Finite Group Databasebeta

← #475661

Subgroups — #475661

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2100C211yesmaximal, characteristic, complemented
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#2094C211yesmaximal, characteristic
4C2×#2093C211yesmaximal, characteristic
5#14700C211yesmaximal, characteristic
6#14687C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11325C212yesmaximal, complemented
9#11051C212yesmaximal
10#11051C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#388—22no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2212yes—
24C2×#382C2212yes—
25C2×C43C2211yescharacteristic
26#2100—24no—
27#2099—24no—
28#2094—24no—
29#2093—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1437—22no—
39C4⋊#95—22no—
40C4⋊#95—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—24no—
66C2×#107—42no—
67C2×#107—42no—
68C2×#106—24no—
69C2×#106—42no—
70C2×#106—42no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—42no—
74C2×#102—42no—
75C2×#102—42no—
76C2×#102—42no—
77#390—22no—
78#390—22no—
79#390—22no—
80#390—24no—
81#390Q814yes—
82#389—22no—
83#389—22no—
84#388—48no—
85#384—22no—
86#384—22no—
87#384—22no—
88#384—22no—
89#382—42no—
90#382—42no—
91#382—42no—
92#382—42no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C43—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—24no—
104#336—24no—
105#336D414yes—
106#336D414yes—
107C25—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4C2×Q812yes—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
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×D414yes—
141C22×C8C2×D414yes—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4C2×Q812yes—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×Q812yes—
204C2×C42#4112yes—
205C2×C42#4114yes—
206#107—48no—
207#106—48no—
208#102—48no—
209#102—48no—
210#95—48no—
211#95—48no—
212#95—44no—
213#95—44no—
214#95—42no—
215C24—22no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24—41no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24C22×D411yescharacteristic
227C2×D4—24no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—24no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
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—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4C22⋊Q814yes—
287C22×C4#12314yes—
288C22×C4#12712yes—
289C22×C4#12712yes—
290C22×C4#13514yes—
291C22×C4AGL(1,ℤ/8)14yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×Q811yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4C2×#4111yescharacteristic
297C22×C4H(5,𝔽2)12yes—
298C22×C4H(5,𝔽2)12yes—
299C22×C42−1+411yescharacteristic
300C22×C42−1+411yescharacteristic
301C22×C42−1+412yes—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C22⋊C4—416no—
338C22⋊C4—88no—
339C22⋊C4—84no—
340C22⋊C4—84no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—48no—
348C42—48no—
349C23—44no—
350C23—44no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—82no—
356C23—82no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—81no—
372C23C2×C22⋊Q811yescharacteristic
373C23C2×#12311yescharacteristic
374C23C2×#12711yescharacteristic
375C23#53511yescharacteristic
376C23#53511yescharacteristic
377C23#53611yescharacteristic
378C23#53611yescharacteristic
379C23C2×#13512yes—
380C23C2×AGL(1,ℤ/8)12yes—
381C23#57512yes—
382C23#57512yes—
383D4—44no—
384D4—44no—
385D4—82no—
386D4—82no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4#49514yes—
435C2×C4#49614yes—
436C2×C4#53512yes—
437C2×C4#53512yes—
438C2×C4#53912yes—
439C2×C4#53912yes—
440C2×C4#54212yes—
441C2×C4#54412yes—
442C2×C4#54711yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4#54712yes—
445C2×C4#54712yes—
446C2×C4D4×Q811yescharacteristic
447C2×C4D4×Q811yescharacteristic
448C2×C4#55312yes—
449C2×C4#55412yes—
450C2×C4#56312yes—
451C2×C4#56312yes—
452C8—44no—
453C8—44no—
454C22—84no—
455C22—84no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—162no—
464C22—162no—
465C22#248711yescharacteristic
466C22#250111yescharacteristic
467C22#254211yescharacteristic
468C22#262311yescharacteristic
469C22#262811yescharacteristic
470C22#265711yescharacteristic
471C22#282111yescharacteristic
472C22#307612yes—
473C22#307712yes—
474C22#309112yes—
475C22#309412yes—
476C22C2×#49511yescharacteristic
477C22C2×#49611yescharacteristic
478C22#315311yescharacteristic
479C22#315311yescharacteristic
480C22#320712yes—
481C22#320712yes—
482C22#321912yes—
483C22#321912yes—
484C22#323212yes—
485C22#323312yes—
486C22#323412yes—
487C22#323512yes—
488C4—24no—
489C4—24no—
490C4—44no—
491C4—44no—
492C4—82no—
493C4—161no—
494C4—161no—
495C4—162no—
496C4—162no—
497C4#271812yes—
498C4#271812yes—
499C4#272012yes—
500C4#272012yes—
501C2—162no—
502C2#1706411yescharacteristic
503C2#2768312yes—
504C2#2791211yescharacteristic
505C2#2791811yescharacteristic
506C2#2851011yescharacteristic
507C2#2852111yescharacteristic
508C2#2991412yes—
509C2#2998712yes—
510C2#3026011yescharacteristic
511C2#3026611yescharacteristic
512C2#3284112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
1. C2×#2100C2semidirect
2. C2×#2099C2semidirect
6. #14687C2semidirect
7. #11325C2semidirect
8. #11325C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21002
C2×#20992
C2×#20942
C2×#20932
#147002
#146872
#113252
#110512
#107952