Finite Group Databasebeta

← #493980

Subgroups — #493980

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34208C211yesmaximal, characteristic
2#20731C212yesmaximal, complemented
3#12863C212yesmaximal, complemented
4#12804C212yesmaximal, complemented
5#12801C212yesmaximal, complemented
6#12701C212yesmaximal
7#12631C212yesmaximal, complemented
8C4×#355C212yesmaximal, complemented
9C2×#532C2211yescharacteristic
10C2×#520C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2921—22no—
13#2364—22no—
14#2364—22no—
15#2352—22no—
16#2352—22no—
17#2328C411yescharacteristic
18#2328C411yescharacteristic
19#2327C2211yescharacteristic
20#2327C2211yescharacteristic
21#2311C411yescharacteristic
22#2311C2211yescharacteristic
23C4×#124C411yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26C4×#124C2211yescharacteristic
27C2×#382C411yescharacteristic
28C4×#117C411yescharacteristic
29C2×#356C2214yes—
30C2×#355C2214yes—
31#1821—22no—
32#1801C2214yes—
33#1800—22no—
34#1785C2214yes—
35C4×#101C2214yes—
36C4×#100—22no—
37#1394C2214yes—
38#1394C2214yes—
39C22×#41C2×C411yescharacteristic
40C22×#41C2311yescharacteristic
41#532—24no—
42#532—24no—
43C2×C4⋊Q8C2311yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2311yescharacteristic
47C2×#121—22no—
48#520—44no—
49#520—44no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2311yescharacteristic
53C2×C4×Q8C2311yescharacteristic
54C2×C4×D4—22no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D412yes—
63C2×#117—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117D412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4D412yes—
71C22×C42C2×C411yescharacteristic
72#433—48no—
73#423—24no—
74C2×#101C2318yes—
75C2×#100—24no—
76#407—24no—
77C4×M16—24no—
78#391C2×C411yescharacteristic
79#391C2×C411yescharacteristic
80#391C2311yescharacteristic
81#391C2311yescharacteristic
82#390C2×C411yescharacteristic
83#390C2×C411yescharacteristic
84#390C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#388—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#386—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#382—22no—
101#382C2×C411yescharacteristic
102#382C2×C411yescharacteristic
103#382C2×C411yescharacteristic
104#382C2×C411yescharacteristic
105#380—22no—
106#380—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115#376C2×C411yescharacteristic
116#376C2×C411yescharacteristic
117C2×#95—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120#356—28no—
121#355—28no—
122#343—24no—
123#343—24no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C22×Q8C22×C411yescharacteristic
133C22×Q8C2411yescharacteristic
134C22×D4—21no—
135C22×D4—22no—
136C22×D4C22×C411yescharacteristic
137C22×D4C22×C411yescharacteristic
138C22×D4C2×D412yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D412yes—
149C2×M16—28no—
150C4⋊Q8—22no—
151C4⋊Q8#4114yes—
152#127—22no—
153#127—22no—
154#124—22no—
155#124—22no—
156#124—22no—
157#124#4114yes—
158#121—216no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×D4—216no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—44no—
167C4×D4—44no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—44no—
171C4×D4—44no—
172#117—24no—
173#117—28no—
174#117—44no—
175#117—42no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C22⋊C411yescharacteristic
204C2×C22⋊C4C22⋊C411yescharacteristic
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C4⋊C8—48no—
225#101—28no—
226#101—28no—
227#100—48no—
228C22⋊C8—48no—
229#97—44no—
230C4×C8—44no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C24—21no—
244C24C2×C22⋊C412yes—
245C24C2×C22⋊C411yescharacteristic
246#41—48no—
247#41—48no—
248#41—48no—
249#41—48no—
250C2×Q8—21no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8#11712yes—
256C2×Q8C2×#4112yes—
257C2×Q8C2×#4112yes—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—28no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—24no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4#11712yes—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D412yes—
316M16—48no—
317M16—48no—
318C2×C8—48no—
319C4⋊C4—28no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4#9914yes—
333C22⋊C4—28no—
334C22⋊C4—28no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C42—24no—
347C42—21no—
348C42—21no—
349C42—24no—
350C42—24no—
351C42—22no—
352C42—22no—
353C42—41no—
354C42—41no—
355C42—42no—
356C42—42no—
357C42—81no—
358C42—81no—
359C42—81no—
360C42—81no—
361C42#11712yes—
362C42#11712yes—
363C23—21no—
364C23—21no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23#41012yes—
378C23C22×C22⋊C411yescharacteristic
379Q8—42no—
380Q8—42no—
381Q8—42no—
382Q8—42no—
383D4—48no—
384D4—42no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—48no—
389D4—88no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—84no—
430C2×C4—162no—
431C2×C4—162no—
432C2×C4—162no—
433C2×C4—162no—
434C2×C4—162no—
435C2×C4—162no—
436C2×C4C2×#9912yes—
437C2×C4C2×#9912yes—
438C2×C4C2×#9912yes—
439C2×C4C2×#11711yescharacteristic
440C2×C4C2×#11711yescharacteristic
441C2×C4#52514yes—
442C8—88no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—44no—
450C22—41no—
451C22—41no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—82no—
457C22—82no—
458C22—81no—
459C22—81no—
460C22#228011yescharacteristic
461C22C22×#9911yescharacteristic
462C22C2×#41011yescharacteristic
463C4—22no—
464C4—22no—
465C4—24no—
466C4—22no—
467C4—21no—
468C4—21no—
469C4—42no—
470C4—42no—
471C4—41no—
472C4—41no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—81no—
480C2—21no—
481C2—21no—
482C2—42no—
483C2—82no—
484C2—82no—
485C2#2072211yescharacteristic
486C2#3337011yescharacteristic
487C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20731C2semidirect
3. #12863C2semidirect
4. #12804C2semidirect
5. #12801C2semidirect
7. #12631C2semidirect
8. C4×#355C2semidirect

Maximal subgroups

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

subgroupindex
#342082
#207312
#128632
#128042
#128012
#127012
#126312
C4×#3552