Finite Group Databasebeta

← #386851

Subgroups — #386851

Order 512.

816 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21175C212yesmaximal
2C2×#1945C211yesmaximal, characteristic
3#21039C212yesmaximal
4#11339C212yesmaximal
5#11249C212yesmaximal
6#11242C212yesmaximal
7#10892C212yesmaximal
8#10859C211yesmaximal, characteristic
9#10842C211yesmaximal, characteristic
10C22×#108C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C2×#428C2212yes—
13C2×#427C2211yescharacteristic
14C2×#427C2211yescharacteristic
15C2×#424C2212yes—
16C22×C4×C8C2211yescharacteristic
17#2317C2212yes—
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20#2274C2212yes—
21#1949—22no—
22#1949—22no—
23#1948—22no—
24#1947—22no—
25#1945C418yescomplemented
26#1921—22no—
27#1920—22no—
28#1919—22no—
29#1918—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32#1736C2212yes—
33#1736C2212yes—
34#1736C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39#1729C2212yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C2×#117C2312yes—
45C2×#117C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C42C2311yescharacteristic
49C2×#108—22no—
50C2×#108—41no—
51C2×#108—41no—
52C2×#108—41no—
53C2×#108—41no—
54C2×#108C2×C414yes—
55#428—42no—
56#428—42no—
57#427—41no—
58#427—41no—
59#427C2×C414yes—
60#427Q814yes—
61C2×#107—22no—
62C2×#107D412yes—
63C2×#107D411yescharacteristic
64C2×#107D411yescharacteristic
65C2×#106—22no—
66C2×#106D411yescharacteristic
67C2×#106D411yescharacteristic
68C2×#106D412yes—
69#424—24no—
70#424C2×C418yes—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4×C8—41no—
74C2×C4×C8—41no—
75C2×C4×C8C2×C414yes—
76#389—22no—
77#389—22no—
78#385—42no—
79#384—22no—
80#382—22no—
81#382—22no—
82#382—41no—
83#382—41no—
84#382Q814yes—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C22⋊C4—42no—
90C2×#95—41no—
91C2×#95—41no—
92#343—42no—
93#343—42no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#339—24no—
105#339C2×C418yes—
106#336—22no—
107#336—22no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C2411yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16C22×C414yes—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8C22×C412yes—
135C22×C8C22×C412yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C22×C8C2×Q812yes—
141C22×C8#4112yes—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117C22×C414yes—
149#117C2×Q814yes—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×Q814yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42C22×C412yes—
178C2×C42C2×Q812yes—
179C2×C42#4112yes—
180#108—44no—
181#108—44no—
182#107—42no—
183#107—44no—
184#107C4⋊C414yes—
185#107C4⋊C414yes—
186#106—41no—
187#106—41no—
188#106—44no—
189#106C4⋊C418yes—
190C4⋊C8—28no—
191C4⋊C8—28no—
192C4×C8—44no—
193#95—44no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—42no—
198#95—44no—
199C24H(5,𝔽2)11yescharacteristic
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4C23×C411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C22×Q811yescharacteristic
223C22×C4C22×Q811yescharacteristic
224C22×C4C2×#4111yescharacteristic
225C22×C4C2×#4111yescharacteristic
226C22×C4H(5,𝔽2)11yescharacteristic
227C22×C4H(5,𝔽2)11yescharacteristic
228C22×C4H(5,𝔽2)11yescharacteristic
229C22×C42−1+411yescharacteristic
230C22×C42−1+411yescharacteristic
231C22×C42−1+411yescharacteristic
232C22×C42−1+411yescharacteristic
233M16—24no—
234M16—24no—
235M16—42no—
236M16—42no—
237M16—42no—
238M16—42no—
239M16—42no—
240M16—42no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—24no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8C2×C4⋊C412yes—
264C2×C8C2×C4⋊C412yes—
265C2×C8C4×D412yes—
266C2×C8C4×D412yes—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—162no—
282C4⋊C4—162no—
283C4⋊C4C2×C4⋊C414yes—
284C4⋊C4C2×C4⋊C414yes—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—24no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—24no—
294C42—42no—
295C42—42no—
296C42—162no—
297C42—162no—
298C42C4×Q814yes—
299C23—22no—
300C23—22no—
301C23#52011yescharacteristic
302C23#53811yescharacteristic
303C23#54311yescharacteristic
304C23D4211yescharacteristic
305C23#55411yescharacteristic
306C23#56211yescharacteristic
307C23#56311yescharacteristic
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—162no—
334C2×C4—162no—
335C2×C4C22×C4⋊C411yescharacteristic
336C2×C4C2×C4×D411yescharacteristic
337C2×C4C2×C4×Q811yescharacteristic
338C2×C4#51811yescharacteristic
339C2×C4#51911yescharacteristic
340C2×C4#52011yescharacteristic
341C2×C4#53411yescharacteristic
342C2×C4#53711yescharacteristic
343C2×C4#53711yescharacteristic
344C2×C4#53711yescharacteristic
345C2×C4#53811yescharacteristic
346C2×C4#53911yescharacteristic
347C2×C4#53911yescharacteristic
348C2×C4#54411yescharacteristic
349C2×C4#54411yescharacteristic
350C2×C4#54411yescharacteristic
351C2×C4#54711yescharacteristic
352C2×C4#54711yescharacteristic
353C2×C4D4×Q811yescharacteristic
354C2×C4D4×Q811yescharacteristic
355C2×C4D4×Q811yescharacteristic
356C2×C4#55011yescharacteristic
357C2×C4#55611yescharacteristic
358C2×C4#55711yescharacteristic
359C2×C4#55711yescharacteristic
360C2×C4Q8211yescharacteristic
361C2×C4#56311yescharacteristic
362C2×C4#56411yescharacteristic
363C8—22no—
364C8—22no—
365C8—22no—
366C8—22no—
367C8—44no—
368C8—82no—
369C22—22no—
370C22—22no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—22no—
377C22#232911yescharacteristic
378C22D4×C4⋊C411yescharacteristic
379C22#238211yescharacteristic
380C22#268211yescharacteristic
381C22#268311yescharacteristic
382C22#280511yescharacteristic
383C22#280611yescharacteristic
384C22#329012yes—
385C22#329012yes—
386C4—44no—
387C4—44no—
388C4—44no—
389C4—84no—
390C4—82no—
391C4—82no—
392C4—162no—
393C4#231611yescharacteristic
394C4#232511yescharacteristic
395C4Q8×C4⋊C411yescharacteristic
396C4#236511yescharacteristic
397C4#272611yescharacteristic
398C4#272611yescharacteristic
399C4#272711yescharacteristic
400C4#273411yescharacteristic
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2#1444111yescharacteristic
406C2#2269011yescharacteristic
407C2#2269111yescharacteristic
408C2#3122912yes—
409C2#3125512yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
25. #1945C4semidirect

Maximal subgroups

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

subgroupindex
#211752
C2×#19452
#210392
#113392
#112492
#112422
#108922
#108592
#108422