Finite Group Databasebeta

← #298654

Subgroups — #298654

Order 512.

1094 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic
2#33043C211yesmaximal, characteristic
3#25902C211yesmaximal, characteristic
4#25856C211yesmaximal, characteristic
5#25813C211yesmaximal, characteristic
6#25813C211yesmaximal, characteristic
7C2×#1710C212yesmaximal, complemented
8#10313C212yesmaximal, complemented
9#10313C212yesmaximal, complemented
10#9821C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#500C2212yes—
15C2×#500C2212yes—
16C2×#487—28no—
17C2×#487—22no—
18C2×#487C2211yescharacteristic
19C2×#487C2212yes—
20C2×#487C2212yes—
21C2×#487C2212yes—
22C2×#487C2212yes—
23C22×#103—22no—
24C22×C4×C8C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2057—22no—
28#2057—22no—
29#2043—22no—
30#2024—22no—
31#1990C2211yescharacteristic
32#1990C2212yes—
33#1990C2212yes—
34#1990C2212yes—
35#1956C2212yes—
36#1956C2211yescharacteristic
37#1933—22no—
38#1892—22no—
39#1872—22no—
40#1852—22no—
41#1852—22no—
42#1834C2211yescharacteristic
43#1787—22no—
44C4×C22⋊C8C2211yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46#1710—28no—
47C2×C82C2212yes—
48#1273C2212yes—
49C23×Q8—21no—
50C23×Q8—21no—
51C23×C8D412yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C42C2311yescharacteristic
73#500—24no—
74#500—24no—
75#500—42no—
76#500—42no—
77#487—216no—
78#487—24no—
79#487—24no—
80#487—24no—
81#487—24no—
82#487—24no—
83#487—24no—
84#487—42no—
85#487—42no—
86#487D418yes—
87C2×#107—24no—
88C2×#107—24no—
89C2×#103—216no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103—24no—
95C2×#103—22no—
96C2×#103—24no—
97C2×C22⋊C8D412yes—
98C2×C22⋊C8D412yes—
99C2×C4×C8—24no—
100C2×C4×C8D412yes—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2312yes—
103C2×C4×C8C2312yes—
104C2×C4×C8C2311yescharacteristic
105#391—21no—
106#391—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#340—24no—
118#340—24no—
119#340—22no—
120#340—24no—
121#340—22no—
122#336D412yes—
123#336D412yes—
124C82—24no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—44no—
131C22×Q8—42no—
132C22×Q8—44no—
133C22×Q8—41no—
134C22×Q8—41no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—82no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C22×C8—24no—
144C22×C8—24no—
145C22×C8—42no—
146C22×C8C2×D412yes—
147C22×C8C2×D414yes—
148C22×C8C2×D412yes—
149C22×C8C2×D414yes—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165C4⋊Q8—44no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8C2×D414yes—
169C4⋊Q8C2×D414yes—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2411yescharacteristic
205#107—48no—
206#107—48no—
207#103—216no—
208#103—216no—
209#103—48no—
210#103—48no—
211#103—44no—
212#103—44no—
213#103—84no—
214C22⋊C8—24no—
215C22⋊C8Q1618yes—
216C4×C8—24no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8C2×D414yes—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227C24#12711yescharacteristic
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—416no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—42no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—816no—
247C2×Q8—82no—
248C2×Q8—88no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×Q8—82no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4#12711yescharacteristic
272C22×C4#12711yescharacteristic
273C22×C4#12711yescharacteristic
274C22×C4C2×Q1614yes—
275C22×C4#13514yes—
276C22×C4#13514yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C2×C8—24no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—48no—
291C2×C8#12014yes—
292C2×C8C2×Q1618yes—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C42—21no—
321C42—21no—
322C42—22no—
323C42—22no—
324C42C22×D412yes—
325C42H(5,𝔽2)12yes—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23#49412yes—
331C23#49512yes—
332C23#49512yes—
333C23C2×#12711yescharacteristic
334Q8—48no—
335Q8—48no—
336Q8—88no—
337Q8—88no—
338Q8—84no—
339Q8—84no—
340Q8—82no—
341Q8—82no—
342Q8—84no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—88no—
356C2×C4—88no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4#49412yes—
374C2×C4#49512yes—
375C2×C4#49512yes—
376C2×C4C2×#12011yescharacteristic
377C2×C4#53511yescharacteristic
378C2×C4D4212yes—
379C2×C4D4212yes—
380C2×C4C22×Q1612yes—
381C2×C4C2×#13512yes—
382C2×C4C2×#13512yes—
383C2×C4#57512yes—
384C2×C4#57712yes—
385C2×C4#57812yes—
386C8—24no—
387C8—28no—
388C8—42no—
389C8—44no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22#171014yescomplemented
397C22#243311yescharacteristic
398C22C2×#49411yescharacteristic
399C22C2×#49511yescharacteristic
400C22C2×#49511yescharacteristic
401C22#315311yescharacteristic
402C22#315411yescharacteristic
403C22#315611yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—84no—
408C4—84no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4—161no—
413C4#300212yes—
414C4#321112yes—
415C4#328214yes—
416C4D4×Q1614yes—
417C2—24no—
418C2C2×#171011yescharacteristic
419C2#1962811yescharacteristic
420C2#1963411yescharacteristic
421C2#1965711yescharacteristic
422C2#2572411yescharacteristic
423C2#2577611yescharacteristic
424C2#2578011yescharacteristic
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×#1710C2semidirect
8. #10313C2semidirect
9. #10313C2semidirect
10. #9821C2semidirect
396. C22#1710semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
#330432
#259022
#258562
#258132
C2×#17102
#103132
#98212
C8×C22⋊C82