Finite Group Databasebeta

← #298671

Subgroups — #298671

Order 512.

1221 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#500C211yesmaximal, characteristic, complemented
2#33170C211yesmaximal, characteristic
3#25903C211yesmaximal, characteristic
4#25856C211yesmaximal, characteristic, complemented
5#25811C211yesmaximal, characteristic, complemented
6#25718C211yesmaximal, characteristic
7C2×#1714C212yesmaximal, complemented
8#10314C212yesmaximal, complemented
9#10308C212yesmaximal, complemented
10#9836C212yesmaximal, complemented
11#8144C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#500—28no—
15C2×#500—22no—
16C2×#500C2212yes—
17C2×#500C2212yes—
18C2×#498C2212yes—
19C2×#492C2211yescharacteristic
20C2×#492C2212yes—
21C2×#487C2212yes—
22C2×#486C2212yes—
23C2×#465—22no—
24C2×#448—22no—
25C22×#107—22no—
26C22×C4×C8C2211yescharacteristic
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2058—22no—
30#2043—22no—
31#2023—22no—
32#2013—22no—
33#1990C2212yes—
34#1990C2212yes—
35#1968—22no—
36#1957C2211yescharacteristic
37#1956C2212yes—
38#1956C2211yescharacteristic
39#1956C2212yes—
40#1874—22no—
41#1853—22no—
42#1844C2211yescharacteristic
43#1787—22no—
44C4×C22⋊C8C2211yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46#1714—28no—
47C2×#322C2212yes—
48#1273C2212yes—
49C23×Q8—21no—
50C23×Q8—21no—
51C22×SD16—22no—
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—216no—
78#500—24no—
79#500—24no—
80#500—24no—
81#500D418yescomplemented
82#498—24no—
83#498—42no—
84#492—24no—
85#492—24no—
86#492—42no—
87#487—24no—
88#487—42no—
89#486—24no—
90#486—42no—
91#465—48no—
92#448—48no—
93C2×#107—216no—
94C2×#107—24no—
95C2×#107—24no—
96C2×#106—24no—
97C2×#106—22no—
98C2×#103—24no—
99C2×#103—22no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#102—24no—
103C2×#102—22no—
104C2×C22⋊C8D412yes—
105C2×C22⋊C8D412yes—
106C2×C4×C8—24no—
107C2×C4×C8D412yes—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2312yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2311yescharacteristic
112#391—21no—
113#390—21no—
114#390—21no—
115#390—21no—
116#389—21no—
117#389—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#380—21no—
121#380—21no—
122#340—24no—
123#340—24no—
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—44no—
150C2×SD16—44no—
151C2×SD16—44no—
152C2×SD16—84no—
153C22×C8—24no—
154C22×C8—24no—
155C22×C8SD1618yescomplemented
156C22×C8C2×D414yes—
157C22×C8C2×D414yes—
158C22×C8C2×D412yes—
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—216no—
218#107—216no—
219#106—48no—
220#106—44no—
221#103—48no—
222#103—48no—
223#103—44no—
224#103—84no—
225#102—48no—
226#102—44no—
227C22⋊C8—24no—
228C22⋊C8—24no—
229C4×C8—24no—
230C4×C8—22no—
231C4×C8—22no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8C2×D414yes—
236#95—41no—
237#95—41no—
238C24—41no—
239C24#12711yescharacteristic
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—42no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—816no—
255C2×Q8—816no—
256C2×Q8—82no—
257C2×Q8—82no—
258C2×Q8—82no—
259C2×Q8—82no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4#12711yescharacteristic
292C22×C4#12711yescharacteristic
293C22×C4#12711yescharacteristic
294C22×C4C2×SD1614yes—
295C22×C4AGL(1,ℤ/8)14yes—
296C22×C4#13714yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300SD16—816no—
301SD16—88no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—216no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8#12014yes—
312C2×C8C2×SD1614yes—
313C2×C8C2×SD1614yes—
314C2×C8#13714yes—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—21no—
341C42—21no—
342C42—22no—
343C42—22no—
344C42C22×D412yes—
345C42H(5,𝔽2)12yes—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23#49212yes—
358C23#49612yes—
359C23#49712yes—
360C23C2×#12711yescharacteristic
361Q8—48no—
362Q8—48no—
363Q8—88no—
364Q8—88no—
365Q8—84no—
366Q8—84no—
367Q8—82no—
368Q8—82no—
369D4—82no—
370D4—82no—
371D4—84no—
372D4—84no—
373D4—164no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—88no—
387C2×C4—88no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4#49212yes—
402C2×C4#49612yes—
403C2×C4#49712yes—
404C2×C4C2×#12011yescharacteristic
405C2×C4#53511yescharacteristic
406C2×C4D4212yes—
407C2×C4D4212yes—
408C2×C4C22×SD1612yes—
409C2×C4C2×AGL(1,ℤ/8)12yes—
410C2×C4C2×#13712yes—
411C2×C4C2×#13712yes—
412C2×C4#57612yes—
413C2×C4#57812yes—
414C8—24no—
415C8—44no—
416C8—44no—
417C8#44818yescomplemented
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22#171414yescomplemented
432C22#243311yescharacteristic
433C22C2×#49211yescharacteristic
434C22C2×#49611yescharacteristic
435C22C2×#49711yescharacteristic
436C22#315211yescharacteristic
437C22#315511yescharacteristic
438C22#315611yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—84no—
443C4—84no—
444C4—161no—
445C4—161no—
446C4—161no—
447C4C2×#44812yes—
448C4#320612yes—
449C4#326414yes—
450C4#327014yes—
451C2—24no—
452C2—321no—
453C2C2×#171411yescharacteristic
454C2#1965511yescharacteristic
455C2#1967511yescharacteristic
456C2#1968311yescharacteristic
457C2#2571811yescharacteristic
458C2#2578811yescharacteristic
459C2#2578911yescharacteristic
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×#500C2semidirect
4. #25856C2semidirect
5. #25811C2semidirect
7. C2×#1714C2semidirect
8. #10314C2semidirect
9. #10308C2semidirect
10. #9836C2semidirect
11. #8144C2semidirect
81. #500D4semidirect
155. C22×C8SD16semidirect
417. C8#448semidirect
431. C22#1714semidirect

Maximal subgroups

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

subgroupindex
C22×#5002
#331702
#259032
#258562
#258112
#257182
C2×#17142
#103142
#103082
#98362
#81442