Finite Group Databasebeta

← #481559

Subgroups — #481559

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35738C211yesmaximal, characteristic
2#20399C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#11740C214yesmaximal, complemented
5#11734C214yesmaximal, complemented
6#11723C212yesmaximal, complemented
7#11697C212yesmaximal
8C2×#554C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2982—21no—
11C22×#101—21no—
12#2873—21no—
13#2634C2212yes—
14#2628C2211yescharacteristic
15#2611C412yes—
16#2598C412yes—
17#2568C411yescharacteristic
18#2536C411yescharacteristic
19#2477—22no—
20#2465—22no—
21#2421—21no—
22#2409—21no—
23#2398C2212yes—
24#2392C2211yescharacteristic
25#2360—21no—
26#2356C411yescharacteristic
27#2352—21no—
28C4×#101—22no—
29C4×#101C2214yes—
30#1742C2212yes—
31#1461—21no—
32#1461C2214yes—
33#1458—22no—
34#1458C2212yes—
35#1353C2218yes—
36#1346C2214yes—
37#1345C2214yes—
38C23×Q8D411yescharacteristic
39C22×M16—22no—
40#554—44no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125C2×C412yes—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—22no—
46#519—44no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D412yes—
50C2×C4×Q8D411yescharacteristic
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8C2312yes—
53C2×C4×D4—21no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C411yescharacteristic
56C2×#117C2×C412yes—
57C2×#117D412yes—
58C2×#117D411yescharacteristic
59C2×#117C2311yescharacteristic
60C22×C4⋊C4—21no—
61C8×Q8—44no—
62#431—24no—
63#431—22no—
64C2×#101—28no—
65C2×#101—24no—
66#405—44no—
67C4×M16—24no—
68#400—22no—
69#400C2×C412yes—
70#400C2×C412yes—
71#400C2312yes—
72#399C2×C412yes—
73#398C2×C412yes—
74#398C2311yescharacteristic
75#397—22no—
76#397C2×C411yescharacteristic
77#396—21no—
78#395—22no—
79#395—21no—
80#394—21no—
81#394C2×C412yes—
82#394C2311yescharacteristic
83#393—22no—
84#393—21no—
85#392C2×C411yescharacteristic
86#391—21no—
87#391—21no—
88#391—22no—
89#389—22no—
90#389—21no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386—22no—
99#386—22no—
100#385—22no—
101#385—22no—
102#383C2×C412yes—
103#382—21no—
104#382C2×C411yescharacteristic
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—22no—
112#376C2×C412yes—
113#343—22no—
114#343C2314yes—
115#324—24no—
116#324C2314yes—
117#323—22no—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C2×D412yes—
125C22×D4—21no—
126C22×D4C22×C411yescharacteristic
127C23×C4—21no—
128C23×C4—22no—
129C23×C4C22⋊C412yes—
130C23×C4C2×D411yescharacteristic
131C2×M16—28no—
132C2×M16—24no—
133C2×M16—42no—
134C2×M16—44no—
135C22×C8—42no—
136C4⋊Q8—42no—
137#125—44no—
138C22⋊Q8—48no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—44no—
144C4×Q8—44no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×D4—44no—
148C4×D4—42no—
149C4×D4—44no—
150#117—24no—
151#117—42no—
152#117—44no—
153#117—44no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C4⋊C4C2411yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22×C411yescharacteristic
184C2×C22⋊C4C2×D411yescharacteristic
185C2×C42—21no—
186C2×C42—22no—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C4⋊C8—44no—
195C4⋊C8—42no—
196#101—216no—
197C22⋊C8—24no—
198#97—44no—
199C4×C8—44no—
200C4×C8—44no—
201C4×C8—42no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—21no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—22no—
215#95C22×C411yescharacteristic
216#95C22×C412yes—
217#95C22×C411yescharacteristic
218C24—21no—
219C24C2×C22⋊C411yescharacteristic
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×Q8—48no—
229C2×Q8—48no—
230C2×Q8—42no—
231C2×Q8—82no—
232C2×Q8#10414yes—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—81no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4#13514yes—
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265M16—28no—
266M16—48no—
267M16—44no—
268C2×C8—44no—
269C2×C8—48no—
270C2×C8—44no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—82no—
287C4⋊C4—81no—
288C4⋊C4#9914yes—
289C22⋊C4—24no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C22⋊C4—81no—
300C22⋊C4—82no—
301C42—24no—
302C42—24no—
303C42—22no—
304C42—24no—
305C42—22no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—82no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#41612yes—
322C23C22×C22⋊C411yescharacteristic
323Q8—28no—
324Q8—48no—
325Q8—44no—
326Q8—162no—
327D4—42no—
328D4—42no—
329D4—84no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—48no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—84no—
367C2×C4—164no—
368C2×C4C2×#9914yes—
369C2×C4C2×#9912yes—
370C2×C4#41612yes—
371C2×C4C2×#10414yes—
372C2×C4C2×#10412yes—
373C2×C4C2×#13512yes—
374C8—48no—
375C8—84no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22C22×#9911yescharacteristic
391C22C2×#41611yescharacteristic
392C22C22×#10411yescharacteristic
393C4—24no—
394C4—22no—
395C4—24no—
396C4—41no—
397C4—42no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—42no—
402C4—81no—
403C4—82no—
404C4—81no—
405C4—82no—
406C4—81no—
407C4#150114yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—82no—
412C2C2×#150111yescharacteristic
413C2#3334911yescharacteristic
414C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#392C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20399C2semidirect
3. #17352C2semidirect
4. #11740C2semidirect
5. #11734C2semidirect
6. #11723C2semidirect

Maximal subgroups

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

subgroupindex
#357382
#203992
#173522
#117402
#117342
#117232
#116972