Finite Group Databasebeta

← #415345

Subgroups — #415345

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic
4#14283C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11384C212yesmaximal
7#11268C212yesmaximal
8#11267C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2660—24no—
18#2660—24no—
19C2×#398—22no—
20C2×#395—22no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1992—24no—
28#1991—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1441—22no—
40#1441—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—42no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—24no—
58C2×#125—42no—
59C2×#125Q814yescomplemented
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#97—22no—
77C2×#97—22no—
78C2×#97—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82#398—216no—
83#395—216no—
84#384—22no—
85#384—22no—
86#384—22no—
87#384—22no—
88#384—22no—
89#384D414yes—
90#382—22no—
91#382—22no—
92#382—22no—
93#382D414yes—
94#376—22no—
95#376—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D412yes—
103C2×#95D412yes—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336Q818yescomplemented
114C22×Q8—24no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—82no—
118C22×Q8—82no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
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×C8C2×Q814yes—
143C22×C8C2×Q814yes—
144C4⋊Q8—48no—
145C4⋊Q8—48no—
146#125—28no—
147#125—28no—
148#125—48no—
149#125—48no—
150#125—44no—
151#125—44no—
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—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×Q814yes—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×Q814yes—
210C2×C42C2×Q812yes—
211C2×C42#4114yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215#97—44no—
216C4×C8—44no—
217#95—28no—
218#95—28no—
219#95—24no—
220#95—24no—
221#95—82no—
222#95—82no—
223#95—82no—
224#95—82no—
225C24C22×D411yescharacteristic
226C2×Q8—48no—
227C2×Q8—88no—
228C2×Q8—88no—
229C2×Q8—84no—
230C2×Q8—84no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
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—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
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#12014yes—
284C22×C4C22⋊Q814yes—
285C22×C4#12412yes—
286C22×C4#12412yes—
287C22×C4#12414yes—
288C22×C4C4⋊Q814yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×Q811yescharacteristic
292C22×C4C22×Q812yes—
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4H(5,𝔽2)12yes—
298C22×C42−1+412yes—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—416no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—88no—
334C4⋊C4—88no—
335C4⋊C4—88no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42C22⋊Q814yes—
341C42C22⋊Q814yes—
342C42#12414yes—
343C42#12414yes—
344C23C2×#12011yescharacteristic
345C23C2×C22⋊Q811yescharacteristic
346C23C2×#12411yescharacteristic
347C23C2×#12411yescharacteristic
348C23C2×C4⋊Q811yescharacteristic
349C23#53511yescharacteristic
350C23#53611yescharacteristic
351C23#57612yes—
352C23#57712yes—
353C23#57712yes—
354C23#57812yes—
355Q8—816no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
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#39318yes—
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4C2×C22⋊Q811yescharacteristic
416C2×C4C2×#12411yescharacteristic
417C2×C4C2×#12411yescharacteristic
418C2×C4#53312yes—
419C2×C4#53312yes—
420C2×C4#54212yes—
421C2×C4#54212yes—
422C2×C4#54612yes—
423C2×C4#54612yes—
424C2×C4D4×Q812yes—
425C2×C4D4×Q812yes—
426C2×C4#55412yes—
427C2×C4#55612yes—
428C2×C4#55712yes—
429C2×C4Q8212yes—
430C8—44no—
431C8—44no—
432C22C2×#39311yescharacteristic
433C22#243411yescharacteristic
434C22#243511yescharacteristic
435C22#256711yescharacteristic
436C22#256811yescharacteristic
437C22#264011yescharacteristic
438C22#286211yescharacteristic
439C22#301212yes—
440C22#301312yes—
441C22#308312yes—
442C22#308412yes—
443C22#313612yes—
444C22#313612yes—
445C22#313712yes—
446C22#313712yes—
447C22#316312yes—
448C22#316412yes—
449C22#322011yescharacteristic
450C22#322111yescharacteristic
451C22#322211yescharacteristic
452C22#322311yescharacteristic
453C22#324112yes—
454C22#324212yes—
455C4—28no—
456C4—42no—
457C4—42no—
458C4—44no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—82no—
463C4—84no—
464C4—162no—
465C4#245112yes—
466C4#245112yes—
467C4#249212yes—
468C4#249212yes—
469C2#1519111yescharacteristic
470C2#2345812yes—
471C2#2582212yes—
472C2#2586711yescharacteristic
473C2#2586811yescharacteristic
474C2#2921511yescharacteristic
475C2#2921611yescharacteristic
476C2#2922812yes—
477C2#3020911yescharacteristic
478C2#3021011yescharacteristic
479C2#3329212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
59. C2×#125Q8semidirect
113. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#19922
C2×#19912
#142832
#113852
#113842
#112682
#112672
#108432
#108412
#106942