#23552C212yesmaximal, complemented
9#20846C212yesmaximal
10#20702C212yesmaximal
11#20627C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#428C2214yes—
17C2×#423C2212yes—
18C2×#407C2212yes—
19#2628—22no—
20#2628—22no—
21#2628C2214yes—
22#2628C2214yes—
23C2×#400—21no—
24C2×#399—21no—
25C2×#395—21no—
26C2×#393—21no—
27#2317—21no—
28#2317—21no—
29#2317C2212yes—
30#2317C2212yes—
31#2316C2214yes—
32#2315—21no—
33#2315—21no—
34#2296—22no—
35#2296C2212yes—
36#2296C2212yes—
37C2×#389C2212yes—
38#2285—22no—
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41#2101C2218yescomplemented
42#2101C2218yescomplemented
43#2088C2218yescomplemented
44#2088C2218yescomplemented
45#2087C2218yescomplemented
46#2086C2218yescomplemented
47#2058C2218yescomplemented
48#2057C2218yescomplemented
49#1854C2218yes—
50#1853C2214yes—
51#1852C2214yes—
52#1825C2218yes—
53#1824C2214yes—
54#1823C2214yes—
55#1798C2214yes—
56#1798C2214yes—
57#1736C2214yes—
58#1736C2214yes—
59#1736C2214yes—
60#1728C2212yes—
61C23×Q8—21no—
62C24×C4C2311yescharacteristic
63C22×M16C2314yes—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—44no—
66C2×C4⋊Q8C2314yes—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—24no—
70C2×#117—21no—
71C2×#117—21no—
72C2×#117—44no—
73C2×#117—44no—
74C2×#117D412yes—
75C2×#117D412yes—
76C2×#117Q818yescomplemented
77C2×#117C2312yes—
78C2×#117C2312yes—
79C2×#117C2312yes—
80C2×#117C2312yes—
81C2×#117C2312yes—
82C2×#117C2312yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4D411yescharacteristic
96C22×C22⋊C4D411yescharacteristic
97C22×C42—21no—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101#428—48no—
102C2×#107C2318yes—
103C2×#106C2318yes—
104#423—44no—
105C2×C4⋊C8C2314yes—
106C2×C4⋊C8C2314yes—
107#407—28no—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110#400—24no—
111#400—24no—
112#399—24no—
113#399—24no—
114#395—24no—
115#395—24no—
116#393—24no—
117#393—24no—
118#389—24no—
119#389—24no—
120#389—24no—
121#389C2314yes—
122#389C2314yes—
123#389C2314yes—
124#389C2314yes—
125#389C2318yes—
126#385—22no—
127#385—22no—
128#385—22no—
129#385—22no—
130#385—22no—
131#385—22no—
132#385—22no—
133#385—22no—
134#384—22no—
135#384—22no—
136#382—22no—
137#382—22no—
138#382—22no—
139#382C2314yes—
140#380—22no—
141#380—22no—
142#380—24no—
143#380—24no—
144#380C2314yes—
145#380C2314yes—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—21no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#340C2318yes—
157#340C2318yes—
158#336C2318yes—
159C25C2×D411yescharacteristic
160C22×Q8—24no—
161C22×Q8—44no—
162C22×Q8—84no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×Q814yes—
178C23×C4#4114yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C2×M16—28no—
183C2×M16—28no—
184C2×M16—28no—
185C22×C8C2418yes—
186C4⋊Q8—44no—
187C4⋊Q8—44no—
188C4⋊Q8—48no—
189#117—28no—
190#117—28no—
191#117—24no—
192#117—24no—
193#117—28no—
194#117—44no—
195#117—44no—
196#117—42no—
197#117—42no—
198#117—42no—
199#117—42no—
200#117—44no—
201#117—44no—
202#117—44no—
203#117C2×Q818yes—
204#117C2×Q818yescomplemented
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—84no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×Q818yescomplemented
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2412yes—
232C2×C4⋊C4C2414yes—
233C2×C4⋊C4C2414yes—
234C2×C4⋊C4C2414yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—84no—
246C2×C22⋊C4C2×D412yes—
247C2×C22⋊C4C2×D412yes—
248C2×C22⋊C4C2×D412yes—
249C2×C22⋊C4C2×D412yes—
250C2×C42—28no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—44no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2412yes—
261C2×C42C2412yes—
262#107—48no—
263#106—48no—
264C4⋊C8—44no—
265C4⋊C8—44no—
266C22⋊C8—28no—
267C22⋊C8—28no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277C24—22no—
278C24—22no—
279C24—42no—
280C24—42no—
281C24C22⋊Q814yes—
282C24C22×D411yescharacteristic
283C24C22×D411yescharacteristic
284C24C22×D411yescharacteristic
285C2×Q8—48no—
286C2×Q8—88no—
287C2×Q8—816no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—216no—
295C22×C4—216no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—28no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4C22⋊Q814yes—
327C22×C4C22⋊Q814yes—
328C22×C4C22⋊Q814yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×Q812yes—
337C22×C4C22×Q812yes—
338C22×C4C22×Q812yes—
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4H(5,𝔽2)14yes—
343C22×C42−1+414yes—
344C22×C4C2511yescharacteristic
345M16—416no—
346C2×C8—28no—
347C2×C8—28no—
348C2×C8—28no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4—24no—
354C4⋊C4—28no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C4⋊C4—88no—
369C4⋊C4C22×Q818yes—
370C22⋊C4—28no—
371C22⋊C4—28no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C42—28no—
381C42—28no—
382C42—24no—
383C42—24no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—44no—
389C42—44no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—28no—
401C23—48no—
402C23—82no—
403C23—82no—
404C23—84no—
405C23C2×C22⋊Q812yes—
406C23C2×C22⋊Q812yes—
407C23C2×C22⋊Q812yes—
408C23#53412yes—
409C23#53612yes—
410C23C23×D411yescharacteristic
411Q8—816no—
412C2×C4—28no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—88no—
449C2×C4—88no—
450C2×C4—88no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4C2×C22⋊Q812yes—
454C2×C4C2×C22⋊Q812yes—
455C2×C4C2×C22⋊Q812yes—
456C2×C4#53412yes—
457C2×C4#53612yes—
458C2×C4#53918yes—
459C2×C4#54418yes—
460C2×C4C23×Q811yescharacteristic
461C2×C4C22×#4111yescharacteristic
462C2×C4C2×H(5,𝔽2)12yes—
463C2×C4C2×2−1+412yes—
464C8—48no—
465C22—24no—
466C22—22no—
467C22—22no—
468C22—24no—
469C22—24no—
470C22—24no—
471C22—22no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—42no—
478C22—42no—
479C22—88no—
480C22#262714yes—
481C22C22×C22⋊Q811yescharacteristic
482C22C2×#53411yescharacteristic
483C22C2×#53611yescharacteristic
484C22#358512yes—
485C22#358612yes—
486C4—22no—
487C4—22no—
488C4—48no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—82no—
494C4—84no—
495C4—82no—
496C4—84no—
497C4#262714yes—
498C4C2×#53912yes—
499C4C2×#54412yes—
500C2—24no—
501C2—42no—
502C2—42no—
503C2C2×#262711yescharacteristic
504C2#6049312yes—
505C2#6075311yescharacteristic
506C2#6075411yescharacteristic
507C2#6087012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29977C2semidirect
4. #29976C2semidirect
5. #29975C2semidirect
6. #23746C2semidirect
7. #23670C2semidirect
8. #23552C2semidirect
41. #2101C22semidirect
42. #2101C22semidirect
43. #2088C22semidirect
44. #2088C22semidirect
45. #2087C22semidirect
46. #2086C22semidirect
47. #2058C22semidirect
48. #2057C22semidirect
76. C2×#117Q8semidirect
204. #117C2×Q8semidirect
229. C2×C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
#340522
#299772
#299762
#299752
#237462
#236702
#235522
#208462
#207022
#206272