Finite Group Databasebeta

← #164854

Subgroups — #164854

Order 512.

811 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#36463C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic, complemented
3#22012C211yesmaximal, characteristic, complemented
4#7718C212yesmaximal, complemented
5#7714C212yesmaximal, complemented
6#7679C211yesmaximal, characteristic, complemented
7#7679C211yesmaximal, characteristic, complemented
8#7647C212yesmaximal, complemented
9#7636C211yesmaximal, characteristic, complemented
10#7599C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12#2858C2211yescharacteristic
13#2854C2212yes—
14#2563C411yescharacteristic
15#2557—22no—
16#2541C411yescharacteristic
17#2513—21no—
18#2490C412yes—
19#2459—22no—
20#2445—21no—
21#2420C412yes—
22#2369C411yescharacteristic
23#2366—21no—
24#2366C411yescharacteristic
25#2357—21no—
26C4×#125C2211yescharacteristic
27C4×#121C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1924—22no—
34#1924—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127—21no—
55C2×#127C2×C411yescharacteristic
56C2×#125C2×C411yescharacteristic
57C2×#124—21no—
58C2×#124—21no—
59C2×#123—22no—
60C2×#121C2×C412yes—
61C2×#121C2×C412yes—
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C22×C42C2312yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—21no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#397—22no—
82#397—22no—
83#394—22no—
84#394—22no—
85#393—22no—
86#392—22no—
87#391—22no—
88#390—21no—
89#390—21no—
90#390—21no—
91#390—22no—
92#390—21no—
93#390—21no—
94#390C2×C412yes—
95#390C2×C412yes—
96#390C2312yes—
97#389—21no—
98#389C2×C412yes—
99#389C2311yescharacteristic
100#389C2311yescharacteristic
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#387—21no—
106#387—21no—
107#386—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#382—21no—
112#382C2×C411yescharacteristic
113#381—22no—
114#380C2×C412yes—
115#380C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—22no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22⋊C411yescharacteristic
146C22×D4C22×C412yes—
147C23×C4—22no—
148C23×C4C22×C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—42no—
158#127—42no—
159#127—44no—
160#125—22no—
161#125—41no—
162#124—81no—
163#124—81no—
164#124—81no—
165#124—81no—
166#123—82no—
167#123—82no—
168#121—24no—
169#121—42no—
170#121—82no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—44no—
175C4×D4—82no—
176C4×D4—82no—
177C4×D4—82no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C22⋊C412yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—82no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2411yescharacteristic
220C4⋊C8—42no—
221C4⋊C8—42no—
222C4⋊C8—82no—
223C4⋊C8—82no—
224C22⋊C8—82no—
225C22⋊C8—82no—
226C22⋊C8—82no—
227C22⋊C8—82no—
228#97—81no—
229#97—81no—
230C4×C8—42no—
231C4×C8—42no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245C24—22no—
246C24C2×C22⋊C412yes—
247C2×Q8—42no—
248C2×Q8—81no—
249C2×Q8—81no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—81no—
265C2×D4—81no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—82no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4#13112yes—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—42no—
361C42—42no—
362C42—41no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—82no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23C22×C22⋊C411yescharacteristic
382Q8—162no—
383D4—44no—
384D4—82no—
385D4—82no—
386D4—88no—
387D4—84no—
388D4—162no—
389D4—164no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—164no—
436C2×C4#41312yes—
437C2×C4#42112yes—
438C2×C4#42112yes—
439C2×C4C2×#13111yescharacteristic
440C2×C4#56811yescharacteristic
441C2×C4#57611yescharacteristic
442C2×C4#57611yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C8—81no—
446C8—81no—
447C8—82no—
448C8—82no—
449C8—81no—
450C8—81no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22#288011yescharacteristic
466C22#288311yescharacteristic
467C22C2×#41311yescharacteristic
468C22#289711yescharacteristic
469C22#289711yescharacteristic
470C22C2×#42111yescharacteristic
471C22C2×#42111yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—41no—
480C4—41no—
481C4—42no—
482C4—41no—
483C4—42no—
484C4—42no—
485C4—82no—
486C4—81no—
487C4—81no—
488C4—81no—
489C4#153514yes—
490C2—82no—
491C2—162no—
492C2#1742911yescharacteristic
493C2#1752211yescharacteristic
494C2#1752611yescharacteristic
495C2#1759911yescharacteristic
496C2#1759911yescharacteristic
497C2C2×#153511yescharacteristic
498C2#1777611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2541C4
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. #22012C2semidirect
4. #7718C2semidirect
5. #7714C2semidirect
6. #7679C2semidirect
7. #7679C2semidirect
8. #7647C2semidirect
9. #7636C2semidirect
10. #7599C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#364632
#220122
#77182
#77142
#76792
#76472
#76362
#75992
#75682