Finite Group Databasebeta

← #192156

Subgroups — #192156

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic
2#21941C212yesmaximal
3C22×#324C211yesmaximal, characteristic
4#13923C211yesmaximal, characteristic
5#8857C212yesmaximal, complemented
6#8856C212yesmaximal, complemented
7#8855C214yesmaximal, complemented
8#8012C212yesmaximal
9C23×C4⋊C4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2345—22no—
12#2345—22no—
13#2344—24no—
14#2316C2212yes—
15#2313—21no—
16#2313C2212yes—
17#2280—21no—
18#2280—21no—
19#2280C2212yes—
20C2×#380C2211yescharacteristic
21#2274—21no—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1854—24no—
24#1853C2214yes—
25#1852C2214yes—
26#1808—22no—
27#1807—22no—
28C2×#324C418yescomplemented
29C2×#324C2212yes—
30C2×#324C2211yescharacteristic
31C2×#324C2211yescharacteristic
32#1441C2212yes—
33#1441C2212yes—
34#1436—21no—
35#1436—21no—
36C22⋊#95—21no—
37C22⋊#95—21no—
38#1294C2214yes—
39#1294C2214yes—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117D414yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4Q814yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C42—21no—
60C2×#107—24no—
61C2×#106—24no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8C2×C418yes—
65C2×C22⋊C8C2312yes—
66C2×C22⋊C8C2312yes—
67#389—24no—
68#389—24no—
69#387—24no—
70#387—24no—
71#387—24no—
72#387—24no—
73#381—22no—
74#381—22no—
75#381—22no—
76#380—22no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380D412yes—
85#380D412yes—
86#380C2314yes—
87#379—22no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—24no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95C2312yes—
103#340—24no—
104#340—24no—
105#324C2×C418yes—
106#324C2×C414yes—
107#324C2×C414yes—
108C25C2×D411yescharacteristic
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—42no—
119C23×C4—82no—
120C23×C4—82no—
121C23×C4C4⋊C414yes—
122C23×C4C4⋊C414yes—
123C23×C4C22×C412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×Q812yes—
132C23×C4#4114yes—
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—28no—
137C22×C8—84no—
138#117—44no—
139#117—44no—
140#117—84no—
141#117—84no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4C4⋊C414yes—
161C2×C4⋊C4C4⋊C414yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×Q812yes—
167C2×C4⋊C4C2×Q812yes—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—42no—
183C2×C42—82no—
184#107—84no—
185#106—84no—
186C22⋊C8—28no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8C22×C414yes—
190C22⋊C8C22×C414yes—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—24no—
197#95—22no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—44no—
202#95—42no—
203#95—44no—
204#95—82no—
205#95—82no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—42no—
211C24C2×C4⋊C412yes—
212C24#12011yescharacteristic
213C24#12011yescharacteristic
214C24#12312yes—
215C24C22×D411yescharacteristic
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4—82no—
264C22×C4—84no—
265C22×C4C2×C4⋊C412yes—
266C22×C4C2×C4⋊C412yes—
267C22×C4C2×C4⋊C412yes—
268C22×C4C2×C4⋊C412yes—
269C22×C4C2×C4⋊C412yes—
270C22×C4C4×D414yes—
271C22×C4C4×D414yes—
272C22×C4#12012yes—
273C22×C4C22⋊Q814yes—
274C22×C4C22⋊Q814yes—
275C22×C4#12312yes—
276C22×C4#13514yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D412yes—
279C22×C4C22×Q811yescharacteristic
280C22×C4C2×#4112yes—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—88no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—48no—
301C22⋊C4—48no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C42—48no—
309C42—84no—
310C42—84no—
311C42—84no—
312C23—22no—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—24no—
318C23—24no—
319C23—24no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—42no—
327C23—42no—
328C23—44no—
329C23—44no—
330C23—44no—
331C23—44no—
332C23#38512yes—
333C23#38512yes—
334C23#45412yes—
335C23#45612yes—
336C23C22×C4⋊C411yescharacteristic
337C23C2×#12011yescharacteristic
338C23C2×#12311yescharacteristic
339C23C2×#13512yes—
340C23#57512yes—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—84no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4#38514yes—
394C2×C4#42714yes—
395C2×C4C2×C4×D412yes—
396C2×C4C2×C22⋊Q812yes—
397C8—44no—
398C8—44no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—24no—
408C22—22no—
409C22—22no—
410C22—24no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—48no—
415C22—44no—
416C22—44no—
417C22—42no—
418C22—42no—
419C22#185714yes—
420C22C2×#38511yescharacteristic
421C22C2×#42711yescharacteristic
422C22#291111yescharacteristic
423C22#300212yes—
424C22C2×#45411yescharacteristic
425C22C2×#45611yescharacteristic
426C22#302612yes—
427C22#309112yes—
428C4—24no—
429C4—22no—
430C4—22no—
431C4—24no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—84no—
436C4—84no—
437C4—84no—
438C4—162no—
439C2—22no—
440C2—24no—
441C2—21no—
442C2—21no—
443C2—42no—
444C2#1837812yes—
445C2#1837912yes—
446C2#2084311yescharacteristic
447C2C2×#185711yescharacteristic
448C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8857C2semidirect
6. #8856C2semidirect
7. #8855C2semidirect
28. C2×#324C4semidirect

Maximal subgroups

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

subgroupindex
#341782
#219412
C22×#3242
#139232
#88572
#88562
#88552
#80122