Finite Group Databasebeta

← #213793

Subgroups — #213793

Order 512.

1054 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#34208C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#9105C212yesmaximal, complemented
5#9104C212yesmaximal, complemented
6#8964C212yesmaximal, complemented
7#8961C212yesmaximal, complemented
8#8730C212yesmaximal, complemented
9#8324C212yesmaximal, complemented
10C2×#532C411yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#520C2211yescharacteristic
13#2983C2212yes—
14#2975C2212yes—
15#2364—22no—
16#2364—22no—
17#2352—22no—
18#2352—22no—
19#2328C411yescharacteristic
20#2328C2211yescharacteristic
21#2327C411yescharacteristic
22#2327C2211yescharacteristic
23#2311C411yescharacteristic
24#2311C411yescharacteristic
25C4×#124C411yescharacteristic
26C4×#124C411yescharacteristic
27C4×#124C2211yescharacteristic
28C4×#124C2211yescharacteristic
29C2×#382C411yescharacteristic
30C4×#117C2211yescharacteristic
31#1479C2214yes—
32#1478C2214yes—
33#1466—22no—
34#1465—22no—
35#1464—22no—
36#1463—22no—
37#1461C2212yes—
38#1460C2212yes—
39#1458C2212yes—
40#1458C2212yes—
41#1276C2214yes—
42#1274C2214yes—
43C22×#41C2×C411yescharacteristic
44C22×#41C2×C411yescharacteristic
45#532—24no—
46#532—24no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#127C2×C411yescharacteristic
49C2×#124C2×C411yescharacteristic
50C2×#124C2×C411yescharacteristic
51C2×#121—22no—
52#520—44no—
53#520—44no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×C4×D4C2311yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C2×#117C2×C411yescharacteristic
69C2×#117D412yes—
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4D412yes—
75C22×C42C2×C411yescharacteristic
76#446—44no—
77#436—44no—
78#432—24no—
79#431C2314yes—
80#431C2314yes—
81#407—24no—
82#391C2×C411yescharacteristic
83#391C2×C411yescharacteristic
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#390C2×C411yescharacteristic
88#390C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#382—22no—
105#382C2×C411yescharacteristic
106#382C2×C411yescharacteristic
107#382C2×C411yescharacteristic
108#382C2×C411yescharacteristic
109#380—22no—
110#380—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119#376C2×C411yescharacteristic
120#376C2311yescharacteristic
121C2×#95—22no—
122C43C2×C411yescharacteristic
123C43C2311yescharacteristic
124Q8⋊C8—44no—
125D4⋊C8—44no—
126#324—22no—
127#324—22no—
128#324—24no—
129#323—22no—
130#323—22no—
131#323—24no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—42no—
136C2×#41—42no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41M1614yescomplemented
140C22×Q8C22×C411yescharacteristic
141C22×Q8C22×C411yescharacteristic
142C22×D4—21no—
143C22×D4—22no—
144C22×D4C22⋊C412yes—
145C22×D4C22×C411yescharacteristic
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D412yes—
157C2×M16—44no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160#127—22no—
161#127—22no—
162#124—22no—
163#124—22no—
164#124—22no—
165#124—22no—
166#121—216no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×D4—216no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—44no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—44no—
178C4×D4—44no—
179C4×D4—44no—
180#117—24no—
181#117—42no—
182#117—44no—
183#117—44no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C2×D412yes—
213C2×C22⋊C4C2×D411yescharacteristic
214C2×C22⋊C4C2×D411yescharacteristic
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C22⋊C412yes—
224C2×C42M1614yescomplemented
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2411yescharacteristic
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—44no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—44no—
238#97—44no—
239C4×C8—24no—
240C4×C8—24no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253C24—21no—
254C24C2×C22⋊C412yes—
255C24C2×C22⋊C411yescharacteristic
256#41—48no—
257#41—48no—
258#41—48no—
259#41—48no—
260C2×Q8—22no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8C2×M1612yes—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—28no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4C2×M1612yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×M1612yes—
322C22×C4C2×M1612yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326M16—164no—
327C2×C8—44no—
328C2×C8—44no—
329C4⋊C4—28no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C22⋊C4—28no—
344C22⋊C4—28no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—24no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—22no—
357C42—24no—
358C42—21no—
359C42—21no—
360C42—24no—
361C42—22no—
362C42—22no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—81no—
367C42—81no—
368C42—81no—
369C42—81no—
370C42C2×M1612yes—
371C42C2×M1612yes—
372C42H(5,𝔽2)14yes—
373C23—21no—
374C23—21no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23#40712yes—
388C23C22×C22⋊C411yescharacteristic
389Q8—24no—
390Q8—42no—
391Q8—42no—
392Q8—42no—
393D4—24no—
394D4—48no—
395D4—48no—
396D4—42no—
397D4—42no—
398D4—42no—
399D4—88no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—48no—
417C2×C4—48no—
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—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—162no—
441C2×C4—162no—
442C2×C4—162no—
443C2×C4—162no—
444C2×C4—162no—
445C2×C4#40712yes—
446C2×C4#51812yes—
447C2×C4#53412yes—
448C2×C4#53412yes—
449C2×C4C22×M1611yescharacteristic
450C2×C4C22×M1611yescharacteristic
451C2×C4#56812yes—
452C8—84no—
453C8—84no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—44no—
461C22—41no—
462C22—41no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—82no—
468C22—82no—
469C22—81no—
470C22—81no—
471C22#230811yescharacteristic
472C22C2×#40711yescharacteristic
473C22#287911yescharacteristic
474C4—22no—
475C4—22no—
476C4—22no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—41no—
481C4—42no—
482C4—41no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—81no—
489C4#297312yes—
490C4#297512yes—
491C2—21no—
492C2—21no—
493C2—42no—
494C2—82no—
495C2—82no—
496C2#2174211yescharacteristic
497C2#2178411yescharacteristic
498C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17372C2semidirect
3. #17370C2semidirect
4. #9105C2semidirect
5. #9104C2semidirect
6. #8964C2semidirect
7. #8961C2semidirect
8. #8730C2semidirect
9. #8324C2semidirect
139. C2×#41M16semidirect
224. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
#342082
#173722
#173702
#91052
#91042
#89642
#89612
#87302
#83242