Finite Group Databasebeta

← #481717

Subgroups — #481717

Order 512.

1195 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36022C211yesmaximal, characteristic
2#20354C211yesmaximal, characteristic, complemented
3#17369C211yesmaximal, characteristic, complemented
4#11757C214yesmaximal, complemented
5#11755C214yesmaximal, complemented
6#11718C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#555C411yescharacteristic, complemented
9C2×#518—21no—
10C2×#518C2211yescharacteristic
11#2975—21no—
12#2635C412yescomplemented
13#2628C2211yescharacteristic
14#2542C411yescharacteristic, complemented
15#2539C2212yes—
16#2535C411yescharacteristic, complemented
17#2482C2212yes—
18#2472—22no—
19#2461—22no—
20#2422—21no—
21#2408—21no—
22#2395C412yescomplemented
23#2392C2211yescharacteristic
24#2363—21no—
25#2357C411yescharacteristic, complemented
26#2354—21no—
27#1740—21no—
28#1740C2212yes—
29C2×#342—22no—
30C2×#342C2214yes—
31#1465—22no—
32#1462C2214yes—
33#1460—21no—
34#1458C2212yes—
35#1352C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C23×D4D411yescharacteristic
39#555—44no—
40C2×C4⋊Q8C2×C411yescharacteristic, complemented
41C2×#124—22no—
42C2×#123—22no—
43C2×C22⋊Q8—22no—
44C2×#121—21no—
45C2×#121—21no—
46C2×#121C2×C412yescomplemented
47C2×#120—22no—
48#518—28no—
49#518—44no—
50#518—44no—
51C2×C4×D4—21no—
52C2×C4×D4—44no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4C2×C411yescharacteristic, complemented
57C2×C4×D4D412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×#117—21no—
61C2×#117—22no—
62C2×#117C2×C412yescomplemented
63C2×#117D411yescharacteristic
64C2×#117C2311yescharacteristic
65C22×C22⋊C4—24no—
66C22×C22⋊C4D412yes—
67C22×C22⋊C4C2312yes—
68#436—44no—
69#431—22no—
70#407—24no—
71#400C2×C412yescomplemented
72#399C2×C412yescomplemented
73#399C2×C412yescomplemented
74#399C2312yes—
75#398C2×C411yescharacteristic, complemented
76#397—22no—
77#397C2×C412yescomplemented
78#397C2311yescharacteristic
79#396—21no—
80#395—21no—
81#394—22no—
82#394—22no—
83#394—21no—
84#394C2×C411yescharacteristic, complemented
85#393—21no—
86#392—22no—
87#392C2×C412yescomplemented
88#392C2311yescharacteristic
89#390—21no—
90#390—21no—
91#390—22no—
92#390—22no—
93#389—21no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#386—22no—
99#386—22no—
100#385—22no—
101#384—21no—
102#382C2×C411yescharacteristic, complemented
103#381C2×C412yescomplemented
104#380—21no—
105#380—21no—
106#379—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C411yescharacteristic, complemented
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4C2×C412yescomplemented
113#342—22no—
114#342—24no—
115#342C2314yes—
116#324—22no—
117#323—24no—
118#323C2314yes—
119C25C2×D412yes—
120C22×Q8—21no—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4C22⋊C412yes—
129C22×D4C22⋊C411yescharacteristic
130C22×D4C22⋊C411yescharacteristic
131C22×D4C22×C411yescharacteristic
132C23×C4—21no—
133C23×C4—44no—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22×C412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C2×M16—44no—
140C4⋊Q8—42no—
141#124—48no—
142#123—84no—
143C22⋊Q8—84no—
144#121—44no—
145#121—44no—
146#121—44no—
147#120—84no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—44no—
154C4×D4—44no—
155C4×D4—432no—
156C4×D4—42no—
157C4×D4—42no—
158#117—28no—
159#117—42no—
160#117—42no—
161#117—44no—
162#117—44no—
163#117—44no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C22⋊C4—216no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—48no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22×C412yes—
201C2×C22⋊C4C2×D411yescharacteristic
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×D412yes—
204C2×C22⋊C4C2411yescharacteristic
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—42no—
209C2×C42C22⋊C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C4⋊C8—42no—
213C22⋊C8—24no—
214C22⋊C8—44no—
215C4×C8—42no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95C22×C411yescharacteristic
224#95C22×C412yes—
225#95C22×C411yescharacteristic
226C24—24no—
227C24—22no—
228C24—22no—
229C24—24no—
230C24—21no—
231C24—82no—
232C24—82no—
233C24C2×C22⋊C412yes—
234C24C2×C22⋊C412yes—
235C24C22×D411yescharacteristic
236C2×Q8—42no—
237C2×Q8—82no—
238C2×D4—216no—
239C2×D4—216no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—81no—
254C2×D4—81no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—82no—
259C2×D4#9914yes—
260C2×D4#10414yes—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—416no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—164no—
282C22×C4—164no—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4AGL(1,ℤ/8)12yes—
290C22×C4#13712yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293M16—164no—
294C2×C8—44no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—28no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—42no—
322C22⋊C4—416no—
323C22⋊C4—44no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—81no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C42—22no—
335C42—22no—
336C42—42no—
337C42—44no—
338C42—48no—
339C42—42no—
340C42—82no—
341C23—28no—
342C23—28no—
343C23—21no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—21no—
348C23—22no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—84no—
363C23—84no—
364C23—162no—
365C23—162no—
366C23—164no—
367C23C2×#9914yes—
368C23C2×#10414yes—
369C23C22×C22⋊C411yescharacteristic
370C23C2×AGL(1,ℤ/8)11yescharacteristic
371C23C2×#13711yescharacteristic
372Q8—162no—
373D4—216no—
374D4—28no—
375D4—28no—
376D4—82no—
377D4—82no—
378D4—82no—
379D4—164no—
380D4—164no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4—162no—
419C2×C4—162no—
420C2×C4—1616no—
421C2×C4C2×#9912yes—
422C2×C4#41712yes—
423C2×C4#41712yes—
424C2×C4C2×#10412yes—
425C8—84no—
426C22—21no—
427C22—21no—
428C22—24no—
429C22—24no—
430C22—21no—
431C22—21no—
432C22—24no—
433C22—24no—
434C22—41no—
435C22—41no—
436C22—44no—
437C22—41no—
438C22—42no—
439C22—41no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—88no—
450C22—88no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22C22×#9911yescharacteristic
455C22C2×#41711yescharacteristic
456C22C22×#10411yescharacteristic
457C4—22no—
458C4—42no—
459C4—41no—
460C4—42no—
461C4—41no—
462C4—41no—
463C4—42no—
464C4—41no—
465C4—81no—
466C4—82no—
467C4—88no—
468C4—82no—
469C4—81no—
470C4#150414yes—
471C2—21no—
472C2—21no—
473C2—41no—
474C2—44no—
475C2—44no—
476C2—41no—
477C2—161no—
478C2C2×#150411yescharacteristic
479C2#3335011yescharacteristic
480C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36022C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20354C2semidirect
3. #17369C2semidirect
4. #11757C2semidirect
5. #11755C2semidirect
6. #11718C2semidirect
7. #11691C2semidirect
8. C2×#555C4semidirect
12. #2635C4semidirect
14. #2542C4semidirect
16. #2535C4semidirect
22. #2395C4semidirect
25. #2357C4semidirect
40. C2×C4⋊Q8C2×C4semidirect
46. C2×#121C2×C4semidirect
56. C2×C4×D4C2×C4semidirect
62. C2×#117C2×C4semidirect
71. #400C2×C4semidirect
72. #399C2×C4semidirect
73. #399C2×C4semidirect
75. #398C2×C4semidirect
77. #397C2×C4semidirect
84. #394C2×C4semidirect
87. #392C2×C4semidirect
102. #382C2×C4semidirect
103. #381C2×C4semidirect
109. C4×C4⋊C4C2×C4semidirect
112. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#360222
#203542
#173692
#117572
#117552
#117182
#116912