Finite Group Databasebeta

← #241300

Subgroups — #241300

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5#10578C214yesmaximal, complemented
6#10576C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#428—21no—
13C2×#423C2211yescharacteristic
14#2571—22no—
15#2571—22no—
16#2571—22no—
17#2571C2214yes—
18C2×#400—22no—
19C2×#389—21no—
20C2×#389—21no—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382—21no—
24C2×#382—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1949—22no—
28#1948C2212yes—
29#1948C2212yes—
30#1948C2212yes—
31#1947—22no—
32#1947C2212yes—
33#1853—21no—
34#1853—21no—
35#1852—22no—
36#1845C2212yes—
37#1834C2212yes—
38C4×#106—22no—
39#1736—21no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×M16—21no—
44C2×#117—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D412yes—
53C22×C42—21no—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#431—22no—
57#431—22no—
58#431Q814yes—
59#431Q814yes—
60#428—44no—
61C2×#107—22no—
62C2×#107—24no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—22no—
66#423—22no—
67#423Q814yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8C2312yes—
76#400—28no—
77#400—28no—
78#389—24no—
79#389—24no—
80#389—24no—
81#389—24no—
82#384—21no—
83#384—21no—
84#384D412yes—
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#384C2312yes—
88#382—24no—
89#382—24no—
90#382—24no—
91#382—24no—
92#382—21no—
93#380—21no—
94#380—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4C2312yes—
104C4×C4⋊C4C2312yes—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×M16—24no—
124C2×M16—42no—
125C2×M16—42no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129#117—42no—
130#117—42no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—84no—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D414yes—
163C2×C22⋊C4—41no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—44no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—41no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×Q812yes—
176C2×C42C2×Q812yes—
177C2×C42C2×Q812yes—
178C2×C42#4112yes—
179C2×C42C2411yescharacteristic
180#107—44no—
181#107—84no—
182#106—44no—
183#106—44no—
184#106—44no—
185C4⋊C8—24no—
186C4⋊C8—24no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8C2×Q814yes—
190C4⋊C8C2×Q814yes—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C4×C8—24no—
194C4×C8—22no—
195C4×C8C2×Q814yes—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—41no—
204#95—41no—
205C24#12011yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4#12011yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4C22⋊Q812yes—
236C22×C4C22⋊Q812yes—
237C22×C4C22⋊Q812yes—
238C22×C4C4⋊Q812yes—
239C22×C4C4⋊Q812yes—
240C22×C4C4⋊Q812yes—
241C22×C4#13512yes—
242C22×C4#13512yes—
243C22×C4AGL(1,ℤ/8)12yes—
244C22×C4#13712yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248M16—48no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—82no—
285C4⋊C4—88no—
286C22⋊C4—82no—
287C42—44no—
288C42—42no—
289C42—42no—
290C42—44no—
291C42—44no—
292C42—82no—
293C42C22⋊Q812yes—
294C42C22⋊Q812yes—
295C42C22⋊Q812yes—
296C42C4⋊Q812yes—
297C42C4⋊Q812yes—
298C42C4⋊Q812yes—
299C42C22×Q811yescharacteristic
300C42C22×Q811yescharacteristic
301C42C22×Q811yescharacteristic
302C42C2×#4111yescharacteristic
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23#39512yes—
308C23#44912yes—
309C23#45212yes—
310C23C2×#12011yescharacteristic
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—88no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4#39512yes—
348C2×C4#39512yes—
349C2×C4#39512yes—
350C2×C4#44912yes—
351C2×C4#45212yes—
352C2×C4#47814yes—
353C2×C4#47914yes—
354C2×C4#49914yes—
355C2×C4#50114yes—
356C2×C4C2×C22⋊Q811yescharacteristic
357C2×C4C2×C22⋊Q811yescharacteristic
358C2×C4C2×C22⋊Q811yescharacteristic
359C2×C4C2×C4⋊Q811yescharacteristic
360C2×C4C2×C4⋊Q811yescharacteristic
361C2×C4C2×C4⋊Q811yescharacteristic
362C2×C4C2×#13511yescharacteristic
363C2×C4C2×#13511yescharacteristic
364C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
365C2×C4C2×#13711yescharacteristic
366C2×C4#57512yes—
367C2×C4#57512yes—
368C8—44no—
369C8—44no—
370C8—44no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22#205914yes—
378C22C2×#39511yescharacteristic
379C22C2×#44911yescharacteristic
380C22C2×#45211yescharacteristic
381C22#300511yescharacteristic
382C22#300511yescharacteristic
383C22#301911yescharacteristic
384C22#302011yescharacteristic
385C4—22no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—84no—
390C4—82no—
391C4—82no—
392C4—82no—
393C4—82no—
394C4—161no—
395C4—161no—
396C4#205914yes—
397C4C2×#47811yescharacteristic
398C4C2×#47911yescharacteristic
399C4#308012yes—
400C4C2×#49911yescharacteristic
401C4C2×#50111yescharacteristic
402C4#316612yes—
403C2—24no—
404C2#1875011yescharacteristic
405C2#1876311yescharacteristic
406C2#1877311yescharacteristic
407C2#1878011yescharacteristic
408C2C2×#205911yescharacteristic
409C2#2355311yescharacteristic
410C2#2356111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #10578C2semidirect
6. #10576C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
#219822
#213592
#211702
#105782
#105762
#80432
#79012
#78922