Finite Group Databasebeta

← #409873

Subgroups — #409873

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2099C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic
4C2×#2024C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic
6#14703C211yesmaximal, characteristic
7#11378C212yesmaximal
8#11122C212yesmaximal
9#10926C212yesmaximal
10#10861C212yesmaximal
11C4×#336C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C4×C8C2212yes—
18C2×#391—22no—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×C43C2211yescharacteristic
25#2099—24no—
26#2096—24no—
27#2058—24no—
28#2024—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1436—22no—
37C4⋊#95—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42#1434—22no—
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#106—24no—
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—42no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×C4×C8—24no—
78C2×C4×C8—24no—
79C2×C4×C8—42no—
80#391—22no—
81#391—22no—
82#391—22no—
83#391—42no—
84#391—48no—
85#391Q814yescomplemented
86#389—22no—
87#389—22no—
88#384—22no—
89#384—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95—21no—
99C43—42no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#336—24no—
105#336D414yes—
106#336D414yes—
107C22×Q8—24no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—42no—
118C22×Q8—44no—
119C22×Q8—42no—
120C22×Q8—44no—
121C22×Q8C2×Q812yes—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
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—22no—
143C22×C8C2×D414yes—
144C22×C8C2×D414yes—
145C22×C8#4114yes—
146C22×C8#4114yes—
147C4⋊Q8—48no—
148C4⋊Q8—48no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
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—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C2×Q812yes—
183C2×C4⋊C4#4114yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42SD1618yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×Q812yes—
201C2×C42#4112yes—
202#106—48no—
203#106—48no—
204#103—48no—
205#103—48no—
206C4×C8—48no—
207#95—28no—
208#95—44no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—42no—
213#95—48no—
214#95—48no—
215#95—44no—
216C24C22×D411yescharacteristic
217C2×Q8—24no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—24no—
221C2×Q8—48no—
222C2×Q8—416no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—84no—
236C2×Q8—88no—
237C2×Q8—84no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4C22⋊Q814yes—
287C22×C4#12712yes—
288C22×C4#12712yes—
289C22×C4C2×SD1614yes—
290C22×C4C2×SD1614yes—
291C22×C4C2×SD1614yes—
292C22×C4#13514yes—
293C22×C4#13714yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4H(5,𝔽2)12yes—
302C22×C42−1+411yescharacteristic
303C22×C42−1+411yescharacteristic
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8#12418yes—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—48no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—48no—
343C42—48no—
344C42—42no—
345C42—42no—
346C23C2×C22⋊Q811yescharacteristic
347C23#52511yescharacteristic
348C23C2×#12711yescharacteristic
349C23#53211yescharacteristic
350C23#53211yescharacteristic
351C23#53411yescharacteristic
352C23#53611yescharacteristic
353C23C22×SD1612yes—
354C23C2×#13512yes—
355C23C2×#13712yes—
356C23#57512yes—
357Q8—44no—
358Q8—44no—
359Q8—816no—
360Q8—82no—
361Q8—82no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4#47518yes—
416C2×C4#49214yes—
417C2×C4#49514yes—
418C2×C4C2×#12412yes—
419C2×C4#52912yes—
420C2×C4#53212yes—
421C2×C4#53312yes—
422C2×C4#53912yes—
423C2×C4#54212yes—
424C2×C4#54612yes—
425C2×C4#54612yes—
426C2×C4#54711yescharacteristic
427C2×C4#54711yescharacteristic
428C2×C4D4×Q811yescharacteristic
429C2×C4D4×Q811yescharacteristic
430C2×C4#55112yes—
431C2×C4#55212yes—
432C2×C4#55412yes—
433C2×C4#55612yes—
434C8—28no—
435C8—44no—
436C22#239611yescharacteristic
437C22#253911yescharacteristic
438C22#254211yescharacteristic
439C22#256411yescharacteristic
440C22#260411yescharacteristic
441C22#266311yescharacteristic
442C22#285811yescharacteristic
443C22C2×#47512yes—
444C22#307612yes—
445C22#310112yes—
446C22#310812yes—
447C22C2×#49211yescharacteristic
448C22C2×#49511yescharacteristic
449C22#315211yescharacteristic
450C22#315311yescharacteristic
451C22#316712yes—
452C22#316712yes—
453C22#317212yes—
454C22#318112yes—
455C22#318712yes—
456C22#319012yes—
457C22#322912yes—
458C22#323312yes—
459C4—24no—
460C4—24no—
461C4—28no—
462C4—44no—
463C4—82no—
464C4—84no—
465C4—82no—
466C4—161no—
467C4—162no—
468C4—161no—
469C4#248812yes—
470C4#249212yes—
471C4#252112yes—
472C4#255612yes—
473C2#1506511yescharacteristic
474C2#2377612yes—
475C2#2830211yescharacteristic
476C2#2830611yescharacteristic
477C2#2851011yescharacteristic
478C2#2851211yescharacteristic
479C2#2915912yes—
480C2#2951212yes—
481C2#3035912yes—
482C2#3323911yescharacteristic
483C2#3325011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
85. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20992
C2×#20962
C2×#20582
C2×#20242
#147142
#147032
#113782
#111222
#109262
#108612
C4×#3362