Finite Group Databasebeta

← #481885

Subgroups — #481885

Order 512.

937 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35739C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#17370C211yesmaximal, characteristic
4#11817C214yesmaximal, complemented
5#11815C214yesmaximal, complemented
6#11724C212yesmaximal, complemented
7#11721C212yesmaximal, complemented
8C2×D4×Q8C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2975—21no—
11#2879—21no—
12#2628C2211yescharacteristic
13#2627C2212yes—
14#2581C411yescharacteristic
15#2567C411yescharacteristic
16#2564C412yes—
17#2451—21no—
18#2449—22no—
19#2440—21no—
20#2438—22no—
21#2409C412yes—
22#2396C2211yescharacteristic
23C2×#395C2212yes—
24#2349C411yescharacteristic
25D4×C4⋊C4—21no—
26#2339—21no—
27#1822—22no—
28#1821C2214yes—
29#1742C2212yes—
30C2×#343—21no—
31#1465—22no—
32#1463C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1354C2214yes—
36#1354C2218yes—
37#1353C2214yes—
38C22×#41D411yescharacteristic
39C23×Q8C2×C412yes—
40C22×M16—22no—
41D4×Q8—44no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8C2×C412yes—
47#520—44no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×C4×D4D411yescharacteristic
57C2×#117D412yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62#436—44no—
63#431—22no—
64#408—44no—
65#407—24no—
66#407—24no—
67#400—21no—
68#400C2×C412yes—
69#400C2311yescharacteristic
70#399—21no—
71#399C2×C411yescharacteristic
72#398—22no—
73#398—22no—
74#398—21no—
75#396—21no—
76#395—22no—
77#395—22no—
78#395C2×C412yes—
79#395C2×C412yes—
80#395C2×C412yes—
81#395C2312yes—
82#395C2311yescharacteristic
83#393C2×C412yes—
84#393C2×C411yescharacteristic
85#391—21no—
86#391—21no—
87#391—22no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—21no—
92#387—22no—
93#387—22no—
94#386—22no—
95#386—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#384—21no—
101#382C2×C411yescharacteristic
102#381—22no—
103#380—21no—
104#380—22no—
105#380C2×C412yes—
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C22⋊C4—21no—
109C2×#95—22no—
110C2×#95C2×C412yes—
111#343—24no—
112#343—22no—
113#343C2314yes—
114#324—24no—
115#324C2314yes—
116#323—22no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—42no—
120C22×Q8—22no—
121C22×Q8—44no—
122C22×Q8—84no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C411yescharacteristic
125C22×Q8C22×C412yes—
126C22×D4C22⋊C412yes—
127C22×D4C22⋊C411yescharacteristic
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C22⋊C412yes—
131C23×C4C22×C412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C2×M16—24no—
136C2×M16—42no—
137C2×M16—44no—
138C2×M16—44no—
139C2×M16—44no—
140C22×C8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—44no—
143C22⋊Q8—48no—
144C22⋊Q8—44no—
145C22⋊Q8—84no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—44no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—44no—
155#117—42no—
156#117—44no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—84no—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2411yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22×C411yescharacteristic
189C2×C22⋊C4C2×D411yescharacteristic
190C2×C42—21no—
191C2×C42—21no—
192C2×C42C22⋊C412yes—
193C2×C42C22⋊C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C4⋊C8—42no—
197C22⋊C8—24no—
198C22⋊C8—48no—
199C22⋊C8—44no—
200C4×C8—42no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95C22×C411yescharacteristic
215#95C22×C412yes—
216#95C22×C411yescharacteristic
217C24C2×C22⋊C412yes—
218C24C2×C22⋊C411yescharacteristic
219#41—48no—
220#41—48no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—88no—
232C2×Q8—88no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—42no—
238C2×D4#9914yes—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4#13714yes—
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272M16—44no—
273M16—44no—
274M16—84no—
275M16—164no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—84no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—82no—
309C42—22no—
310C42—22no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C23—21no—
316C23—22no—
317C23—22no—
318C23—21no—
319C23—42no—
320C23—42no—
321C23—41no—
322C23—41no—
323C23C2×#9914yes—
324C23#41912yes—
325C23C22×C22⋊C411yescharacteristic
326Q8—42no—
327Q8—42no—
328Q8—44no—
329Q8—88no—
330Q8—84no—
331D4—28no—
332D4—48no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—48no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—88no—
374C2×C4—88no—
375C2×C4—162no—
376C2×C4C2×#9912yes—
377C2×C4#41912yes—
378C2×C4#42112yes—
379C2×C4C2×#13712yes—
380C8—84no—
381C8—84no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—24no—
386C22—22no—
387C22—24no—
388C22—42no—
389C22—44no—
390C22—44no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22C22×#9911yescharacteristic
395C22C2×#41911yescharacteristic
396C22C2×#42111yescharacteristic
397C4—22no—
398C4—42no—
399C4—41no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—81no—
408C4—82no—
409C4—82no—
410C4—81no—
411C4#151114yes—
412C2—21no—
413C2—21no—
414C2—44no—
415C2—42no—
416C2C2×#151111yescharacteristic
417C2#3334811yescharacteristic
418C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11817C2semidirect
5. #11815C2semidirect
6. #11724C2semidirect
7. #11721C2semidirect

Maximal subgroups

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

subgroupindex
#357392
#206952
#173702
#118172
#118152
#117242
#117212