Finite Group Databasebeta

← #381127

Subgroups — #381127

Order 512.

1137 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35618C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic, complemented
3#20393C211yesmaximal, characteristic, complemented
4#11225C212yesmaximal, complemented
5#11149C212yesmaximal, complemented
6#11056C212yesmaximal, complemented
7#10986C212yesmaximal
8#10949C212yesmaximal, complemented
9#10892C212yesmaximal, complemented
10C2×#537C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2905C2212yes—
13C2×#421C2214yes—
14C2×#413C2212yes—
15C2×#410C2212yes—
16#2504C412yescomplemented
17#2503C412yescomplemented
18#2500C412yescomplemented
19#2499C412yescomplemented
20#2444—24no—
21#2421—22no—
22#2420—22no—
23#2317C2211yescharacteristic
24#2311C2211yescharacteristic
25#2297C2211yescharacteristic
26#2297C2211yescharacteristic
27C2×#382C2211yescharacteristic
28#2274C2212yes—
29#1801—22no—
30#1800—22no—
31C4×#99—24no—
32C2×#343C2212yes—
33#1740C2212yes—
34#1736C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C22×#41C2311yescharacteristic
39C22×M16C2314yes—
40#537—44no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8C2×C412yes—
43C2×#121—21no—
44C2×#121—22no—
45C2×#121C2×C412yes—
46#520—44no—
47C2×C4×Q8D412yes—
48C2×C4×D4D414yes—
49C2×C4×D4D412yes—
50C2×#117C2×C412yes—
51C2×#117C2314yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#424—44no—
59#423—24no—
60#421—28no—
61#421—44no—
62C2×#104—44no—
63#413—44no—
64C2×#101—22no—
65C2×#100—22no—
66#410—28no—
67C2×#99—24no—
68#400—24no—
69#397—24no—
70#397—24no—
71#394—24no—
72#391—21no—
73#391C2×C412yes—
74#391C2×C411yescharacteristic
75#391C2×C411yescharacteristic
76#390—21no—
77#390C2×C412yes—
78#390C2×C411yescharacteristic
79#390C2×C411yescharacteristic
80#389—22no—
81#388—24no—
82#387—24no—
83#386—22no—
84#386—24no—
85#386—22no—
86#386—22no—
87#385—22no—
88#384C2×C412yes—
89#382—22no—
90#382C2×C412yes—
91#382C2×C412yes—
92#382C2×C412yes—
93#381—24no—
94#380—22no—
95C4×C4⋊C4—44no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4C2×C412yes—
100C2×#95—22no—
101#343—24no—
102#342—24no—
103#340—44no—
104#339—28no—
105H(3,ℤ/4)—44no—
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—24no—
134C22×C8—24no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137#121—44no—
138#121—48no—
139#121—44no—
140C4×Q8—44no—
141C4×D4—44no—
142C4×D4—48no—
143#117—24no—
144#117—22no—
145#117—24no—
146#117—22no—
147#117—24no—
148#117—24no—
149#117—44no—
150#117#4118yes—
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—44no—
159C2×C4⋊C4C22⋊C414yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C2×D412yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4C22⋊C414yes—
175C2×C22⋊C4C2×D412yes—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—21no—
179C2×C42—44no—
180C2×C42C22⋊C414yes—
181C2×C42C22×C412yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D412yes—
185C4⋊C8—48no—
186#104—416no—
187#101—44no—
188#100—44no—
189#99—216no—
190#95—24no—
191#95—24no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—24no—
197#95—24no—
198#95—44no—
199C24—21no—
200C24C2×C22⋊C412yes—
201C24C22×D411yescharacteristic
202#41—44no—
203#41—44no—
204#41—44no—
205#41—44no—
206#41—44no—
207#41—44no—
208#41—44no—
209#41—44no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—44no—
231C2×D4—84no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—84no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C414yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C2×#4114yes—
261C22×C4H(5,𝔽2)14yes—
262C22×C4H(5,𝔽2)12yes—
263C22×C42−1+412yes—
264M16—44no—
265M16—44no—
266M16—44no—
267M16—44no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—28no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—44no—
283C4⋊C4—84no—
284C22⋊C4—24no—
285C22⋊C4—28no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—82no—
292C22⋊C4—84no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C42—24no—
297C42—22no—
298C42—22no—
299C42—24no—
300C42—28no—
301C42—44no—
302C42—42no—
303C42—44no—
304C42—164no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23C22×C22⋊C411yescharacteristic
318C23#52512yes—
319C23#53412yes—
320C23#53712yes—
321Q8—42no—
322Q8—42no—
323Q8—44no—
324D4—48no—
325D4—44no—
326D4—42no—
327D4—42no—
328D4—84no—
329D4—164no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—48no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—48no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—164no—
362C2×C4—162no—
363C2×C4C4×#4112yes—
364C2×C4#51812yes—
365C2×C4#52012yes—
366C2×C4#53212yes—
367C2×C4#53412yes—
368C2×C4#53511yescharacteristic
369C2×C4#53611yescharacteristic
370C2×C4#53814yes—
371C2×C4#53914yes—
372C2×C4#54014yes—
373C2×C4#54214yes—
374C8—84no—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#229711yescharacteristic
392C22#230811yescharacteristic
393C22#231111yescharacteristic
394C22#263014yes—
395C4—44no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—88no—
403C4—82no—
404C4—81no—
405C4#233514yes—
406C4#266112yes—
407C4#266312yes—
408C2—21no—
409C2—21no—
410C2—24no—
411C2—82no—
412C2—161no—
413C2#1428711yescharacteristic
414C2#2178411yescharacteristic
415C2#2178511yescharacteristic
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. #20640C2semidirect
3. #20393C2semidirect
4. #11225C2semidirect
5. #11149C2semidirect
6. #11056C2semidirect
8. #10949C2semidirect
9. #10892C2semidirect
16. #2504C4semidirect
17. #2503C4semidirect
18. #2500C4semidirect
19. #2499C4semidirect

Maximal subgroups

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

subgroupindex
#356182
#206402
#203932
#112252
#111492
#110562
#109862
#109492
#108922