Finite Group Databasebeta

← #192515

Subgroups — #192515

Order 512.

803 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60291C211yesmaximal, characteristic
2#21979C211yesmaximal, characteristic, complemented
3#17817C211yesmaximal, characteristic, complemented
4#17567C211yesmaximal, characteristic
5#17492C211yesmaximal, characteristic
6#8877C214yesmaximal, complemented
7#8876C214yesmaximal, complemented
8#8070C211yesmaximal, characteristic, complemented
9#7933C211yesmaximal, characteristic, complemented
10C2×#529C2211yescharacteristic
11#2995—21no—
12#2995—21no—
13#2992C2212yes—
14#2992C2212yes—
15#2983—21no—
16#2974—22no—
17#2973—21no—
18#2968—22no—
19#2967—22no—
20#2924—22no—
21#2920C2211yescharacteristic
22#2920C2211yescharacteristic
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26#2283C2211yescharacteristic
27#1940—22no—
28#1851—22no—
29#1843C2212yescomplemented
30#1832C2212yescomplemented
31#1574—22no—
32#1573C2214yescomplemented
33#1567—21no—
34#1566—21no—
35#1521C414yes—
36#1520C412yes—
37#1519C412yes—
38#1500C2214yes—
39#1495—22no—
40#1487—21no—
41#1486—21no—
42#1483C2214yes—
43#1300C2214yescomplemented
44#1283C2212yescomplemented
45#1283C2212yescomplemented
46C22×M16—21no—
47#529—21no—
48#529—21no—
49#529—21no—
50#529—21no—
51#529C2×C412yes—
52#529D412yes—
53#529Q812yes—
54#529C2312yes—
55C2×#126—21no—
56C2×#125Q811yescharacteristic
57C2×#124Q811yescharacteristic
58C2×#123—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×#121C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4D411yescharacteristic
64C2×#117—21no—
65C22×C42C2311yescharacteristic
66#446—22no—
67#446—22no—
68#446—22no—
69#446—22no—
70#436—22no—
71#436—22no—
72#435—22no—
73#435—22no—
74#435—22no—
75#435—22no—
76#435—22no—
77#435—22no—
78#435—22no—
79#435—22no—
80C8×D4—22no—
81C8×D4—22no—
82#433—22no—
83#433—22no—
84#433—22no—
85#433—22no—
86#432—22no—
87#432—21no—
88#432—21no—
89#432—21no—
90#432—21no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432C2312yes—
95#431—22no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431C2×C412yes—
102#431D412yes—
103#431C2312yes—
104#424—22no—
105#424—22no—
106#423—21no—
107#423—21no—
108#423—21no—
109#423—21no—
110#423—21no—
111#423D411yescharacteristic
112#423D411yescharacteristic
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—24no—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119#407—22no—
120#407—21no—
121#407—21no—
122C2×C22⋊C8—22no—
123C2×C22⋊C8C2×C412yes—
124C2×C22⋊C8D412yes—
125#405—22no—
126#405—22no—
127C4×M16—21no—
128C4×M16C2312yes—
129C2×#97D412yes—
130C2×C4×C8—21no—
131#383C2×C412yes—
132#382—21no—
133#381—21no—
134#380C2×C412yes—
135C4×C4⋊C4—21no—
136C4×C22⋊C4—21no—
137#376C2312yes—
138#336—24no—
139#335—24no—
140#334—24no—
141#328C2×C418yes—
142#327C2×C418yes—
143Q8⋊C8—24no—
144D4⋊C8—24no—
145C22×Q8C4⋊C411yescharacteristic
146C22×D4—21no—
147C22×D4C4⋊C411yescharacteristic
148C23×C4—21no—
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4#4111yescharacteristic
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—22no—
157C2×M16—41no—
158C2×M16—41no—
159C2×M16—82no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166#126—22no—
167#126—22no—
168#126—22no—
169#126—22no—
170#125—21no—
171#125—21no—
172#125C4⋊C412yes—
173#125C2×Q812yes—
174#124—21no—
175#124—21no—
176#124C4⋊C412yes—
177#124C2×Q812yes—
178#123—22no—
179#123—22no—
180#123—22no—
181#123—22no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8C4⋊C412yes—
185C22⋊Q8C22×C412yes—
186#121—21no—
187#121—21no—
188#121C4⋊C412yes—
189#121C22×C412yes—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8C4⋊C412yes—
193C4×Q8C2×D412yes—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4C4⋊C412yes—
201C4×D4C2×D412yes—
202#117—22no—
203#117—22no—
204#117—22no—
205#117—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C4⋊C411yescharacteristic
212C2×C4⋊C4C4⋊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×C42—21no—
220C2×C42—22no—
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q811yescharacteristic
228C2×C42C2×Q811yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—84no—
252C4⋊C8C2×D412yes—
253C4⋊C8C2×D412yes—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C22⋊C8C22×C414yes—
266#97—22no—
267#97—22no—
268#97—22no—
269#97—22no—
270#97—22no—
271#97—22no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—22no—
277C4×C8—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95C22×C414yes—
282C24—21no—
283C24C4×D411yescharacteristic
284C2×Q8—21no—
285C2×Q8—81no—
286C2×Q8C2×C4⋊C412yes—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—81no—
291C2×D4—82no—
292C2×D4C2×C4⋊C412yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4C2×C4⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4C22⋊Q811yescharacteristic
315C22×C4C22⋊Q811yescharacteristic
316C22×C4C22⋊Q811yescharacteristic
317C22×C4C22⋊Q811yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323M16—44no—
324M16—81no—
325M16—81no—
326M16—82no—
327M16—82no—
328M16—82no—
329M16—82no—
330C2×C8—24no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—44no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—82no—
341C2×C8—82no—
342C2×C8—82no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—82no—
354C4⋊C4C2×C4⋊C412yes—
355C4⋊C4C2×C4⋊C412yes—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—82no—
367C42—22no—
368C42—22no—
369C42—42no—
370C42—42no—
371C42C2×C4⋊C411yescharacteristic
372C42C2×C4⋊C411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42#12011yescharacteristic
376C42#12011yescharacteristic
377C42C22⋊Q811yescharacteristic
378C42C22⋊Q811yescharacteristic
379C42C22⋊Q811yescharacteristic
380C42C22⋊Q811yescharacteristic
381C42#12311yescharacteristic
382C42#12311yescharacteristic
383C42C22×D411yescharacteristic
384C42C22×D411yescharacteristic
385C42C22×Q811yescharacteristic
386C42C2×#4111yescharacteristic
387C23—22no—
388C23—22no—
389C23—41no—
390C23—41no—
391C23—42no—
392C23—81no—
393C23—81no—
394C23#38511yescharacteristic
395C23#38511yescharacteristic
396C23C2×C4×D411yescharacteristic
397Q8—42no—
398Q8—81no—
399D4—42no—
400D4—44no—
401D4—81no—
402D4—82no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4#38511yescharacteristic
437C2×C4#38511yescharacteristic
438C2×C4#38511yescharacteristic
439C2×C4#38511yescharacteristic
440C2×C4#38511yescharacteristic
441C2×C4#38511yescharacteristic
442C2×C4C22×C4⋊C411yescharacteristic
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×#12011yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×#12311yescharacteristic
448C8—44no—
449C8—42no—
450C8—42no—
451C8—42no—
452C8—42no—
453C8—82no—
454C8—82no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—82no—
467C22—162no—
468C22C2×#38511yescharacteristic
469C22#295712yes—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—84no—
485C4#302111yescharacteristic
486C4#302211yescharacteristic
487C2—22no—
488C2—41no—
489C2—41no—
490C2—82no—
491C2#2086511yescharacteristic
492C2#2088611yescharacteristic
493C2#2089211yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21979C2semidirect
3. #17817C2semidirect
6. #8877C2semidirect
7. #8876C2semidirect
8. #8070C2semidirect
9. #7933C2semidirect
29. #1843C22semidirect
30. #1832C22semidirect
32. #1573C22semidirect
43. #1300C22semidirect
44. #1283C22semidirect
45. #1283C22semidirect

Maximal subgroups

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

subgroupindex
#602912
#219792
#178172
#175672
#174922
#88772
#88762
#80702
#79332