Finite Group Databasebeta

← #382810

Subgroups — #382810

Order 512.

1394 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34065C211yesmaximal, characteristic
2#20683C211yesmaximal, characteristic, complemented
3#13852C211yesmaximal, characteristic, complemented
4#11226C212yesmaximal, complemented
5#11201C212yesmaximal, complemented
6#11075C212yesmaximal, complemented
7#10971C212yesmaximal
8#10808C212yesmaximal, complemented
9#10790C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#104C2214yes—
14C2×#407C2212yes—
15#2324C414yescomplemented
16#2323C414yescomplemented
17#2322—22no—
18#2321—22no—
19#2297C2211yescharacteristic
20#2296C2211yescharacteristic
21#2293—22no—
22#2290—22no—
23#2280C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2212yes—
27#1818—22no—
28#1818—22no—
29C2×#339C2214yes—
30C2×#339C2214yes—
31C2×#339C2214yes—
32#1728C2212yes—
33#1728C2212yes—
34#1441—22no—
35#1439—22no—
36C4×#95C2212yes—
37C22×#41C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C22×M16C2314yes—
40#525—24no—
41#525—24no—
42C2×#123—21no—
43C2×C22⋊Q8C2×C412yes—
44C2×#121C2×C412yes—
45C2×#120—21no—
46C4×#41—44no—
47C2×C4×Q8C2×C412yes—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4C2×C412yes—
51C2×#117—21no—
52C2×#117—44no—
53C2×#117C2×C414yes—
54C2×#117D412yes—
55C2×#117D412yes—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4D412yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42—21no—
64C22×C42C2314yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#104—28no—
68C2×#104—44no—
69C2×#104—44no—
70#407—44no—
71C2×C22⋊C8—24no—
72#391C2×C412yes—
73#390—24no—
74#390C2×C412yes—
75#389—24no—
76#389C2×C414yes—
77#388—24no—
78#388—24no—
79#388—24no—
80#387—24no—
81#386—22no—
82#386—24no—
83#385—22no—
84#385—24no—
85#381—21no—
86#380C2×C414yes—
87#379—21no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—44no—
90C4×C4⋊C4C2×C414yes—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—44no—
95#376—21no—
96C2×#95—24no—
97C2×#95—22no—
98C2×#95—22no—
99C43—21no—
100#339—28no—
101#339—44no—
102#339—44no—
103#336—22no—
104#336—22no—
105#336—24no—
106C25C2×D411yescharacteristic
107C2×#41—22no—
108C2×#41—24no—
109C2×#41—22no—
110C2×#41—22no—
111C2×#41#4114yes—
112C22×Q8C22×C411yescharacteristic
113C22×D4—21no—
114C22×D4—22no—
115C22×D4C22×C411yescharacteristic
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4C22×C412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C22×C8—24no—
137#123—28no—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140#121—24no—
141#121—24no—
142#120—28no—
143C4×Q8—44no—
144C4×D4—216no—
145C4×D4—44no—
146C4×D4—44no—
147C4×D4—44no—
148#117—22no—
149#117—24no—
150#117—22no—
151#117—22no—
152#117—48no—
153#117—44no—
154#117—44no—
155#117—44no—
156#117#4114yes—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4C22⋊C414yes—
165C2×C4⋊C4C22⋊C414yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4C22⋊C414yes—
182C2×C22⋊C4C22⋊C414yes—
183C2×C42—21no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—44no—
197C2×C42C22×C412yes—
198C2×C42C22×C412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202#104—416no—
203C22⋊C8—48no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—48no—
209#95—44no—
210#95—44no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—22no—
216C24—44no—
217C24C2×C22⋊C412yes—
218C24C22×D411yescharacteristic
219#41—28no—
220#41—28no—
221#41—44no—
222#41—44no—
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8#11712yes—
227C2×D4—24no—
228C2×D4—28no—
229C2×D4—21no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—84no—
236C2×D4#11712yes—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—84no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C414yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4C2×#4112yes—
282C22×C4H(5,𝔽2)14yes—
283C22×C4H(5,𝔽2)12yes—
284C22×C42−1+412yes—
285M16—44no—
286M16—44no—
287M16—44no—
288M16—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C4⋊C4—28no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—416no—
300C4⋊C4—84no—
301C4⋊C4#11714yes—
302C22⋊C4—28no—
303C22⋊C4—28no—
304C22⋊C4—24no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—416no—
309C22⋊C4—42no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—84no—
313C22⋊C4—84no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—22no—
318C42—24no—
319C42—22no—
320C42—44no—
321C42—42no—
322C42—42no—
323C42—44no—
324C42—44no—
325C42—44no—
326C42—44no—
327C42—84no—
328C23—24no—
329C23—22no—
330C23—22no—
331C23—24no—
332C23—24no—
333C23—22no—
334C23—22no—
335C23—44no—
336C23—44no—
337C23—41no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—44no—
342C23—84no—
343C23—82no—
344C23—82no—
345C23C22×C22⋊C411yescharacteristic
346C23#52512yes—
347C23#53412yes—
348C23#53511yescharacteristic
349C23#53611yescharacteristic
350Q8—24no—
351Q8—42no—
352D4—24no—
353D4—44no—
354D4—44no—
355D4—42no—
356D4—88no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—88no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4C2×#11711yescharacteristic
395C2×C4C2×#11711yescharacteristic
396C2×C4#51812yes—
397C2×C4#52012yes—
398C2×C4#52512yes—
399C2×C4#53412yes—
400C2×C4#53712yes—
401C2×C4#53814yes—
402C2×C4#53914yes—
403C2×C4#54014yes—
404C2×C4#54214yes—
405C8—84no—
406C22—24no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—44no—
415C22—42no—
416C22—44no—
417C22—44no—
418C22—84no—
419C22—84no—
420C22—82no—
421C22—82no—
422C22#228011yescharacteristic
423C22#230811yescharacteristic
424C22#231111yescharacteristic
425C22#263512yes—
426C22#263812yes—
427C22#290114yes—
428C4—42no—
429C4—42no—
430C4—44no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—88no—
435C4—82no—
436C4—82no—
437C4#231812yes—
438C4#231912yes—
439C4#263014yes—
440C2—22no—
441C2—22no—
442C2—82no—
443C2—82no—
444C2#1434411yescharacteristic
445C2#2074912yes—
446C2#2177512yes—
447C2#2190112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20683C2semidirect
3. #13852C2semidirect
4. #11226C2semidirect
5. #11201C2semidirect
6. #11075C2semidirect
8. #10808C2semidirect
9. #10790C2semidirect
15. #2324C4semidirect
16. #2323C4semidirect

Maximal subgroups

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

subgroupindex
#340652
#206832
#138522
#112262
#112012
#110752
#109712
#108082
#107902