Finite Group Databasebeta

← #495428

Subgroups — #495428

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#20637C211yesmaximal, characteristic
3C4×#413C211yesmaximal, characteristic
4#13302C214yesmaximal, complemented
5#13301C214yesmaximal, complemented
6#12587C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C2×#423—21no—
10C2×#413C2212yes—
11C2×#413C2212yes—
12C2×C4×M16—21no—
13#2857—21no—
14#2857—21no—
15#2628—21no—
16#2628—21no—
17#2568C2212yes—
18#2567C2212yes—
19#2563C412yes—
20#2542C412yes—
21#2487—21no—
22#2487—21no—
23#2409—21no—
24#2409—21no—
25#2317C412yes—
26#2311C2212yes—
27#2285C411yescharacteristic
28#1801C2212yes—
29#1801C2212yes—
30#1800—21no—
31#1800—21no—
32C4×#101C2212yes—
33C4×#101C2212yes—
34C4×#100—21no—
35C4×#100—21no—
36#1742—21no—
37#1742—21no—
38#1394C22116yes—
39C22×#41C2311yescharacteristic
40C23×Q8—21no—
41C22×M16—22no—
42C22×M16—22no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2×C412yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C4×#41—41no—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C412yes—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C412yes—
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74#423—22no—
75#423—22no—
76#423—41no—
77#423—41no—
78C2×C4⋊C8—41no—
79C2×C4⋊C8—41no—
80#413—24no—
81#413—28no—
82#413Q818yescomplemented
83C2×#101C2314yes—
84C2×#101C2314yes—
85C2×#100—22no—
86C2×#100—22no—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16—41no—
90C4×M16—41no—
91C2×#97—41no—
92C2×C4×C8—41no—
93#400—22no—
94#400—22no—
95#399—22no—
96#399—22no—
97#395—22no—
98#395—22no—
99#393—22no—
100#393—22no—
101#391—22no—
102#391—22no—
103#391C2314yes—
104#390C2×C414yes—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#386—22no—
110#386—22no—
111#385—22no—
112#385—22no—
113#384—21no—
114#384C2×C414yes—
115#382—21no—
116#382C2×C414yes—
117#380—22no—
118#380—22no—
119#380—21no—
120#380—21no—
121#380—22no—
122#380—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125#343—22no—
126#343—22no—
127#343—22no—
128#343—22no—
129C2×#41—41no—
130C2×#41—41no—
131C2×#41—42no—
132C2×#41C2×Q814yes—
133C2×#41#4114yes—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—84no—
139C22×Q8C2411yescharacteristic
140C22×D4C22×C411yescharacteristic
141C22×D4C2×D411yescharacteristic
142C22×D4C2×D411yescharacteristic
143C23×C4—22no—
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×M16—28no—
154C2×M16—24no—
155C2×M16—24no—
156C2×M16—44no—
157C2×M16—42no—
158C2×M16—42no—
159C2×M16—44no—
160C2×M16—44no—
161C22×C8—42no—
162C22×C8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C22⋊Q8—84no—
174C22⋊Q8—84no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191#117—41no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—41no—
197#117—44no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22×C414yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4C22⋊C411yescharacteristic
221C2×C22⋊C4C22⋊C411yescharacteristic
222C2×C22⋊C4C22⋊C411yescharacteristic
223C2×C22⋊C4C22⋊C411yescharacteristic
224C2×C22⋊C4C22⋊C411yescharacteristic
225C2×C22⋊C4C22⋊C411yescharacteristic
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C414yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253#101—24no—
254#101—28no—
255#101C2×Q818yescomplemented
256#100—28no—
257#100—44no—
258#97—42no—
259#97—42no—
260C4×C8—42no—
261C4×C8—42no—
262#95—22no—
263#95—22no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271#41—44no—
272#41—44no—
273#41—44no—
274#41—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—84no—
283C2×Q8—88no—
284C2×Q8—88no—
285C2×Q8C22×Q812yes—
286C2×Q8C2×#4112yes—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4C4×Q814yes—
294C2×D4C22⋊Q814yes—
295C2×D4#12412yes—
296C2×D4C4⋊Q812yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—84no—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C4×Q814yes—
338C22×C4C22⋊Q814yes—
339C22×C4#12412yes—
340C22×C4C4⋊Q812yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344M16—28no—
345M16—44no—
346M16—48no—
347M16—44no—
348M16—44no—
349M16—84no—
350C2×C8—44no—
351C2×C8—48no—
352C2×C8—44no—
353C2×C8—84no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—44no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—44no—
399C42#9911yescharacteristic
400C42#9911yescharacteristic
401C42#9911yescharacteristic
402C42#9911yescharacteristic
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23#39112yes—
416C23#39112yes—
417C23#39112yes—
418C23C22×C22⋊C411yescharacteristic
419Q8—42no—
420Q8—42no—
421Q8—84no—
422Q8—88no—
423Q8—84no—
424D4—44no—
425D4—44no—
426D4—42no—
427D4—42no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—88no—
475C2×C4#39112yes—
476C2×C4C2×#9911yescharacteristic
477C2×C4C2×#9911yescharacteristic
478C2×C4C2×#9911yescharacteristic
479C2×C4C2×#9911yescharacteristic
480C2×C4C2×#9911yescharacteristic
481C2×C4C2×#9911yescharacteristic
482C2×C4#41012yes—
483C2×C4C2×C4×Q812yes—
484C2×C4C2×C22⋊Q812yes—
485C2×C4C2×#12411yescharacteristic
486C2×C4C2×C4⋊Q811yescharacteristic
487C8—48no—
488C8—84no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—81no—
505C22—81no—
506C22C2×#39111yescharacteristic
507C22C22×#9911yescharacteristic
508C22C2×#41011yescharacteristic
509C4—22no—
510C4—21no—
511C4—22no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—84no—
522C4—84no—
523C4#199612yes—
524C4#199612yes—
525C4#213412yes—
526C4#213512yes—
527C2—21no—
528C2—21no—
529C2—42no—
530C2—42no—
531C2—42no—
532C2C2×#199611yescharacteristic
533C2C2×#213411yescharacteristic
534C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13302C2semidirect
5. #13301C2semidirect
82. #413Q8semidirect
255. #101C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352792
#206372
C4×#4132
#133022
#133012
#125872