Finite Group Databasebeta

← #415651

Subgroups — #415651

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1941C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊C8C211yesmaximal, characteristic
3#14697C212yesmaximal
4#11327C211yesmaximal, characteristic, complemented
5#11257C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11095C211yesmaximal, characteristic
8#11095C212yesmaximal
9#11095C211yesmaximal, characteristic
10#10880C212yesmaximal, complemented
11#10861C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2212yes—
15C22×#106C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#385—22no—
23C2×#380C2212yes—
24C2×#380C2212yes—
25#1941—24no—
26C4×C22⋊C8—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1436—22no—
37#1436—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C22⋊#95—22no—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C42—22no—
56C22×C42—22no—
57C22×C42—42no—
58C22×C42—42no—
59C22×C42D412yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#106—22no—
63C2×#106—24no—
64C2×#106—22no—
65C2×#106—24no—
66C2×#106—24no—
67C2×#106—42no—
68C2×#106—42no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8Q814yes—
74C2×C22⋊C8Q814yes—
75C2×C4×C8—22no—
76C2×C4×C8—42no—
77C2×C4×C8—42no—
78C2×C4×C8D414yes—
79#389—24no—
80#389—24no—
81#389—42no—
82#389—42no—
83#385—48no—
84#380—24no—
85#380—24no—
86#380—42no—
87#380—42no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96#340—24no—
97#340—22no—
98#340—22no—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#336—22no—
104#336—22no—
105C25C2×D411yescharacteristic
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4SD1618yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×Q812yes—
129C23×C4C2×Q812yes—
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C22×C8C2×Q812yes—
146C22×C8C2×Q812yes—
147C22×C8C2×Q812yes—
148C22×C8C2×Q812yes—
149C22×C8#4112yes—
150C22×C8#4112yes—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4#4114yes—
180C2×C4⋊C4#4114yes—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—44no—
195C2×C42—44no—
196C2×C42—48no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C2×D412yes—
200C2×C42#4112yes—
201#106—28no—
202#106—48no—
203#106—44no—
204C22⋊C8—28no—
205C22⋊C8—28no—
206C4×C8—44no—
207C4×C8—44no—
208#95—28no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216C24—21no—
217C24—22no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24C4⋊Q812yes—
224C24C2×SD1614yes—
225C24C22×D411yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—216no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4#12114yes—
294C22×C4C4⋊Q812yes—
295C22×C4C2×SD1614yes—
296C22×C4C2×SD1614yes—
297C22×C4#13514yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×Q811yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4112yes—
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4112yes—
306C22×C4H(5,𝔽2)12yes—
307C22×C4H(5,𝔽2)12yes—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8C22⋊Q814yes—
327C2×C8C22⋊Q814yes—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—88no—
345C4⋊C4—88no—
346C22⋊C4—416no—
347C22⋊C4—88no—
348C42—22no—
349C42—22no—
350C42—24no—
351C42—48no—
352C42—41no—
353C42—41no—
354C42—42no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23#49814yes—
376C23C2×#12111yescharacteristic
377C23#52512yes—
378C23C2×C4⋊Q811yescharacteristic
379C23#53211yescharacteristic
380C23#53412yes—
381C23C22×SD1612yes—
382C23C2×#13512yes—
383C23#57512yes—
384C23#57512yes—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—432no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#46518yes—
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4C2×C22⋊Q811yescharacteristic
441C2×C4#52511yescharacteristic
442C2×C4#52912yes—
443C2×C4#53211yescharacteristic
444C2×C4#53311yescharacteristic
445C2×C4#53311yescharacteristic
446C2×C4#53411yescharacteristic
447C2×C4#53511yescharacteristic
448C2×C4#53812yes—
449C2×C4#55112yes—
450C2×C4#55312yes—
451C2×C4#55312yes—
452C2×C4#55412yes—
453C2×C4#55412yes—
454C2×C4#55412yes—
455C2×C4#55412yes—
456C2×C4#55912yes—
457C8—24no—
458C8—24no—
459C8—42no—
460C8—42no—
461C22—28no—
462C22—24no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—42no—
467C22—44no—
468C22—44no—
469C22—42no—
470C22#239211yescharacteristic
471C22#258312yes—
472C22#260212yes—
473C22#261711yescharacteristic
474C22#285711yescharacteristic
475C22C2×#46512yes—
476C22#305412yes—
477C22#310112yes—
478C22#310212yes—
479C22#310212yes—
480C22#310312yes—
481C22C2×#49811yescharacteristic
482C22#316011yescharacteristic
483C22#316612yes—
484C22#316712yes—
485C22#318311yescharacteristic
486C22#318311yescharacteristic
487C22#318712yes—
488C22#318712yes—
489C22#319112yes—
490C22#319112yes—
491C4—28no—
492C4—44no—
493C4—42no—
494C4—84no—
495C4—84no—
496C4—162no—
497C4—162no—
498C4#256411yescharacteristic
499C4#256611yescharacteristic
500C4#257911yescharacteristic
501C4#258011yescharacteristic
502C4#258712yes—
503C4#258912yes—
504C2—44no—
505C2#1520411yescharacteristic
506C2#2370012yes—
507C2#2929712yes—
508C2#2930012yes—
509C2#2946412yes—
510C2#2946612yes—
511C2#2973412yes—
512C2#3320711yescharacteristic
513C2#3321111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1941C2semidirect
4. #11327C2semidirect
5. #11257C2semidirect
6. #11220C2semidirect
10. #10880C2semidirect
11. #10861C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19412
C2×C4×C22⋊C82
#146972
#113272
#112572
#112202
#110952
#108802
#108612