Finite Group Databasebeta

← #164860

Subgroups — #164860

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36303C211yesmaximal, characteristic
2#22000C212yesmaximal, complemented
3#7730C212yesmaximal, complemented
4#7722C212yesmaximal, complemented
5#7690C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7643C211yesmaximal, characteristic, complemented
8#7629C211yesmaximal, characteristic, complemented
9#7601C211yesmaximal, characteristic, complemented
10#7595C211yesmaximal, characteristic, complemented
11C2×#1272C211yesmaximal, characteristic, complemented
12C2×D4×Q8C411yescharacteristic
13C2×#547C411yescharacteristic
14#2862C2211yescharacteristic
15#2850C2211yescharacteristic
16#2850C2211yescharacteristic
17#2568C411yescharacteristic
18#2547C411yescharacteristic
19#2503C411yescharacteristic
20#2503C411yescharacteristic
21#2492—22no—
22#2488—22no—
23#2488—22no—
24#2436C411yescharacteristic
25#2436C411yescharacteristic
26#2339—22no—
27C4×#124C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1924—22no—
32#1917—22no—
33#1869—22no—
34#1850—22no—
35#1839C2212yes—
36#1831C2212yes—
37#1817C2212yes—
38#1797C2212yes—
39#1749C2212yes—
40C2×#336C2212yes—
41#1272C2212yes—
42#1272C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1272C2211yescharacteristic
47#1272C2211yescharacteristic
48#1272C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51C22×#41—21no—
52C23×Q8C2×C411yescharacteristic
53C23×Q8C2×C411yescharacteristic
54C23×C8—22no—
55D4×Q8—44no—
56#547—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#123—21no—
64C2×#123—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×#121C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4D412yes—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—22no—
87#400—22no—
88#400—22no—
89#400—22no—
90#397—22no—
91#393—22no—
92#393—22no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391—22no—
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2311yescharacteristic
104#391C2311yescharacteristic
105#391C2311yescharacteristic
106#390C2311yescharacteristic
107#388—22no—
108#388—22no—
109#388—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#382—22no—
115#382C2×C411yescharacteristic
116#382C2×C411yescharacteristic
117#382C2×C411yescharacteristic
118#381—22no—
119#381—22no—
120#380—22no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D412yes—
123C4×C22⋊C4—22no—
124C4×C22⋊C4C2×C411yescharacteristic
125C4×C22⋊C4C2×C411yescharacteristic
126C4×C22⋊C4D412yes—
127C4×C22⋊C4D412yes—
128C2×#95—22no—
129C2×#95—22no—
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2312yes—
139#336C2312yes—
140C2×#41—48no—
141C2×#41—44no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—81no—
152C22×Q8—81no—
153C22×Q8C22×C411yescharacteristic
154C22×Q8C22×C411yescharacteristic
155C22×Q8C22×C411yescharacteristic
156C22×D4—21no—
157C22×D4—21no—
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—44no—
174C22×C8—82no—
175C22×C8—82no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—44no—
179#124—22no—
180#124—41no—
181#124—81no—
182#124—81no—
183#123—44no—
184#123—44no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190#121—42no—
191#121—42no—
192C4×Q8—42no—
193C4×D4—24no—
194C4×D4—42no—
195C4×D4—44no—
196C4×D4—44no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4C22⋊C411yescharacteristic
223C2×C22⋊C4C22⋊C411yescharacteristic
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C42—21no—
227C2×C42—81no—
228C2×C42—81no—
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2411yescharacteristic
238C4⋊C8—82no—
239C4⋊C8—82no—
240C22⋊C8—44no—
241C22⋊C8—82no—
242C22⋊C8—82no—
243#97—82no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259C24—21no—
260C24C2×C22⋊C411yescharacteristic
261C24C2×C22⋊C411yescharacteristic
262#41—816no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—44no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—81no—
281C2×Q8—81no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×Q8—84no—
286C2×Q8#13112yes—
287C2×D4—24no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—44no—
296C2×D4#10014yes—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D412yes—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—88no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—24no—
371C22⋊C4—42no—
372C22⋊C4—44no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C42—21no—
383C42—21no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—42no—
388C42—42no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—42no—
393C42—42no—
394C23—21no—
395C23—21no—
396C23—22no—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23C2×#10012yes—
407C23C2×#10012yes—
408C23#42114yes—
409C23C22×C22⋊C411yescharacteristic
410Q8—42no—
411Q8—42no—
412Q8—81no—
413Q8—81no—
414Q8—81no—
415Q8—81no—
416Q8—84no—
417Q8—84no—
418Q8—84no—
419D4—28no—
420D4—81no—
421D4—81no—
422D4—84no—
423D4—84no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4C2×#10012yes—
477C2×C4C2×#13111yescharacteristic
478C2×C4#56811yescharacteristic
479C2×C4#57712yes—
480C2×C4#57812yes—
481C8—82no—
482C8—82no—
483C8—82no—
484C8—82no—
485C22—22no—
486C22—22no—
487C22—24no—
488C22—24no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—82no—
497C22—82no—
498C22—81no—
499C22#153314yes—
500C22#288011yescharacteristic
501C22#288311yescharacteristic
502C22C22×#10011yescharacteristic
503C22#289812yes—
504C22C2×#42112yes—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—42no—
513C4—41no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—81no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C2—24no—
524C2—42no—
525C2—161no—
526C2#1742111yescharacteristic
527C2#1752412yes—
528C2#1759812yes—
529C2C2×#153311yescharacteristic
530C2#1777911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
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. #22000C2semidirect
3. #7730C2semidirect
4. #7722C2semidirect
5. #7690C2semidirect
6. #7649C2semidirect
7. #7643C2semidirect
8. #7629C2semidirect
9. #7601C2semidirect
10. #7595C2semidirect
11. C2×#1272C2semidirect

Maximal subgroups

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

subgroupindex
#363032
#220002
#77302
#77222
#76902
#76492
#76432
#76292
#76012
#75952
C2×#12722