Finite Group Databasebeta

← #495419

Subgroups — #495419

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35333C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3C4×#410C211yesmaximal, characteristic
4#13300C218yesmaximal, complemented
5#12583C214yesmaximal
6C22×#125C411yescharacteristic, complemented
7C2×C4×#41C2211yescharacteristic
8C2×#410C2212yes—
9C2×#410C2212yes—
10#2851—21no—
11#2851—21no—
12#2629—21no—
13#2629—21no—
14#2579—22no—
15#2579—22no—
16#2566C2214yes—
17#2565C414yescomplemented
18#2317C412yescomplemented
19#2311C2212yes—
20#2285—21no—
21#2285C411yescharacteristic, complemented
22C4×#117—21no—
23#1799—22no—
24#1799C2214yes—
25C4×#99—22no—
26C4×#99C2214yes—
27#1740—21no—
28#1740—21no—
29#1393C22116yes—
30C22×#41C2311yescharacteristic
31C2×#125—21no—
32C2×#125—21no—
33C2×#125—41no—
34C2×#125—41no—
35C2×#125—42no—
36C2×#125C2×C414yes—
37C4×#41—41no—
38C4×#41—41no—
39C4×#41—42no—
40C2×C4×Q8C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C412yes—
48C2×#117D411yescharacteristic
49C2×#117D411yescharacteristic
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2×C412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55#410—24no—
56#410—28no—
57#410Q818yes—
58C2×#99—22no—
59C2×#99—22no—
60C2×#99C2314yes—
61C2×#99C2314yes—
62#398—24no—
63#398—24no—
64#396—24no—
65#396—24no—
66#391C2×C414yes—
67#390C2314yes—
68#389—22no—
69#389—22no—
70#387—24no—
71#387—24no—
72#386—22no—
73#386—22no—
74#385—24no—
75#385—24no—
76#385—22no—
77#385—22no—
78#384—21no—
79#384—41no—
80#384C2×C414yes—
81#382—21no—
82#382—41no—
83#382C2×C414yes—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—24no—
88C4×C4⋊C4—42no—
89C4×C4⋊C4—42no—
90C4×C4⋊C4C2×C414yes—
91C4×C22⋊C4—24no—
92#376—41no—
93C43—41no—
94#342—24no—
95#342—24no—
96C2×#41—41no—
97C2×#41—41no—
98C2×#41—42no—
99C2×#41C2×Q814yes—
100C2×#41#4114yes—
101C22×Q8C22×C411yescharacteristic
102C22×D4C2×D411yescharacteristic
103C22×D4C2×D411yescharacteristic
104C22×D4C2411yescharacteristic
105C23×C4—22no—
106C23×C4C22⋊C412yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112#125—44no—
113#125—44no—
114#125—42no—
115#125—42no—
116#125—41no—
117#125—41no—
118#125—42no—
119C4×Q8—42no—
120C4×Q8—42no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126C4×D4—42no—
127#117—28no—
128#117—41no—
129#117—41no—
130#117—41no—
131#117—41no—
132#117—41no—
133#117—41no—
134#117—44no—
135#117—44no—
136#117—44no—
137#117—44no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C414yes—
155C2×C4⋊C4C22×C412yes—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C2×D412yes—
163C2×C22⋊C4C2×D412yes—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—44no—
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C412yes—
180C2×C42C22⋊C412yes—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C22×C414yes—
184#99—28no—
185#99—24no—
186#99—28no—
187#99—44no—
188#99C2×Q818yes—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—44no—
195#95—44no—
196C24C2×C22⋊C411yescharacteristic
197C24C22×D411yescharacteristic
198C24C22×D411yescharacteristic
199#41—44no—
200#41—44no—
201#41—44no—
202#41—44no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C4×Q814yes—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4C22⋊Q814yes—
213C2×D4#12412yes—
214C2×D4C4⋊Q812yes—
215C2×D4C22×Q812yes—
216C2×D4C2×#4112yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—84no—
240C22×C4—82no—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C4×Q814yes—
249C22×C4C22⋊Q814yes—
250C22×C4#12412yes—
251C22×C4C4⋊Q812yes—
252C22×C4C23×C411yescharacteristic
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C4⋊C4—48no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—84no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—88no—
269C22⋊C4—28no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—84no—
278C22⋊C4—84no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—44no—
292C42—48no—
293C42—84no—
294C42—88no—
295C42#9911yescharacteristic
296C42#9911yescharacteristic
297C42#9912yes—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23#39112yes—
308C23C22×C22⋊C411yescharacteristic
309C23C2×C22⋊Q812yes—
310C23C2×#12411yescharacteristic
311C23C2×C4⋊Q811yescharacteristic
312Q8—44no—
313D4—44no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—42no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—88no—
353C2×C4—88no—
354C2×C4—84no—
355C2×C4—88no—
356C2×C4#39112yes—
357C2×C4#39112yes—
358C2×C4#39112yes—
359C2×C4C2×#9912yes—
360C2×C4C2×#9912yes—
361C2×C4C2×#9911yescharacteristic
362C2×C4C2×#9911yescharacteristic
363C2×C4#41012yes—
364C2×C4C2×C4×Q812yes—
365C22—22no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22C2×#39111yescharacteristic
381C22C22×#9911yescharacteristic
382C22C2×#41011yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—84no—
392C4—88no—
393C4—84no—
394C4—164no—
395C4#199612yes—
396C4#199612yes—
397C2—21no—
398C2—21no—
399C2—42no—
400C2—41no—
401C2—42no—
402C2—41no—
403C2C2×#199611yescharacteristic
404C2#3335812yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13300C2semidirect
6. C22×#125C4semidirect
17. #2565C4semidirect
18. #2317C4semidirect
21. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#353332
#206322
C4×#4102
#133002
#125832