Finite Group Databasebeta

← #413241

Subgroups — #413241

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic
2C2×#2061C212yesmaximal
3C2×#2060C212yesmaximal
4#21171C211yesmaximal, characteristic
5#20800C211yesmaximal, characteristic
6#14354C211yesmaximal, characteristic
7#11340C212yesmaximal
8#11339C212yesmaximal
9#10858C211yesmaximal, characteristic
10#10840C211yesmaximal, characteristic
11#10701C211yesmaximal, characteristic
12C2×#544C2211yescharacteristic
13C2×#544C2211yescharacteristic
14C2×#428C2212yes—
15C2×#428C2212yes—
16C2×#428C2212yes—
17C2×#423C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2862—21no—
20#2859—21no—
21#2660—21no—
22#2660—21no—
23#2550—22no—
24#2550—22no—
25#2549—24no—
26#2317C2211yescharacteristic
27#2317C2211yescharacteristic
28#2285C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#384C2212yes—
31C2×#382C2212yes—
32C2×#382C2211yescharacteristic
33C4×#117C2211yescharacteristic
34C2×#376C2211yescharacteristic
35#2061—28no—
36#2060—28no—
37#1948—22no—
38#1947—22no—
39#1847—21no—
40#1847—21no—
41#1845—21no—
42#1844—21no—
43C2×#343C2212yes—
44C2×#343C2212yes—
45#1736C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50#1444—24no—
51C22×M16C2312yes—
52C22×M16C2312yes—
53#544—24no—
54#544—24no—
55#544—42no—
56#544—42no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×#125—21no—
62C2×#125—21no—
63C2×#125—21no—
64C2×#125—21no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—24no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#428—24no—
79#428—24no—
80#428—24no—
81#428—24no—
82#428—42no—
83#428—42no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106—22no—
90#423—22no—
91#423Q814yescomplemented
92C2×C4⋊C8—22no—
93C4×M16—22no—
94C4×M16Q814yescomplemented
95C2×#97—21no—
96C2×C4×C8—21no—
97#398—24no—
98#398—24no—
99#395—24no—
100#395—24no—
101#391—24no—
102#391—24no—
103#389—24no—
104#389—24no—
105#389—22no—
106#389—22no—
107#385—22no—
108#385—22no—
109#384—21no—
110#384—21no—
111#384—24no—
112#384—21no—
113#384—21no—
114#384—22no—
115#384—42no—
116#384D411yescharacteristic
117#384D411yescharacteristic
118#382—21no—
119#382—21no—
120#382—21no—
121#382—21no—
122#382—22no—
123#382—24no—
124#382—42no—
125#382D411yescharacteristic
126#382D411yescharacteristic
127#380—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D412yes—
131C4×C22⋊C4—22no—
132#376—21no—
133#376—22no—
134#376Q814yescomplemented
135C2×#95—22no—
136C2×#95—24no—
137C2×#95D414yes—
138C43—21no—
139#343—24no—
140#343—24no—
141#340—24no—
142#339—24no—
143#339—24no—
144#339D418yes—
145#339Q818yescomplemented
146C22×Q8—22no—
147C22×Q8—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2411yescharacteristic
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16C2×D414yes—
163C2×M16C2×Q814yes—
164C2×M16C2×Q814yes—
165C2×M16C2×Q814yes—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174#125—44no—
175#125—44no—
176#125—42no—
177#125—42no—
178#125—42no—
179#125—42no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117C2×D414yes—
189#117C2×Q814yes—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—84no—
218C2×C4⋊C4—88no—
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D412yes—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42C2×D414yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×Q814yes—
244C2×C42C2×Q814yes—
245C2×C42C2×Q812yes—
246C2×C42C2×Q812yes—
247C2×C42C2×Q812yes—
248C2×C42#4112yes—
249#107—44no—
250#107—44no—
251#107—44no—
252#106—44no—
253#106—44no—
254#106—44no—
255C4⋊C8—28no—
256#97—24no—
257C4×C8—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—28no—
267#95—48no—
268C24C22×D411yescharacteristic
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—84no—
272C2×Q8—84no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—48no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4#12012yes—
298C22×C4#12012yes—
299C22×C4C22⋊Q814yes—
300C22×C4C4⋊Q814yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D412yes—
304C22×C4C22×Q811yescharacteristic
305C22×C4C22×Q811yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C22×Q812yes—
308C22×C4C2×#4111yescharacteristic
309C22×C4H(5,𝔽2)12yes—
310C22×C42−1+412yes—
311M16—24no—
312M16—24no—
313M16—24no—
314M16—24no—
315M16—24no—
316M16—42no—
317M16—42no—
318M16C4⋊Q818yes—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C4⋊C4—816no—
355C4⋊C4#12712yes—
356C4⋊C4#12712yes—
357C4⋊C4C4⋊Q814yes—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—82no—
372C42—82no—
373C42C22⋊Q812yes—
374C42C22⋊Q812yes—
375C42C22⋊Q814yes—
376C42C4⋊Q818yes—
377C42C4⋊Q814yes—
378C42C4⋊Q812yes—
379C42C4⋊Q812yes—
380C23—22no—
381C23—22no—
382C23C2×#12011yescharacteristic
383C23C2×C22⋊Q811yescharacteristic
384C23C2×C4⋊Q811yescharacteristic
385C23D4212yes—
386C23D4×Q812yes—
387Q8—164no—
388Q8—164no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—28no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—48no—
401C2×C4—44no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4—84no—
425C2×C4#39514yes—
426C2×C4#39514yes—
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×C22⋊Q811yescharacteristic
429C2×C4C2×#12711yescharacteristic
430C2×C4C2×C4⋊Q811yescharacteristic
431C2×C4C2×C4⋊Q811yescharacteristic
432C2×C4C2×C4⋊Q812yes—
433C2×C4C2×C4⋊Q812yes—
434C2×C4C2×C4⋊Q811yescharacteristic
435C2×C4#53511yescharacteristic
436C2×C4#53611yescharacteristic
437C2×C4#54412yes—
438C2×C4#54412yes—
439C2×C4D4212yes—
440C2×C4#54712yes—
441C2×C4D4×Q812yes—
442C2×C4D4×Q812yes—
443C2×C4#55412yes—
444C2×C4Q8212yes—
445C8—28no—
446C8—44no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—24no—
452C22C2×#39511yescharacteristic
453C22#243311yescharacteristic
454C22#243411yescharacteristic
455C22#244012yes—
456C22#255112yes—
457C22#302112yes—
458C22#302212yes—
459C4—24no—
460C4—24no—
461C4—48no—
462C4—42no—
463C4—42no—
464C4—44no—
465C4—81no—
466C4—82no—
467C4—81no—
468C4—81no—
469C4—82no—
470C4—81no—
471C4—162no—
472C4#244012yes—
473C4#254211yescharacteristic
474C4#255112yes—
475C4#256711yescharacteristic
476C4#266011yescharacteristic
477C4#286811yescharacteristic
478C2—22no—
479C2—22no—
480C2#1514311yescharacteristic
481C2#2356212yes—
482C2#2575111yescharacteristic
483C2#2575211yescharacteristic
484C2#2583312yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
91. #423Q8semidirect
94. C4×M16Q8semidirect
134. #376Q8semidirect
145. #339Q8semidirect

Maximal subgroups

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

subgroupindex
#353302
C2×#20612
C2×#20602
#211712
#208002
#143542
#113402
#113392
#108582
#108402
#107012