Finite Group Databasebeta

← #192161

Subgroups — #192161

Order 512.

1484 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#34181C211yesmaximal, characteristic
2#21982C212yesmaximal, complemented
3C2×#1460C211yesmaximal, characteristic
4#13906C211yesmaximal, characteristic
5#8853C218yesmaximal, complemented
6#8014C212yesmaximal, complemented
7C2×#518—21no—
8C22×C4×D4C2211yescharacteristic
9C2×#431C2212yes—
10#2349—24no—
11D4×C4⋊C4—22no—
12D4×C4⋊C4—22no—
13#2316—21no—
14#2315C2214yes—
15C2×#390—21no—
16#2285C2212yes—
17C2×#384C2211yescharacteristic
18C2×#380—22no—
19C2×#380—22no—
20C2×C4×C4⋊C4C2211yescharacteristic
21#1947—24no—
22#1852C2218yes—
23#1845—22no—
24#1834—22no—
25#1460C418yescomplemented
26C2×#323C2214yes—
27#1439C2214yes—
28C4⋊#95—21no—
29#1294C2218yes—
30C23×D4D411yescharacteristic
31C24×C4D412yes—
32#518—44no—
33#518—44no—
34C2×C4×D4—24no—
35C2×C4×D4—24no—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4—24no—
39C2×C4×D4C2×C414yes—
40C2×C4×D4Q814yes—
41C2×#117—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—24no—
50C22×C22⋊C4—24no—
51C22×C22⋊C4C2312yes—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#431—24no—
55#431C2×C418yescomplemented
56C2×#107—28no—
57C2×C4⋊C8—22no—
58C2×C22⋊C8C2314yes—
59C2×C4×C8—22no—
60#390—44no—
61#389—24no—
62#389—24no—
63#385—28no—
64#385—28no—
65#384—22no—
66#384—22no—
67#384D412yes—
68#384D412yes—
69#382—22no—
70#380—24no—
71#380—24no—
72#380—24no—
73#380—22no—
74#380—48no—
75#380C2314yes—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—24no—
78C4×C4⋊C4D414yes—
79C2×#95—24no—
80C2×#95—24no—
81C2×#95C2312yes—
82#340—28no—
83#323C2×C4116yescomplemented
84C25C2×D412yes—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—22no—
88C22×D4—22no—
89C22×D4C4⋊C414yes—
90C22×D4D814yes—
91C22×D4Q1614yes—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—22no—
99C23×C4—44no—
100C23×C4—44no—
101C23×C4C4⋊C418yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2411yescharacteristic
106C22×C8—24no—
107C4×D4—28no—
108C4×D4—28no—
109C4×D4—28no—
110C4×D4—28no—
111C4×D4—416no—
112C4×D4—416no—
113#117—44no—
114#117—44no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—84no—
135C2×C4⋊C4—84no—
136C2×C4⋊C4C4⋊C414yes—
137C2×C4⋊C4C2×D414yes—
138C2×C4⋊C4C2×D414yes—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—48no—
145C2×C22⋊C4—48no—
146C2×C22⋊C4—48no—
147C2×C22⋊C4—84no—
148C2×C22⋊C4C22×C414yes—
149C2×C22⋊C4C2×Q814yes—
150C2×C42—24no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42—42no—
154C2×C42C22×C412yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×Q812yes—
158C2×C42#4114yes—
159#107—88no—
160C4⋊C8—28no—
161C22⋊C8—24no—
162C22⋊C8C22×C418yes—
163C4×C8—28no—
164#95—28no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—48no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—22no—
173C24—42no—
174C24—42no—
175C24—42no—
176C24—42no—
177C24C2×C4⋊C414yes—
178C24C2×D814yes—
179C24C2×Q1614yes—
180C24C22×D411yescharacteristic
181C2×D4—28no—
182C2×D4—28no—
183C2×D4—28no—
184C2×D4—28no—
185C2×D4—28no—
186C2×D4—28no—
187C2×D4#107116yescomplemented
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—28no—
193C22×C4—28no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—416no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—48no—
211C22×C4—48no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—88no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—84no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4C2×C4⋊C412yes—
225C22×C4C2×C4⋊C414yes—
226C22×C4C2×C4⋊C412yes—
227C22×C4#12011yescharacteristic
228C22×C4#12012yes—
229C22×C4#12011yescharacteristic
230C22×C4#12312yes—
231C22×C4#12312yes—
232C22×C4C2×D812yes—
233C22×C4C2×Q1612yes—
234C22×C4AGL(1,ℤ/8)12yes—
235C22×C4#13712yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D412yes—
238C22×C4C22×Q811yescharacteristic
239C22×C4C2×#4112yes—
240C2×C8—28no—
241C2×C8—44no—
242C2×C8—44no—
243C4⋊C4—28no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—48no—
247C4⋊C4—48no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—82no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—82no—
257C22⋊C4—28no—
258C22⋊C4—28no—
259C22⋊C4—416no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—816no—
263C42—48no—
264C42—84no—
265C42C4×D414yes—
266C42C4×D414yes—
267C42C22⋊Q814yes—
268C42C22⋊Q814yes—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—24no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23—24no—
278C23—24no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—82no—
290C23—82no—
291C23—82no—
292C23—82no—
293C23—82no—
294C23—82no—
295C23C2×#10718yes—
296C23C22×C4⋊C411yescharacteristic
297C23C2×#12011yescharacteristic
298C23C2×#12311yescharacteristic
299C23C22×D811yescharacteristic
300C23C22×Q1611yescharacteristic
301C23C2×AGL(1,ℤ/8)11yescharacteristic
302C23C2×#13711yescharacteristic
303D4—232no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—416no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—88no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—168no—
339C2×C4—168no—
340C2×C4#38512yes—
341C2×C4#38514yes—
342C2×C4#38512yes—
343C2×C4C2×#10714yes—
344C2×C4#42814yes—
345C2×C4#44714yes—
346C2×C4#45114yes—
347C2×C4#45312yes—
348C2×C4#45512yes—
349C2×C4#48014yes—
350C2×C4#48414yes—
351C2×C4C2×C4×D412yes—
352C2×C4C2×C22⋊Q812yes—
353C8—48no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—28no—
367C22—48no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—44no—
373C22—44no—
374C22—84no—
375C22—84no—
376C22—84no—
377C22C2×#38511yescharacteristic
378C22C22×#10711yescharacteristic
379C22C2×#42811yescharacteristic
380C22C2×#44711yescharacteristic
381C22C2×#45111yescharacteristic
382C22C2×#45311yescharacteristic
383C22C2×#45511yescharacteristic
384C22#302711yescharacteristic
385C22#302811yescharacteristic
386C22C2×#48011yescharacteristic
387C22C2×#48411yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—44no—
392C4—84no—
393C4—84no—
394C4—84no—
395C4—162no—
396C4#185218yes—
397C4#185614yes—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—48no—
403C2#1838711yescharacteristic
404C2#1839011yescharacteristic
405C2#1839411yescharacteristic
406C2#1839711yescharacteristic
407C2C2×#185211yescharacteristic
408C2C2×#185611yescharacteristic
409C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21982C2semidirect
5. #8853C2semidirect
6. #8014C2semidirect
25. #1460C4semidirect
55. #431C2×C4semidirect
83. #323C2×C4semidirect
187. C2×D4#107semidirect

Maximal subgroups

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

subgroupindex
#341812
#219822
C2×#14602
#139062
#88532
#80142