Finite Group Databasebeta

← #316261

Subgroups — #316261

Order 512.

742 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27680C211yesmaximal, characteristic
2#24302C211yesmaximal, characteristic
3C8×C22⋊Q8C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5#10601C212yesmaximal, complemented
6#10483C212yesmaximal, complemented
7#9567C212yesmaximal, complemented
8#9216C212yesmaximal, complemented
9#8724C211yesmaximal, characteristic
10#8596C211yesmaximal, characteristic
11#8342C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#477C2212yes—
14C2×C8×Q8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C4×C22⋊Q8C2211yescharacteristic
17C2×#384C2211yescharacteristic
18#2099—21no—
19#2096—21no—
20#2057—21no—
21#2024—21no—
22#1990—21no—
23#1990—21no—
24#1990C2212yes—
25#1990C2212yes—
26#1987C2212yes—
27#1984C2211yescharacteristic
28#1948C2212yes—
29#1939C2211yescharacteristic
30#1916—21no—
31#1872—21no—
32#1872—21no—
33#1853—21no—
34#1852—21no—
35#1850—21no—
36#1803C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C8×C4⋊C4—21no—
41C8×C4⋊C4C2211yescharacteristic
42C8×C22⋊C4—21no—
43C2×#340—21no—
44C2×#340—21no—
45C2×#335C2212yes—
46C2×#333C2212yes—
47C2×#333C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×Q8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8C2312yes—
60C2×C22⋊Q8D411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C42C2311yescharacteristic
68#477—24no—
69#477—42no—
70C8×Q8—42no—
71C2×#107—22no—
72C2×#106—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×#103—22no—
80C2×#103—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—42no—
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#391D411yescharacteristic
92#389D411yescharacteristic
93#387—21no—
94#385—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384C2312yes—
99#380D411yescharacteristic
100C4×C4⋊C4—21no—
101C4×C4⋊C4C2311yescharacteristic
102C4×C22⋊C4—21no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#335—42no—
119#333—24no—
120#333—42no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—41no—
124C22×Q8—82no—
125C22×Q8—82no—
126C22×Q8C2×D411yescharacteristic
127C23×C4—41no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4#4111yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C4⋊Q8—24no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—44no—
151C4⋊Q8—44no—
152C4⋊Q8—42no—
153C4⋊Q8#4114yes—
154C22⋊Q8—22no—
155C22⋊Q8Q1614yes—
156C4×Q8—22no—
157C4×Q8—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42C2411yescharacteristic
191#107—44no—
192#106—82no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201#103—44no—
202#103—82no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—41no—
209C4×C8—41no—
210C4×C8—42no—
211C4×C8—42no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217C24#12111yescharacteristic
218C2×Q8—21no—
219C2×Q8—22no—
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—816no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8C2×Q1612yes—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4C2×Q1612yes—
255C22×C4#13512yes—
256C22×C4#13512yes—
257C22×C4#13712yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8#13712yes—
286C2×C8#13712yes—
287C4⋊C4—21no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4C2×Q1612yes—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C42—21no—
311C42—21no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42C2×#4112yes—
319C42H(5,𝔽2)12yes—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23#46412yes—
325C23#46712yes—
326C23#47112yes—
327C23C2×#12111yescharacteristic
328Q8—24no—
329Q8—42no—
330Q8—84no—
331Q8—84no—
332Q8—88no—
333C2×C4—28no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—88no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—164no—
368C2×C4#46412yes—
369C2×C4#46712yes—
370C2×C4#47212yes—
371C2×C4#52511yescharacteristic
372C2×C4#53411yescharacteristic
373C2×C4#54612yes—
374C2×C4#55112yes—
375C2×C4C22×Q1611yescharacteristic
376C2×C4C2×#13511yescharacteristic
377C2×C4C2×#13511yescharacteristic
378C2×C4C2×#13711yescharacteristic
379C2×C4C2×#13711yescharacteristic
380C2×C4C2×#13711yescharacteristic
381C2×C4#57712yes—
382C8—22no—
383C8—22no—
384C8—41no—
385C8—41no—
386C8—81no—
387C8—81no—
388C8—82no—
389C8—82no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22#168414yes—
397C22#253411yescharacteristic
398C22C2×#46411yescharacteristic
399C22#303911yescharacteristic
400C22C2×#46711yescharacteristic
401C22#305711yescharacteristic
402C22C2×#47111yescharacteristic
403C22C2×#47211yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—81no—
411C4—81no—
412C4—84no—
413C4—162no—
414C4#310811yescharacteristic
415C4#311311yescharacteristic
416C4#319011yescharacteristic
417C4#319511yescharacteristic
418C4#329912yes—
419C4#330112yes—
420C2—24no—
421C2C2×#168411yescharacteristic
422C2#1921211yescharacteristic
423C2#1927211yescharacteristic
424C2#1935711yescharacteristic
425C2#2809511yescharacteristic
426C2#2812411yescharacteristic
427C2#2813411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. #10601C2semidirect
6. #10483C2semidirect
7. #9567C2semidirect
8. #9216C2semidirect

Maximal subgroups

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

subgroupindex
#276802
#243022
C8×C22⋊Q82
C2×#19902
#106012
#104832
#95672
#92162
#87242
#85962
#83422