Finite Group Databasebeta

← #480572

Subgroups — #480572

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35574C211yesmaximal, characteristic
2#17370C211yesmaximal, characteristic
3#17353C211yesmaximal, characteristic
4#11580C214yesmaximal, complemented
5#11577C214yesmaximal, complemented
6#11454C212yesmaximal, complemented
7#11452C212yesmaximal, complemented
8C22×C4⋊Q8C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2982—21no—
11#2975—21no—
12#2579C412yes—
13#2579C2212yes—
14#2452—21no—
15#2452—21no—
16#2451—21no—
17#2451—21no—
18#2440—22no—
19#2439—22no—
20#2409C412yes—
21#2409C2212yes—
22#2317C2211yescharacteristic
23#2316C411yescharacteristic
24#2311C411yescharacteristic
25#2297C411yescharacteristic
26#2296C2211yescharacteristic
27#1465C2214yes—
28#1463—22no—
29#1461—21no—
30#1460—21no—
31#1460C2214yes—
32#1458—22no—
33#1458C2212yes—
34#1458C2212yes—
35#1326C2214yes—
36#1326C2218yes—
37#1326C2214yes—
38C22×#41C2×C411yescharacteristic
39C23×Q8C2×C412yes—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—44no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46#520—44no—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4C2×C412yes—
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4C2312yes—
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117D412yes—
56C2×#117C2311yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4D412yes—
61C22×C42C2×C411yescharacteristic
62C8×Q8—44no—
63#436—44no—
64#431—22no—
65#431—24no—
66#431—22no—
67#407—24no—
68#398—22no—
69#398—22no—
70#396—22no—
71#396—22no—
72#395—22no—
73#395—22no—
74#393—22no—
75#393—22no—
76#391—21no—
77#391—21no—
78#391—22no—
79#391—22no—
80#390—21no—
81#390—21no—
82#389—22no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#387—22no—
90#387—22no—
91#386C2×C412yes—
92#386C2×C412yes—
93#386C2×C412yes—
94#386C2×C412yes—
95#385—22no—
96#385—22no—
97#385C2×C412yes—
98#385C2×C411yescharacteristic
99#385C2×C411yescharacteristic
100#385C2312yes—
101#385C2311yescharacteristic
102#385C2311yescharacteristic
103#384—21no—
104#382—21no—
105#380—22no—
106#380C2×C412yes—
107#380C2×C412yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C22⋊C4—22no—
111#324—22no—
112#324—22no—
113#324—24no—
114#323—24no—
115#323C2314yes—
116#323C2314yes—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—42no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—84no—
124C22×Q8C22⋊C411yescharacteristic
125C22×D4C22⋊C411yescharacteristic
126C22×D4C22×C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C2×D412yes—
135C2×M16—44no—
136C4⋊Q8—48no—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139C22⋊Q8—84no—
140C22⋊Q8—84no—
141C4×Q8—24no—
142C4×Q8—42no—
143C4×D4—24no—
144C4×D4—42no—
145C4×D4—44no—
146C4×D4—44no—
147#117—24no—
148#117—24no—
149#117—42no—
150#117—44no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22×C411yescharacteristic
182C2×C22⋊C4C22×C411yescharacteristic
183C2×C22⋊C4C22×C412yes—
184C2×C22⋊C4C22×C411yescharacteristic
185C2×C22⋊C4C2×D412yes—
186C2×C22⋊C4C2411yescharacteristic
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—42no—
192C2×C42C22⋊C412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C4⋊C8—42no—
197C4⋊C8—44no—
198C4⋊C8—42no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—44no—
202C4×C8—42no—
203C4×C8—44no—
204C4×C8—42no—
205#95—22no—
206#95—22no—
207#95—21no—
208#95—21no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95—21no—
216#95—22no—
217C24C2×C22⋊C412yes—
218C24C23×C411yescharacteristic
219#41—416no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—88no—
229C2×Q8—88no—
230C2×Q8#10114yes—
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—42no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4#13114yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270M16—164no—
271C2×C8—44no—
272C2×C8—44no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4#9914yes—
294C22⋊C4—24no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—24no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C42—22no—
306C42—24no—
307C42—22no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—42no—
321C23—42no—
322C23C22×C22⋊C411yescharacteristic
323C23C2×#13112yes—
324Q8—28no—
325Q8—88no—
326Q8—88no—
327D4—48no—
328D4—44no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—28no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—48no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—162no—
372C2×C4#40812yes—
373C2×C4#40812yes—
374C2×C4C2×#9914yes—
375C2×C4C2×#9912yes—
376C2×C4C2×#10114yes—
377C2×C4C2×#10112yes—
378C8—84no—
379C8—84no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—42no—
384C22—42no—
385C22—44no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22C2×#40811yescharacteristic
393C22C22×#9911yescharacteristic
394C22C22×#10111yescharacteristic
395C4—24no—
396C4—22no—
397C4—24no—
398C4—41no—
399C4—42no—
400C4—42no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—42no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—82no—
409C4#147214yes—
410C2—21no—
411C2—21no—
412C2—42no—
413C2—82no—
414C2C2×#147211yescharacteristic
415C2#3334811yescharacteristic
416C2#3334911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11580C2semidirect
5. #11577C2semidirect
6. #11454C2semidirect
7. #11452C2semidirect

Maximal subgroups

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

subgroupindex
#355742
#173702
#173532
#115802
#115772
#114542
#114522