Finite Group Databasebeta

← #164398

Subgroups — #164398

Order 512.

845 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36453C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic, complemented
3#20558C211yesmaximal, characteristic, complemented
4#7701C212yesmaximal, complemented
5#7683C211yesmaximal, characteristic, complemented
6#7680C211yesmaximal, characteristic, complemented
7#7655C212yesmaximal, complemented
8#7635C211yesmaximal, characteristic, complemented
9#7620C212yesmaximal, complemented
10#7586C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic, complemented
12C2×C8×D4—21no—
13#2859C2211yescharacteristic
14#2855C2212yes—
15#2576—22no—
16#2538C411yescharacteristic
17#2537C411yescharacteristic
18#2494—21no—
19#2494C412yes—
20#2460—22no—
21#2427C412yes—
22#2420—21no—
23#2361—21no—
24#2355C411yescharacteristic
25D4×C4⋊C4C411yescharacteristic
26#2345—21no—
27C4×#124C2211yescharacteristic
28C4×#121C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1817C2212yes—
34#1797C2212yes—
35#1770—21no—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1752—22no—
39#1749C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C23×C8—22no—
54C2×#125—21no—
55C2×#124C2×C411yescharacteristic
56C2×C22⋊Q8C2×C412yes—
57C2×#121—21no—
58C2×#121C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C22×C4⋊C4—21no—
67C22×C42C2312yes—
68C8×D4—28no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×#97—21no—
76C2×#97—22no—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80#401—22no—
81#397—22no—
82#397—22no—
83#396—22no—
84#394—22no—
85#394—22no—
86#391C2×C412yes—
87#391C2311yescharacteristic
88#390—21no—
89#390C2311yescharacteristic
90#389—21no—
91#389—21no—
92#389—21no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2312yes—
96#388—22no—
97#388—21no—
98#388—21no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#386—22no—
105#386—22no—
106#385—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#382—21no—
111#382—21no—
112#382C2×C411yescharacteristic
113#381—22no—
114#380—21no—
115#380—21no—
116#380—22no—
117#380C2×C412yes—
118#380C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—22no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141C22×Q8C22×C411yescharacteristic
142C22×D4—22no—
143C22×D4C22⋊C411yescharacteristic
144C22×D4C22×C411yescharacteristic
145C23×C4—21no—
146C23×C4—22no—
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—28no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—82no—
158C22×C8—82no—
159#125—81no—
160#125—81no—
161#124—22no—
162#124—41no—
163C22⋊Q8—82no—
164#121—24no—
165#121—42no—
166#121—44no—
167C4×Q8—42no—
168C4×D4—24no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—82no—
173C4×D4—82no—
174C4×D4—82no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—82no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—24no—
219C4⋊C8—42no—
220C4⋊C8—82no—
221C22⋊C8—28no—
222C22⋊C8—82no—
223C22⋊C8—82no—
224#97—81no—
225#97—81no—
226#97—82no—
227C4×C8—24no—
228C4×C8—42no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244C24—21no—
245C24C2×C22⋊C412yes—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—82no—
258C2×D4#10414yes—
259C2×D4#13112yes—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—88no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—82no—
327C4⋊C4—81no—
328C4⋊C4—82no—
329C4⋊C4—81no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—42no—
356C42—41no—
357C42—42no—
358C42—42no—
359C42—82no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23C2×#10414yes—
374C23C22×C22⋊C411yescharacteristic
375Q8—81no—
376Q8—81no—
377D4—28no—
378D4—42no—
379D4—42no—
380D4—88no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4—164no—
428C2×C4#41312yes—
429C2×C4C2×#10412yes—
430C2×C4#42112yes—
431C2×C4C2×#13111yescharacteristic
432C2×C4#56811yescharacteristic
433C2×C4#57611yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57711yescharacteristic
436C2×C4#57811yescharacteristic
437C8—42no—
438C8—44no—
439C8—42no—
440C8—81no—
441C8—82no—
442C8—81no—
443C22—24no—
444C22—24no—
445C22—24no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22#288011yescharacteristic
458C22#288311yescharacteristic
459C22C2×#41311yescharacteristic
460C22#289711yescharacteristic
461C22#289811yescharacteristic
462C22C22×#10411yescharacteristic
463C22C2×#42111yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—42no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—42no—
475C4—42no—
476C4—81no—
477C4—82no—
478C4—81no—
479C4—82no—
480C4—81no—
481C4#153214yes—
482C2—44no—
483C2—82no—
484C2#1742411yescharacteristic
485C2#1751711yescharacteristic
486C2#1752311yescharacteristic
487C2#1754811yescharacteristic
488C2#1759911yescharacteristic
489C2C2×#153211yescharacteristic
490C2#1778111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22012C2semidirect
3. #20558C2semidirect
4. #7701C2semidirect
5. #7683C2semidirect
6. #7680C2semidirect
7. #7655C2semidirect
8. #7635C2semidirect
9. #7620C2semidirect
10. #7586C2semidirect
11. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364532
#220122
#205582
#77012
#76832
#76802
#76552
#76352
#76202
#75862
#75672