Finite Group Databasebeta

← #216201

Subgroups — #216201

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2652C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic
3#21902C211yesmaximal, characteristic
4#9880C212yesmaximal, complemented
5#9650C212yesmaximal, complemented
6#9081C212yesmaximal
7#9081C212yesmaximal
8#8562C212yesmaximal, complemented
9#8562C212yesmaximal, complemented
10C22×C22⋊C8C2212yes—
11#2652—24no—
12#2652—24no—
13C2×#400C411yescharacteristic
14C2×#400C411yescharacteristic
15C2×#400C411yescharacteristic
16C2×#398C411yescharacteristic
17C2×#397C411yescharacteristic
18C2×#397C411yescharacteristic
19C2×#397C411yescharacteristic
20C2×#396C411yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#387C2211yescharacteristic
23C2×#387C2211yescharacteristic
24C2×#387C2211yescharacteristic
25C2×#385C2211yescharacteristic
26C2×#385C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1923—24no—
29#1850—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1779C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#324C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C24×C4C2311yescharacteristic
40C23×C8—22no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—44no—
66C2×C22⋊C8C2314yes—
67C2×C22⋊C8C2314yes—
68C2×C4×C8—22no—
69#400—24no—
70#400—24no—
71#400—24no—
72#398—24no—
73#397—24no—
74#397—24no—
75#397—24no—
76#396—24no—
77#389—22no—
78#389—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#385—22no—
86#385—22no—
87#385—22no—
88#385—22no—
89C4×C22⋊C4D414yes—
90C4×C22⋊C4D414yes—
91C2×#95C2×C411yescharacteristic
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#324—44no—
112#324—44no—
113C25C22×C411yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—81no—
117C23×C4—81no—
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2411yescharacteristic
134C22×C8—24no—
135C22×C8—24no—
136C22×C8—44no—
137C22×C8—82no—
138C22×C8—82no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
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⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4—81no—
159C2×C4⋊C4—81no—
160C2×C4⋊C4—81no—
161C2×C4⋊C4—81no—
162C2×C4⋊C4—81no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—81no—
172C2×C22⋊C4—81no—
173C2×C22⋊C4C22⋊C414yes—
174C2×C22⋊C4C22⋊C414yes—
175C2×C42—81no—
176C2×C42—81no—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C4⋊C8—82no—
182C4⋊C8—82no—
183C4⋊C8—82no—
184C22⋊C8—24no—
185C22⋊C8—24no—
186C22⋊C8—24no—
187C22⋊C8—24no—
188C22⋊C8—44no—
189C22⋊C8—82no—
190C22⋊C8—82no—
191C4×C8—82no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95C22⋊C414yes—
209#95C22⋊C414yes—
210C24—22no—
211C24—22no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24C2×C22⋊C412yes—
217C24#13114yes—
218C24C23×C411yescharacteristic
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4#13114yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4H(5,𝔽2)12yes—
285C22×C42−1+412yes—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—88no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C22⋊C4—28no—
308C22⋊C4—28no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—28no—
328C23—44no—
329C23—44no—
330C23—82no—
331C23—82no—
332C23—82no—
333C23—82no—
334C23#40814yes—
335C23C22×C22⋊C411yescharacteristic
336C23#52012yes—
337C23C2×#13112yes—
338C23C2×#13112yes—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—84no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4#40814yes—
400C2×C4#53511yescharacteristic
401C2×C4#53611yescharacteristic
402C2×C4#53712yes—
403C8—84no—
404C8—84no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—24no—
412C22—24no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—88no—
419C22#231111yescharacteristic
420C22C2×#40811yescharacteristic
421C22C2×#40811yescharacteristic
422C22#288311yescharacteristic
423C22#288311yescharacteristic
424C22#288911yescharacteristic
425C22#288911yescharacteristic
426C22#299414yes—
427C4—21no—
428C4—22no—
429C4—21no—
430C4—24no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—81no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—81no—
443C2—24no—
444C2—44no—
445C2#1738711yescharacteristic
446C2#1738711yescharacteristic
447C2#1738711yescharacteristic
448C2#1738811yescharacteristic
449C2#2184811yescharacteristic
450C2#2187611yescharacteristic
451C2#2188311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9880C2semidirect
5. #9650C2semidirect
8. #8562C2semidirect
9. #8562C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26522
#219302
#219022
#98802
#96502
#90812
#85622