Finite Group Databasebeta

← #403675

Subgroups — #403675

Order 512.

1284 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×Q8C211yesmaximal, characteristic
2C2×#2088C212yesmaximal
3C2×#1941C211yesmaximal, characteristic
4#15202C211yesmaximal, characteristic
5#15158C211yesmaximal, characteristic
6#11231C212yesmaximal
7#11228C211yesmaximal, characteristic
8#11227C211yesmaximal, characteristic
9#10880C212yesmaximal
10#10880C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×C8×Q8—28no—
14C2×C8×Q8—24no—
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2212yes—
19C22×C4×C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21C2×#395—21no—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26#2088—28no—
27#1941—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36#1446—21no—
37#1444—21no—
38#1444—22no—
39#1441—22no—
40#1441—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×Q8—24no—
50C2×C4×Q8—22no—
51C2×C4×Q8—24no—
52C2×C4×Q8—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C8×Q8—232no—
68C2×#106—22no—
69C2×#106—22no—
70C2×#106—22no—
71C2×#106—24no—
72C2×#106—22no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8D418yes—
78C2×C4×C8—24no—
79C2×C4×C8—24no—
80C2×C4×C8—24no—
81C2×C4×C8—24no—
82C2×C4×C8D414yes—
83#395—44no—
84#389—24no—
85#389—24no—
86#389—22no—
87#389—22no—
88#389—24no—
89#389—22no—
90#389—24no—
91#389—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—22no—
108#340—22no—
109#340—24no—
110#340—24no—
111#340Q818yescomplemented
112C22×Q8—24no—
113C22×Q8—22no—
114C22×Q8—24no—
115C22×Q8—22no—
116C22×Q8SD1614yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×D414yes—
141C22×C8C2×D414yes—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8C2×Q814yes—
145C4×Q8—28no—
146C4×Q8—216no—
147C4×Q8—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C2×Q814yes—
193C2×C4⋊C4C2×Q814yes—
194C2×C4⋊C4#4112yes—
195C2×C4⋊C4#4112yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42C2×D414yes—
203C2×C42C2×D412yes—
204C2×C42#4114yes—
205C2×C42#4112yes—
206#106—28no—
207#106—44no—
208C4⋊C8—28no—
209C4⋊C8—216no—
210C4×C8—216no—
211C4×C8—28no—
212#95—44no—
213#95—42no—
214#95—44no—
215#95—42no—
216#95—44no—
217#95—42no—
218#95—44no—
219#95—42no—
220#95—81no—
221#95—81no—
222#95—82no—
223#95—82no—
224C24C22×D411yescharacteristic
225C2×Q8—28no—
226C2×Q8—28no—
227C2×Q8—216no—
228C2×Q8—28no—
229C2×Q8—216no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4#12112yes—
283C22×C4#12114yes—
284C22×C4#12114yes—
285C22×C4#12112yes—
286C22×C4C2×SD1614yes—
287C22×C4C2×SD1612yes—
288C22×C4#13514yes—
289C22×C4#13512yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D412yes—
292C22×C4C22×Q812yes—
293C22×C4C2×#4112yes—
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C42−1+412yes—
298C22×C42−1+411yescharacteristic
299C22×C42−1+411yescharacteristic
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8#12014yes—
316C2×C8C4⋊Q814yes—
317C4⋊C4—216no—
318C4⋊C4—28no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—84no—
342C4⋊C4—88no—
343C4⋊C4—84no—
344C4⋊C4—88no—
345C42—24no—
346C42—22no—
347C42—22no—
348C42—24no—
349C42—24no—
350C42—24no—
351C42—24no—
352C23C2×#12112yes—
353C23C2×#12111yescharacteristic
354C23#54811yescharacteristic
355C23#54811yescharacteristic
356C23D4×Q812yes—
357C23C22×SD1611yescharacteristic
358C23C2×#13512yes—
359C23C2×#13511yescharacteristic
360C23#57611yescharacteristic
361C23#57712yes—
362C23#57811yescharacteristic
363Q8—232no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4#46514yes—
423C2×C4#46518yes—
424C2×C4C2×#12011yescharacteristic
425C2×C4#52511yescharacteristic
426C2×C4#52511yescharacteristic
427C2×C4#52512yes—
428C2×C4C2×C4⋊Q811yescharacteristic
429C2×C4#53312yes—
430C2×C4#53312yes—
431C2×C4#53611yescharacteristic
432C2×C4#53611yescharacteristic
433C2×C4#54112yes—
434C2×C4#54112yes—
435C2×C4#54712yes—
436C2×C4#54712yes—
437C2×C4D4×Q812yes—
438C2×C4D4×Q812yes—
439C2×C4#55211yescharacteristic
440C2×C4#55211yescharacteristic
441C2×C4#55712yes—
442C8—24no—
443C8—24no—
444C8—44no—
445C22#242912yes—
446C22#242912yes—
447C22#253611yescharacteristic
448C22#254611yescharacteristic
449C22#284711yescharacteristic
450C22C2×#46511yescharacteristic
451C22C2×#46512yes—
452C22#305011yescharacteristic
453C22#305012yes—
454C22#305711yescharacteristic
455C22#306111yescharacteristic
456C22#306212yes—
457C22#306312yes—
458C22#335911yescharacteristic
459C22#336012yes—
460C22#336111yescharacteristic
461C22#336211yescharacteristic
462C22#336411yescharacteristic
463C22#336512yes—
464C22Q8×SD1612yes—
465C22#338012yes—
466C22#338112yes—
467C22#338312yes—
468C4—28no—
469C4—24no—
470C4—42no—
471C4—44no—
472C4—82no—
473C4—82no—
474C4—84no—
475C4—84no—
476C4—161no—
477C4—161no—
478C4#239911yescharacteristic
479C4#240611yescharacteristic
480C4#243411yescharacteristic
481C4#246212yes—
482C4#246212yes—
483C4#263411yescharacteristic
484C2#1488011yescharacteristic
485C2#2547012yes—
486C2#2547212yes—
487C2#2547312yes—
488C2#2547412yes—
489C2#2818611yescharacteristic
490C2#2818911yescharacteristic
491C2#2867711yescharacteristic
492C2#2867911yescharacteristic
493C2#3292911yescharacteristic
494C2#3293811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
111. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C8×Q82
C2×#20882
C2×#19412
#152022
#151582
#112312
#112282
#112272
#108802
#108202