Finite Group Databasebeta

← #204644

Subgroups — #204644

Order 512.

1144 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#35180C211yesmaximal, characteristic
2#20534C211yesmaximal, characteristic, complemented
3C2×#1515C211yesmaximal, characteristic, complemented
4C2×#1514C211yesmaximal, characteristic
5C2×#1506C211yesmaximal, characteristic, complemented
6C2×#1505C211yesmaximal, characteristic
7#13964C211yesmaximal, characteristic, complemented
8#9682C214yesmaximal, complemented
9#9681C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14#2850C2212yes—
15#2850C2212yes—
16#2639—22no—
17#2592—22no—
18#2568C412yescomplemented
19#2563C412yescomplemented
20#2562—22no—
21#2548C412yescomplemented
22#2547C412yescomplemented
23#2311—22no—
24C2×#383C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1771—22no—
28#1771—22no—
29#1749C2212yes—
30C4×C22⋊C8C2212yes—
31#1515—24no—
32#1514—24no—
33#1506—24no—
34#1505—24no—
35C2×#328C2214yes—
36C2×#328C2214yes—
37C2×#327C2214yes—
38C2×#327C2214yes—
39#1441—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42#1434C2212yes—
43C22×#41—21no—
44#533—22no—
45#533—22no—
46#533—22no—
47#533—22no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#125—21no—
57C2×#124—21no—
58C2×#124C2×C412yes—
59C2×#124C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2312yes—
62C2×#121—21no—
63C2×#121C2312yes—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C412yes—
66C2×C4×D4—21no—
67C2×C4×D4C2×C412yes—
68C2×#117—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D412yes—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#432D414yes—
78#432D414yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C22⋊C8C2312yes—
82C2×C22⋊C8C2312yes—
83C2×#97—21no—
84C2×#97—21no—
85C2×#97—22no—
86C2×C4×C8—22no—
87#400—24no—
88#397—24no—
89#394—24no—
90#393—24no—
91#391—24no—
92#391C2×C412yes—
93#391C2×C412yes—
94#390—24no—
95#390C2×C412yes—
96#390C2×C412yes—
97#388—24no—
98#388—24no—
99#386—24no—
100#386—24no—
101#383—22no—
102#383—22no—
103#382C2×C412yes—
104#382C2×C412yes—
105#380—24no—
106#380—24no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D412yes—
111#376—22no—
112#376—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95D412yes—
120C2×#95C2312yes—
121C2×#95C2312yes—
122#336—22no—
123#336—22no—
124#328—28no—
125#328—28no—
126#327—28no—
127#327—28no—
128C2×#41—44no—
129C2×#41—44no—
130C2×#41—42no—
131C2×#41—42no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8C22×C412yes—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C22×C412yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155#127—22no—
156#127—22no—
157#125—42no—
158#125—42no—
159#124—42no—
160#124—42no—
161#124—82no—
162#124—82no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169#121—22no—
170#121—22no—
171#121—22no—
172#121—22no—
173#121—42no—
174#121—42no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—42no—
182C4×D4—42no—
183#117—42no—
184#117—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C22⋊C414yes—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C42—22no—
208C2×C42—44no—
209C2×C42—42no—
210C2×C42—81no—
211C2×C42—81no—
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C4⋊C8—24no—
225C4⋊C8—24no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8C2×D414yes—
229C22⋊C8C2×D414yes—
230#97—24no—
231#97—24no—
232#97—82no—
233C4×C8—82no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—82no—
248#95C22⋊C414yes—
249#95C22⋊C414yes—
250C24—22no—
251C24C22×D411yescharacteristic
252#41—84no—
253#41—84no—
254#41—88no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—44no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—82no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—82no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—48no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4C2×C22⋊C414yes—
308C22×C4C2×C22⋊C414yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4#12412yes—
314C22×C4#12412yes—
315C22×C4#12712yes—
316C22×C4#12712yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—88no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—48no—
357C42—44no—
358C42C4×D412yes—
359C42C4×D412yes—
360C42C4×D412yes—
361C42C4×D412yes—
362C42#12112yes—
363C42#12112yes—
364C42#12112yes—
365C42#12112yes—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—42no—
370C23—42no—
371C23—44no—
372C23—44no—
373C23—42no—
374C23C22×C22⋊C411yescharacteristic
375C23C2×#12411yescharacteristic
376C23C2×#12711yescharacteristic
377C23#57512yes—
378C23#57512yes—
379Q8—82no—
380Q8—82no—
381Q8—84no—
382Q8—84no—
383D4—82no—
384D4—82no—
385D4—84no—
386D4—84no—
387D4—82no—
388D4—82no—
389D4—84no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—88no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4#39012yes—
440C2×C4#39012yes—
441C2×C4#39012yes—
442C2×C4#39012yes—
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C8—44no—
448C8—44no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—21no—
456C22—21no—
457C22—82no—
458C22—82no—
459C22—84no—
460C22—84no—
461C22—82no—
462C22C2×#39011yescharacteristic
463C22#288311yescharacteristic
464C22#288411yescharacteristic
465C22#289612yes—
466C22#290112yes—
467C22#314011yescharacteristic
468C22#314011yescharacteristic
469C22#315312yes—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—44no—
479C4—44no—
480C4—84no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4—162no—
485C2—22no—
486C2—21no—
487C2—21no—
488C2—162no—
489C2#1762412yes—
490C2#1762912yes—
491C2#2120411yescharacteristic
492C2#2121411yescharacteristic
493C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20534C2semidirect
3. C2×#1515C2semidirect
5. C2×#1506C2semidirect
7. #13964C2semidirect
8. #9682C2semidirect
9. #9681C2semidirect
18. #2568C4semidirect
19. #2563C4semidirect
21. #2548C4semidirect
22. #2547C4semidirect

Maximal subgroups

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

subgroupindex
#351802
#205342
C2×#15152
C2×#15142
C2×#15062
C2×#15052
#139642
#96822
#96812