Finite Group Databasebeta

← #440881

Subgroups — #440881

Order 512.

1210 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2074C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4#15235C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11373C211yesmaximal, characteristic
7#11267C212yesmaximal
8#11190C212yesmaximal
9#10928C211yesmaximal, characteristic
10#10875C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#97C2211yescharacteristic
17#2567—24no—
18#2567—24no—
19C2×#395—22no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389—22no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#2074—24no—
28#1836—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1443—22no—
39#1442—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—42no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—24no—
54C2×C4×Q8—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—42no—
75C2×#97—24no—
76C2×#97—22no—
77#395—216no—
78#391—22no—
79#391—22no—
80#391—22no—
81#391D414yes—
82#389—216no—
83#384—22no—
84#384—22no—
85#384—22no—
86#384D414yes—
87#382—22no—
88#382—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#340—42no—
106#336—24no—
107#336—22no—
108#336—24no—
109C22×Q8—24no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—82no—
119C22×Q8—82no—
120C22×Q8C2×D412yes—
121C22×Q8#4114yes—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
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×C8—22no—
145C22×C8—22no—
146C22×C8#4114yes—
147C4⋊Q8—48no—
148C4⋊Q8—48no—
149C4×Q8—48no—
150C4×Q8—48no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
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—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42#4114yes—
205#103—48no—
206#103—48no—
207#97—28no—
208#95—28no—
209#95—28no—
210#95—24no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—82no—
215#95—82no—
216#95—82no—
217C24C22×D411yescharacteristic
218C2×Q8—22no—
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—24no—
222C2×Q8—24no—
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—48no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—88no—
235C2×Q8—88no—
236C2×Q8—84no—
237C2×Q8—84no—
238C2×Q8#12414yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
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—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4#12014yes—
293C22×C4#12412yes—
294C22×C4#12412yes—
295C22×C4#13714yes—
296C22×C4#13714yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4112yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4H(5,𝔽2)12yes—
303C22×C42−1+412yes—
304C22×C42−1+412yes—
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—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—216no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
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—
338C42—22no—
339C42—22no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—48no—
344C42—41no—
345C42—41no—
346C42—42no—
347C23C2×#12011yescharacteristic
348C23C2×#12411yescharacteristic
349C23#53511yescharacteristic
350C23#54712yes—
351C23#54812yes—
352C23C2×#13712yes—
353C23C2×#13712yes—
354C23#57612yes—
355C23#57712yes—
356Q8—28no—
357Q8—44no—
358Q8—44no—
359Q8—44no—
360Q8—816no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4#49114yes—
419C2×C4#52511yescharacteristic
420C2×C4C2×#12411yescharacteristic
421C2×C4#52912yes—
422C2×C4#53611yescharacteristic
423C2×C4#53611yescharacteristic
424C2×C4#53912yes—
425C2×C4#54112yes—
426C2×C4#54212yes—
427C2×C4#54612yes—
428C2×C4#54612yes—
429C2×C4#54712yes—
430C2×C4#54812yes—
431C2×C4#55112yes—
432C2×C4#55212yes—
433C2×C4#55212yes—
434C2×C4#56312yes—
435C8—44no—
436C8—82no—
437C22#240611yescharacteristic
438C22#243511yescharacteristic
439C22#243611yescharacteristic
440C22#248912yes—
441C22#272312yes—
442C22#300412yes—
443C22#301212yes—
444C22C2×#49111yescharacteristic
445C22#313612yes—
446C22#313911yescharacteristic
447C22#321812yes—
448C22#322011yescharacteristic
449C22#322211yescharacteristic
450C22#334412yes—
451C22#334612yes—
452C22#335612yes—
453C22#335712yes—
454C22#336912yes—
455C22#336912yes—
456C22#337512yes—
457C22#337512yes—
458C4—24no—
459C4—42no—
460C4—44no—
461C4—44no—
462C4—44no—
463C4—82no—
464C4—82no—
465C4—82no—
466C4—162no—
467C4—162no—
468C4#247712yes—
469C4#248812yes—
470C4#269012yes—
471C4#269812yes—
472C2#1567011yescharacteristic
473C2#2410912yes—
474C2#2586111yescharacteristic
475C2#2586711yescharacteristic
476C2#2594212yes—
477C2#2773812yes—
478C2#2774612yes—
479C2#3244212yes—
480C2#3244412yes—
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×#25672
C2×#20742
C2×#18362
#152352
#113842
#113732
#112672
#111902
#109282
#108752
#107892