Finite Group Databasebeta

← #403432

Subgroups — #403432

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic, complemented
2C2×#2094C211yesmaximal, characteristic
3C2×#2025C211yesmaximal, characteristic
4#14260C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic, complemented
6#11373C211yesmaximal, characteristic, complemented
7#11327C212yesmaximal, complemented
8#11066C212yesmaximal
9#10931C212yesmaximal
10#10908C212yesmaximal, complemented
11#10796C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#106C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16#2330—24no—
17#2330—24no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389—22no—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#387—22no—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×#376C2211yescharacteristic
28#2094—24no—
29#2025—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1440—22no—
40#1439—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#125—24no—
47C2×#125—24no—
48C2×#125—22no—
49C2×#125D414yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#106—24no—
68C2×#106—24no—
69C2×#106—42no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—42no—
76#390—22no—
77#390—22no—
78#390—22no—
79#390—42no—
80#390Q814yes—
81#389—216no—
82#389—24no—
83#389—24no—
84#389—42no—
85#387—216no—
86#384—22no—
87#384D414yes—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90#376—22no—
91#376—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#336—24no—
105#336—22no—
106#336—22no—
107#336—24no—
108C25—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4C2×Q812yes—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2411yescharacteristic
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×C8—22no—
141C22×C8—22no—
142C22×C8#4114yes—
143#125—216no—
144#125—216no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
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—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C4⋊C4#4114yes—
179C2×C4⋊C4#4114yes—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—28no—
201C2×C42—82no—
202C2×C42—82no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207#106—48no—
208#102—48no—
209#102—48no—
210#95—28no—
211#95—28no—
212#95—24no—
213#95—24no—
214#95—42no—
215#95—82no—
216#95—82no—
217C24—22no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24C22×D411yescharacteristic
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
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—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4#12014yes—
297C22×C4C22⋊Q814yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4112yes—
302C22×C4C2×#4112yes—
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4H(5,𝔽2)12yes—
306C22×C42−1+411yescharacteristic
307C22×C42−1+411yescharacteristic
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—232no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—88no—
337C4⋊C4—88no—
338C4⋊C4—84no—
339C4⋊C4#12318yes—
340C22⋊C4—88no—
341C22⋊C4—88no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C42—24no—
345C42—24no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—42no—
351C42—42no—
352C42—88no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—81no—
365C23—82no—
366C23—82no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23C2×#12011yescharacteristic
377C23C2×C22⋊Q811yescharacteristic
378C23#52511yescharacteristic
379C23#52511yescharacteristic
380C23#53211yescharacteristic
381C23#53511yescharacteristic
382C23#53611yescharacteristic
383C23#57612yes—
384C23#57712yes—
385C23#57712yes—
386C23#57712yes—
387D4—44no—
388D4—44no—
389D4—82no—
390D4—82no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—28no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4C2×#12312yes—
445C2×C4#52512yes—
446C2×C4#52912yes—
447C2×C4#53312yes—
448C2×C4#54212yes—
449C2×C4#54212yes—
450C2×C4#54612yes—
451C2×C4#54811yescharacteristic
452C2×C4#54811yescharacteristic
453C2×C4D4×Q811yescharacteristic
454C2×C4D4×Q811yescharacteristic
455C2×C4#55012yes—
456C2×C4#55212yes—
457C2×C4#55412yes—
458C2×C4#55612yes—
459C2×C4#55912yes—
460C8—44no—
461C8—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—84no—
466C22—84no—
467C22—161no—
468C22—161no—
469C22—162no—
470C22—161no—
471C22—161no—
472C22—162no—
473C22#239611yescharacteristic
474C22#240911yescharacteristic
475C22#243111yescharacteristic
476C22#258011yescharacteristic
477C22#260411yescharacteristic
478C22#263211yescharacteristic
479C22#284911yescharacteristic
480C22#301012yes—
481C22#301212yes—
482C22#308112yes—
483C22#308412yes—
484C22#311912yes—
485C22#311912yes—
486C22#312512yes—
487C22#312512yes—
488C22#317412yes—
489C22#317912yes—
490C22#322011yescharacteristic
491C22#322111yescharacteristic
492C22#322211yescharacteristic
493C22#322211yescharacteristic
494C22#324112yes—
495C22#324412yes—
496C4—44no—
497C4—44no—
498C4—48no—
499C4—81no—
500C4—81no—
501C4—84no—
502C4—82no—
503C4—161no—
504C4—161no—
505C4#242312yes—
506C4#244812yes—
507C4#245812yes—
508C4#246512yes—
509C2—162no—
510C2#1487511yescharacteristic
511C2#2379012yes—
512C2#2430912yes—
513C2#2560711yescharacteristic
514C2#2561111yescharacteristic
515C2#2926511yescharacteristic
516C2#2926911yescharacteristic
517C2#2952912yes—
518C2#3007111yescharacteristic
519C2#3007411yescharacteristic
520C2#3301312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2330C2semidirect
5. #11385C2semidirect
6. #11373C2semidirect
7. #11327C2semidirect
10. #10908C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23302
C2×#20942
C2×#20252
#142602
#113852
#113732
#113272
#110662
#109312
#109082
#107962