Finite Group Databasebeta

← #413273

Subgroups — #413273

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2062C214yesmaximal
3#14130C211yesmaximal, characteristic
4#13880C212yesmaximal
5#11344C214yesmaximal
6#10854C212yesmaximal
7C4×#342C211yesmaximal, characteristic
8C2×#543C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10#2857—21no—
11#2857—21no—
12#2659—21no—
13#2659—21no—
14#2549—24no—
15#2440—22no—
16#2440—22no—
17#2315C2212yes—
18C2×#389—21no—
19C2×#385C2212yes—
20C2×#385C2218yes—
21C2×#384—22no—
22C2×#380C2211yescharacteristic
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#2062—216no—
27C2×#342C2218yes—
28C2×#342C2212yes—
29C2×#342C2212yes—
30#1444—21no—
31#1444—24no—
32#1442—21no—
33C4⋊#95—21no—
34C4⋊#95—22no—
35C22⋊#95C2212yes—
36C24×C4C2312yes—
37C24×C4C2311yescharacteristic
38#543—28no—
39#543—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—28no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—44no—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4D412yes—
56C22×C22⋊C4C2314yes—
57C22×C22⋊C4C2312yes—
58C22×C22⋊C4C2311yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C22⋊C4C2314yes—
61C22×C42—22no—
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64#398—24no—
65#398—24no—
66#395—24no—
67#395—24no—
68#391—22no—
69#391—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—44no—
75#385—24no—
76#385—24no—
77#385—216no—
78#385—28no—
79#385—48no—
80#384—21no—
81#384—21no—
82#384—48no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380Q814yescomplemented
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4Q818yescomplemented
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—28no—
100C2×#95—22no—
101C2×#95D411yescharacteristic
102C2×#95D411yescharacteristic
103C2×#95D414yes—
104#342—216no—
105#342D418yes—
106#342Q818yescomplemented
107C25C2411yescharacteristic
108C22×Q8—22no—
109C22×Q8—22no—
110C23×C4—24no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—44no—
119C23×C4—82no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D414yes—
123C23×C4C2×D414yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q814yes—
131C23×C4C2×Q812yes—
132C23×C4#4112yes—
133C4⋊Q8—82no—
134C4⋊Q8—82no—
135C22⋊Q8—416no—
136C22⋊Q8—416no—
137C22⋊Q8—44no—
138C22⋊Q8—44no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—416no—
168C2×C4⋊C4—416no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—88no—
172C2×C4⋊C4—82no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4C2×D414yes—
188C2×C22⋊C4C2×D418yes—
189C2×C22⋊C4C2×Q814yes—
190C2×C22⋊C4C2×Q814yes—
191C2×C22⋊C4C2×Q818yes—
192C2×C22⋊C4C2×Q818yes—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—28no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42—42no—
201C2×C42—42no—
202#95—24no—
203#95—24no—
204#95—22no—
205#95—22no—
206#95—24no—
207#95—24no—
208#95—28no—
209#95—24no—
210#95—416no—
211#95—44no—
212#95—44no—
213#95—82no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24C22×D411yescharacteristic
219C24C22×D412yes—
220C24C22×D412yes—
221C24C22×Q812yes—
222C24C22×Q811yescharacteristic
223C24C22×Q812yes—
224C24C2×#4111yescharacteristic
225C24H(5,𝔽2)12yes—
226C242−1+412yes—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—82no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—28no—
242C22×C4—28no—
243C22×C4—28no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—48no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—48no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—88no—
294C22×C4—88no—
295C22×C4#12011yescharacteristic
296C22×C4#12012yes—
297C22×C4#12011yescharacteristic
298C22×C4C22⋊Q814yes—
299C22×C4C22⋊Q812yes—
300C22×C4C22⋊Q812yes—
301C22×C4C22⋊Q814yes—
302C22×C4#12712yes—
303C22×C4#12712yes—
304C22×C4C4⋊Q812yes—
305C22×C4C4⋊Q814yes—
306C22×C4C4⋊Q812yes—
307C22×C4C4⋊Q812yes—
308C22×C4C4⋊Q812yes—
309C22×C4C4⋊Q814yes—
310C4⋊C4—48no—
311C4⋊C4—44no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—832no—
327C4⋊C4—88no—
328C22⋊C4—28no—
329C22⋊C4—28no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—28no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—48no—
337C22⋊C4—44no—
338C22⋊C4C4⋊Q8116yescomplemented
339C42—416no—
340C42—42no—
341C42—42no—
342C42—81no—
343C42—81no—
344C23—21no—
345C23—22no—
346C23—21no—
347C23—22no—
348C23—21no—
349C23—22no—
350C23—21no—
351C23—22no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23C2×#12011yescharacteristic
367C23C2×C22⋊Q811yescharacteristic
368C23C2×C22⋊Q812yes—
369C23C2×#12711yescharacteristic
370C23C2×C4⋊Q811yescharacteristic
371C23C2×C4⋊Q811yescharacteristic
372C23C2×C4⋊Q812yes—
373C23C2×C4⋊Q814yes—
374C23#53511yescharacteristic
375C23#53611yescharacteristic
376C23#54414yes—
377C23D4214yes—
378C23#54712yes—
379C23D4×Q814yes—
380C23D4×Q812yes—
381C23#55412yes—
382C23Q8212yes—
383Q8—164no—
384Q8—164no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—28no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—28no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—216no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—416no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—88no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—88no—
447C2×C4—88no—
448C2×C4#39512yes—
449C2×C4#39514yes—
450C2×C4#39512yes—
451C2×C4B(4,𝔽2)12yes—
452C2×C4#45812yes—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—21no—
462C22—24no—
463C22—21no—
464C22—21no—
465C22—24no—
466C22—21no—
467C22—21no—
468C22—24no—
469C22—21no—
470C22—21no—
471C22—24no—
472C22—21no—
473C22—44no—
474C22—44no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22C2×#39511yescharacteristic
480C22#243311yescharacteristic
481C22#243411yescharacteristic
482C22#244014yes—
483C22#254211yescharacteristic
484C22#255114yes—
485C22#256711yescharacteristic
486C22#266011yescharacteristic
487C22#286811yescharacteristic
488C22C2×B(4,𝔽2)11yescharacteristic
489C22C2×#45811yescharacteristic
490C22#302711yescharacteristic
491C22#302811yescharacteristic
492C4—22no—
493C4—24no—
494C4—416no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—84no—
500C4—84no—
501C4—84no—
502C4—84no—
503C4#206214yescomplemented
504C2—24no—
505C2—21no—
506C2—21no—
507C2—21no—
508C2—21no—
509C2—24no—
510C2#1514311yescharacteristic
511C2C2×#206211yescharacteristic
512C2#2356611yescharacteristic
513C2#2577111yescharacteristic
514C2#2577211yescharacteristic
515C2#2585211yescharacteristic
516C2#2585311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
90. #380Q8semidirect
94. C4×C22⋊C4Q8semidirect
106. #342Q8semidirect
338. C22⋊C4C4⋊Q8semidirect
503. C4#2062semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#20622
#141302
#138802
#113442
#108542
C4×#3422