Finite Group Databasebeta

← #298720

Subgroups — #298720

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#494C211yesmaximal, characteristic
2#33043C211yesmaximal, characteristic
3#25902C211yesmaximal, characteristic
4#25813C211yesmaximal, characteristic
5#25813C211yesmaximal, characteristic
6#25811C211yesmaximal, characteristic
7C2×#1723C212yesmaximal, complemented
8#10318C212yesmaximal, complemented
9#10309C212yesmaximal, complemented
10#9835C212yesmaximal, complemented
11#8145C211yesmaximal, characteristic
12C23×Q16—22no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×#500C2212yes—
16C2×#500C2212yes—
17C2×#498C2212yes—
18C2×#494—28no—
19C2×#494—22no—
20C2×#494C2212yes—
21C2×#494C2212yes—
22C2×#487C2211yescharacteristic
23C2×#487C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2058—22no—
28#2057—22no—
29#2057—22no—
30#2024—22no—
31#1990C2211yescharacteristic
32#1990C2212yes—
33#1990C2212yes—
34#1990C2212yes—
35#1990C2212yes—
36#1956C2211yescharacteristic
37#1933—22no—
38#1892—22no—
39#1872—22no—
40#1853—22no—
41#1852—22no—
42#1852—22no—
43#1834C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46#1723—28no—
47C2×#322C2212yes—
48#1273C2212yes—
49C23×Q8—21no—
50C23×Q8—21no—
51C22×Q16—216no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16—48no—
55C22×Q16—48no—
56C23×C8D412yes—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—42no—
66C2×C4⋊Q8—42no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2312yes—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C42C2311yescharacteristic
78#500—24no—
79#500—24no—
80#500—42no—
81#500—42no—
82#498—24no—
83#498—42no—
84#494—216no—
85#494—24no—
86#494—24no—
87#494—24no—
88#494D418yes—
89#487—24no—
90#487—24no—
91#487—42no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#106—24no—
95C2×#103—24no—
96C2×#103—22no—
97C2×#103—22no—
98C2×C22⋊C8D412yes—
99C2×C22⋊C8D412yes—
100C2×C4×C8—24no—
101C2×C4×C8D412yes—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2312yes—
104C2×C4×C8C2312yes—
105C2×C4×C8C2311yescharacteristic
106#391—21no—
107#391—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—22no—
122#340—24no—
123#340—22no—
124#336D412yes—
125#336D412yes—
126#322—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—44no—
140C22×Q8—44no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×Q16—216no—
147C2×Q16—216no—
148C2×Q16—464no—
149C2×Q16—48no—
150C2×Q16—48no—
151C2×Q16—48no—
152C2×Q16—48no—
153C22×C8—24no—
154C22×C8—24no—
155C22×C8—24no—
156C22×C8C2×D414yes—
157C22×C8C2×D412yes—
158C22×C8C2×D414yes—
159C22×C8C2×D412yes—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—44no—
173C4⋊Q8—44no—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8C2×D414yes—
179C4⋊Q8C2×D414yes—
180C22⋊Q8—44no—
181C22⋊Q8—44no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2411yescharacteristic
214#107—48no—
215#107—48no—
216#106—48no—
217#103—48no—
218#103—44no—
219#103—44no—
220C22⋊C8—24no—
221C22⋊C8Q1618yescomplemented
222C4×C8—24no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8C2×D414yes—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233C24#12711yescharacteristic
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—416no—
239C2×Q8—416no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—82no—
254C2×Q8—82no—
255C2×Q8—88no—
256C2×Q8—88no—
257C2×Q8—82no—
258C2×Q8—82no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4#12711yescharacteristic
279C22×C4#12711yescharacteristic
280C22×C4#12711yescharacteristic
281C22×C4C2×Q1614yes—
282C22×C4AGL(1,ℤ/8)14yes—
283C22×C4#13714yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287Q16—432no—
288Q16—432no—
289C2×C8—216no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8#12014yes—
300C2×C8C2×Q1618yescomplemented
301C2×C8AGL(1,ℤ/8)14yes—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C42—21no—
330C42—21no—
331C42—22no—
332C42—22no—
333C42C22×D412yes—
334C42H(5,𝔽2)12yes—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#49412yes—
340C23#49612yes—
341C23#49712yes—
342C23C2×#12711yescharacteristic
343Q8—48no—
344Q8—48no—
345Q8—88no—
346Q8—88no—
347Q8—84no—
348Q8—84no—
349Q8—82no—
350Q8—82no—
351Q8—84no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—88no—
365C2×C4—88no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4#49412yes—
383C2×C4#49612yes—
384C2×C4#49712yes—
385C2×C4C2×#12011yescharacteristic
386C2×C4#53511yescharacteristic
387C2×C4D4212yes—
388C2×C4D4212yes—
389C2×C4C22×Q1612yes—
390C2×C4C2×AGL(1,ℤ/8)12yes—
391C2×C4C2×AGL(1,ℤ/8)12yes—
392C2×C4C2×#13712yes—
393C2×C4#57812yes—
394C2×C4#57812yes—
395C8—24no—
396C8—24no—
397C8—28no—
398C8—44no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22#172314yescomplemented
406C22#243311yescharacteristic
407C22C2×#49411yescharacteristic
408C22C2×#49611yescharacteristic
409C22C2×#49711yescharacteristic
410C22#315111yescharacteristic
411C22#315611yescharacteristic
412C22#315611yescharacteristic
413C4—22no—
414C4—22no—
415C4—22no—
416C4—84no—
417C4—84no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#300312yes—
423C4#321712yes—
424C4#327014yes—
425C4D4×Q1614yes—
426C2—24no—
427C2C2×#172311yescharacteristic
428C2#1963211yescharacteristic
429C2#1968411yescharacteristic
430C2#1969911yescharacteristic
431C2#2572511yescharacteristic
432C2#2578011yescharacteristic
433C2#2578911yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
7. C2×#1723C2semidirect
8. #10318C2semidirect
9. #10309C2semidirect
10. #9835C2semidirect
221. C22⋊C8Q16semidirect
300. C2×C8C2×Q16semidirect
405. C22#1723semidirect

Maximal subgroups

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

subgroupindex
C22×#4942
#330432
#259022
#258132
#258112
C2×#17232
#103182
#103092
#98352
#81452