Finite Group Databasebeta

← #413834

Subgroups — #413834

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic
2C2×#1943C212yesmaximal
3C2×#1941C211yesmaximal, characteristic
4#15202C211yesmaximal, characteristic
5#15158C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic
7#11228C212yesmaximal
8#11227C212yesmaximal
9#11227C211yesmaximal, characteristic
10#10863C212yesmaximal
11#10862C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#446—28no—
14C2×#446—24no—
15C22×#107C2212yes—
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#97C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#395—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27#1943—28no—
28#1941—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36#1446—21no—
37#1444—22no—
38#1444—21no—
39#1441—22no—
40#1441—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
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—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67#446—232no—
68C2×#107—24no—
69C2×#107—24no—
70C2×#107Q818yescomplemented
71C2×#106—22no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—24no—
75C2×#106Q814yescomplemented
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—24no—
81C2×#97—24no—
82C2×#97—24no—
83C2×#97D418yes—
84C2×C4×C8—24no—
85C2×C4×C8D414yes—
86#395—44no—
87#389—22no—
88#389—22no—
89#389—24no—
90#389—24no—
91#389—22no—
92#389—24no—
93#389—22no—
94#389—24no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107#340—22no—
108#340—22no—
109#340—24no—
110#340—24no—
111#340—42no—
112#340—42no—
113#340—42no—
114C22×Q8—24no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—42no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8C2×D412yes—
141C22×C8C2×D414yes—
142C22×C8C2×D414yes—
143C22×C8C2×D412yes—
144C22×C8C2×Q812yes—
145C22×C8C2×Q814yes—
146C22×C8#4112yes—
147C4×Q8—28no—
148C4×Q8—216no—
149C4×Q8—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
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—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C2×Q812yes—
195C2×C4⋊C4C2×Q812yes—
196C2×C4⋊C4C2×Q814yes—
197C2×C4⋊C4C2×Q814yes—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C2×D412yes—
205C2×C42C2×D414yes—
206C2×C42#4114yes—
207C2×C42#4112yes—
208#107—216no—
209#106—28no—
210#106—44no—
211C4⋊C8—28no—
212C4⋊C8—216no—
213#97—216no—
214C4×C8—28no—
215#95—44no—
216#95—44no—
217#95—42no—
218#95—44no—
219#95—44no—
220#95—42no—
221#95—42no—
222#95—42no—
223#95—81no—
224#95—81no—
225#95—82no—
226#95—82no—
227C24C22×D411yescharacteristic
228C2×Q8—28no—
229C2×Q8—48no—
230C2×Q8—48no—
231C2×Q8—44no—
232C2×Q8—44no—
233C22×C4—22no—
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—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4#12112yes—
288C22×C4#12114yes—
289C22×C4#12114yes—
290C22×C4#12112yes—
291C22×C4AGL(1,ℤ/8)12yes—
292C22×C4#13712yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D412yes—
295C22×C4C22×Q812yes—
296C22×C4C22×Q811yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4112yes—
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C42−1+412yes—
302C22×C42−1+411yescharacteristic
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8#12014yes—
316C2×C8C22⋊Q814yes—
317C2×C8C22⋊Q818yes—
318C2×C8C4⋊Q818yes—
319C2×C8C4⋊Q814yes—
320C4⋊C4—28no—
321C4⋊C4—216no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—88no—
345C4⋊C4—84no—
346C4⋊C4—88no—
347C4⋊C4—84no—
348C42—22no—
349C42—24no—
350C42—22no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—24no—
355C23C2×#12111yescharacteristic
356C23C2×#12112yes—
357C23D4×Q812yes—
358C23D4×Q811yescharacteristic
359C23#55011yescharacteristic
360C23C2×AGL(1,ℤ/8)11yescharacteristic
361C23C2×#13711yescharacteristic
362C23#57512yes—
363C23#57611yescharacteristic
364C23#57712yes—
365C23#57811yescharacteristic
366Q8—416no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
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—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#46814yes—
427C2×C4C2×#12011yescharacteristic
428C2×C4C2×C22⋊Q811yescharacteristic
429C2×C4C2×C22⋊Q812yes—
430C2×C4#52511yescharacteristic
431C2×C4#52512yes—
432C2×C4C2×C4⋊Q811yescharacteristic
433C2×C4C2×C4⋊Q812yes—
434C2×C4#53611yescharacteristic
435C2×C4#53612yes—
436C2×C4#53711yescharacteristic
437C2×C4#53712yes—
438C2×C4#55411yescharacteristic
439C2×C4#55412yes—
440C2×C4#55611yescharacteristic
441C2×C4#55612yes—
442C2×C4#55712yes—
443C2×C4#55712yes—
444C2×C4#55712yes—
445C8—42no—
446C8—44no—
447C8#39518yescomplemented
448C22#253811yescharacteristic
449C22#253812yes—
450C22#254611yescharacteristic
451C22#254612yes—
452C22#284711yescharacteristic
453C22C2×#46811yescharacteristic
454C22#305411yescharacteristic
455C22#305512yes—
456C22#305612yes—
457C22#305711yescharacteristic
458C22#306111yescharacteristic
459C22#306212yes—
460C22#306312yes—
461C22#338412yes—
462C22#338511yescharacteristic
463C22#338611yescharacteristic
464C22#338712yes—
465C22#338812yes—
466C22#338912yes—
467C22#339012yes—
468C22#339811yescharacteristic
469C22#339911yescharacteristic
470C22#340112yes—
471C4—24no—
472C4—44no—
473C4—42no—
474C4—44no—
475C4—82no—
476C4—82no—
477C4—84no—
478C4—84no—
479C4—161no—
480C4—161no—
481C4C2×#39511yescharacteristic
482C4#258111yescharacteristic
483C4#258112yes—
484C4#258411yescharacteristic
485C4#258412yes—
486C4#261611yescharacteristic
487C2#1515811yescharacteristic
488C2#2827011yescharacteristic
489C2#2827211yescharacteristic
490C2#2827612yes—
491C2#2827812yes—
492C2#2868011yescharacteristic
493C2#2868411yescharacteristic
494C2#2868512yes—
495C2#2868612yes—
496C2#3293511yescharacteristic
497C2#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
70. C2×#107Q8semidirect
75. C2×#106Q8semidirect
447. C8#395semidirect

Maximal subgroups

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

subgroupindex
C22×#4462
C2×#19432
C2×#19412
#152022
#151582
#112282
#112272
#108632
#108622