Finite Group Databasebeta

← #164468

Subgroups — #164468

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36457C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic, complemented
3#20573C211yesmaximal, characteristic, complemented
4#7699C212yesmaximal, complemented
5#7679C211yesmaximal, characteristic, complemented
6#7679C211yesmaximal, characteristic, complemented
7#7659C212yesmaximal, complemented
8#7635C211yesmaximal, characteristic, complemented
9#7618C212yesmaximal, complemented
10#7584C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic, complemented
12C2×Q82C411yescharacteristic
13C2×#550C411yescharacteristic
14C2×#446—21no—
15#2858C2212yes—
16#2858C2211yescharacteristic
17#2563C411yescharacteristic
18#2563C412yes—
19#2548C412yes—
20#2548C411yescharacteristic
21#2491—22no—
22#2491—21no—
23#2491—21no—
24#2491—22no—
25#2341—21no—
26#2341—21no—
27C4×#124C2211yescharacteristic
28C4×#124C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1771—22no—
36C8×C4⋊C4—21no—
37#1767C2211yescharacteristic
38#1767C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40#1745C2212yes—
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1270C2211yescharacteristic
47#1270C2212yes—
48#1270C2212yes—
49#1270C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2212yes—
53C22×#41—21no—
54Q82—44no—
55#550—44no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#127C2×C412yes—
61C2×#127C2×C411yescharacteristic
62C2×#124—22no—
63C2×#124—21no—
64C2×#124—21no—
65C2×#124—22no—
66C2×#124C2×C411yescharacteristic
67C2×#124C2×C412yes—
68C2×#121—22no—
69C2×#121—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8C2×C412yes—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C412yes—
78#446—28no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×#97—22no—
84C2×#97—21no—
85C2×#97—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—22no—
89#394—22no—
90#394—22no—
91#394—22no—
92#393—22no—
93#393—22no—
94#393—22no—
95#391C2311yescharacteristic
96#390—22no—
97#390—21no—
98#390—21no—
99#390—22no—
100#390—21no—
101#390—21no—
102#390—22no—
103#390—22no—
104#390C2×C412yes—
105#390C2×C412yes—
106#390C2×C412yes—
107#390C2311yescharacteristic
108#390C2312yes—
109#388—22no—
110#388—21no—
111#388—21no—
112#388—22no—
113#382C2×C411yescharacteristic
114#382C2×C412yes—
115#380—22no—
116#380—21no—
117#380—21no—
118#380—22no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C43C2312yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143C2×#41—44no—
144C2×#41—44no—
145C2×#41—44no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22×C411yescharacteristic
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4C22×C411yescharacteristic
155C22×D4C22×C412yes—
156C23×C4—22no—
157C23×C4—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C4⋊Q8—44no—
165C4⋊Q8—42no—
166C4⋊Q8—48no—
167C4⋊Q8—44no—
168#127—44no—
169#127—42no—
170#124—22no—
171#124—24no—
172#124—41no—
173#124—42no—
174#124—82no—
175#124—81no—
176#124—81no—
177#124—82no—
178#124—82no—
179#124—82no—
180#124—81no—
181#124—81no—
182#121—48no—
183#121—44no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—42no—
187C4×Q8—44no—
188C4×Q8—42no—
189C4×Q8—44no—
190C4×D4—42no—
191C4×D4—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C412yes—
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2411yescharacteristic
236C4⋊C8—24no—
237C4⋊C8—28no—
238C4⋊C8—42no—
239C4⋊C8—82no—
240#97—28no—
241#97—81no—
242#97—81no—
243C4×C8—24no—
244C4×C8—42no—
245C4×C8—82no—
246#95—21no—
247#95—22no—
248#95—21no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255C24—21no—
256C24—22no—
257#41—816no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—44no—
265C2×Q8—82no—
266C2×Q8—82no—
267C2×Q8—81no—
268C2×Q8—81no—
269C2×Q8#10414yes—
270C2×Q8#13112yes—
271C2×D4—21no—
272C2×D4—22no—
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—81no—
290C2×D4—81no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4#10014yes—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C42—22no—
364C42—21no—
365C42—22no—
366C42—21no—
367C42—22no—
368C42—21no—
369C42—21no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—21no—
374C42—21no—
375C42—24no—
376C42—22no—
377C42—22no—
378C42—24no—
379C42—22no—
380C42—42no—
381C42—41no—
382C42—44no—
383C42—42no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—42no—
388C23—41no—
389C23—42no—
390C23—41no—
391C23—42no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—42no—
397C23—41no—
398C23C22×C22⋊C411yescharacteristic
399Q8—28no—
400Q8—42no—
401Q8—42no—
402Q8—164no—
403Q8—162no—
404D4—81no—
405D4—82no—
406D4—81no—
407D4—82no—
408D4—88no—
409D4—84no—
410D4—164no—
411D4—162no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—88no—
450C2×C4—84no—
451C2×C4C2×#10014yes—
452C2×C4C2×#10012yes—
453C2×C4C2×#10414yes—
454C2×C4C2×#10412yes—
455C2×C4#42112yes—
456C2×C4C2×#13111yescharacteristic
457C2×C4#56811yescharacteristic
458C2×C4#57611yescharacteristic
459C2×C4#57611yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57711yescharacteristic
462C8—42no—
463C8—44no—
464C8—42no—
465C8—81no—
466C8—82no—
467C8—81no—
468C22—81no—
469C22—82no—
470C22—81no—
471C22—82no—
472C22—81no—
473C22—82no—
474C22—82no—
475C22—81no—
476C22—81no—
477C22—82no—
478C22—82no—
479C22—81no—
480C22—82no—
481C22—81no—
482C22#288011yescharacteristic
483C22#288411yescharacteristic
484C22C22×#10011yescharacteristic
485C22#289711yescharacteristic
486C22#289711yescharacteristic
487C22C22×#10411yescharacteristic
488C22C2×#42111yescharacteristic
489C4—21no—
490C4—21no—
491C4—24no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—24no—
497C4—42no—
498C4—41no—
499C4—41no—
500C4—42no—
501C4—41no—
502C4—42no—
503C4—81no—
504C4—82no—
505C4—81no—
506C4#152714yes—
507C2—161no—
508C2—162no—
509C2#1742511yescharacteristic
510C2#1752011yescharacteristic
511C2#1752611yescharacteristic
512C2#1755011yescharacteristic
513C2#1760211yescharacteristic
514C2C2×#152711yescharacteristic
515C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#550C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22022C2semidirect
3. #20573C2semidirect
4. #7699C2semidirect
5. #7679C2semidirect
6. #7679C2semidirect
7. #7659C2semidirect
8. #7635C2semidirect
9. #7618C2semidirect
10. #7584C2semidirect
11. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364572
#220222
#205732
#76992
#76792
#76592
#76352
#76182
#75842
#75672