Finite Group Databasebeta

← #298723

Subgroups — #298723

Order 512.

1229 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic, complemented
2#33042C211yesmaximal, characteristic
3#25902C211yesmaximal, characteristic, complemented
4#25813C211yesmaximal, characteristic, complemented
5#25718C211yesmaximal, characteristic
6#25718C211yesmaximal, characteristic
7C2×#1714C212yesmaximal, complemented
8#10315C212yesmaximal, complemented
9#10314C212yesmaximal, complemented
10#9828C212yesmaximal, complemented
11#8145C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#500C2212yes—
15C2×#498—28no—
16C2×#498—22no—
17C2×#498C2212yes—
18C2×#498C2212yes—
19C2×#492C2212yes—
20C2×#492C2212yes—
21C2×#487C2212yes—
22C2×#486C2211yescharacteristic
23C2×#448—22no—
24C2×#448—22no—
25C22×#106—22no—
26C22×C4×C8C2211yescharacteristic
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2057—22no—
30#2024—22no—
31#2013—22no—
32#2013—22no—
33#1990C2211yescharacteristic
34#1990C2212yes—
35#1990C2212yes—
36#1957C2211yescharacteristic
37#1956C2212yes—
38#1956C2212yes—
39#1934—22no—
40#1893—22no—
41#1872—22no—
42#1852—22no—
43#1834C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46#1714—28no—
47C2×#322C2212yes—
48#1273C2212yes—
49C23×Q8—21no—
50C23×Q8—21no—
51C22×SD16—24no—
52C22×SD16—24no—
53C23×C8D412yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—42no—
63C2×C4⋊Q8—42no—
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#127C2311yescharacteristic
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C42C2311yescharacteristic
77#500—24no—
78#500—42no—
79#498—216no—
80#498—24no—
81#498—24no—
82#498—24no—
83#498D418yescomplemented
84#492—24no—
85#492—24no—
86#492—42no—
87#492—42no—
88#487—24no—
89#487—42no—
90#486—24no—
91#448—48no—
92#448—48no—
93C2×#107—24no—
94C2×#106—216no—
95C2×#106—24no—
96C2×#106—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#102—22no—
101C2×#102—22no—
102C2×C22⋊C8D412yes—
103C2×C22⋊C8D412yes—
104C2×C4×C8—24no—
105C2×C4×C8D412yes—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2312yes—
108C2×C4×C8C2312yes—
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#390—21no—
112#390—21no—
113#390—21no—
114#389—21no—
115#389—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#380—21no—
119#380—21no—
120#340—24no—
121#340—22no—
122#340—24no—
123#340—22no—
124#336D412yes—
125#336D412yes—
126#322—24no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—44no—
131C22×Q8—44no—
132C22×Q8—41no—
133C22×Q8—41no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—82no—
137C22×Q8—82no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×SD16—48no—
148C2×SD16—48no—
149C2×SD16—48no—
150C2×SD16—48no—
151C2×SD16—84no—
152C2×SD16—84no—
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—44no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173C4⋊Q8—44no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8C2×D414yes—
177C4⋊Q8C2×D414yes—
178#127—22no—
179#127—22no—
180C22⋊Q8—44no—
181#121—44no—
182#121—44no—
183#121—44no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
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—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2411yescharacteristic
217#107—48no—
218#106—216no—
219#106—216no—
220#103—48no—
221#103—48no—
222#103—48no—
223#102—44no—
224#102—44no—
225C22⋊C8—24no—
226C22⋊C8SD1618yescomplemented
227C4×C8—24no—
228C4×C8—22no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—22no—
232C4×C8—22no—
233C4×C8C2×D414yes—
234#95—41no—
235#95—41no—
236C24—41no—
237C24#12711yescharacteristic
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—42no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—42no—
251C2×Q8—816no—
252C2×Q8—816no—
253C2×Q8—82no—
254C2×Q8—82no—
255C2×Q8—82no—
256C2×Q8—82no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4#12711yescharacteristic
289C22×C4#12711yescharacteristic
290C22×C4#12711yescharacteristic
291C22×C4C2×SD1614yes—
292C22×C4AGL(1,ℤ/8)14yes—
293C22×C4#13714yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297SD16—816no—
298SD16—816no—
299C2×C8—216no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8#12014yes—
310C2×C8C2×SD1618yes—
311C2×C8#13714yes—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C42—21no—
339C42—21no—
340C42—22no—
341C42—22no—
342C42C22×D412yes—
343C42H(5,𝔽2)12yes—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23#49212yes—
356C23#49612yes—
357C23#49712yes—
358C23C2×#12711yescharacteristic
359Q8—48no—
360Q8—48no—
361Q8—88no—
362Q8—88no—
363Q8—84no—
364Q8—84no—
365Q8—84no—
366D4—82no—
367D4—82no—
368D4—84no—
369D4—84no—
370D4—164no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—88no—
384C2×C4—88no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#49212yes—
399C2×C4#49612yes—
400C2×C4#49712yes—
401C2×C4C2×#12011yescharacteristic
402C2×C4#53511yescharacteristic
403C2×C4D4212yes—
404C2×C4D4212yes—
405C2×C4C22×SD1612yes—
406C2×C4C2×AGL(1,ℤ/8)12yes—
407C2×C4C2×#13712yes—
408C2×C4C2×#13712yes—
409C2×C4#57712yes—
410C2×C4#57712yes—
411C8—24no—
412C8—24no—
413C8—28no—
414C8—44no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22#171414yescomplemented
429C22#243311yescharacteristic
430C22C2×#49211yescharacteristic
431C22C2×#49611yescharacteristic
432C22C2×#49711yescharacteristic
433C22#315211yescharacteristic
434C22#315411yescharacteristic
435C22#315411yescharacteristic
436C4—22no—
437C4—22no—
438C4—22no—
439C4—84no—
440C4—84no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4#300412yes—
445C4#321812yes—
446C4#326514yes—
447C4D4×SD1614yes—
448C2—24no—
449C2—321no—
450C2C2×#171411yescharacteristic
451C2#1962211yescharacteristic
452C2#1967211yescharacteristic
453C2#1970211yescharacteristic
454C2#2572611yescharacteristic
455C2#2577411yescharacteristic
456C2#2578311yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
1. C22×#498C2semidirect
3. #25902C2semidirect
4. #25813C2semidirect
7. C2×#1714C2semidirect
8. #10315C2semidirect
9. #10314C2semidirect
10. #9828C2semidirect
11. #8145C2semidirect
83. #498D4semidirect
226. C22⋊C8SD16semidirect
428. C22#1714semidirect

Maximal subgroups

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

subgroupindex
C22×#4982
#330422
#259022
#258132
#257182
C2×#17142
#103152
#103142
#98282
#81452