Finite Group Databasebeta

← #467151

Subgroups — #467151

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic
3C2×#1918C211yesmaximal, characteristic
4#15156C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11384C212yesmaximal
7#11357C211yesmaximal, characteristic
8#11346C211yesmaximal, characteristic
9#11292C212yesmaximal
10#11219C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×#107C2211yescharacteristic
15#2551—24no—
16#2551—24no—
17C2×#400—21no—
18C2×#395—22no—
19C2×#389—21no—
20C2×#389C2212yes—
21C2×#389C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#2099—24no—
26#1918—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1443—22no—
41#1441—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8—22no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—28no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—44no—
57C2×C4⋊Q8—44no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#107—22no—
75C2×#107—22no—
76C2×#107Q814yescomplemented
77#400—44no—
78#395—216no—
79#389—28no—
80#389—24no—
81#389—24no—
82#389—22no—
83#389—22no—
84#384—22no—
85#384D414yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D414yes—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—22no—
111#336—42no—
112#336—42no—
113C22×Q8—28no—
114C22×Q8—44no—
115C22×Q8—44no—
116C22×Q8—44no—
117C22×Q8—44no—
118C22×Q8—82no—
119C22×Q8—82no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8C2×Q812yes—
148C4⋊Q8—48no—
149C4⋊Q8—48no—
150C4⋊Q8—48no—
151C4⋊Q8—432no—
152C4⋊Q8—48no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4D814yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×Q812yes—
201C2×C4⋊C4C2×Q812yes—
202C2×C4⋊C4#4112yes—
203C2×C4⋊C4#4114yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217#107—28no—
218#95—28no—
219#95—24no—
220#95—24no—
221#95—42no—
222#95—44no—
223#95—42no—
224#95—44no—
225#95—44no—
226#95—82no—
227#95—81no—
228C24C22×D411yescharacteristic
229C2×Q8—416no—
230C2×Q8—416no—
231C2×Q8—88no—
232C2×Q8—88no—
233C2×Q8—88no—
234C2×Q8—88no—
235C2×Q8—88no—
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—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4#12012yes—
293C22×C4#12012yes—
294C22×C4C2×D814yes—
295C22×C4C2×D812yes—
296C22×C4AGL(1,ℤ/8)12yes—
297C22×C4#13714yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4112yes—
303C22×C4H(5,𝔽2)12yes—
304C22×C42−1+412yes—
305C22×C42−1+412yes—
306C22×C42−1+411yescharacteristic
307C22×C42−1+411yescharacteristic
308C2×C8—22no—
309C2×C8—24no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—28no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—28no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—416no—
337C4⋊C4—416no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—48no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4C22⋊Q814yes—
353C4⋊C4#12414yes—
354C42—22no—
355C42—22no—
356C42—24no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—48no—
362C23C2×#12011yescharacteristic
363C23#53612yes—
364C23#54612yes—
365C23#54811yescharacteristic
366C23D4×Q811yescharacteristic
367C23C22×D811yescharacteristic
368C23C2×AGL(1,ℤ/8)11yescharacteristic
369C23C2×#13712yes—
370C23#57711yescharacteristic
371C23#57712yes—
372C23#57811yescharacteristic
373Q8—832no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4#44714yes—
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4C2×#12411yescharacteristic
438C2×C4#53511yescharacteristic
439C2×C4#53611yescharacteristic
440C2×C4#53611yescharacteristic
441C2×C4#53611yescharacteristic
442C2×C4#53612yes—
443C2×C4#54112yes—
444C2×C4#54112yes—
445C2×C4#54212yes—
446C2×C4#54412yes—
447C2×C4#54612yes—
448C2×C4#54811yescharacteristic
449C2×C4D4×Q811yescharacteristic
450C2×C4#55612yes—
451C2×C4#55712yes—
452C2×C4#56312yes—
453C2×C4#56312yes—
454C8—42no—
455C8—81no—
456C8—82no—
457C22#240911yescharacteristic
458C22#243511yescharacteristic
459C22#261411yescharacteristic
460C22#271312yes—
461C22#271612yes—
462C22C2×#44711yescharacteristic
463C22#300411yescharacteristic
464C22#300911yescharacteristic
465C22#301311yescharacteristic
466C22#322612yes—
467C22#323812yes—
468C22#324212yes—
469C22#324412yes—
470C22#329412yes—
471C22#331212yes—
472C22#332012yes—
473C22#332412yes—
474C22#335811yescharacteristic
475C22#336711yescharacteristic
476C22#336912yes—
477C22Q8×D811yescharacteristic
478C22#338411yescharacteristic
479C22#338712yes—
480C4—24no—
481C4—44no—
482C4—44no—
483C4—44no—
484C4—44no—
485C4—42no—
486C4—82no—
487C4—84no—
488C4—161no—
489C4—162no—
490C4#256811yescharacteristic
491C4#262811yescharacteristic
492C4#264211yescharacteristic
493C4#271312yes—
494C4#271612yes—
495C4#282111yescharacteristic
496C2#1656011yescharacteristic
497C2#2432611yescharacteristic
498C2#2434511yescharacteristic
499C2#2588511yescharacteristic
500C2#2589411yescharacteristic
501C2#2963811yescharacteristic
502C2#2964811yescharacteristic
503C2#3213012yes—
504C2#3215012yes—
505C2#3222612yes—
506C2#3224812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
76. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#20992
C2×#19182
#151562
#113842
#113572
#113462
#112922
#112192
#108062