Finite Group Databasebeta

← #472113

Subgroups — #472113

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2083C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic, complemented
6#14700C211yesmaximal, characteristic
7#11378C212yesmaximal, complemented
8#11377C212yesmaximal, complemented
9#11123C212yesmaximal
10#11121C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#391—22no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388—22no—
23C2×#384C2211yescharacteristic
24C2×C43C2211yescharacteristic
25#2083—24no—
26#2071—24no—
27#2058—24no—
28#2023—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1437—22no—
38C4⋊#95—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C42D412yes—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—24no—
71C2×#107—42no—
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—42no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—42no—
80C2×#102—42no—
81#391—48no—
82#390—22no—
83#390—22no—
84#390—22no—
85#390—22no—
86#390—42no—
87#389—22no—
88#389—22no—
89#388—48no—
90#384—22no—
91#384—22no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C43—42no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#336—42no—
103#336—42no—
104#336D414yes—
105#336D414yes—
106C25—21no—
107C22×Q8—24no—
108C22×Q8—42no—
109C22×Q8—44no—
110C22×Q8—42no—
111C22×Q8—44no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4#4112yes—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
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×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8C2×D414yes—
145C22×C8C2×D414yes—
146C4⋊Q8—48no—
147C4⋊Q8—48no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4#4114yes—
182C2×C4⋊C4#4112yes—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211#107—48no—
212#106—48no—
213#102—48no—
214#102—48no—
215#95—44no—
216#95—44no—
217#95—48no—
218#95—42no—
219#95—48no—
220C24—22no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24C22×D411yescharacteristic
232C2×Q8—48no—
233C2×Q8—416no—
234C2×Q8—84no—
235C2×Q8—88no—
236C2×Q8—84no—
237C2×D4—24no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4#12712yes—
297C22×C4#12712yes—
298C22×C4#13514yes—
299C22×C4AGL(1,ℤ/8)14yes—
300C22×C4AGL(1,ℤ/8)14yes—
301C22×C4#13714yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4112yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4H(5,𝔽2)12yes—
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C42−1+412yes—
310C22×C42−1+412yes—
311C22×C42−1+411yescharacteristic
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—48no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C22⋊C4—416no—
347C22⋊C4—88no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C42—24no—
351C42—24no—
352C42—48no—
353C42—48no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C23—44no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—82no—
364C23—82no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—81no—
380C23—81no—
381C23#52511yescharacteristic
382C23#52511yescharacteristic
383C23C2×#12711yescharacteristic
384C23#53411yescharacteristic
385C23#53611yescharacteristic
386C23#53611yescharacteristic
387C23#53711yescharacteristic
388C23C2×#13512yes—
389C23C2×AGL(1,ℤ/8)12yes—
390C23C2×AGL(1,ℤ/8)12yes—
391C23C2×#13712yes—
392Q8—816no—
393D4—44no—
394D4—44no—
395D4—82no—
396D4—82no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—82no—
429C2×C4—84no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4#49514yes—
445C2×C4#49614yes—
446C2×C4#53612yes—
447C2×C4#53612yes—
448C2×C4#53912yes—
449C2×C4#54112yes—
450C2×C4#54212yes—
451C2×C4#54212yes—
452C2×C4#54611yescharacteristic
453C2×C4#54612yes—
454C2×C4#54612yes—
455C2×C4#54611yescharacteristic
456C2×C4#54811yescharacteristic
457C2×C4#54811yescharacteristic
458C2×C4#55212yes—
459C2×C4#55612yes—
460C2×C4#56212yes—
461C2×C4#56312yes—
462C8—44no—
463C8—44no—
464C22—82no—
465C22—84no—
466C22—82no—
467C22—82no—
468C22—84no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—162no—
474C22—162no—
475C22#253911yescharacteristic
476C22#254011yescharacteristic
477C22#259911yescharacteristic
478C22#263711yescharacteristic
479C22#263811yescharacteristic
480C22#265811yescharacteristic
481C22#281411yescharacteristic
482C22#310512yes—
483C22#310712yes—
484C22#310912yes—
485C22#311212yes—
486C22C2×#49511yescharacteristic
487C22C2×#49611yescharacteristic
488C22#315111yescharacteristic
489C22#315211yescharacteristic
490C22#318812yes—
491C22#319312yes—
492C22#322812yes—
493C22#322812yes—
494C22#323812yes—
495C22#323912yes—
496C22#324812yes—
497C22#325312yes—
498C4—24no—
499C4—24no—
500C4—44no—
501C4—44no—
502C4—84no—
503C4—82no—
504C4—161no—
505C4—161no—
506C4—162no—
507C4#268712yes—
508C4#269012yes—
509C4#270012yes—
510C4#271612yes—
511C2—162no—
512C2#1681211yescharacteristic
513C2#2830311yescharacteristic
514C2#2831411yescharacteristic
515C2#2837211yescharacteristic
516C2#2838111yescharacteristic
517C2#2938912yes—
518C2#3014612yes—
519C2#3017312yes—
520C2#3027811yescharacteristic
521C2#3028511yescharacteristic
522C2#3273912yes—
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×#2083C2semidirect
3. C2×#2058C2semidirect
5. #14714C2semidirect
7. #11378C2semidirect
8. #11377C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20832
C2×#20712
C2×#20582
C2×#20232
#147142
#147002
#113782
#113772
#111232
#111212
#107952