Finite Group Databasebeta

← #480273

Subgroups — #480273

Order 512.

1580 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34174C211yesmaximal, characteristic
2#17371C212yesmaximal, complemented
3#11503C214yesmaximal, complemented
4#11501C214yesmaximal, complemented
5#11427C212yesmaximal, complemented
6#11425C212yesmaximal, complemented
7C23×#41C411yescharacteristic
8C2×#519C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2983—22no—
11#2341—22no—
12#2341—24no—
13#2341—22no—
14#2309C412yes—
15#2309C411yescharacteristic
16#2309C2211yescharacteristic
17#2309C2212yes—
18#2280C412yes—
19#2280C411yescharacteristic
20#2280C2211yescharacteristic
21#2280C2212yes—
22#1467—24no—
23#1464C2218yes—
24#1461—22no—
25#1458C2214yes—
26#1317C2214yes—
27#1316C2214yes—
28#1316C2218yes—
29C22×#41—22no—
30C22×#41—21no—
31C22×#41—21no—
32C22×#41—24no—
33C22×#41—24no—
34C22×#41—44no—
35C22×#41—44no—
36C23×Q8C2×C411yescharacteristic
37C23×D4C2×C412yes—
38C23×D4C2×C411yescharacteristic
39C24×C4C2×C412yes—
40C24×C4C2×C411yescharacteristic
41#519—44no—
42#519—44no—
43C2×C4×Q8—22no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8D412yes—
47C2×C4×Q8D414yes—
48C2×C4×Q8D412yes—
49C2×#117C2×C412yes—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C412yes—
53C2×#117C2×C412yes—
54C2×#117C2×C411yescharacteristic
55C2×#117C2311yescharacteristic
56C2×#117C2312yes—
57#446—48no—
58#432—28no—
59#431—24no—
60#390—24no—
61#390—22no—
62#390—22no—
63#390—24no—
64#390—22no—
65#390—24no—
66#390—22no—
67#390—24no—
68#381C2×C411yescharacteristic
69#381C2×C412yes—
70#381C2311yescharacteristic
71#380—24no—
72#380—22no—
73#380—22no—
74#380—24no—
75#379C2×C411yescharacteristic
76#379C2×C412yes—
77#379C2×C412yes—
78#379C2×C412yes—
79#379C2×C412yes—
80#379C2312yes—
81#379C2311yescharacteristic
82C4×C22⋊C4—22no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—24no—
86#324—24no—
87#324—28no—
88#323C2318yes—
89C25C22×C412yes—
90C25C22×C411yescharacteristic
91C2×#41—216no—
92C2×#41—28no—
93C2×#41—28no—
94C2×#41—432no—
95C2×#41—432no—
96C2×#41—48no—
97C2×#41—44no—
98C2×#41—48no—
99C2×#41—44no—
100C2×#41—44no—
101C2×#41—44no—
102C22×Q8—24no—
103C22×Q8—24no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8C22⋊C412yes—
107C22×Q8C22⋊C411yescharacteristic
108C22×Q8C22⋊C411yescharacteristic
109C22×D4—24no—
110C22×D4—24no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—44no—
117C22×D4—42no—
118C22×D4—44no—
119C22×D4—42no—
120C22×D4M1614yes—
121C23×C4—24no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—44no—
129C23×C4—42no—
130C23×C4—44no—
131C23×C4—42no—
132C23×C4M1614yes—
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C4×Q8—28no—
137C4×Q8—44no—
138C4×Q8—48no—
139C4×Q8—48no—
140C4×Q8—44no—
141C4×Q8—44no—
142#117—42no—
143#117—44no—
144#117—44no—
145#117—42no—
146#117—44no—
147#117—42no—
148#117—42no—
149#117—44no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D414yes—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4C22×C411yescharacteristic
166C2×C22⋊C4C22×C412yes—
167C2×C22⋊C4C22×C411yescharacteristic
168C2×C22⋊C4C22×C411yescharacteristic
169C2×C22⋊C4C22×C412yes—
170C2×C22⋊C4C2411yescharacteristic
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42C22⋊C412yes—
176C2×C42C22⋊C414yes—
177C2×C42C2×D412yes—
178C2×C42C2×D414yes—
179C4⋊C8—44no—
180C4⋊C8—48no—
181C22⋊C8—28no—
182#97—48no—
183C4×C8—44no—
184#95—22no—
185#95—24no—
186#95—22no—
187C24—22no—
188C24—21no—
189C24—21no—
190C24—22no—
191C24—42no—
192C24—41no—
193C24—42no—
194C24—41no—
195C24—84no—
196C24—82no—
197C24C2×M1614yes—
198C24C2×M1612yes—
199C24C23×C411yescharacteristic
200#41—464no—
201#41—432no—
202#41—432no—
203C2×Q8—216no—
204C2×Q8—216no—
205C2×Q8—24no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8#9914yes—
213C2×Q8#10014yes—
214C2×D4—28no—
215C2×D4—28no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—416no—
219C2×D4—416no—
220C2×D4—42no—
221C2×D4—44no—
222C2×D4—42no—
223C2×D4—44no—
224C2×D4—48no—
225C2×D4—48no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—44no—
229C2×D4—42no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—416no—
246C22×C4—416no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C414yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×M1612yes—
266C22×C4C2×M1612yes—
267C22×C4C2×M1612yes—
268C22×C4C22×D412yes—
269C2×C8—48no—
270C4⋊C4—28no—
271C4⋊C4—24no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—84no—
285C22⋊C4—84no—
286C22⋊C4—84no—
287C22⋊C4—82no—
288C22⋊C4—84no—
289C22⋊C4—82no—
290C22⋊C4—84no—
291C42—24no—
292C42—28no—
293C42—42no—
294C42—44no—
295C42—42no—
296C42—44no—
297C42—44no—
298C42—42no—
299C23—21no—
300C23—21no—
301C23—24no—
302C23—22no—
303C23—42no—
304C23—44no—
305C23—42no—
306C23—44no—
307C23—42no—
308C23—42no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—42no—
313C23—41no—
314C23—42no—
315C23—41no—
316C23—84no—
317C23—82no—
318C23—82no—
319C23—84no—
320C23—88no—
321C23—84no—
322C23—88no—
323C23—84no—
324C23C22×C22⋊C411yescharacteristic
325C23C22×M1611yescharacteristic
326C23C22×M1611yescharacteristic
327Q8—216no—
328Q8—28no—
329Q8—28no—
330D4—48no—
331D4—416no—
332D4—416no—
333D4—48no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—48no—
349C2×C4—416no—
350C2×C4—416no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4#40712yes—
366C2×C4#40712yes—
367C2×C4C2×#9914yes—
368C2×C4C2×#9912yes—
369C2×C4C2×#10014yes—
370C2×C4C2×#10012yes—
371C8—88no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—42no—
378C22—44no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—44no—
384C22—42no—
385C22—84no—
386C22—82no—
387C22—88no—
388C22—84no—
389C22—84no—
390C22—88no—
391C22—84no—
392C22—82no—
393C22C2×#40711yescharacteristic
394C22C22×#9911yescharacteristic
395C22C22×#10011yescharacteristic
396C4—22no—
397C4—24no—
398C4—24no—
399C4—42no—
400C4—42no—
401C4—44no—
402C4—42no—
403C4—81no—
404C4—82no—
405C4—81no—
406C4#146214yes—
407C2—21no—
408C2—21no—
409C2—41no—
410C2—41no—
411C2—84no—
412C2—82no—
413C2C2×#146211yescharacteristic
414C2#3335112yes—
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×#41C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #11503C2semidirect
4. #11501C2semidirect
5. #11427C2semidirect
6. #11425C2semidirect

Maximal subgroups

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

subgroupindex
#341742
#173712
#115032
#115012
#114272
#114252