Finite Group Databasebeta

← #215230

Subgroups — #215230

Order 512.

944 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#36645C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3#20778C211yesmaximal, characteristic
4#9705C212yesmaximal, complemented
5#9702C212yesmaximal, complemented
6#9701C214yesmaximal, complemented
7#8343C214yesmaximal
8C2×#533C2212yes—
9C2×#432—21no—
10#2868C2211yescharacteristic
11#2868C2211yescharacteristic
12#2862C2212yes—
13#2859—21no—
14#2859—21no—
15#2857—21no—
16#2857—21no—
17#2857C412yes—
18#2857C412yes—
19C4×C4⋊Q8—21no—
20C4×C4⋊Q8—21no—
21C4×C4⋊Q8C412yes—
22C4×C4⋊Q8C412yes—
23#2285—21no—
24#2285—21no—
25C2×C43C2211yescharacteristic
26#1842C2212yes—
27#1839—21no—
28#1831—21no—
29#1831C2212yes—
30#1817—21no—
31#1797—21no—
32#1797C2212yes—
33#1797C2212yes—
34#1767—21no—
35#1767C2212yes—
36#1767C2212yes—
37#1749—21no—
38#1745—21no—
39C2×#333C2218yes—
40C2×#332C2218yes—
41#533—48no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2312yes—
47C2×C4⋊Q8C2312yes—
48C2×#125—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C414yes—
51C2×C4×Q8—22no—
52C2×C4×Q8C2×C414yes—
53C2×#117—22no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61#432—44no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2314yes—
65C2×C4⋊C8C2314yes—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×#97—22no—
69C2×#97—22no—
70#391—22no—
71#391—22no—
72#391C2×C414yes—
73#391C2×C414yes—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384C2×C412yes—
89#384C2×C412yes—
90#382—22no—
91#382—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C43C2×C411yescharacteristic
101C43C2×C411yescharacteristic
102C43D411yescharacteristic
103C43D411yescharacteristic
104C43D411yescharacteristic
105C43D411yescharacteristic
106C43C2311yescharacteristic
107C43C2311yescharacteristic
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114#333—48no—
115#332—48no—
116C22×Q8—22no—
117C22×Q8C22×C414yes—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C22×C8—24no—
126C22×C8—24no—
127C4⋊Q8—28no—
128C4⋊Q8—28no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134#125—48no—
135C22⋊Q8—48no—
136C22⋊Q8—48no—
137C4×Q8—48no—
138C4×Q8—48no—
139#117—48no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22×C414yes—
154C2×C22⋊C4—24no—
155C2×C42—24no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2411yescharacteristic
192C4⋊C8—24no—
193C4⋊C8—28no—
194C4⋊C8—24no—
195C4⋊C8—44no—
196C4⋊C8—44no—
197C22⋊C8—44no—
198C22⋊C8—44no—
199#97—44no—
200#97—44no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—24no—
205C24C2×C22⋊C411yescharacteristic
206C2×Q8—28no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—44no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4#13714yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C4⋊C4—28no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C22⋊C4—48no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—21no—
276C42—22no—
277C42—22no—
278C42—21no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42—48no—
299C42AGL(1,ℤ/8)12yes—
300C42#13712yes—
301C42#13714yes—
302C42H(5,𝔽2)14yes—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23#41912yes—
310C23C22×C22⋊C411yescharacteristic
311Q8—88no—
312Q8—84no—
313Q8—84no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4#41312yes—
357C2×C4#41312yes—
358C2×C4#41712yes—
359C2×C4#41712yes—
360C2×C4#41912yes—
361C2×C4#41912yes—
362C2×C4#41912yes—
363C2×C4#51812yes—
364C2×C4#53512yes—
365C2×C4#53512yes—
366C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4C2×#13712yes—
369C2×C4C2×#13712yes—
370C2×C4#57512yes—
371C8—84no—
372C8—84no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22#230911yescharacteristic
381C22C2×#41311yescharacteristic
382C22C2×#41311yescharacteristic
383C22C2×#41911yescharacteristic
384C22C2×#41911yescharacteristic
385C22C2×#41711yescharacteristic
386C22#289611yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—84no—
400C4—84no—
401C4#321412yes—
402C4#321612yes—
403C4#321814yes—
404C2—42no—
405C2#1773411yescharacteristic
406C2#1774811yescharacteristic
407C2#1774911yescharacteristic
408C2#1774911yescharacteristic
409C2#2180011yescharacteristic
410C2#2180911yescharacteristic
411C2#2181111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9705C2semidirect
5. #9702C2semidirect
6. #9701C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#208102
#207782
#97052
#97022
#97012
#83432