Finite Group Databasebeta

← #481617

Subgroups — #481617

Order 512.

1065 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2C4×#413C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#11782C214yesmaximal, complemented
5#11780C214yesmaximal, complemented
6#11700C214yesmaximal, complemented
7C2×#536—21no—
8C22×C4⋊Q8C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2920—21no—
11C2×#413—21no—
12C2×C4×M16—21no—
13#2857—21no—
14#2857—21no—
15#2621C412yes—
16#2621C2212yes—
17#2547—21no—
18#2547C412yes—
19#2546—21no—
20#2546C2212yes—
21#2486—21no—
22#2486—21no—
23#2486C412yes—
24#2486C2212yes—
25#2310—21no—
26#2285C411yescharacteristic
27C4×#101—21no—
28C4×#101C2212yes—
29C4×#100—21no—
30C4×#100C2212yes—
31#1742C2212yes—
32#1467C2212yes—
33#1466—21no—
34#1465—21no—
35#1465C2212yes—
36#1458—21no—
37#1458C2212yes—
38#1346C22116yes—
39C22×#41D411yescharacteristic
40C23×Q8—21no—
41C22×M16—22no—
42#536—48no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C412yes—
52C2×#124—22no—
53C2×#123—22no—
54C2×#123—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C4×#41—41no—
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117C2×C411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C412yes—
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78#432—22no—
79#432—41no—
80#431—22no—
81#431—41no—
82#423—41no—
83#413—28no—
84C2×#101—22no—
85C2×#100—22no—
86#407—22no—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16—41no—
90C4×M16—41no—
91C4×M16—41no—
92C2×#97—41no—
93C2×C4×C8—41no—
94#398—22no—
95#398—22no—
96#398C2×C414yes—
97#398C2314yes—
98#397—22no—
99#397—22no—
100#397C2×C414yes—
101#397C2×C414yes—
102#391—22no—
103#391—22no—
104#391—22no—
105#391—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#384—21no—
111#382C2×C412yes—
112#381—22no—
113#381—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—21no—
118#380—22no—
119#380C2×C412yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C412yes—
122#343—22no—
123#343C2314yes—
124#324—22no—
125#324C2314yes—
126#323—22no—
127#323—22no—
128C2×#41—24no—
129C2×#41—24no—
130C2×#41—41no—
131C2×#41—41no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—44no—
137C22×Q8—44no—
138C22×Q8C2×D411yescharacteristic
139C22×D4C22⋊C411yescharacteristic
140C22×D4C22⋊C411yescharacteristic
141C22×D4C2×D411yescharacteristic
142C23×C4—22no—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—28no—
153C2×M16—24no—
154C2×M16—42no—
155C2×M16—44no—
156C2×M16—42no—
157C22×C8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—48no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169#124—48no—
170#123—48no—
171#123—48no—
172C22⋊Q8—48no—
173C22⋊Q8—48no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×D4—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—
190#117—41no—
191#117—41no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—44no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22×C414yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2411yescharacteristic
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4C22⋊C411yescharacteristic
220C2×C22⋊C4C22⋊C411yescharacteristic
221C2×C22⋊C4C22×C411yescharacteristic
222C2×C22⋊C4C22×C411yescharacteristic
223C2×C22⋊C4C2×D411yescharacteristic
224C2×C22⋊C4C2×D411yescharacteristic
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—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C4⋊C8—42no—
249C4⋊C8—42no—
250#101—28no—
251#100—28no—
252C22⋊C8—24no—
253C22⋊C8—42no—
254#97—42no—
255#97—42no—
256#97—42no—
257C4×C8—42no—
258C4×C8—42no—
259C4×C8—42no—
260#95—22no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95C22×C414yes—
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C411yescharacteristic
269#41—48no—
270#41—48no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—88no—
282C2×Q8—88no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
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—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4#10414yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
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×C4AGL(1,ℤ/8)14yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339M16—28no—
340M16—48no—
341M16—44no—
342M16—84no—
343C2×C8—44no—
344C2×C8—48no—
345C2×C8—44no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—44no—
370C22⋊C4—48no—
371C22⋊C4—48no—
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#9911yescharacteristic
392C42#9911yescharacteristic
393C42#9911yescharacteristic
394C42#9911yescharacteristic
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23#41812yes—
409C23C2×#10412yes—
410C23C22×C22⋊C411yescharacteristic
411Q8—44no—
412Q8—88no—
413Q8—84no—
414Q8—84no—
415D4—44no—
416D4—44no—
417D4—44no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—21no—
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—42no—
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—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—88no—
464C2×C4C2×#9911yescharacteristic
465C2×C4C2×#9911yescharacteristic
466C2×C4C2×#9911yescharacteristic
467C2×C4C2×#9911yescharacteristic
468C2×C4C2×#9911yescharacteristic
469C2×C4C2×#9911yescharacteristic
470C2×C4#41812yes—
471C2×C4C2×#10412yes—
472C2×C4C2×#10412yes—
473C2×C4C2×AGL(1,ℤ/8)12yes—
474C8—48no—
475C8—84no—
476C22—21no—
477C22—21no—
478C22—22no—
479C22—22no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—81no—
492C22—81no—
493C22C22×#9911yescharacteristic
494C22C2×#41811yescharacteristic
495C22C22×#10411yescharacteristic
496C4—21no—
497C4—21no—
498C4—22no—
499C4—22no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—84no—
508C4—84no—
509C4#150712yes—
510C4#150712yes—
511C4#212312yes—
512C4#213512yes—
513C2—21no—
514C2—21no—
515C2—42no—
516C2—42no—
517C2—81no—
518C2C2×#150711yescharacteristic
519C2C2×#212311yescharacteristic
520C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11782C2semidirect
5. #11780C2semidirect
6. #11700C2semidirect

Maximal subgroups

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

subgroupindex
#353162
C4×#4132
#173612
#117822
#117802
#117002