Finite Group Databasebeta

← #215688

Subgroups — #215688

Order 512.

784 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3#20777C211yesmaximal, characteristic
4#9901C212yesmaximal, complemented
5#9901C212yesmaximal, complemented
6#9899C212yesmaximal, complemented
7#9897C212yesmaximal, complemented
8#8727C212yesmaximal
9#8726C212yesmaximal
10#2864—21no—
11#2864—21no—
12#2859C2211yescharacteristic
13#2859C2211yescharacteristic
14#2859C2211yescharacteristic
15#2859C2211yescharacteristic
16#2851—21no—
17#2851—21no—
18#2851C412yes—
19#2851C412yes—
20C4×#125—21no—
21C4×#125—21no—
22C4×#125C412yes—
23C4×#125C412yes—
24#2296—21no—
25#2296—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#1839—21no—
30#1839—21no—
31#1831C2212yes—
32#1831C2212yes—
33#1817—21no—
34#1817—21no—
35#1797C2212yes—
36#1797C2212yes—
37C4×C4⋊C8C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C22⋊C8—21no—
40C4×C22⋊C8—21no—
41#1744—21no—
42#1744—21no—
43#1296C2214yes—
44#1296C2214yes—
45#1296C2214yes—
46#1296C2214yes—
47C2×#125—21no—
48C2×#125—21no—
49C2×#125C2×C412yes—
50C2×#125C2×C412yes—
51C2×#117—21no—
52C2×#117—21no—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8C2314yes—
61C2×C4⋊C8C2314yes—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C4×C8—22no—
65C2×C4×C8—22no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#385—22no—
91#385—22no—
92#385—22no—
93#385—22no—
94#384C2311yescharacteristic
95#384C2311yescharacteristic
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98#382—21no—
99#382—21no—
100#382—21no—
101#382—21no—
102#382C2×C412yes—
103#382C2×C412yes—
104#380—21no—
105#380—21no—
106#380C2×C412yes—
107#380C2×C412yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141#125—44no—
142#125—44no—
143#125—44no—
144#125—44no—
145#117—44no—
146#117—44no—
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—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2411yescharacteristic
206C4⋊C8—28no—
207C4⋊C8—28no—
208C22⋊C8—44no—
209C22⋊C8—44no—
210C4×C8—44no—
211C4×C8—44no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224C24C2×C22⋊C411yescharacteristic
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4#13514yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C2×C8—44no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—44no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—44no—
319C42—44no—
320C42#13514yes—
321C422−1+414yes—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23#41612yes—
329C23C22×C22⋊C411yescharacteristic
330C2×C4—21no—
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—28no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4#41312yes—
380C2×C4#41312yes—
381C2×C4#41612yes—
382C2×C4#41612yes—
383C2×C4#41612yes—
384C2×C4#51912yes—
385C2×C4#53612yes—
386C2×C4#53612yes—
387C2×C4C2×#13512yes—
388C2×C4C2×#13512yes—
389C2×C4#57512yes—
390C2×C4#57512yes—
391C8—84no—
392C8—84no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22#231011yescharacteristic
401C22C2×#41311yescharacteristic
402C22C2×#41311yescharacteristic
403C22C2×#41611yescharacteristic
404C22C2×#41611yescharacteristic
405C22#289611yescharacteristic
406C22#289611yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4—82no—
423C4#323214yes—
424C4#323314yes—
425C2—42no—
426C2#1770011yescharacteristic
427C2#1771311yescharacteristic
428C2#1771611yescharacteristic
429C2#1775611yescharacteristic
430C2#2183111yescharacteristic
431C2#2183611yescharacteristic
432C2#2183711yescharacteristic
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. #9901C2semidirect
5. #9901C2semidirect
6. #9899C2semidirect
7. #9897C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#207772
#99012
#98992
#98972
#87272
#87262