Finite Group Databasebeta

← #497510

Subgroups — #497510

Order 512.

939 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38954C211yesmaximal, characteristic
2D4×#101C212yesmaximal
3#13301C212yesmaximal, complemented
4#13094C214yesmaximal, complemented
5#12862C212yesmaximal, complemented
6#12802C211yesmaximal, characteristic, complemented
7C4×#356C211yesmaximal, characteristic, complemented
8#12586C212yesmaximal, complemented
9C2×#556C411yescharacteristic
10C2×D4×Q8C2211yescharacteristic
11D4×M16—22no—
12C22×#101C2214yes—
13#2862C411yescharacteristic
14#2698—24no—
15#2621C412yes—
16#2568C2211yescharacteristic
17#2546C411yescharacteristic
18#2499C412yes—
19#2451—22no—
20#2447—22no—
21#2434C2212yes—
22#2310C2212yes—
23C4×C4⋊Q8C2211yescharacteristic
24#2293C411yescharacteristic
25C2×#356C2212yes—
26C2×#356C2212yes—
27#1966—22no—
28#1901—24no—
29#1801C2212yes—
30#1784C2214yes—
31C4×#101C2212yes—
32#1394C2212yes—
33#1394C2214yes—
34#1394C2214yes—
35#1394C2211yescharacteristic
36#1394C2211yescharacteristic
37C23×Q8C2312yes—
38C22×M16—24no—
39#556—42no—
40#556—42no—
41D4×Q8—42no—
42D4×Q8—42no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8D412yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#126—22no—
49C2×#124—21no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8D414yes—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4D412yes—
58C2×#117C2×C412yes—
59C2×#117C2×C412yes—
60#436—42no—
61#435—44no—
62C8×D4—42no—
63#423—22no—
64C2×#101—28no—
65C2×#101—24no—
66C2×#101—24no—
67C2×#101—44no—
68C2×#101C2314yes—
69C2×#101C2312yes—
70C2×#101C2312yes—
71#407—24no—
72C4×M16—22no—
73#400—22no—
74#398—24no—
75#398—22no—
76#397—24no—
77#397—24no—
78#395—24no—
79#391—24no—
80#391—22no—
81#391—22no—
82#391C2×C411yescharacteristic
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2311yescharacteristic
87#391C2312yes—
88#390—21no—
89#389—22no—
90#389—21no—
91#389—22no—
92#388—22no—
93#387—24no—
94#385—24no—
95#384C2×C411yescharacteristic
96#382—22no—
97#382C2×C412yes—
98#381—24no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C411yescharacteristic
101C4×C22⋊C4C2×C412yes—
102C43C2×C411yescharacteristic
103#356—24no—
104#356—42no—
105#343—22no—
106#343—22no—
107C22×Q8—24no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8—44no—
112C22×Q8—82no—
113C22×Q8C22×C411yescharacteristic
114C22×Q8C22×C412yes—
115C22×Q8C2×D414yes—
116C22×Q8C2411yescharacteristic
117C22×D4C22⋊C411yescharacteristic
118C23×C4C22×C412yes—
119C23×C4C2×D414yes—
120C2×M16—22no—
121C2×M16—24no—
122C2×M16—22no—
123C2×M16—44no—
124C2×M16—48no—
125C2×M16—84no—
126C22×C8—44no—
127C4⋊Q8—22no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—82no—
137#126—48no—
138#124—42no—
139#124—42no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—24no—
154C4×D4—22no—
155#117—44no—
156#117—82no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C414yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C2×D412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C414yes—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C4⋊C8—42no—
189C4⋊C8—42no—
190#101—22no—
191#101—22no—
192#101—24no—
193#101—24no—
194#101—22no—
195#101—48no—
196#101—84no—
197#101#4114yes—
198C22⋊C8—44no—
199C22⋊C8—44no—
200#97—42no—
201C4×C8—42no—
202#95—22no—
203#95—22no—
204#95—21no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95—24no—
211C24C2×C22⋊C412yes—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—44no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—84no—
230C2×Q8—84no—
231C2×Q8—84no—
232C2×Q8—84no—
233C2×Q8C2×#4112yes—
234C2×Q8H(5,𝔽2)11yescharacteristic
235C2×Q82−1+411yescharacteristic
236C2×D4—42no—
237C2×D4#9912yes—
238C2×D4#9912yes—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4C2×C22⋊C414yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D412yes—
261M16—24no—
262M16—42no—
263M16—42no—
264M16—44no—
265M16—44no—
266M16—42no—
267M16—88no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—44no—
271C2×C8—88no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—82no—
290C4⋊C4—84no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—82no—
298C42—22no—
299C42—22no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—41no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—44no—
309C42—41no—
310C42—41no—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23C2×#9914yes—
316C23C22×C22⋊C411yescharacteristic
317Q8—42no—
318Q8—41no—
319Q8—41no—
320Q8—84no—
321Q8—84no—
322Q8—82no—
323Q8—82no—
324Q8—81no—
325Q8—81no—
326Q8—82no—
327Q8—82no—
328D4—24no—
329D4—42no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4C2×#9911yescharacteristic
368C2×C4C2×#9911yescharacteristic
369C2×C4C4×#4111yescharacteristic
370C2×C4#52011yescharacteristic
371C2×C4#54612yes—
372C2×C4#54712yes—
373C8—44no—
374C8—84no—
375C8—82no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—24no—
380C22—24no—
381C22—82no—
382C22—82no—
383C22#233911yescharacteristic
384C22C22×#9911yescharacteristic
385C22#288311yescharacteristic
386C4—21no—
387C4—22no—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—42no—
392C4—41no—
393C4—42no—
394C4—42no—
395C4—44no—
396C4—81no—
397C4—82no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4#213312yes—
402C2—21no—
403C2—21no—
404C2—44no—
405C2#2247611yescharacteristic
406C2C2×#213311yescharacteristic
407C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×D4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13301C2semidirect
4. #13094C2semidirect
5. #12862C2semidirect
6. #12802C2semidirect
7. C4×#356C2semidirect
8. #12586C2semidirect

Maximal subgroups

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

subgroupindex
#389542
D4×#1012
#133012
#130942
#128622
#128022
C4×#3562
#125862