Finite Group Databasebeta

← #215244

Subgroups — #215244

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#21924C212yesmaximal
3#9709C212yesmaximal, complemented
4#9701C212yesmaximal, complemented
5#9697C212yesmaximal, complemented
6#9693C212yesmaximal, complemented
7#8345C212yesmaximal
8#8342C211yesmaximal, characteristic
9#8342C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#2862C2212yes—
13#2862C2212yes—
14#2642C414yes—
15#2568—22no—
16#2568—22no—
17#2503—22no—
18#2503—22no—
19#2310C412yes—
20#2310C412yes—
21C4×#117C2211yescharacteristic
22#1924—22no—
23#1923—22no—
24#1917—22no—
25#1916—22no—
26#1842C2212yes—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31C4×C4⋊C8C2212yes—
32C2×#333C2211yescharacteristic
33C2×#333C2212yes—
34C2×#333C2211yescharacteristic
35C2×#333C2212yes—
36C2×#333C2211yescharacteristic
37C2×#333C2211yescharacteristic
38C2×#332C2212yes—
39C2×#332C2212yes—
40C2×#332C2212yes—
41C2×#332C2212yes—
42C23×Q8C2×C412yes—
43#533—41no—
44#533—42no—
45#533—41no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—42no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×#125C2×C412yes—
55C2×C22⋊Q8—22no—
56C2×C4×Q8—22no—
57C2×#117C2×C412yes—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C22×C4⋊C4—22no—
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×C4×C8—22no—
72#400—24no—
73#400—24no—
74#400—24no—
75#400—24no—
76#391—24no—
77#391—22no—
78#391—22no—
79#391—22no—
80#391—22no—
81#391—22no—
82#391—22no—
83#391C2×C414yes—
84#391C2×C414yes—
85#391C2×C414yes—
86#391C2×C414yes—
87#382—22no—
88#382—22no—
89#382—22no—
90#382—22no—
91#381—24no—
92#381—22no—
93#381—22no—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C4×C22⋊C4D412yes—
97C4×C22⋊C4D412yes—
98#376C2×C412yes—
99C43C2312yes—
100#336—22no—
101#336—22no—
102#336—22no—
103#336—22no—
104#336—22no—
105#336—22no—
106#333—41no—
107#333—42no—
108#333—41no—
109#333—42no—
110#333—41no—
111#333—41no—
112#332—42no—
113#332—42no—
114#332—42no—
115#332—42no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8C22×C412yes—
123C22×Q8C22×C412yes—
124C23×C4—22no—
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D412yes—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—44no—
144C4⋊Q8—44no—
145#125—42no—
146#125—42no—
147C22⋊Q8—44no—
148C22⋊Q8—44no—
149C4×Q8—44no—
150C4×Q8—44no—
151#117—42no—
152#117—42no—
153#117—42no—
154#117—42no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22×C414yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22⋊C411yescharacteristic
169C2×C22⋊C4C22⋊C411yescharacteristic
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C412yes—
176C2×C42C22×C412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2411yescharacteristic
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—82no—
192C4⋊C8—82no—
193C22⋊C8—82no—
194C22⋊C8—82no—
195#97—82no—
196C4×C8—82no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95C22⋊C412yes—
206#95C22⋊C412yes—
207#95C22⋊C412yes—
208#95C22⋊C412yes—
209C24C2×C22⋊C411yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—44no—
227C2×Q8—88no—
228C2×Q8—88no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D412yes—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C22⋊C4—24no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—44no—
282C42—21no—
283C42—21no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—42no—
295C42—42no—
296C42—44no—
297C42H(5,𝔽2)11yescharacteristic
298C42H(5,𝔽2)12yes—
299C42H(5,𝔽2)11yescharacteristic
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23#41312yes—
305C23C22×C22⋊C411yescharacteristic
306Q8—82no—
307Q8—82no—
308Q8—82no—
309Q8—82no—
310Q8—88no—
311Q8—82no—
312Q8—82no—
313Q8—82no—
314Q8—82no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—88no—
350C2×C4#41312yes—
351C2×C4#51811yescharacteristic
352C2×C4#51811yescharacteristic
353C2×C4#53512yes—
354C2×C4#53512yes—
355C2×C4#57712yes—
356C2×C4#57712yes—
357C2×C4#57812yes—
358C2×C4#57812yes—
359C8—82no—
360C8—82no—
361C8—82no—
362C8—82no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22#230911yescharacteristic
369C22C2×#41311yescharacteristic
370C22C2×#41311yescharacteristic
371C22#289812yes—
372C22#289812yes—
373C4—21no—
374C4—21no—
375C4—22no—
376C4—42no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—82no—
382C4—82no—
383C4—84no—
384C4#322112yes—
385C4#322312yes—
386C4#322512yes—
387C4#322512yes—
388C2—42no—
389C2#1777812yes—
390C2#1778011yescharacteristic
391C2#1778011yescharacteristic
392C2#2179811yescharacteristic
393C2#2181412yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9709C2semidirect
4. #9701C2semidirect
5. #9697C2semidirect
6. #9693C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#219242
#97092
#97012
#96972
#96932
#83452
#83422