Finite Group Databasebeta

← #381264

Subgroups — #381264

Order 512.

1313 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36641C211yesmaximal, characteristic
2#20752C211yesmaximal, characteristic, complemented
3C42×C4⋊C4C211yesmaximal, characteristic, complemented
4#11226C212yesmaximal, complemented
5#11209C212yesmaximal, complemented
6#11075C212yesmaximal
7#11053C212yesmaximal
8#10923C212yesmaximal
9#10878C212yesmaximal, complemented
10C2×#423C2212yes—
11C22×#104C2214yes—
12#2863C412yescomplemented
13#2863C412yescomplemented
14#2860C412yescomplemented
15#2860C412yescomplemented
16#2852—22no—
17#2852—22no—
18C4×#126—22no—
19C4×#126—22no—
20#2297C2211yescharacteristic
21#2297C2211yescharacteristic
22#2297C2211yescharacteristic
23#2296C2211yescharacteristic
24#2296C2211yescharacteristic
25#2296C2211yescharacteristic
26C2×C4×C4⋊C4—22no—
27C2×C4×C4⋊C4C2212yes—
28C2×C43C2212yes—
29C2×C43C2211yescharacteristic
30#1831—22no—
31#1831—22no—
32C2×#339C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1728C2212yes—
36#1728C2212yes—
37C4×#95—22no—
38C4×#95C2212yes—
39C22×#41C2311yescharacteristic
40C22×M16C2314yes—
41C2×#126—22no—
42C2×#126—22no—
43C2×#117C2311yescharacteristic
44C2×#117C2311yescharacteristic
45C2×#117C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C42—24no—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42C2314yes—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54#423—44no—
55C2×C4⋊C8—24no—
56C2×#104—28no—
57C2×#104—44no—
58C2×#104—44no—
59#391C2×C411yescharacteristic
60#391C2×C411yescharacteristic
61#391C2×C412yes—
62#391C2×C412yes—
63#390C2×C411yescharacteristic
64#390C2×C411yescharacteristic
65#390C2×C412yes—
66#390C2×C412yes—
67#389C2×C412yes—
68#389C2×C412yes—
69#389C2×C412yes—
70#389C2×C412yes—
71#389C2×C412yes—
72#389C2×C412yes—
73#388—22no—
74#388—22no—
75#388—22no—
76#388—22no—
77#388—22no—
78#388—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#386—21no—
86#386—22no—
87#386—22no—
88#386—21no—
89#385—21no—
90#385—22no—
91#385—22no—
92#385—21no—
93#383—22no—
94#383—22no—
95C4×C4⋊C4—216no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—28no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—22no—
105C2×#95—22no—
106C2×#95—24no—
107C43—44no—
108C43C2×C412yes—
109C43C2×C412yes—
110C43D412yes—
111C43D412yes—
112#339—28no—
113#339—44no—
114#339—44no—
115#336—24no—
116#336—22no—
117#336—22no—
118C2×#41—24no—
119C2×#41—22no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—22no—
123C22×Q8C22×C411yescharacteristic
124C22×D4—21no—
125C22×D4C22×C411yescharacteristic
126C23×C4—24no—
127C23×C4—21no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
139C22×C8—24no—
140#126—48no—
141#126—48no—
142#117—24no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—216no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—28no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—216no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—24no—
183C2×C42—44no—
184C2×C42—48no—
185C2×C42—48no—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C22⋊C412yes—
191C2×C42C22⋊C412yes—
192C2×C42C22⋊C412yes—
193C2×C42C22⋊C412yes—
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42#4118yes—
203C4⋊C8—48no—
204#104—416no—
205#95—216no—
206#95—28no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219C24—21no—
220C24C22×D411yescharacteristic
221#41—44no—
222#41—44no—
223#41—44no—
224#41—44no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—42no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—42no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—216no—
242C22×C4—22no—
243C22×C4—28no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C2×#4114yes—
269C22×C4H(5,𝔽2)12yes—
270C22×C4H(5,𝔽2)12yes—
271C22×C42−1+412yes—
272C22×C42−1+412yes—
273M16—44no—
274M16—44no—
275M16—44no—
276M16—44no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C4⋊C4—232no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—42no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C42—24no—
302C42—24no—
303C42—22no—
304C42—22no—
305C42—24no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—24no—
311C42—24no—
312C42—24no—
313C42—24no—
314C42—24no—
315C42—28no—
316C42—28no—
317C42—42no—
318C42—42no—
319C42—432no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—42no—
324C42—44no—
325C42—44no—
326C42#10418yescomplemented
327C23—22no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—41no—
334C23C22×C22⋊C411yescharacteristic
335C23#52512yes—
336C23#53511yescharacteristic
337C23#53611yescharacteristic
338C23#53712yes—
339Q8—44no—
340D4—44no—
341D4—82no—
342D4—82no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—216no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—48no—
362C2×C4—416no—
363C2×C4—416no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4C2×#10414yes—
383C2×C4C2×#10414yes—
384C2×C4C2×#10414yes—
385C2×C4#42114yes—
386C2×C4C4×#4112yes—
387C2×C4#52012yes—
388C2×C4#52012yes—
389C2×C4#53212yes—
390C2×C4#53511yescharacteristic
391C2×C4#53611yescharacteristic
392C2×C4#53712yes—
393C2×C4#53814yes—
394C2×C4#53914yes—
395C2×C4#54114yes—
396C2×C4#54214yes—
397C8—84no—
398C22—24no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#229711yescharacteristic
409C22#231111yescharacteristic
410C22#231111yescharacteristic
411C22#263612yes—
412C22#263712yes—
413C22C22×#10412yes—
414C22C2×#42112yes—
415C4—24no—
416C4—24no—
417C4—24no—
418C4—24no—
419C4—416no—
420C4—42no—
421C4—42no—
422C4—82no—
423C4—82no—
424C4—81no—
425C4—81no—
426C4—81no—
427C4#233414yes—
428C4#266212yes—
429C4#266412yes—
430C2—22no—
431C2—22no—
432C2—161no—
433C2#1429111yescharacteristic
434C2#2141511yescharacteristic
435C2#2141611yescharacteristic
436C2#2185612yes—
437C2#2189912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20752C2semidirect
3. C42×C4⋊C4C2semidirect
4. #11226C2semidirect
5. #11209C2semidirect
9. #10878C2semidirect
12. #2863C4semidirect
13. #2863C4semidirect
14. #2860C4semidirect
15. #2860C4semidirect
326. C42#104semidirect

Maximal subgroups

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

subgroupindex
#366412
#207522
C42×C4⋊C42
#112262
#112092
#110752
#110532
#109232
#108782