Finite Group Databasebeta

← #382788

Subgroups — #382788

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34065C211yesmaximal, characteristic
2#20627C211yesmaximal, characteristic, complemented
3#13853C211yesmaximal, characteristic, complemented
4#11209C212yesmaximal, complemented
5#11201C212yesmaximal, complemented
6#11053C212yesmaximal, complemented
7#10971C212yesmaximal
8#10808C212yesmaximal, complemented
9#10790C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C2×#423C2212yes—
14C22×#104C2214yes—
15C2×#407C2212yes—
16#2324C414yescomplemented
17#2323C414yescomplemented
18#2322—22no—
19#2321—22no—
20#2297C2211yescharacteristic
21#2296C2211yescharacteristic
22#2293—22no—
23#2290—22no—
24#2280C2211yescharacteristic
25C4×#117C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27#1798—22no—
28#1798—22no—
29C2×#339C2214yes—
30C2×#339C2214yes—
31C2×#339C2214yes—
32#1728C2212yes—
33#1445—22no—
34#1440—22no—
35C4×#95C2212yes—
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⋊C4D412yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D412yes—
62C22×C42—21no—
63C22×C42C2314yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#423—44no—
67C2×C4⋊C8—24no—
68C2×#104—28no—
69C2×#104—44no—
70C2×#104—44no—
71#407—44no—
72C2×C22⋊C8—24no—
73#391C2×C412yes—
74#390—24no—
75#390C2×C412yes—
76#389—24no—
77#389C2×C414yes—
78#388—24no—
79#388—24no—
80#388—24no—
81#387—24no—
82#386—22no—
83#386—24no—
84#385—22no—
85#385—24no—
86#381—21no—
87#380C2×C414yes—
88#379—21no—
89C4×C4⋊C4—22no—
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—24no—
98C2×#95—22no—
99C43—21no—
100#339—28no—
101#339—44no—
102#339—44no—
103#336—22no—
104#336—22no—
105C25C2×D411yescharacteristic
106C2×#41—22no—
107C2×#41—24no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41#4114yes—
111C22×Q8C22×C411yescharacteristic
112C22×D4—21no—
113C22×D4—22no—
114C22×D4C22×C411yescharacteristic
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—22no—
123C23×C4C22×C412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2411yescharacteristic
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C22×C8—24no—
136#123—28no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139#121—24no—
140#121—24no—
141#120—28no—
142C4×Q8—44no—
143C4×D4—216no—
144C4×D4—44no—
145C4×D4—44no—
146C4×D4—44no—
147#117—22no—
148#117—24no—
149#117—22no—
150#117—22no—
151#117—48no—
152#117—44no—
153#117—44no—
154#117—44no—
155#117#4114yes—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4C22⋊C414yes—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—48no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4C22⋊C414yes—
179C2×C22⋊C4C22⋊C414yes—
180C2×C42—21no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—44no—
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C4⋊C8—48no—
200#104—416no—
201C22⋊C8—48no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—48no—
207#95—48no—
208#95—44no—
209C24—21no—
210C24—22no—
211C24—22no—
212C24—22no—
213C24—44no—
214C24C2×C22⋊C412yes—
215C24C22×D411yescharacteristic
216#41—28no—
217#41—28no—
218#41—44no—
219#41—44no—
220C2×Q8—21no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8#11712yes—
224C2×D4—24no—
225C2×D4—28no—
226C2×D4—21no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—84no—
232C2×D4#11712yes—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—28no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—84no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C414yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4112yes—
278C22×C4C2×#4112yes—
279C22×C4H(5,𝔽2)14yes—
280C22×C4H(5,𝔽2)12yes—
281C22×C42−1+412yes—
282M16—44no—
283M16—44no—
284M16—44no—
285M16—44no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—44no—
289C4⋊C4—28no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—84no—
297C4⋊C4#11714yes—
298C22⋊C4—28no—
299C22⋊C4—28no—
300C22⋊C4—24no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—416no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—84no—
308C22⋊C4—84no—
309C42—24no—
310C42—24no—
311C42—24no—
312C42—24no—
313C42—22no—
314C42—22no—
315C42—44no—
316C42—42no—
317C42—41no—
318C42—41no—
319C42—44no—
320C42—44no—
321C42—44no—
322C42—44no—
323C42—84no—
324C23—24no—
325C23—22no—
326C23—22no—
327C23—24no—
328C23—24no—
329C23—22no—
330C23—22no—
331C23—44no—
332C23—44no—
333C23—41no—
334C23—41no—
335C23—44no—
336C23—44no—
337C23—84no—
338C23—84no—
339C23C22×C22⋊C411yescharacteristic
340C23#52512yes—
341C23#53412yes—
342C23#53712yes—
343Q8—24no—
344Q8—42no—
345D4—24no—
346D4—48no—
347D4—42no—
348D4—88no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—88no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4C2×#11711yescharacteristic
388C2×C4C2×#11711yescharacteristic
389C2×C4#51812yes—
390C2×C4#52012yes—
391C2×C4#52512yes—
392C2×C4#53412yes—
393C2×C4#53511yescharacteristic
394C2×C4#53611yescharacteristic
395C2×C4#53814yes—
396C2×C4#53914yes—
397C2×C4#54014yes—
398C2×C4#54214yes—
399C8—84no—
400C22—24no—
401C22—22no—
402C22—22no—
403C22—24no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—44no—
408C22—42no—
409C22—44no—
410C22—44no—
411C22—88no—
412C22—82no—
413C22—82no—
414C22#228011yescharacteristic
415C22#230811yescharacteristic
416C22#231111yescharacteristic
417C22#263014yes—
418C22#290114yes—
419C4—42no—
420C4—42no—
421C4—44no—
422C4—42no—
423C4—44no—
424C4—42no—
425C4—88no—
426C4—82no—
427C4—82no—
428C4#231812yes—
429C4#231912yes—
430C4#263512yes—
431C4#263812yes—
432C2—24no—
433C2—82no—
434C2—82no—
435C2#1434411yescharacteristic
436C2#2074912yes—
437C2#2177512yes—
438C2#2185812yes—
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. #20627C2semidirect
3. #13853C2semidirect
4. #11209C2semidirect
5. #11201C2semidirect
6. #11053C2semidirect
8. #10808C2semidirect
9. #10790C2semidirect
16. #2324C4semidirect
17. #2323C4semidirect

Maximal subgroups

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

subgroupindex
#340652
#206272
#138532
#112092
#112012
#110532
#109712
#108082
#107902