Finite Group Databasebeta

← #6445522

Subgroups — #6445522

Order 512.

3196 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62843C211yesmaximal, characteristic
2#35090C212yesmaximal, complemented
3#34199C211yesmaximal, characteristic
4#24353C218yesmaximal, complemented
5#24351C218yesmaximal, complemented
6#23565C212yesmaximal, complemented
7C2×#2030C212yesmaximal, complemented
8#21333C212yesmaximal, complemented
9C2×#1890C212yesmaximal
10#20889C212yesmaximal
11#20633C211yesmaximal, characteristic
12#3501—216no—
13#3501—24no—
14#3501C2218yes—
15#3458—22no—
16C2×#543—22no—
17C2×#543C2212yes—
18C2×#543C2212yes—
19C22×#123C2211yescharacteristic
20C2×#518C2212yes—
21C2×#518C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#99C2212yes—
24#2855C2218yescomplemented
25#2627C2218yescomplemented
26#2438—216no—
27#2359—24no—
28#2344—22no—
29#2342—22no—
30#2322C2214yes—
31#2315C2211yescharacteristic
32#2315C2214yes—
33#2315C2212yes—
34#2308C2211yescharacteristic
35C4×#123C2214yes—
36C2×#381C2214yes—
37C2×#381C2211yescharacteristic
38#2273C2211yescharacteristic
39#2062C22116yescomplemented
40#2030C2218yescomplemented
41#2030C2218yescomplemented
42#1996C22116yescomplemented
43#1890C2218yes—
44#1890C2218yes—
45#1867C22116yes—
46#1799C2218yes—
47C2×#342C2214yes—
48C2×#342C2212yes—
49C2×#342C2214yes—
50C23×D4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52#554—232no—
53#554—216no—
54#554—232no—
55#554—216no—
56D4×Q8—216no—
57D4×Q8—216no—
58#544—216no—
59#544—28no—
60#543—216no—
61#543—24no—
62#543—24no—
63#543—28no—
64#543—28no—
65#543—28no—
66#543—24no—
67#543C2318yes—
68#533—216no—
69#533—28no—
70C2×#123—22no—
71C2×#123—24no—
72C2×#123Q818yescomplemented
73C2×#123C2312yes—
74C2×#123C2312yes—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8D4116yes—
80C2×C22⋊Q8C2318yes—
81#518—24no—
82#518—28no—
83#518—28no—
84#518C2318yes—
85C2×C4×D4—28no—
86C2×C4×D4D414yes—
87C2×C4×D4C2314yes—
88C2×#117—22no—
89C2×#117—22no—
90C2×#117—22no—
91C2×#117—28no—
92C2×#117D412yes—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4D412yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C42C2312yes—
106C2×#99—28no—
107C2×#99C2318yes—
108#400—28no—
109#398—216no—
110#395—28no—
111#391—28no—
112#389—24no—
113#389—28no—
114#388—24no—
115#388—24no—
116#387—216no—
117#387—24no—
118#387—24no—
119#386—22no—
120#386C2314yes—
121#385—22no—
122#385—28no—
123#385—24no—
124#385—28no—
125#385C2314yes—
126#385C23116yes—
127#381—21no—
128#381—22no—
129#381—28no—
130#381C2312yes—
131#381C2318yes—
132#381C2318yes—
133#381C2312yes—
134#380—22no—
135#380—28no—
136#380C2318yes—
137#379—21no—
138C4×C4⋊C4—24no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4C2314yes—
143C2×#95—22no—
144C2×#95—28no—
145#342C2318yes—
146#342C23116yes—
147#342C23116yes—
148C25C2×D411yescharacteristic
149C25C2411yescharacteristic
150C22×Q8—28no—
151C22×Q8—24no—
152C22×Q8C2×D418yes—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4C2×D412yes—
156C22×D4C2×Q814yes—
157C22×D4#4114yes—
158C22×D4C2412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D414yes—
170C23×C4C2×D418yes—
171C23×C4C2×D412yes—
172C23×C4C2×Q814yescomplemented
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C23×C4C2412yes—
177C4⋊Q8—232no—
178C4⋊Q8—28no—
179C4⋊Q8—28no—
180#125—232no—
181#125—216no—
182#123—216no—
183#123—24no—
184#123—24no—
185#123C2×Q818yescomplemented
186#123C2×Q818yescomplemented
187C22⋊Q8—232no—
188C22⋊Q8—216no—
189C22⋊Q8—216no—
190C22⋊Q8—232no—
191C22⋊Q8—232no—
192C22⋊Q8—216no—
193C22⋊Q8—216no—
194C22⋊Q8—216no—
195C22⋊Q8—232no—
196C22⋊Q8—216no—
197C22⋊Q8—232no—
198C22⋊Q8—232no—
199C4×Q8—216no—
200C4×Q8—216no—
201C4×D4—232no—
202C4×D4—232no—
203C4×D4—216no—
204C4×D4—216no—
205#117—28no—
206#117—28no—
207#117—28no—
208#117—48no—
209#117—48no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D4116yes—
225C2×C4⋊C4C2×D418yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4C2×D412yes—
250C2×C22⋊C4C2×D414yes—
251C2×C22⋊C4C2×D418yes—
252C2×C22⋊C4C2×D412yes—
253C2×C22⋊C4C2×D412yes—
254C2×C22⋊C4C2×D418yes—
255C2×C22⋊C4C2×Q814yescomplemented
256C2×C22⋊C4C2412yes—
257C2×C22⋊C4C2418yes—
258C2×C22⋊C4C2412yes—
259C2×C22⋊C4C2418yes—
260C2×C22⋊C4C2412yes—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—28no—
264C2×C42—24no—
265C2×C42C2×D414yes—
266#99—216no—
267#95—24no—
268#95—24no—
269#95—216no—
270#95—216no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—216no—
275#95—28no—
276C24—22no—
277C24—22no—
278C24—82no—
279C24—82no—
280C24C22⋊Q814yes—
281C24C22×D412yes—
282C24C22×D411yescharacteristic
283C24C22×D411yescharacteristic
284C24C22×D414yes—
285C24C22×Q812yes—
286C24C2×#4112yes—
287C242−1+414yes—
288C24C2511yescharacteristic
289C2×Q8—216no—
290C2×Q8—216no—
291C2×Q8—88no—
292C2×Q8—168no—
293C2×Q8—164no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—216no—
297C2×D4—216no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4C22⋊Q818yes—
301C2×D4C22×Q814yes—
302C2×D4C2×#4114yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—216no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—216no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—22no—
324C22×C4—28no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—48no—
332C22×C4—42no—
333C22×C4—48no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—84no—
338C22×C4—82no—
339C22×C4—88no—
340C22×C4—82no—
341C22×C4—88no—
342C22×C4—164no—
343C22×C4C22⋊Q814yes—
344C22×C4C22×D412yes—
345C22×C4C22×D414yes—
346C22×C4C22×D4116yes—
347C22×C4C22×D414yes—
348C22×C4C22×D418yes—
349C22×C4C22×Q812yes—
350C22×C4C22×Q812yes—
351C4⋊C4—232no—
352C4⋊C4—216no—
353C4⋊C4—216no—
354C4⋊C4—232no—
355C4⋊C4—216no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C4⋊C4—48no—
360C4⋊C4—88no—
361C4⋊C4—84no—
362C4⋊C4—816no—
363C22⋊C4—216no—
364C22⋊C4—28no—
365C22⋊C4—24no—
366C22⋊C4—28no—
367C22⋊C4—28no—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—28no—
371C22⋊C4—28no—
372C22⋊C4—24no—
373C22⋊C4—22no—
374C22⋊C4—28no—
375C22⋊C4—28no—
376C22⋊C4—24no—
377C22⋊C4—28no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—48no—
382C22⋊C4—84no—
383C22⋊C4#12018yes—
384C22⋊C4#12018yes—
385C22⋊C4C22×Q814yes—
386C42—216no—
387C42—28no—
388C42—48no—
389C42—84no—
390C23—22no—
391C23—28no—
392C23—22no—
393C23—48no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—84no—
399C23—82no—
400C23—82no—
401C23—84no—
402C23—84no—
403C23—162no—
404C23C2×#12012yes—
405C23C2×#12012yes—
406C23C2×C22⋊Q814yes—
407C23C2×C22⋊Q812yes—
408C23C2×C22⋊Q812yes—
409C23#53612yes—
410C23#53612yes—
411C23#54818yes—
412C23D4×Q818yes—
413C23C23×D411yescharacteristic
414C23C23×D412yes—
415C23C23×Q811yescharacteristic
416C23C22×#4111yescharacteristic
417C23C2×2−1+412yes—
418Q8—816no—
419Q8—816no—
420D4—216no—
421D4—28no—
422D4—28no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—22no—
426C2×C4—216no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—48no—
439C2×C4—48no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—416no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—416no—
449C2×C4—48no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—48no—
453C2×C4—88no—
454C2×C4—88no—
455C2×C4—84no—
456C2×C4—88no—
457C2×C4—84no—
458C2×C4—816no—
459C2×C4—162no—
460C2×C4—168no—
461C2×C4C2×#12018yes—
462C2×C4C2×C22⋊Q814yes—
463C22—24no—
464C22—22no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—42no—
469C22—42no—
470C22—48no—
471C22—42no—
472C22—42no—
473C22—48no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—44no—
478C22—44no—
479C22—82no—
480C22—88no—
481C22#240914yes—
482C22#240914yes—
483C22C22×#12011yescharacteristic
484C22C22×C22⋊Q811yescharacteristic
485C22C2×#53611yescharacteristic
486C22C2×#54812yes—
487C22C2×D4×Q812yes—
488C4—24no—
489C4—28no—
490C4—44no—
491C4—42no—
492C4—42no—
493C4—44no—
494C4—416no—
495C4—88no—
496C4—82no—
497C4—82no—
498C2—21no—
499C2—22no—
500C2—21no—
501C2—24no—
502C2—24no—
503C2—22no—
504C2—44no—
505C2C2×#240911yescharacteristic
506C2#6040112yes—
subgroupG/H
center, socleC22C22×#120
commutator, Frattini, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35090C2semidirect
4. #24353C2semidirect
5. #24351C2semidirect
6. #23565C2semidirect
7. C2×#2030C2semidirect
8. #21333C2semidirect
24. #2855C22semidirect
25. #2627C22semidirect
39. #2062C22semidirect
40. #2030C22semidirect
41. #2030C22semidirect
42. #1996C22semidirect
72. C2×#123Q8semidirect
172. C23×C4C2×Q8semidirect
185. #123C2×Q8semidirect
186. #123C2×Q8semidirect
255. C2×C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#628432
#350902
#341992
#243532
#243512
#235652
C2×#20302
#213332
C2×#18902
#208892
#206332