Finite Group Databasebeta

← #116287

Subgroups — #116287

Order 512.

665 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21306C211yesmaximal, characteristic
2#21146C211yesmaximal, characteristic
3C2×#1503C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7#7882C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423—21no—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1986—22no—
17#1986—22no—
18#1940—22no—
19#1939—22no—
20#1733—21no—
21#1733—21no—
22#1728—21no—
23#1503—22no—
24#1503C412yescomplemented
25#1503C412yescomplemented
26C2×#328—22no—
27#1493—22no—
28#1487—22no—
29#1481—22no—
30#1303—22no—
31#1294—22no—
32#1278—22no—
33C22×M16—21no—
34C22×M16D411yescharacteristic
35C22×M16D411yescharacteristic
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533D412yes—
44C2×C4⋊Q8Q811yescharacteristic
45C2×C4⋊Q8Q811yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—42no—
69#423—42no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D412yes—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C2×#97—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#384—21no—
86#382—21no—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C2×#95D412yes—
97#339—24no—
98#339—24no—
99#336—24no—
100#328—24no—
101#328—24no—
102Q8⋊C8—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4—22no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—21no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16SD1612yes—
126C2×M16Q1612yes—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—41no—
131C22×C8—41no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8C4⋊C412yes—
139C4⋊Q8C4⋊C412yes—
140#125—22no—
141#125—22no—
142#125—41no—
143#125—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—41no—
160#117—22no—
161#117—22no—
162#117—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166#117—41no—
167#117—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42C22×C411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×Q811yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—48no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217#97—41no—
218#97—41no—
219C4×C8—22no—
220C4×C8—22no—
221C4×C8—41no—
222C4×C8—41no—
223C4×C8—41no—
224C4×C8—41no—
225#95—22no—
226#95—22no—
227#95—41no—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230C24#12111yescharacteristic
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—82no—
236C2×Q8—82no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—82no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12611yescharacteristic
266C22×C4C2×SD1611yescharacteristic
267C22×C4C2×Q1611yescharacteristic
268C22×C4#13511yescharacteristic
269C22×C4#13511yescharacteristic
270C22×C4AGL(1,ℤ/8)11yescharacteristic
271C22×C4#13711yescharacteristic
272C22×C4#13711yescharacteristic
273C22×C4#13711yescharacteristic
274M16—22no—
275M16—22no—
276M16—22no—
277M16—22no—
278M16—41no—
279M16—41no—
280M16—41no—
281M16—41no—
282M16—81no—
283M16—81no—
284M16—81no—
285M16—81no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—82no—
298C2×C8—82no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—81no—
317C4⋊C4—81no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42C2×C4⋊C411yescharacteristic
333C42#11711yescharacteristic
334C42C4×D411yescharacteristic
335C42C4×D411yescharacteristic
336C42#12011yescharacteristic
337C42#12111yescharacteristic
338C42#12111yescharacteristic
339C42C22⋊Q811yescharacteristic
340C42C22⋊Q811yescharacteristic
341C42C22⋊Q811yescharacteristic
342C42#12311yescharacteristic
343C42#12311yescharacteristic
344C42#12411yescharacteristic
345C42#12411yescharacteristic
346C42#12611yescharacteristic
347C42#12711yescharacteristic
348C23—41no—
349C23—41no—
350C23#38611yescharacteristic
351C23#39411yescharacteristic
352C23#39611yescharacteristic
353C23#46111yescharacteristic
354C23#46211yescharacteristic
355C23#46311yescharacteristic
356C23#46411yescharacteristic
357Q8—44no—
358Q8—82no—
359Q8—82no—
360Q8—82no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4#38011yescharacteristic
389C2×C4#38511yescharacteristic
390C2×C4#38511yescharacteristic
391C2×C4#38811yescharacteristic
392C2×C4#38811yescharacteristic
393C2×C4#39011yescharacteristic
394C2×C4#39211yescharacteristic
395C2×C4#39311yescharacteristic
396C2×C4#39411yescharacteristic
397C2×C4#39611yescharacteristic
398C2×C4#39711yescharacteristic
399C2×C4#39911yescharacteristic
400C2×C4C4×SD1611yescharacteristic
401C2×C4C4×Q1611yescharacteristic
402C2×C4#44011yescharacteristic
403C2×C4#44111yescharacteristic
404C2×C4#44811yescharacteristic
405C2×C4#44911yescharacteristic
406C2×C4#45011yescharacteristic
407C2×C4#45111yescharacteristic
408C2×C4#45111yescharacteristic
409C2×C4#45211yescharacteristic
410C2×C4#45211yescharacteristic
411C2×C4#45211yescharacteristic
412C2×C4#46511yescharacteristic
413C2×C4#46711yescharacteristic
414C2×C4#46811yescharacteristic
415C2×C4#47011yescharacteristic
416C2×C4#47511yescharacteristic
417C2×C4#47711yescharacteristic
418C2×C4#47911yescharacteristic
419C2×C4#47911yescharacteristic
420C2×C4#48711yescharacteristic
421C2×C4#48811yescharacteristic
422C2×C4#48911yescharacteristic
423C2×C4#49111yescharacteristic
424C8—42no—
425C8—42no—
426C8—82no—
427C8—82no—
428C8—82no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22#143911yescharacteristic
440C22#187711yescharacteristic
441C22#187811yescharacteristic
442C22#204211yescharacteristic
443C22#204311yescharacteristic
444C22#206611yescharacteristic
445C22#206811yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—44no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—162no—
458C4#186112yes—
459C4#186311yescharacteristic
460C4#186511yescharacteristic
461C4#193311yescharacteristic
462C4#193511yescharacteristic
463C4#200011yescharacteristic
464C4#200111yescharacteristic
465C4#202411yescharacteristic
466C4#202611yescharacteristic
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2#1106711yescharacteristic
474C2#1108011yescharacteristic
475C2#1109411yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1503C4semidirect
25. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#213062
#211462
C2×#15032
#174862
#80952
#80142
#78822