Finite Group Databasebeta

← #415128

Subgroups — #415128

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1991C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic
3C2×#1939C211yesmaximal, characteristic
4C2×C4×#107C211yesmaximal, characteristic
5#14160C211yesmaximal, characteristic
6#13954C211yesmaximal, characteristic
7#11228C212yesmaximal
8#11227C212yesmaximal
9#11219C212yesmaximal
10#11219C212yesmaximal
11#10850C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1991—24no—
24#1989—24no—
25#1939—24no—
26C4×#107—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1444—21no—
38#1444—21no—
39#1441—21no—
40#1441—21no—
41#1436—22no—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125Q814yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—24no—
72C2×#107Q818yes—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—42no—
78C2×C4⋊C8—42no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8D414yes—
82#389—24no—
83#389—24no—
84#389—42no—
85#389—42no—
86#382—22no—
87#382—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4D414yes—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95D412yes—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—42no—
113#340—42no—
114#336—24no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×Q814yes—
143#125—28no—
144#125—28no—
145#125—44no—
146#125—44no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×Q812yes—
191C2×C4⋊C4C2×Q812yes—
192C2×C4⋊C4C2×Q814yes—
193C2×C4⋊C4C2×Q814yes—
194C2×C4⋊C4#4114yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—42no—
203C2×C42—81no—
204C2×C42—81no—
205C2×C42—81no—
206C2×C42—81no—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×Q812yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213#107—216no—
214C4⋊C8—48no—
215C4⋊C8—44no—
216C4×C8—28no—
217#95—24no—
218#95—24no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—42no—
223#95—42no—
224#95—42no—
225#95—44no—
226#95—82no—
227#95—82no—
228C24C22×D411yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4#12112yes—
285C22×C4#12112yes—
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4C22⋊Q812yes—
289C22×C4C22⋊Q812yes—
290C22×C4#13514yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×Q811yescharacteristic
294C22×C4C22×Q812yes—
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4112yes—
299C22×C4H(5,𝔽2)12yes—
300C22×C42−1+412yes—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8C4⋊Q814yes—
318C4⋊C4—28no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—44no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—88no—
344C4⋊C4—88no—
345C4⋊C4C22⋊Q818yes—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—84no—
351C42—84no—
352C42#12114yes—
353C42C22⋊Q814yes—
354C42C22⋊Q814yes—
355C42#12614yes—
356C23C2×#12111yescharacteristic
357C23C2×#12111yescharacteristic
358C23C2×C22⋊Q811yescharacteristic
359C23#54612yes—
360C23D4×Q812yes—
361C23C2×#13512yes—
362C23#57512yes—
363C23#57611yescharacteristic
364C23#57611yescharacteristic
365C23#57811yescharacteristic
366C23#57811yescharacteristic
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—28no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
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—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4#39614yes—
428C2×C4#39614yes—
429C2×C4C2×#12111yescharacteristic
430C2×C4C2×C22⋊Q811yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×C22⋊Q812yes—
433C2×C4C2×#12611yescharacteristic
434C2×C4C2×C4⋊Q811yescharacteristic
435C2×C4#53211yescharacteristic
436C2×C4#53312yes—
437C2×C4#53511yescharacteristic
438C2×C4#53611yescharacteristic
439C2×C4#53712yes—
440C2×C4#54212yes—
441C2×C4#55212yes—
442C2×C4#55412yes—
443C2×C4#55712yes—
444C2×C4#55712yes—
445C2×C4Q8212yes—
446C2×C4#55912yes—
447C8—24no—
448C8—42no—
449C8—82no—
450C22C2×#39611yescharacteristic
451C22#241712yes—
452C22#245112yes—
453C22#254411yescharacteristic
454C22#254611yescharacteristic
455C22#303512yes—
456C22#304111yescharacteristic
457C22#304411yescharacteristic
458C22#305011yescharacteristic
459C22#305411yescharacteristic
460C22#306111yescharacteristic
461C22#306111yescharacteristic
462C22#307612yes—
463C22#308311yescharacteristic
464C22#308311yescharacteristic
465C22#329712yes—
466C22#330012yes—
467C22#332212yes—
468C22#332712yes—
469C22#338012yes—
470C22#338312yes—
471C22#338912yes—
472C22#339012yes—
473C4—24no—
474C4—24no—
475C4—44no—
476C4—48no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—84no—
481C4—161no—
482C4—161no—
483C4#251112yes—
484C4#256511yescharacteristic
485C4#256711yescharacteristic
486C4#258212yes—
487C4#262011yescharacteristic
488C4#285711yescharacteristic
489C2#1518411yescharacteristic
490C2#2360911yescharacteristic
491C2#2361111yescharacteristic
492C2#2482012yes—
493C2#2482112yes—
494C2#2644812yes—
495C2#2645112yes—
496C2#2860111yescharacteristic
497C2#2860311yescharacteristic
498C2#2867911yescharacteristic
499C2#2868011yescharacteristic
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

Maximal subgroups

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

subgroupindex
C2×#19912
C2×#19892
C2×#19392
C2×C4×#1072
#141602
#139542
#112282
#112272
#112192
#108502