Finite Group Databasebeta

← #164240

Subgroups — #164240

Order 512.

861 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36302C211yesmaximal, characteristic
2#22636C211yesmaximal, characteristic, complemented
3#20430C211yesmaximal, characteristic, complemented
4#7696C212yesmaximal, complemented
5#7684C211yesmaximal, characteristic
6#7676C212yesmaximal, complemented
7#7664C211yesmaximal, characteristic
8#7628C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7574C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×#424—21no—
13C2×#405—21no—
14#2861C2212yes—
15#2856C2211yescharacteristic
16#2552C411yescharacteristic
17#2548C411yescharacteristic
18#2531—22no—
19#2513—21no—
20#2500—21no—
21#2490C412yes—
22#2476—22no—
23#2427C412yes—
24#2366C411yescharacteristic
25#2363—21no—
26#2355C411yescharacteristic
27#2354—21no—
28C4×C22⋊Q8C2211yescharacteristic
29#2293C2212yes—
30C4×#117C2211yescharacteristic
31#1939—21no—
32#1924—22no—
33#1831C2212yes—
34#1797C2211yescharacteristic
35#1770—21no—
36#1768—22no—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C22⋊C8C2211yescharacteristic
40#1745C2212yes—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×M16—21no—
54C23×C8—21no—
55C2×#127C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×#124C2×C412yes—
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×#121—21no—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×#117—22no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C42C2311yescharacteristic
69#424—28no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76#405—44no—
77C2×#97—22no—
78C2×#97—22no—
79C2×C4×C8—22no—
80C2×C4×C8—21no—
81#398—22no—
82#397—22no—
83#397—22no—
84#394—22no—
85#394—22no—
86#394—22no—
87#391C2×C412yes—
88#391C2311yescharacteristic
89#390—21no—
90#390—21no—
91#390—21no—
92#390—22no—
93#390—21no—
94#390—21no—
95#390C2×C412yes—
96#390C2×C412yes—
97#390C2312yes—
98#389—21no—
99#389C2×C412yes—
100#389C2311yescharacteristic
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#387—21no—
109#387—22no—
110#386—22no—
111#386—22no—
112#385—21no—
113#384—21no—
114#384C2×C411yescharacteristic
115#382—21no—
116#380—22no—
117#380C2×C411yescharacteristic
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130#376D412yes—
131C43C2312yes—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C412yes—
147C23×C4—22no—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—24no—
152C2×M16—42no—
153C2×M16—42no—
154C22×C8—24no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—42no—
162C22×C8—42no—
163#127—42no—
164#125—82no—
165#124—82no—
166C22⋊Q8—22no—
167C22⋊Q8—41no—
168#121—42no—
169#121—42no—
170#121—81no—
171#121—81no—
172#121—82no—
173C4×Q8—42no—
174C4×D4—24no—
175C4×D4—42no—
176C4×D4—44no—
177#117—22no—
178#117—22no—
179#117—24no—
180#117—82no—
181#117—82no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—42no—
216C2×C42C22×C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—28no—
227C4⋊C8—28no—
228C4⋊C8—42no—
229C4⋊C8—82no—
230C22⋊C8—42no—
231C22⋊C8—81no—
232#97—44no—
233#97—44no—
234C4×C8—42no—
235C4×C8—44no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249C24—22no—
250C24C2×C22⋊C411yescharacteristic
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—81no—
265C2×D4—81no—
266C2×D4—82no—
267C2×D4—82no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—82no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4AGL(1,ℤ/8)12yes—
294C22×C4#13712yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298M16—48no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—48no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—82no—
331C4⋊C4—81no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4#10414yes—
335C4⋊C4#13112yes—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—44no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—81no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—24no—
361C42—42no—
362C42—41no—
363C42—41no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—82no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23#41712yes—
382C23C22×C22⋊C411yescharacteristic
383Q8—81no—
384Q8—81no—
385D4—82no—
386D4—82no—
387D4—84no—
388D4—162no—
389D4—164no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—88no—
435C2×C4#41012yes—
436C2×C4#41712yes—
437C2×C4C2×#10414yes—
438C2×C4C2×#10412yes—
439C2×C4C2×#13111yescharacteristic
440C2×C4#56811yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4#57611yescharacteristic
444C2×C4#57711yescharacteristic
445C8—42no—
446C8—42no—
447C8—42no—
448C8—44no—
449C8—81no—
450C8—82no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22#288011yescharacteristic
466C22C2×#41011yescharacteristic
467C22#288811yescharacteristic
468C22C2×#41711yescharacteristic
469C22#289711yescharacteristic
470C22C22×#10411yescharacteristic
471C22#290111yescharacteristic
472C4—21no—
473C4—21no—
474C4—24no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—41no—
484C4—42no—
485C4—81no—
486C4—82no—
487C4—81no—
488C4—81no—
489C4#150914yes—
490C2—42no—
491C2—162no—
492C2#1741811yescharacteristic
493C2#1751511yescharacteristic
494C2#1751711yescharacteristic
495C2C2×#150911yescharacteristic
496C2#1761011yescharacteristic
497C2#1765511yescharacteristic
498C2#1776211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22636C2semidirect
3. #20430C2semidirect
4. #7696C2semidirect
6. #7676C2semidirect
8. #7628C2semidirect
9. #7617C2semidirect
10. #7574C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#363022
#226362
#204302
#76962
#76842
#76762
#76642
#76282
#76172
#75742
#75662