Finite Group Databasebeta

← #381137

Subgroups — #381137

Order 512.

1175 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35530C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#20397C211yesmaximal, characteristic, complemented
4#11210C212yesmaximal, complemented
5#11149C212yesmaximal, complemented
6#11072C212yesmaximal, complemented
7#10986C212yesmaximal
8#10959C212yesmaximal, complemented
9#10889C212yesmaximal, complemented
10C2×#537C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#421C2214yes—
13C2×#413C2212yes—
14C2×#410C2212yes—
15#2873C2212yes—
16#2502C412yescomplemented
17#2501C412yescomplemented
18#2500C412yescomplemented
19#2499C412yescomplemented
20#2443—24no—
21#2423—22no—
22#2422—22no—
23#2317C2211yescharacteristic
24#2316C2212yes—
25#2311C2211yescharacteristic
26#2297C2211yescharacteristic
27#2297C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#1799—24no—
30C4×#101—22no—
31C4×#100—22no—
32C2×#343C2212yes—
33#1740C2212yes—
34#1736C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C22×#41C2311yescharacteristic
39C22×M16C2314yes—
40#537—44no—
41C2×#123—24no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8C2×C412yes—
44C2×#121—21no—
45C2×#121—22no—
46C2×#121C2×C412yes—
47#520—44no—
48C2×C4×Q8D412yes—
49C2×C4×D4D414yes—
50C2×C4×D4D412yes—
51C2×#117C2×C412yes—
52C2×#117C2314yes—
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42C2311yescharacteristic
60#421—28no—
61#421—44no—
62C2×#104—44no—
63#413—28no—
64C2×#101—22no—
65C2×#100—22no—
66#410—44no—
67C2×#99—24no—
68#405—44no—
69C4×M16—24no—
70#399—24no—
71#398—24no—
72#397—24no—
73#394—24no—
74#391—21no—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#391C2×C411yescharacteristic
78#390—21no—
79#390C2×C412yes—
80#390C2×C411yescharacteristic
81#390C2×C411yescharacteristic
82#389—22no—
83#389—44no—
84#388—24no—
85#387—24no—
86#386—22no—
87#386—22no—
88#386—22no—
89#385—24no—
90#385—22no—
91#384—22no—
92#384C2×C412yes—
93#384C2×C412yes—
94#384C2×C412yes—
95#382C2×C412yes—
96#380—22no—
97#380—24no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4C2×C412yes—
100C4×C22⋊C4C2×C412yes—
101#343—24no—
102#342—24no—
103#340—44no—
104#339—44no—
105H(3,ℤ/4)—28no—
106C2×#41—24no—
107C2×#41—24no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—24no—
111C2×#41—24no—
112C2×#41—22no—
113C2×#41—22no—
114C22×Q8C22×C411yescharacteristic
115C22×Q8C2×D411yescharacteristic
116C22×D4—21no—
117C22×D4—22no—
118C22×D4C22×C411yescharacteristic
119C22×D4C2×D412yes—
120C22×D4C2×D411yescharacteristic
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C22⋊C414yes—
125C23×C4C22×C412yes—
126C23×C4C2×D412yes—
127C23×C4C2411yescharacteristic
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16#4118yes—
134C22×C8—24no—
135#123—88no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138#121—44no—
139#121—48no—
140#121—44no—
141C4×Q8—44no—
142C4×D4—44no—
143C4×D4—48no—
144#117—24no—
145#117—24no—
146#117—24no—
147#117—24no—
148#117—24no—
149#117—24no—
150#117—44no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—84no—
163C2×C4⋊C4C22⋊C414yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4C22⋊C414yes—
179C2×C22⋊C4C2×D412yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—24no—
184C2×C42C22⋊C414yes—
185C2×C42C22×C412yes—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D412yes—
189#104—416no—
190#101—28no—
191#100—28no—
192#99—48no—
193#97—44no—
194C4×C8—44no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—44no—
202C24—21no—
203C24C2×C22⋊C412yes—
204C24C22×D411yescharacteristic
205#41—44no—
206#41—44no—
207#41—44no—
208#41—44no—
209#41—44no—
210#41—44no—
211#41—44no—
212#41—44no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—44no—
234C2×D4—84no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—84no—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C414yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C2×#4114yes—
264C22×C4H(5,𝔽2)14yes—
265C22×C4H(5,𝔽2)12yes—
266C22×C42−1+412yes—
267M16—28no—
268M16—28no—
269M16—44no—
270M16—44no—
271C2×C8—28no—
272C2×C8—44no—
273C2×C8—44no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C4⋊C4—164no—
289C22⋊C4—24no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—84no—
301C42—24no—
302C42—24no—
303C42—22no—
304C42—22no—
305C42—44no—
306C42—42no—
307C42—44no—
308C42—44no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C22×C22⋊C411yescharacteristic
322C23#53212yes—
323C23#53412yes—
324C23#53511yescharacteristic
325C23#53611yescharacteristic
326Q8—44no—
327Q8—44no—
328D4—44no—
329D4—44no—
330D4—44no—
331D4—44no—
332D4—82no—
333D4—82no—
334D4—164no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—48no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—164no—
367C2×C4—162no—
368C2×C4C4×#4112yes—
369C2×C4#51812yes—
370C2×C4#52012yes—
371C2×C4#52512yes—
372C2×C4#53412yes—
373C2×C4#53712yes—
374C2×C4#53814yes—
375C2×C4#53914yes—
376C2×C4#54014yes—
377C2×C4#54214yes—
378C8—48no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#229711yescharacteristic
397C22#230811yescharacteristic
398C22#231111yescharacteristic
399C22#266112yes—
400C22#266312yes—
401C4—42no—
402C4—42no—
403C4—44no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4—82no—
408C4—82no—
409C4—81no—
410C4—164no—
411C4#233514yes—
412C4#263014yes—
413C2—21no—
414C2—21no—
415C2—22no—
416C2—22no—
417C2—82no—
418C2—161no—
419C2#1428711yescharacteristic
420C2#2178211yescharacteristic
421C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20634C2semidirect
3. #20397C2semidirect
4. #11210C2semidirect
5. #11149C2semidirect
6. #11072C2semidirect
8. #10959C2semidirect
9. #10889C2semidirect
16. #2502C4semidirect
17. #2501C4semidirect
18. #2500C4semidirect
19. #2499C4semidirect

Maximal subgroups

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

subgroupindex
#355302
#206342
#203972
#112102
#111492
#110722
#109862
#109592
#108892