Finite Group Databasebeta

← #214827

Subgroups — #214827

Order 512.

881 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35749C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic
3#17352C211yesmaximal, characteristic
4#9484C212yesmaximal, complemented
5#9394C212yesmaximal, complemented
6#9195C212yesmaximal
7#9033C212yesmaximal
8#8768C212yesmaximal, complemented
9#8390C212yesmaximal, complemented
10C2×#557C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2982C2212yes—
13#2905C2212yes—
14C22×#101—21no—
15#2634C412yes—
16#2621C2211yescharacteristic
17#2611C412yes—
18#2599C412yes—
19#2581C2211yescharacteristic
20#2546C2211yescharacteristic
21#2475—22no—
22#2462—22no—
23#2446—21no—
24#2434—21no—
25#2399C412yes—
26#2396C2211yescharacteristic
27#2365—21no—
28#2362—21no—
29#2356C2211yescharacteristic
30#1801—21no—
31#1801—21no—
32#1801C2212yes—
33#1801C2212yes—
34#1742—21no—
35#1546C2214yes—
36#1487C2214yes—
37#1461—22no—
38#1461C2212yes—
39#1458—22no—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C23×Q8D411yescharacteristic
44C22×M16—22no—
45#557—44no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×#125—22no—
50C2×#125C2×C412yes—
51C2×C22⋊Q8—21no—
52#519—44no—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×#117C2×C412yes—
63C2×#117D411yescharacteristic
64C2×#117D412yes—
65C2×#117C2311yescharacteristic
66C8×Q8—44no—
67#431—24no—
68#431C2314yes—
69#424—44no—
70#423—22no—
71#423C2314yes—
72C2×#101—22no—
73C2×#101—22no—
74C2×#101—44no—
75#400—22no—
76#400C2×C412yes—
77#400C2×C411yescharacteristic
78#399C2×C411yescharacteristic
79#398—22no—
80#398—21no—
81#398—21no—
82#398C2×C412yes—
83#398C2×C412yes—
84#398C2×C412yes—
85#397—21no—
86#397C2×C412yes—
87#396—21no—
88#395—22no—
89#395—22no—
90#395C2×C412yes—
91#395C2×C411yescharacteristic
92#393C2×C411yescharacteristic
93#391—21no—
94#391—21no—
95#391—21no—
96#391—21no—
97#389—22no—
98#389—21no—
99#389—22no—
100#388—22no—
101#388—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#385—22no—
110#385—22no—
111#383C2×C412yes—
112#382C2311yescharacteristic
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2311yescharacteristic
118C4×C22⋊C4—21no—
119#376C2×C412yes—
120#343—22no—
121#343—22no—
122#332—44no—
123Q8⋊C8—44no—
124#324—24no—
125#324—22no—
126#323—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—42no—
132C22×Q8C22⋊C412yes—
133C22×Q8C22×C411yescharacteristic
134C22×Q8C2×D411yescharacteristic
135C22×Q8C2×D411yescharacteristic
136C23×C4C22⋊C412yes—
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—42no—
140C2×M16—44no—
141C2×M16—44no—
142C22×C8—42no—
143C4⋊Q8—24no—
144C4⋊Q8—44no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147#125—44no—
148#125—48no—
149C22⋊Q8—44no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—44no—
154C4×Q8—44no—
155C4×Q8—44no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—44no—
159#117—24no—
160#117—44no—
161#117—44no—
162#117—42no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22×C411yescharacteristic
191C2×C22⋊C4C2×D411yescharacteristic
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C412yes—
199C2×C42C22×C412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203C4⋊C8—24no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—44no—
207C4⋊C8—44no—
208#101—44no—
209#101—44no—
210C22⋊C8—24no—
211C4×C8—24no—
212C4×C8—44no—
213#95—22no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95C22×C411yescharacteristic
229#95C22×C412yes—
230#95C22×C411yescharacteristic
231C24C2×C22⋊C411yescharacteristic
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—48no—
244C2×Q8—48no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8AGL(1,ℤ/8)12yes—
250C2×Q8#13712yes—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4#13514yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274M16—44no—
275M16—44no—
276M16—84no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—84no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—44no—
309C42—24no—
310C42—24no—
311C42—22no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—44no—
316C42—44no—
317C42—42no—
318C42—41no—
319C42—41no—
320C42H(5,𝔽2)14yes—
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23#41612yes—
326C23C22×C22⋊C411yescharacteristic
327Q8—24no—
328Q8—24no—
329Q8—24no—
330Q8—48no—
331Q8—42no—
332Q8—42no—
333Q8—42no—
334Q8—82no—
335Q8—82no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—48no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4#41612yes—
375C2×C4#41714yes—
376C2×C4#51812yes—
377C2×C4#53412yes—
378C2×C4#53412yes—
379C2×C4C2×#13512yes—
380C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
381C2×C4C2×#13711yescharacteristic
382C8—84no—
383C8—84no—
384C22—21no—
385C22—21no—
386C22—22no—
387C22—42no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22#230811yescharacteristic
392C22C2×#41611yescharacteristic
393C22C2×#41711yescharacteristic
394C4—22no—
395C4—24no—
396C4—22no—
397C4—22no—
398C4—42no—
399C4—42no—
400C4—41no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—41no—
405C4—81no—
406C4—82no—
407C4—82no—
408C4—81no—
409C4#318812yes—
410C4#319012yes—
411C2—21no—
412C2—21no—
413C2—42no—
414C2#2176311yescharacteristic
415C2#2178011yescharacteristic
416C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9484C2semidirect
5. #9394C2semidirect
8. #8768C2semidirect
9. #8390C2semidirect

Maximal subgroups

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

subgroupindex
#357492
#206392
#173522
#94842
#93942
#91952
#90332
#87682
#83902