Finite Group Databasebeta

← #495399

Subgroups — #495399

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35333C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3C4×#413C211yesmaximal, characteristic
4#13294C218yesmaximal, complemented
5#12575C214yesmaximal
6C22×#125C411yescharacteristic, complemented
7C2×C4×#41C2211yescharacteristic
8C2×#413C2212yes—
9C2×#410C2212yes—
10C2×C4×M16—21no—
11#2851—21no—
12#2851—21no—
13#2629—21no—
14#2629—21no—
15#2579C414yescomplemented
16#2579C2214yes—
17#2566—22no—
18#2565—22no—
19#2317C412yescomplemented
20#2311C2212yes—
21#2285—21no—
22#2285C411yescharacteristic, complemented
23#1799—22no—
24#1799C2214yes—
25C4×#101—22no—
26C4×#100C2214yes—
27#1742—21no—
28#1740—21no—
29#1390C22116yes—
30C22×#41C2311yescharacteristic
31C22×M16—22no—
32C2×#125—22no—
33C2×#125—21no—
34C2×#125—21no—
35C2×#125—42no—
36C2×#125—41no—
37C2×#125—41no—
38C4×#41—42no—
39C4×#41—41no—
40C4×#41—41no—
41C2×C4×Q8D412yes—
42C2×C4×D4C2×C412yes—
43C2×C4×D4D412yes—
44C2×C4×D4C2312yes—
45C2×#117—22no—
46C2×#117C2×C412yes—
47C2×#117C2×C411yescharacteristic
48C2×#117D411yescharacteristic
49C2×#117D411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C414yes—
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55#413—24no—
56#413Q818yes—
57C2×#101—22no—
58C2×#100C2314yes—
59#410—28no—
60C2×#99—22no—
61C2×#99C2314yes—
62C4×M16—24no—
63C4×M16—42no—
64C2×#97—41no—
65C2×C4×C8—41no—
66#398—24no—
67#398—24no—
68#396—24no—
69#396—24no—
70#391—22no—
71#390—22no—
72#389—22no—
73#389—22no—
74#387—24no—
75#387—24no—
76#386C2×C414yes—
77#386C2314yes—
78#385—24no—
79#385—24no—
80#385C2×C414yes—
81#385C2×C414yes—
82#384—22no—
83#384—21no—
84#384—41no—
85#382—22no—
86#382—21no—
87#382—41no—
88#380—22no—
89#380—22no—
90#380—24no—
91#380C2×C414yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—42no—
94#343—24no—
95#342—24no—
96C2×#41—41no—
97C2×#41—41no—
98C2×#41—42no—
99C2×#41C2×Q814yes—
100C2×#41#4114yes—
101C22×Q8C2×D411yescharacteristic
102C22×D4C22×C411yescharacteristic
103C22×D4C2×D411yescharacteristic
104C22×D4C2411yescharacteristic
105C23×C4C22⋊C412yes—
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C412yes—
108C23×C4C22×C414yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C2×M16—28no—
113C2×M16—24no—
114C2×M16—44no—
115C2×M16—42no—
116C22×C8—42no—
117#125—42no—
118#125—42no—
119#125—44no—
120#125—44no—
121#125—42no—
122#125—41no—
123#125—41no—
124C4×Q8—42no—
125C4×Q8—42no—
126C4×D4—42no—
127C4×D4—42no—
128C4×D4—42no—
129C4×D4—42no—
130C4×D4—42no—
131C4×D4—42no—
132#117—41no—
133#117—41no—
134#117—41no—
135#117—41no—
136#117—41no—
137#117—41no—
138#117—44no—
139#117—44no—
140#117—44no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22×C414yes—
163C2×C22⋊C4C22×C412yes—
164C2×C22⋊C4C2×D412yes—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—42no—
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C412yes—
180C2×C42C22⋊C412yes—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C2×D412yes—
183#101—28no—
184#100—24no—
185#100C2×Q818yes—
186#99—28no—
187#99—44no—
188#97—44no—
189C4×C8—44no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—44no—
196C24C2×C22⋊C411yescharacteristic
197C24C23×C411yescharacteristic
198C24C22×D411yescharacteristic
199#41—44no—
200#41—44no—
201#41—44no—
202#41—44no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C22⋊Q814yes—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4C4×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—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—84no—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C4×Q814yes—
248C22×C4C22⋊Q814yes—
249C22×C4#12412yes—
250C22×C4C4⋊Q812yes—
251C22×C4C22×D411yescharacteristic
252M16—28no—
253M16—44no—
254M16—48no—
255C2×C8—44no—
256C2×C8—48no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—48no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4—88no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—84no—
279C22⋊C4—84no—
280C42—22no—
281C42—22no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—44no—
293C42—84no—
294C42#9912yes—
295C42#9911yescharacteristic
296C42#9911yescharacteristic
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23#39112yes—
307C23C22×C22⋊C411yescharacteristic
308C23C2×C4×Q812yes—
309C23C2×#12411yescharacteristic
310C23C2×C4⋊Q811yescharacteristic
311Q8—44no—
312D4—44no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—42no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—88no—
351C2×C4—84no—
352C2×C4—88no—
353C2×C4#39112yes—
354C2×C4#39112yes—
355C2×C4#39112yes—
356C2×C4C2×#9912yes—
357C2×C4C2×#9912yes—
358C2×C4C2×#9911yescharacteristic
359C2×C4C2×#9911yescharacteristic
360C2×C4#41012yes—
361C2×C4C2×C22⋊Q812yes—
362C8—48no—
363C22—22no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—22no—
368C22—42no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—81no—
377C22—81no—
378C22C2×#39111yescharacteristic
379C22C22×#9911yescharacteristic
380C22C2×#41011yescharacteristic
381C4—22no—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—22no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—84no—
390C4—84no—
391C4—164no—
392C4#199612yes—
393C4#199612yes—
394C2—21no—
395C2—21no—
396C2—41no—
397C2—42no—
398C2—41no—
399C2—42no—
400C2C2×#199611yescharacteristic
401C2#3334712yes—
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ 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. #13294C2semidirect
6. C22×#125C4semidirect
15. #2579C4semidirect
19. #2317C4semidirect
22. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#353332
#206322
C4×#4132
#132942
#125752