Finite Group Databasebeta

← #162798

Subgroups — #162798

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2#20752C211yesmaximal, characteristic
3C4×#423C211yesmaximal, characteristic
4#7697C214yesmaximal, complemented
5#7663C212yesmaximal, complemented
6#7660C212yesmaximal, complemented
7#7632C214yesmaximal
8C2×#423—21no—
9C2×#423—21no—
10C2×C4×M16—21no—
11#2864C412yes—
12#2864C2212yes—
13#2859C411yescharacteristic
14#2859C411yescharacteristic
15#2859C2211yescharacteristic
16#2859C2211yescharacteristic
17#2851—21no—
18#2851—21no—
19#2851—21no—
20#2851—21no—
21C4×#125—21no—
22C4×#125—21no—
23C4×#125—21no—
24C4×#125—21no—
25#2296C412yes—
26#2296C2212yes—
27#2285C411yescharacteristic
28#2285C411yescharacteristic
29C2×C43C2211yescharacteristic
30#1831—21no—
31#1831—21no—
32#1831C2212yes—
33#1831C2212yes—
34#1767—21no—
35#1767C2212yes—
36C4×C4⋊C8—21no—
37C4×C4⋊C8C2212yes—
38#1728—21no—
39#1728C2212yes—
40#1728C2212yes—
41#1271C2218yes—
42#1271C2218yes—
43C22×M16—22no—
44C22×M16—22no—
45C2×#125—21no—
46C2×#125—21no—
47C2×#125—21no—
48C2×#125—21no—
49C2×#117C2×C412yes—
50C2×#117C2×C412yes—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58#423—28no—
59#423—44no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C4×M16—44no—
65C2×#97—22no—
66C2×C4×C8—22no—
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#389C2×C412yes—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2312yes—
82#389C2312yes—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#385—22no—
92#385—22no—
93#385—22no—
94#385—22no—
95#384C2×C411yescharacteristic
96#384C2×C411yescharacteristic
97#384C2×C411yescharacteristic
98#384C2×C411yescharacteristic
99#382—21no—
100#382—21no—
101#382—21no—
102#382—21no—
103#382C2×C412yes—
104#382C2×C412yes—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C43C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336C2314yes—
130#336C2314yes—
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—28no—
139C2×M16—44no—
140C2×M16—44no—
141C2×M16—44no—
142C2×M16—44no—
143C2×M16—44no—
144C22×C8—24no—
145C22×C8—24no—
146#125—44no—
147#125—44no—
148#125—42no—
149#125—42no—
150#125—42no—
151#125—42no—
152#117—44no—
153#117—44no—
154C2×C4⋊C4—21no—
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—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—42no—
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×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2411yescharacteristic
215C4⋊C8—28no—
216C4⋊C8—28no—
217C4⋊C8—44no—
218C4⋊C8—44no—
219#97—44no—
220C4×C8—44no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233C24C2×C22⋊C411yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275M16—416no—
276M16—88no—
277C2×C8—28no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—44no—
282C2×C8—44no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C42—22no—
308C42—22no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—44no—
334C42—44no—
335C42#10114yes—
336C42#10414yes—
337C42AGL(1,ℤ/8)12yes—
338C42AGL(1,ℤ/8)12yes—
339C42#13712yes—
340C42#13712yes—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23#41012yes—
348C23C22×C22⋊C411yescharacteristic
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—28no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4C2×#10112yes—
399C2×C4C2×#10112yes—
400C2×C4C2×#10112yes—
401C2×C4#41712yes—
402C2×C4#41712yes—
403C2×C4#41712yes—
404C2×C4#41712yes—
405C2×C4C2×#10412yes—
406C2×C4C2×#10412yes—
407C2×C4C2×#10412yes—
408C2×C4#42112yes—
409C2×C4#56812yes—
410C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
411C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
412C2×C4C2×#13711yescharacteristic
413C2×C4C2×#13711yescharacteristic
414C8—48no—
415C8—84no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22#287911yescharacteristic
424C22C2×#41011yescharacteristic
425C22C22×#10111yescharacteristic
426C22C2×#41711yescharacteristic
427C22C2×#41711yescharacteristic
428C22C22×#10411yescharacteristic
429C22C2×#42111yescharacteristic
430C4—22no—
431C4—22no—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—41no—
444C4—41no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—82no—
449C4#156114yes—
450C2—42no—
451C2#1739411yescharacteristic
452C2#1749811yescharacteristic
453C2#1750411yescharacteristic
454C2#1754611yescharacteristic
455C2#1758811yescharacteristic
456C2#1764811yescharacteristic
457C2C2×#156111yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7697C2semidirect
5. #7663C2semidirect
6. #7660C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#207522
C4×#4232
#76972
#76632
#76602
#76322