Finite Group Databasebeta

← #215825

Subgroups — #215825

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9874C212yesmaximal, complemented
5#9870C212yesmaximal, complemented
6#9868C212yesmaximal, complemented
7#9864C212yesmaximal, complemented
8#8710C212yesmaximal
9#8707C212yesmaximal
10C2×#533—21no—
11C2×#433—21no—
12#2552C412yes—
13#2551C412yes—
14#2550—22no—
15#2549—22no—
16#2440C412yes—
17#2440C412yes—
18#2399—21no—
19#2399—21no—
20C4×C4⋊Q8C2212yes—
21C4×C4⋊Q8C2212yes—
22#2296—21no—
23#2285C2211yescharacteristic
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1923—22no—
27#1916—22no—
28#1842C2212yes—
29#1832—21no—
30#1798—21no—
31#1797C2212yes—
32#1768—21no—
33#1767C2211yescharacteristic
34C4×C4⋊C8C2211yescharacteristic
35C4×C4⋊C8C2212yes—
36C4×C22⋊C8C2212yes—
37#1744C2211yescharacteristic
38C42×C8C2211yescharacteristic
39#1279C2212yes—
40#1279C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#533—48no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2×C412yes—
51C2×#125—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×#117—21no—
57C2×#117D411yescharacteristic
58C2×#117D411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#433—44no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—21no—
71C2×C4×C8—21no—
72C2×C4×C8—22no—
73C2×C4×C8C2312yes—
74C2×C4×C8C2312yes—
75#398—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#395—22no—
80#395—22no—
81#395—22no—
82#395—22no—
83#391—22no—
84#391—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#385—22no—
94#385—22no—
95#384C2×C412yes—
96#384C2×C412yes—
97#384C2×C412yes—
98#382—21no—
99#380—21no—
100#380C2×C411yescharacteristic
101#380C2×C411yescharacteristic
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4C2312yes—
112C4×C4⋊C4C2312yes—
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4D412yes—
115#376D411yescharacteristic
116#376D411yescharacteristic
117C2×#95—21no—
118C43C2311yescharacteristic
119C43C2311yescharacteristic
120#336—22no—
121#336—21no—
122#336—21no—
123#336—22no—
124#336—22no—
125#336—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C23×C4—22no—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—48no—
141#125—48no—
142C22⋊Q8—48no—
143C22⋊Q8—48no—
144C4×Q8—48no—
145C4×Q8—48no—
146#117—24no—
147#117—44no—
148#117SD1614yes—
149#117Q1614yes—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C22⋊C412yes—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—44no—
182C2×C42C22×C412yes—
183C2×C42C22×C412yes—
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2411yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—44no—
199C4⋊C8—44no—
200C4⋊C8—44no—
201C22⋊C8—44no—
202C22⋊C8—44no—
203#97—42no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—44no—
209C4×C8—42no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223#95C22⋊C412yes—
224C24C2×C22⋊C411yescharacteristic
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—82no—
230C2×Q8—82no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—82no—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×SD1612yes—
254C22×C4C2×Q1612yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C2×C8—24no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4#10318yes—
292C4⋊C4#13114yes—
293C22⋊C4—24no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4#10318yes—
297C42—21no—
298C42—21no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—44no—
311C42—44no—
312C42C2×SD1612yes—
313C42C2×SD1612yes—
314C42C2×Q1612yes—
315C42C2×Q1612yes—
316C42AGL(1,ℤ/8)14yes—
317C422−1+412yes—
318C422−1+412yes—
319C23—21no—
320C23—22no—
321C23—21no—
322C23—22no—
323C23C2×#10312yes—
324C23C22×C22⋊C411yescharacteristic
325Q8—88no—
326Q8—88no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—88no—
362C2×C4C2×#10314yes—
363C2×C4C2×#10314yes—
364C2×C4C2×#10312yes—
365C2×C4#41814yes—
366C2×C4#51911yescharacteristic
367C2×C4#51911yescharacteristic
368C2×C4#53612yes—
369C2×C4#53612yes—
370C2×C4C2×#13112yes—
371C2×C4#56812yes—
372C2×C4C22×SD1611yescharacteristic
373C2×C4C22×Q1611yescharacteristic
374C2×C4C2×AGL(1,ℤ/8)12yes—
375C8—44no—
376C8—82no—
377C8—82no—
378C8—82no—
379C22—22no—
380C22—24no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22#231011yescharacteristic
385C22#288012yes—
386C22C22×#10311yescharacteristic
387C22C2×#41811yescharacteristic
388C22#290112yes—
389C4—22no—
390C4—22no—
391C4—24no—
392C4—24no—
393C4—24no—
394C4—41no—
395C4—41no—
396C4—42no—
397C4—44no—
398C4—82no—
399C4—82no—
400C4—82no—
401C4#298512yes—
402C4#298611yescharacteristic
403C4#298711yescharacteristic
404C4#322712yes—
405C4#323012yes—
406C2—42no—
407C2#1750912yes—
408C2#1751112yes—
409C2#2182411yescharacteristic
410C2#2183411yescharacteristic
411C2#2184611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9874C2semidirect
5. #9870C2semidirect
6. #9868C2semidirect
7. #9864C2semidirect

Maximal subgroups

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

subgroupindex
#349382
#214302
#208122
#98742
#98702
#98682
#98642
#87102
#87072