Finite Group Databasebeta

← #409857

Subgroups — #409857

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2099C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1934C211yesmaximal, characteristic
5#15155C211yesmaximal, characteristic
6#15143C211yesmaximal, characteristic, complemented
7#11373C212yesmaximal, complemented
8#11049C212yesmaximal
9#10925C212yesmaximal
10#10841C212yesmaximal, complemented
11C8×#95C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#102C2212yes—
14C22×#102C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×C4×C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18C2×#399—22no—
19C2×#399—22no—
20C2×#395—22no—
21C2×#390C2212yes—
22C2×#390C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#384C2212yes—
26C2×#384C2212yes—
27#2099—24no—
28#2023—24no—
29#1990—24no—
30#1934—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40#1444—22no—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2312yes—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—42no—
74C2×#102—42no—
75C2×#102Q814yescomplemented
76C2×C4×C8—24no—
77C2×C4×C8—24no—
78C2×C4×C8—42no—
79C2×C4×C8—42no—
80C2×C4×C8D414yes—
81#399—48no—
82#399—48no—
83#395—48no—
84#390—22no—
85#390—22no—
86#390—24no—
87#390—24no—
88#389—22no—
89#389—22no—
90#384—24no—
91#384—22no—
92#384—22no—
93#384—24no—
94#384—24no—
95#384—42no—
96C2×#95—24no—
97C2×#95—21no—
98C2×#95D412yes—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—24no—
106#340—24no—
107#336—24no—
108#336D414yes—
109C25—21no—
110C22×Q8—44no—
111C22×Q8—42no—
112C22×Q8—44no—
113C22×Q8—82no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4C2×Q812yes—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4C2×D414yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8C2×D414yes—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8C2×Q812yes—
149C22×C8#4112yes—
150C4⋊Q8—28no—
151C4⋊Q8—48no—
152C4⋊Q8—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4C2×Q812yes—
192C2×C4⋊C4#4114yes—
193C2×C22⋊C4—48no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—48no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—48no—
199C2×C22⋊C4—44no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42#4114yes—
209C2×C42#4114yes—
210#102—28no—
211#102—48no—
212#102—44no—
213C4×C8—44no—
214C4×C8—48no—
215#95—42no—
216#95—48no—
217#95D818yes—
218#95SD1618yes—
219C24—22no—
220C24—42no—
221C24—41no—
222C24—42no—
223C24—41no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24C22×D411yescharacteristic
229C2×Q8—48no—
230C2×Q8—816no—
231C2×Q8—84no—
232C2×Q8—84no—
233C2×D4—22no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—24no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4#12418yes—
283C22×C4#12712yes—
284C22×C4#12712yes—
285C22×C4C2×D818yes—
286C22×C4C2×D814yes—
287C22×C4C2×SD1618yes—
288C22×C4C2×SD1614yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×Q811yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C22×C4H(5,𝔽2)12yes—
297C22×C42−1+411yescharacteristic
298C22×C42−1+411yescharacteristic
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8C22⋊Q814yes—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—816no—
336C4⋊C4—816no—
337C4⋊C4—84no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C22⋊C4—816no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C42—22no—
344C42—24no—
345C42—22no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—24no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—42no—
354C23—82no—
355C23—82no—
356C23—81no—
357C23—84no—
358C23—84no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—82no—
364C23—84no—
365C23—84no—
366C23—81no—
367C23C2×#12412yes—
368C23C2×#12711yescharacteristic
369C23#54612yes—
370C23#54711yescharacteristic
371C23D4×Q811yescharacteristic
372C23C22×D812yes—
373C23C22×SD1612yes—
374C23#57712yes—
375C23#57812yes—
376Q8—816no—
377D4—42no—
378D4—42no—
379D4—44no—
380D4—81no—
381D4—81no—
382D4—82no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—28no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—44no—
396C2×C4—48no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—84no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—84no—
417C2×C4—81no—
418C2×C4#486116yes—
419C2×C4#49214yes—
420C2×C4#49314yes—
421C2×C4C2×C22⋊Q811yescharacteristic
422C2×C4#52511yescharacteristic
423C2×C4#52912yes—
424C2×C4#53211yescharacteristic
425C2×C4#53211yescharacteristic
426C2×C4#53212yes—
427C2×C4#53312yes—
428C2×C4#53411yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#53912yes—
431C2×C4#54212yes—
432C2×C4#54612yes—
433C2×C4#54711yescharacteristic
434C2×C4D4×Q811yescharacteristic
435C2×C4#55112yes—
436C2×C4#55212yes—
437C2×C4#55412yes—
438C2×C4#55612yes—
439C8—24no—
440C8—42no—
441C8—44no—
442C22—82no—
443C22—82no—
444C22—88no—
445C22—82no—
446C22—164no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22#248812yes—
452C22#249212yes—
453C22#253911yescharacteristic
454C22#254211yescharacteristic
455C22#285811yescharacteristic
456C22C2×#48614yes—
457C22#313714yes—
458C22C2×#49211yescharacteristic
459C22C2×#49311yescharacteristic
460C22#315411yescharacteristic
461C22#315611yescharacteristic
462C22#329414yes—
463C22#329814yes—
464C22#333412yes—
465C22#333512yes—
466C22Q8×D812yes—
467C22Q8×SD1612yes—
468C4—216no—
469C4—24no—
470C4—24no—
471C4—82no—
472C4—84no—
473C4—161no—
474C4—162no—
475C4—161no—
476C4#239611yescharacteristic
477C4#252112yes—
478C4#255612yes—
479C4#256411yescharacteristic
480C4#260411yescharacteristic
481C4#266311yescharacteristic
482C2—162no—
483C2#1506511yescharacteristic
484C2#2769514yes—
485C2#2781314yes—
486C2#2832911yescharacteristic
487C2#2833611yescharacteristic
488C2#2852711yescharacteristic
489C2#2853011yescharacteristic
490C2#3322411yescharacteristic
491C2#3322911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2099C2semidirect
3. C2×#1990C2semidirect
6. #15143C2semidirect
7. #11373C2semidirect
10. #10841C2semidirect
11. C8×#95C2semidirect
75. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20992
C2×#20232
C2×#19902
C2×#19342
#151552
#151432
#113732
#110492
#109252
#108412
C8×#952