Finite Group Databasebeta

← #164755

Subgroups — #164755

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36460C211yesmaximal, characteristic
2#22639C211yesmaximal, characteristic, complemented
3#20774C211yesmaximal, characteristic, complemented
4#7726C212yesmaximal, complemented
5#7684C211yesmaximal, characteristic
6#7672C212yesmaximal, complemented
7#7664C211yesmaximal, characteristic
8#7649C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7584C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12#2860C2212yes—
13#2856C2211yescharacteristic
14#2571C411yescharacteristic
15#2559—22no—
16#2548C411yescharacteristic
17#2526—22no—
18#2512—21no—
19#2500—21no—
20#2426C412yes—
21#2420C412yes—
22#2366C411yescharacteristic
23#2363—21no—
24#2357—21no—
25Q8×C4⋊C4C411yescharacteristic
26C4×#124C2212yes—
27C4×C22⋊Q8C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1923—22no—
31#1840—22no—
32#1839C2212yes—
33#1839C2212yes—
34#1839C2212yes—
35#1797C2211yescharacteristic
36C8×C4⋊C4—21no—
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8C2211yescharacteristic
39#1729—21no—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×M16—21no—
54C23×C8—21no—
55C2×#127C2×C411yescharacteristic
56C2×#126—22no—
57C2×#124C2×C412yes—
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×#121—21no—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C412yes—
64C2×C4×D4—21no—
65C2×C4×D4C2×C412yes—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75#400—22no—
76#397—22no—
77#396—22no—
78#394—22no—
79#394—22no—
80#394—22no—
81#391C2311yescharacteristic
82#390—21no—
83#390—21no—
84#390—21no—
85#390—22no—
86#390—21no—
87#390—21no—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2312yes—
91#389—21no—
92#389—22no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2311yescharacteristic
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#387—21no—
103#386—22no—
104#386—22no—
105#385—21no—
106#384—21no—
107#384C2×C411yescharacteristic
108#382—21no—
109#382C2×C412yes—
110#381—22no—
111#380C2×C411yescharacteristic
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C22⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4D412yes—
124#376D412yes—
125C43C2312yes—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2312yes—
135#336C2311yescharacteristic
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8C22×C411yescharacteristic
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C22×C412yes—
145C23×C4—22no—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—42no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8—42no—
161#127—42no—
162#126—82no—
163#126—82no—
164#124—24no—
165#124—42no—
166C22⋊Q8—22no—
167C22⋊Q8—41no—
168#121—42no—
169#121—42no—
170#121—81no—
171#121—81no—
172#121—82no—
173C4×Q8—42no—
174C4×Q8—82no—
175C4×D4—44no—
176C4×D4—82no—
177#117—22no—
178#117—22no—
179#117—42no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C412yes—
216C2×C42C22×C412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2411yescharacteristic
224C4⋊C8—42no—
225C4⋊C8—44no—
226C22⋊C8—42no—
227C22⋊C8—81no—
228C4×C8—42no—
229C4×C8—82no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243C24—22no—
244C24C2×C22⋊C411yescharacteristic
245C2×Q8—21no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8#13112yes—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—81no—
259C2×D4—81no—
260C2×D4—82no—
261C2×D4—82no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—44no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4#13512yes—
288C22×C4#13512yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—84no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—84no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—82no—
327C4⋊C4—81no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—81no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—82no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23#41612yes—
376C23C22×C22⋊C411yescharacteristic
377Q8—42no—
378Q8—42no—
379D4—82no—
380D4—82no—
381D4—84no—
382D4—162no—
383D4—164no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—164no—
430C2×C4#41012yes—
431C2×C4#41612yes—
432C2×C4#42112yes—
433C2×C4C2×#13111yescharacteristic
434C2×C4#56811yescharacteristic
435C2×C4C2×#13511yescharacteristic
436C2×C4C2×#13511yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57711yescharacteristic
439C8—42no—
440C8—82no—
441C8—81no—
442C8—81no—
443C8—81no—
444C8—82no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22#288011yescharacteristic
460C22C2×#41011yescharacteristic
461C22#288811yescharacteristic
462C22C2×#41611yescharacteristic
463C22#289711yescharacteristic
464C22C2×#42111yescharacteristic
465C22#290111yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—41no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—41no—
479C4—81no—
480C4—82no—
481C4—81no—
482C4—81no—
483C4#152114yes—
484C2—42no—
485C2—162no—
486C2#1741511yescharacteristic
487C2#1751011yescharacteristic
488C2#1752611yescharacteristic
489C2C2×#152111yescharacteristic
490C2#1760711yescharacteristic
491C2#1768111yescharacteristic
492C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22639C2semidirect
3. #20774C2semidirect
4. #7726C2semidirect
6. #7672C2semidirect
8. #7649C2semidirect
9. #7617C2semidirect
10. #7584C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#364602
#226392
#207742
#77262
#76842
#76722
#76642
#76492
#76172
#75842
#75662