Finite Group Databasebeta

← #298658

Subgroups — #298658

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic
2#33043C211yesmaximal, characteristic
3#25813C211yesmaximal, characteristic
4#25811C211yesmaximal, characteristic
5#25811C211yesmaximal, characteristic
6#25722C211yesmaximal, characteristic
7C2×#1707C212yesmaximal, complemented
8#10318C212yesmaximal, complemented
9#10314C212yesmaximal, complemented
10#9821C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#500C2212yes—
15C2×#498C2212yes—
16C2×#498C2212yes—
17C2×#494C2212yes—
18C2×#487—28no—
19C2×#487—22no—
20C2×#487C2211yescharacteristic
21C2×#487C2212yes—
22C2×#487C2212yes—
23C2×#451—22no—
24C22×#103—22no—
25C22×C4×C8C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2058—22no—
29#2058—22no—
30#2057—22no—
31#2016—22no—
32#1990C2211yescharacteristic
33#1990C2212yes—
34#1990C2212yes—
35#1990C2212yes—
36#1956C2211yescharacteristic
37#1956C2212yes—
38#1933—22no—
39#1892—22no—
40#1853—22no—
41#1853—22no—
42#1852—22no—
43#1834C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46#1707—28no—
47C2×C82C2212yes—
48#1273C2212yes—
49C23×Q8—21no—
50C23×Q8—21no—
51C22×Q16—24no—
52C23×C8D412yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8—42no—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#500—24no—
75#500—42no—
76#498—24no—
77#498—24no—
78#498—42no—
79#498—42no—
80#494—24no—
81#494—42no—
82#487—216no—
83#487—24no—
84#487—24no—
85#487—24no—
86#487—24no—
87#487D418yes—
88#451—48no—
89C2×#107—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#103—216no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—24no—
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—22no—
123#336D412yes—
124#336D412yes—
125C82—24no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—44no—
132C22×Q8—42no—
133C22×Q8—44no—
134C22×Q8—41no—
135C22×Q8—41no—
136C22×Q8—44no—
137C22×Q8—44no—
138C22×Q8—82no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×Q16—48no—
145C2×Q16—48no—
146C2×Q16—84no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—42no—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D414yes—
153C22×C8C2×D414yes—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8C2×D414yes—
173C4⋊Q8C2×D414yes—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
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—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209#107—48no—
210#106—48no—
211#106—48no—
212#103—216no—
213#103—216no—
214#103—44no—
215#103—44no—
216#103—48no—
217C22⋊C8SD1618yes—
218C22⋊C8Q1618yescomplemented
219C4×C8—24no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8C2×D414yes—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230C24#12711yescharacteristic
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—416no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—44no—
244C2×Q8—42no—
245C2×Q8—44no—
246C2×Q8—42no—
247C2×Q8—44no—
248C2×Q8—42no—
249C2×Q8—816no—
250C2×Q8—82no—
251C2×Q8—88no—
252C2×Q8—82no—
253C2×Q8—82no—
254C2×Q8—82no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4#12711yescharacteristic
275C22×C4#12711yescharacteristic
276C22×C4#12711yescharacteristic
277C22×C4C2×SD1614yes—
278C22×C4C2×Q1614yes—
279C22×C4#13514yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283Q16—816no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—48no—
294C2×C8#12014yes—
295C2×C8C2×SD1618yes—
296C2×C8C2×Q1618yes—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—21no—
325C42—21no—
326C42—22no—
327C42—22no—
328C42C22×D412yes—
329C42H(5,𝔽2)12yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#49212yes—
335C23#49412yes—
336C23#49512yes—
337C23C2×#12711yescharacteristic
338Q8—48no—
339Q8—48no—
340Q8—88no—
341Q8—88no—
342Q8—84no—
343Q8—84no—
344Q8—82no—
345Q8—82no—
346Q8—84no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—88no—
360C2×C4—88no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
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—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4#49212yes—
378C2×C4#49412yes—
379C2×C4#49512yes—
380C2×C4C2×#12011yescharacteristic
381C2×C4#53511yescharacteristic
382C2×C4D4212yes—
383C2×C4D4212yes—
384C2×C4C22×SD1612yes—
385C2×C4C22×Q1612yes—
386C2×C4C2×#13512yes—
387C2×C4#57512yes—
388C2×C4#57712yes—
389C2×C4#57812yes—
390C8—24no—
391C8—28no—
392C8—28no—
393C8—42no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22#170714yescomplemented
401C22#243311yescharacteristic
402C22C2×#49211yescharacteristic
403C22C2×#49411yescharacteristic
404C22C2×#49511yescharacteristic
405C22#315311yescharacteristic
406C22#315411yescharacteristic
407C22#315611yescharacteristic
408C4—22no—
409C4—22no—
410C4—22no—
411C4—84no—
412C4—84no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4#300212yes—
418C4#321112yes—
419C4D4×SD1614yes—
420C4D4×Q1614yes—
421C2—24no—
422C2C2×#170711yescharacteristic
423C2#1962411yescharacteristic
424C2#1963411yescharacteristic
425C2#1965911yescharacteristic
426C2#2572411yescharacteristic
427C2#2577411yescharacteristic
428C2#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×#1707C2semidirect
8. #10318C2semidirect
9. #10314C2semidirect
10. #9821C2semidirect
218. C22⋊C8Q16semidirect
400. C22#1707semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
#330432
#258132
#258112
#257222
C2×#17072
#103182
#103142
#98212
C8×C22⋊C82