Finite Group Databasebeta

← #496337

Subgroups — #496337

Order 512.

1069 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#20723C211yesmaximal, characteristic
4#13220C214yesmaximal, complemented
5#13217C214yesmaximal, complemented
6#12692C214yesmaximal
7C2×#536C2212yes—
8C22×C4⋊Q8C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10#2920—21no—
11#2857—21no—
12#2857—21no—
13#2621—21no—
14#2621—21no—
15#2547—21no—
16#2547—21no—
17#2546—21no—
18#2546—21no—
19#2486C412yescomplemented
20#2486C412yescomplemented
21#2486C2212yes—
22#2486C2212yes—
23#2310C412yescomplemented
24#2285C411yescharacteristic, complemented
25#2283—21no—
26C2×#352C22116yes—
27#1822—21no—
28#1820—21no—
29#1801—21no—
30#1800C2212yes—
31#1799—21no—
32#1799C2212yes—
33#1785C2212yes—
34#1782C2212yes—
35C4×#101—21no—
36C4×#100C2212yes—
37C4×#99—21no—
38C4×#99C2212yes—
39C22×#41C2311yescharacteristic
40C23×Q8C2×C412yes—
41#536—48no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×#124—22no—
52C2×#123C2×C414yes—
53C2×#123C2314yes—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117C2×C412yes—
69C2×#117C2×C411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C412yes—
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77#432—41no—
78#432—41no—
79#431—41no—
80#431—41no—
81#423—22no—
82C2×#101—22no—
83C2×#100C2314yes—
84C2×#99—22no—
85C2×#99C2314yes—
86#407—22no—
87C4×M16—22no—
88#398—22no—
89#398—22no—
90#398—22no—
91#398—22no—
92#397—22no—
93#397—22no—
94#397—22no—
95#397—22no—
96#391—22no—
97#391—22no—
98#391—22no—
99#391—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#384—21no—
105#383—41no—
106#382—21no—
107#382—41no—
108#381—22no—
109#381C2×C414yes—
110#381C2×C414yes—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380C2×C412yes—
117#380C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—41no—
121C4×C22⋊C4—22no—
122#376—41no—
123#352—416no—
124#343—22no—
125#343—22no—
126#342—22no—
127#342—22no—
128C2×#41—24no—
129C2×#41—41no—
130C2×#41—41no—
131C2×#41—42no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—44no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—44no—
138C22×Q8C2×D411yescharacteristic
139C22×D4C22×C411yescharacteristic
140C22×D4C2×D411yescharacteristic
141C22×D4C2411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C414yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—24no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—48no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164#124—48no—
165#123—216no—
166#123—48no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—41no—
191#117—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4C22⋊C411yescharacteristic
216C2×C22⋊C4C22⋊C411yescharacteristic
217C2×C22⋊C4C22×C414yes—
218C2×C22⋊C4C22×C411yescharacteristic
219C2×C22⋊C4C22×C411yescharacteristic
220C2×C22⋊C4C2×D411yescharacteristic
221C2×C22⋊C4C2×D411yescharacteristic
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
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—22no—
235C2×C42—42no—
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C4⋊C8—42no—
247C4⋊C8—42no—
248#101—44no—
249#100—28no—
250#99—28no—
251#99—44no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—42no—
255C4×C8—42no—
256#95—22no—
257#95—22no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—42no—
266C24C2×C22⋊C411yescharacteristic
267C24C23×C411yescharacteristic
268C24C22×D411yescharacteristic
269#41—48no—
270#41—48no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—88no—
282C2×Q8—88no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4H(5,𝔽2)14yes—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
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—24no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
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×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340M16—44no—
341M16—44no—
342C2×C8—44no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—84no—
361C22⋊C4—28no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—44no—
391C42—84no—
392C42#9911yescharacteristic
393C42#9911yescharacteristic
394C42#9911yescharacteristic
395C42#9911yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23C22×C22⋊C411yescharacteristic
410C23#51812yes—
411C23#53512yes—
412Q8—44no—
413Q8—84no—
414Q8—84no—
415Q8—88no—
416D4—44no—
417D4—44no—
418D4—44no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—88no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4C2×#9911yescharacteristic
471C2×C4C2×#9911yescharacteristic
472C2×C4C2×#9911yescharacteristic
473C2×C4C2×#9911yescharacteristic
474C2×C4C2×#9911yescharacteristic
475C2×C4C2×#9911yescharacteristic
476C2×C4#41012yes—
477C2×C4#53512yes—
478C8—84no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—81no—
495C22—81no—
496C22#230911yescharacteristic
497C22C22×#9911yescharacteristic
498C22C2×#41011yescharacteristic
499C4—22no—
500C4—21no—
501C4—21no—
502C4—22no—
503C4—21no—
504C4—21no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—41no—
512C4—84no—
513C4—84no—
514C4—164no—
515C4#212312yes—
516C4#212312yes—
517C2—21no—
518C2—21no—
519C2—42no—
520C2—42no—
521C2—81no—
522C2#2179311yescharacteristic
523C2C2×#212311yescharacteristic
524C2C2×#212311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13220C2semidirect
5. #13217C2semidirect
8. C22×C4⋊Q8C4semidirect
19. #2486C4semidirect
20. #2486C4semidirect
23. #2310C4semidirect
24. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#353162
#207282
#207232
#132202
#132172
#126922