Finite Group Databasebeta

← #162771

Subgroups — #162771

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3C2×#1767C211yesmaximal, characteristic
4#7697C214yesmaximal, complemented
5#7691C214yesmaximal
6#7625C212yesmaximal, complemented
7#7622C212yesmaximal, complemented
8C22×C4⋊C8—21no—
9C22×#97—21no—
10#2864C2212yes—
11#2863—21no—
12#2863—21no—
13#2862C411yescharacteristic
14#2859C411yescharacteristic
15#2857C411yescharacteristic
16#2857C411yescharacteristic
17#2855—21no—
18#2855—21no—
19#2851C2212yes—
20C4×C4⋊Q8C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×#125C412yes—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25#2295—21no—
26#2295—21no—
27#2283C412yes—
28C2×C43C2211yescharacteristic
29#1839C2212yes—
30#1831C2212yes—
31#1817—21no—
32#1797—21no—
33#1767—21no—
34#1767—21no—
35#1767—21no—
36#1767C2212yes—
37#1767C2212yes—
38#1749C2212yes—
39#1745—21no—
40C2×#336C2212yes—
41#1271C2218yes—
42#1271C2218yes—
43C23×C8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—28no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×#97—22no—
64C2×#97—22no—
65C2×#97—22no—
66C2×#97—44no—
67#391—21no—
68#391—21no—
69#391—21no—
70#391—21no—
71#391—21no—
72#391—21no—
73#391—21no—
74#391—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2312yes—
86#389C2312yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#385—22no—
96#385—22no—
97#384C2×C411yescharacteristic
98#384C2×C411yescharacteristic
99#383—22no—
100#382C2×C412yes—
101#382C2×C412yes—
102#382C2×C412yes—
103#381—22no—
104#380—21no—
105#380—21no—
106#380C2×C412yes—
107#380C2×C412yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C22⋊C4—22no—
115#376—22no—
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336C2314yes—
129#336C2314yes—
130C22×Q8—21no—
131C22×Q8—21no—
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C22×C8—28no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—44no—
143C22×C8—44no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146#125—44no—
147C22⋊Q8—44no—
148C22⋊Q8—44no—
149C4×Q8—44no—
150C4×Q8—42no—
151C4×Q8—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
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×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—28no—
210C4⋊C8—44no—
211C22⋊C8—44no—
212#97—44no—
213#97—44no—
214#97—44no—
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—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229C24C2×C22⋊C411yescharacteristic
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
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—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4#10114yes—
261C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C2×C8—28no—
277C2×C8—416no—
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—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C42—22no—
303C42—22no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—44no—
331C42—44no—
332C42AGL(1,ℤ/8)12yes—
333C42#13712yes—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23C2×#10112yes—
339C23C2×#10412yes—
340C23#42112yes—
341C23C22×C22⋊C411yescharacteristic
342Q8—84no—
343Q8—84no—
344C2×C4—24no—
345C2×C4—24no—
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—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—28no—
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—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
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#41012yes—
397C2×C4C2×#10112yes—
398C2×C4C2×#10112yes—
399C2×C4#41712yes—
400C2×C4#41712yes—
401C2×C4C2×#10412yes—
402C2×C4C2×#10412yes—
403C2×C4#56812yes—
404C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C4C2×#13711yescharacteristic
406C2×C4#57512yes—
407C8—48no—
408C8—84no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22#153114yes—
416C22#287911yescharacteristic
417C22C2×#41011yescharacteristic
418C22C22×#10111yescharacteristic
419C22C2×#41711yescharacteristic
420C22#289611yescharacteristic
421C22C22×#10411yescharacteristic
422C22C2×#42111yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—21no—
428C4—21no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4—82no—
442C2—24no—
443C2#1740411yescharacteristic
444C2#1749211yescharacteristic
445C2#1750211yescharacteristic
446C2#1754611yescharacteristic
447C2#1756011yescharacteristic
448C2C2×#153111yescharacteristic
449C2#1775111yescharacteristic
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
6. #7625C2semidirect
7. #7622C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#207782
C2×#17672
#76972
#76912
#76252
#76222