Finite Group Databasebeta

← #164355

Subgroups — #164355

Order 512.

941 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36463C211yesmaximal, characteristic
2#20567C211yesmaximal, characteristic, complemented
3#20558C211yesmaximal, characteristic, complemented
4#7679C211yesmaximal, characteristic, complemented
5#7679C211yesmaximal, characteristic, complemented
6#7678C212yesmaximal, complemented
7#7674C212yesmaximal, complemented
8#7636C211yesmaximal, characteristic, complemented
9#7633C212yesmaximal, complemented
10#7568C211yesmaximal, characteristic, complemented
11#7565C212yesmaximal, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14#2858C2211yescharacteristic
15#2854C2212yes—
16#2563C411yescharacteristic
17#2557—22no—
18#2541C411yescharacteristic
19#2513—21no—
20#2490C412yes—
21#2459—22no—
22#2445—21no—
23#2420C412yes—
24#2369C411yescharacteristic
25#2366—21no—
26#2366C411yescharacteristic
27#2357—21no—
28C4×#125C2211yescharacteristic
29C4×#121C2212yes—
30C42×D4C2211yescharacteristic
31#1770—21no—
32C8×C4⋊C4—21no—
33#1767C2211yescharacteristic
34#1767C2211yescharacteristic
35#1753—22no—
36#1752—22no—
37#1749C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2211yescharacteristic
40#1744C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×M16—22no—
54C23×C8—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#127—21no—
57C2×#127C2×C411yescharacteristic
58C2×#125C2×C411yescharacteristic
59C2×#124—21no—
60C2×#124—21no—
61C2×#123—22no—
62C2×#121C2×C412yes—
63C2×#121C2×C412yes—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C22×C42C2312yes—
71#436—28no—
72C8×D4—28no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—22no—
78C2×#97—21no—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—22no—
83#397—22no—
84#397—22no—
85#394—22no—
86#394—22no—
87#393—22no—
88#392—22no—
89#391—22no—
90#390—21no—
91#390—21no—
92#390—21no—
93#390—22no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2×C412yes—
98#390C2312yes—
99#389—21no—
100#389C2×C412yes—
101#389C2311yescharacteristic
102#389C2311yescharacteristic
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—21no—
108#387—21no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#382—21no—
114#382C2×C411yescharacteristic
115#381—22no—
116#380C2×C412yes—
117#380C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22⋊C411yescharacteristic
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4C22×C412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C2×M16—28no—
154C2×M16—82no—
155C2×M16—82no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—28no—
163C22×C8—82no—
164C22×C8—82no—
165C4⋊Q8—42no—
166#127—42no—
167#127—44no—
168#125—22no—
169#125—41no—
170#124—81no—
171#124—81no—
172#124—81no—
173#124—81no—
174#123—82no—
175#123—82no—
176#121—24no—
177#121—42no—
178#121—82no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—42no—
182C4×D4—44no—
183C4×D4—82no—
184C4×D4—82no—
185C4×D4—82no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
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⋊C4—22no—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—82no—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2411yescharacteristic
228C4⋊C8—24no—
229C4⋊C8—24no—
230C22⋊C8—28no—
231C22⋊C8—28no—
232#97—81no—
233#97—81no—
234#97—82no—
235C4×C8—24no—
236C4×C8—24no—
237C4×C8—82no—
238#95—22no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—21no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—22no—
252C24C2×C22⋊C412yes—
253C2×Q8—42no—
254C2×Q8—81no—
255C2×Q8—81no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—44no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—81no—
270C2×D4—81no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4#10414yes—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305M16—88no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—88no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—82no—
341C4⋊C4#10414yes—
342C4⋊C4#13112yes—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—42no—
369C42—42no—
370C42—41no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—82no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23C2×#10414yes—
389C23C22×C22⋊C411yescharacteristic
390Q8—162no—
391D4—28no—
392D4—82no—
393D4—82no—
394D4—88no—
395D4—84no—
396D4—162no—
397D4—164no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—164no—
443C2×C4#41312yes—
444C2×C4C2×#10414yes—
445C2×C4C2×#10412yes—
446C2×C4C2×#10412yes—
447C2×C4C2×#13111yescharacteristic
448C2×C4#56811yescharacteristic
449C2×C4#57611yescharacteristic
450C2×C4#57611yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57711yescharacteristic
453C8—42no—
454C8—42no—
455C8—44no—
456C8—42no—
457C8—44no—
458C8—42no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22#288011yescharacteristic
474C22#288311yescharacteristic
475C22C2×#41311yescharacteristic
476C22#289711yescharacteristic
477C22#289711yescharacteristic
478C22C22×#10411yescharacteristic
479C22C22×#10411yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—24no—
487C4—42no—
488C4—41no—
489C4—42no—
490C4—41no—
491C4—41no—
492C4—42no—
493C4—81no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4#152414yes—
498C2—44no—
499C2—162no—
500C2#1742911yescharacteristic
501C2#1751711yescharacteristic
502C2#1752011yescharacteristic
503C2#1754711yescharacteristic
504C2#1754711yescharacteristic
505C2C2×#152411yescharacteristic
506C2#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. #20567C2semidirect
3. #20558C2semidirect
4. #7679C2semidirect
5. #7679C2semidirect
6. #7678C2semidirect
7. #7674C2semidirect
8. #7636C2semidirect
9. #7633C2semidirect
10. #7568C2semidirect
11. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#364632
#205672
#205582
#76792
#76782
#76742
#76362
#76332
#75682
#75652