Finite Group Databasebeta

← #241245

Subgroups — #241245

Order 512.

836 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37561C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#20800C211yesmaximal, characteristic
4C2×#1836C211yesmaximal, characteristic
5#10585C214yesmaximal, complemented
6#10584C214yesmaximal, complemented
7#8034C211yesmaximal, characteristic
8#8033C211yesmaximal, characteristic
9#8029C211yesmaximal, characteristic
10C2×#432C2211yescharacteristic
11C2×#428—22no—
12C2×C4×M16C2211yescharacteristic
13C22×#97C2211yescharacteristic
14#2845C2212yes—
15#2844C2212yes—
16#2666—22no—
17#2660—21no—
18#2659—21no—
19#2647—21no—
20#2646—21no—
21#2570—21no—
22#2550—21no—
23#2442—21no—
24#2440—21no—
25#2438—22no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#2283C2211yescharacteristic
29C2×#382C2211yescharacteristic
30#1948—21no—
31#1947—21no—
32#1853—21no—
33#1852—21no—
34#1847C2214yes—
35#1847C2212yes—
36#1845—21no—
37#1844—21no—
38#1836—22no—
39#1836C2214yes—
40#1836C2212yes—
41C2×#340—21no—
42C2×#340—21no—
43#1295C2212yes—
44#1295C2214yes—
45#1295C2214yes—
46#1295C2212yes—
47C22×M16—21no—
48C23×C8—21no—
49C2×C4⋊Q8—21no—
50C2×#125—21no—
51C2×C22⋊Q8—22no—
52C2×#117D412yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C42C2311yescharacteristic
59#432—22no—
60#432Q814yescomplemented
61#428—48no—
62C2×#107—24no—
63C2×#107—22no—
64C2×#106—24no—
65C2×#106—22no—
66C2×C4⋊C8—21no—
67C2×C22⋊C8—21no—
68C4×M16—22no—
69C4×M16Q814yescomplemented
70C2×#97—21no—
71C2×#97—22no—
72C2×#97Q814yescomplemented
73C2×#97C2312yes—
74C2×#97C2312yes—
75C2×#97C2312yes—
76C2×C4×C8—21no—
77#400—22no—
78#400—22no—
79#400—22no—
80#400—22no—
81#398—22no—
82#398—22no—
83#398—22no—
84#398—22no—
85#398—22no—
86#398—22no—
87#398—22no—
88#398—22no—
89#395—22no—
90#395—22no—
91#395—22no—
92#395—22no—
93#391—22no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—22no—
99#387—22no—
100#387—22no—
101#385—22no—
102#385—22no—
103#384—21no—
104#384—21no—
105#384—21no—
106#384—21no—
107#384D411yescharacteristic
108#384D411yescharacteristic
109#384C2312yes—
110#383—21no—
111#382—21no—
112#382—21no—
113#382—21no—
114#382D412yes—
115#382C2311yescharacteristic
116#382C2312yes—
117#382C2312yes—
118#382C2311yescharacteristic
119#381—21no—
120#380—22no—
121#380—21no—
122#380—21no—
123#380D411yescharacteristic
124#380D411yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C22⋊C4—21no—
130#376—21no—
131C2×#95—21no—
132C2×#95—21no—
133#340—24no—
134#340—24no—
135#340—22no—
136#340—22no—
137C22×Q8—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—24no—
147C22×C8—22no—
148C22×C8—24no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—42no—
152C22×C8—42no—
153C4⋊Q8—82no—
154#125—82no—
155C22⋊Q8—84no—
156#117—44no—
157C2×C4⋊C4—21no—
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—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—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q812yes—
201C2×C42C2×Q812yes—
202C2×C42C2×Q812yes—
203C2×C42#4112yes—
204C2×C42C2411yescharacteristic
205#107—48no—
206#107—82no—
207#106—48no—
208#106—82no—
209C4⋊C8—24no—
210C22⋊C8—24no—
211#97—24no—
212#97—22no—
213#97—22no—
214#97—22no—
215#97C2×Q814yes—
216#97C2×Q814yes—
217#97C2×Q814yes—
218C4×C8—24no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239C24#12011yescharacteristic
240C2×Q8—82no—
241C2×Q8—82no—
242C2×Q8—82no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4C22⋊Q812yes—
271C22×C4C22⋊Q812yes—
272C22×C4C22⋊Q812yes—
273C22×C4C4⋊Q812yes—
274C22×C4C4⋊Q812yes—
275C22×C4C4⋊Q812yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—216no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—48no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C22⋊C4—44no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—42no—
324C42—42no—
325C42—82no—
326C42C22⋊Q812yes—
327C42C22⋊Q812yes—
328C42C22⋊Q812yes—
329C42C4⋊Q812yes—
330C42C4⋊Q812yes—
331C42C4⋊Q812yes—
332C42C22×Q811yescharacteristic
333C42C22×Q811yescharacteristic
334C42C22×Q811yescharacteristic
335C42C2×#4111yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23#39512yes—
341C23#45312yes—
342C23#45612yes—
343C23C2×#12011yescharacteristic
344Q8—164no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—162no—
389C2×C4—162no—
390C2×C4—164no—
391C2×C4#39512yes—
392C2×C4#39512yes—
393C2×C4#39512yes—
394C2×C4#45412yes—
395C2×C4#45512yes—
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×C22⋊Q811yescharacteristic
398C2×C4C2×C22⋊Q811yescharacteristic
399C2×C4C2×C4⋊Q811yescharacteristic
400C2×C4C2×C4⋊Q811yescharacteristic
401C2×C4C2×C4⋊Q811yescharacteristic
402C8—28no—
403C8—44no—
404C8—44no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22#206014yes—
412C22C2×#39511yescharacteristic
413C22C2×#45311yescharacteristic
414C22C2×#45511yescharacteristic
415C22C2×#45411yescharacteristic
416C22C2×#45611yescharacteristic
417C22#302111yescharacteristic
418C22#302211yescharacteristic
419C4—21no—
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—82no—
428C4—82no—
429C4—81no—
430C4—82no—
431C4—81no—
432C4—81no—
433C4—82no—
434C4—81no—
435C4—162no—
436C4#206114yes—
437C2—24no—
438C2#1901511yescharacteristic
439C2#1901611yescharacteristic
440C2#1903211yescharacteristic
441C2#1903511yescharacteristic
442C2C2×#206011yescharacteristic
443C2C2×#206111yescharacteristic
444C2#2356211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10585C2semidirect
6. #10584C2semidirect
60. #432Q8semidirect
69. C4×M16Q8semidirect
72. C2×#97Q8semidirect

Maximal subgroups

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

subgroupindex
#375612
#219872
#208002
C2×#18362
#105852
#105842
#80342
#80332
#80292