Finite Group Databasebeta

← #214466

Subgroups — #214466

Order 512.

1166 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#35178C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic, complemented
3#13942C211yesmaximal, characteristic, complemented
4#9156C212yesmaximal, complemented
5#9148C212yesmaximal, complemented
6#9118C214yesmaximal, complemented
7#8717C212yesmaximal, complemented
8#8332C212yesmaximal
9C2×#536—21no—
10C2×#532C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2212yes—
13#2854C2212yes—
14#2854C2212yes—
15#2618—21no—
16#2618—21no—
17#2548C412yescomplemented
18#2546C412yescomplemented
19#2545—22no—
20#2538C414yescomplemented
21#2536—22no—
22#2311—21no—
23C2×#391—21no—
24#2285C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1985—22no—
28#1985—22no—
29#1817C2212yes—
30#1817C2212yes—
31C2×#327C2214yes—
32C2×#327C2214yes—
33#1441—21no—
34#1441—21no—
35C4⋊#95—21no—
36#1434C2212yes—
37#1434C2212yes—
38#1294C2214yes—
39#1294C2214yes—
40C22×#41—21no—
41C23×Q8D411yescharacteristic
42#536—44no—
43#536—44no—
44#532—42no—
45#532—42no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#127C2×C411yescharacteristic
49C2×#124—22no—
50C2×#124—21no—
51C2×#123—22no—
52C2×#123—22no—
53C2×C22⋊Q8—24no—
54C2×#121—21no—
55C2×#121C2312yes—
56C2×C4×Q8—42no—
57C2×C4×Q8—42no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8D412yes—
61C2×C4×Q8D412yes—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#432—42no—
75#432—42no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8C2314yes—
78C2×#97—22no—
79#398—22no—
80#398—22no—
81#396—24no—
82#396—24no—
83#394—22no—
84#394—22no—
85#391—22no—
86#391—44no—
87#390—22no—
88#390C2×C414yes—
89#390C2×C414yes—
90#389C2×C414yes—
91#389C2×C414yes—
92#388—24no—
93#388—24no—
94#386—22no—
95#386—22no—
96#384C2×C412yes—
97#382—21no—
98#382—42no—
99#380—21no—
100#380—24no—
101#380C2312yes—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—42no—
104C4×C4⋊C4C2×C412yes—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95C2312yes—
111C2×#95C2312yes—
112#336—22no—
113#336—22no—
114#336—24no—
115#327—44no—
116#327—44no—
117C2×#41—42no—
118C2×#41—42no—
119C2×#41—42no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—42no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—41no—
129C22×Q8—41no—
130C22×Q8C22⋊C412yes—
131C22×Q8C22⋊C412yes—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4C22×C412yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—24no—
147C4⋊Q8—42no—
148C4⋊Q8—48no—
149#127—42no—
150#124—44no—
151#124—48no—
152#123—44no—
153#123—44no—
154#123—44no—
155#123—44no—
156C22⋊Q8—416no—
157#121—24no—
158#121—24no—
159#121—42no—
160#121—42no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×D4—44no—
168C4×D4—44no—
169#117—42no—
170#117—42no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C414yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C4⋊C8—44no—
221C22⋊C8—24no—
222C22⋊C8—24no—
223#97—44no—
224#95—22no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95—24no—
229#95—24no—
230#95—44no—
231#95—44no—
232#95—42no—
233#95—44no—
234#95—44no—
235C24—22no—
236C24C22×D411yescharacteristic
237#41—84no—
238#41—84no—
239#41—84no—
240#41—84no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—24no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—82no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4AGL(1,ℤ/8)12yes—
320C22×C4AGL(1,ℤ/8)12yes—
321C22×C4#13712yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
326C2×C8—48no—
327C2×C8—44no—
328C4⋊C4—24no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—82no—
342C4⋊C4—84no—
343C22⋊C4—44no—
344C22⋊C4—48no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—48no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C42—22no—
352C42—24no—
353C42—24no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—44no—
362C42—84no—
363C23—22no—
364C23—22no—
365C23—42no—
366C23—44no—
367C23—42no—
368C23—42no—
369C23—44no—
370C23C22×C22⋊C411yescharacteristic
371C23#53411yescharacteristic
372C23#53411yescharacteristic
373C23C2×AGL(1,ℤ/8)11yescharacteristic
374C23C2×AGL(1,ℤ/8)11yescharacteristic
375C23C2×AGL(1,ℤ/8)11yescharacteristic
376C23C2×#13711yescharacteristic
377Q8—48no—
378Q8—48no—
379Q8—84no—
380D4—84no—
381D4—84no—
382D4—84no—
383D4—82no—
384D4—82no—
385D4—82no—
386D4—82no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—88no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#41712yes—
448C2×C4#41712yes—
449C2×C4#41814yes—
450C2×C4#51812yes—
451C2×C4#53412yes—
452C8—84no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—21no—
460C22—21no—
461C22—82no—
462C22—82no—
463C22—84no—
464C22—84no—
465C22—82no—
466C22#230811yescharacteristic
467C22#288412yes—
468C22C2×#41811yescharacteristic
469C22C2×#41711yescharacteristic
470C22#290112yes—
471C22#319211yescharacteristic
472C22#319211yescharacteristic
473C22#319311yescharacteristic
474C22#319311yescharacteristic
475C4—22no—
476C4—22no—
477C4—24no—
478C4—22no—
479C4—22no—
480C4—42no—
481C4—42no—
482C4—41no—
483C4—42no—
484C4—41no—
485C4—84no—
486C4—82no—
487C4—84no—
488C4—162no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—162no—
494C2#1762812yes—
495C2#1763112yes—
496C2#2175111yescharacteristic
497C2#2176411yescharacteristic
498C2#2177511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21822C2semidirect
3. #13942C2semidirect
4. #9156C2semidirect
5. #9148C2semidirect
6. #9118C2semidirect
7. #8717C2semidirect
17. #2548C4semidirect
18. #2546C4semidirect
20. #2538C4semidirect

Maximal subgroups

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

subgroupindex
#351782
#218222
#139422
#91562
#91482
#91182
#87172
#83322