Finite Group Databasebeta

← #238717

Subgroups — #238717

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#21884C211yesmaximal, characteristic
3#20817C211yesmaximal, characteristic, complemented
4#20647C211yesmaximal, characteristic, complemented
5C2×#1492C211yesmaximal, characteristic
6C22×Q8⋊C8C211yesmaximal, characteristic, complemented
7#10434C214yesmaximal, complemented
8#10433C214yesmaximal, complemented
9#8045C211yesmaximal, characteristic, complemented
10C2×#536—21no—
11C2×#532C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#419—21no—
16C22×C4×C8C2211yescharacteristic
17#2854—22no—
18#2618—22no—
19#2548—21no—
20#2546—21no—
21#2545—22no—
22#2538C2214yes—
23#2536—22no—
24#2311—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#1988—21no—
28#1982—21no—
29#1893—22no—
30#1845C2212yes—
31#1834C2212yes—
32#1808—22no—
33#1805—22no—
34#1803C2214yes—
35#1802C2214yes—
36C4×#107—22no—
37#1736—21no—
38#1492—24no—
39C2×Q8⋊C8—24no—
40C2×Q8⋊C8C2212yes—
41C2×Q8⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44#1296C2214yes—
45C22×#41—21no—
46C23×Q8D411yescharacteristic
47C22×M16—21no—
48C23×C8—21no—
49C23×C8—21no—
50#536—44no—
51#536—44no—
52#532—22no—
53#532—22no—
54#532—22no—
55#532—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8D411yescharacteristic
58C2×#127D411yescharacteristic
59C2×#124—22no—
60C2×#124—21no—
61C2×#123—24no—
62C2×C22⋊Q8—24no—
63C2×#121—21no—
64C2×#121D412yes—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8D412yes—
72C2×C4×Q8C2312yes—
73C2×C4×D4—21no—
74C2×C4×D4C2312yes—
75C2×#117—21no—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82#423—22no—
83#423—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90#419—44no—
91C2×#103—24no—
92C2×#102—24no—
93C2×C4×C8—21no—
94C2×C4×C8—22no—
95C2×C4×C8D414yes—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#398—24no—
99#396—28no—
100#394—24no—
101#391—22no—
102#390—22no—
103#390—24no—
104#389—24no—
105#388—28no—
106#386—24no—
107#384C2312yes—
108#384C2312yes—
109#382—21no—
110#382—21no—
111#380—21no—
112#380—24no—
113#380—21no—
114#380—21no—
115#380D412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120#340—24no—
121#340—24no—
122#340—24no—
123Q8⋊C8—24no—
124Q8⋊C8—24no—
125D4⋊C8—24no—
126D4⋊C8—24no—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—42no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8D814yes—
141C22×Q8C2×D412yes—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C2×D412yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—42no—
153C2×M16—42no—
154C2×M16—42no—
155C22×C8—21no—
156C22×C8—24no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—44no—
161C22×C8—42no—
162C22×C8—82no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—48no—
166#127—22no—
167#127—22no—
168#124—48no—
169#124—42no—
170#124—42no—
171#123—416no—
172C22⋊Q8—48no—
173C22⋊Q8—48no—
174#121—24no—
175#121—24no—
176#121—42no—
177#121—42no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—42no—
193C4×D4—42no—
194#117—42no—
195#117—42no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D414yes—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—41no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—41no—
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235#107—48no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242#103—48no—
243#102—84no—
244C4×C8—24no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8C2×D412yes—
248C4×C8C2×D412yes—
249#95—28no—
250#95—22no—
251#95—22no—
252#95—41no—
253#95—41no—
254C24—22no—
255C24#12011yescharacteristic
256#41—84no—
257#41—84no—
258#41—84no—
259#41—84no—
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—28no—
267C2×Q8—28no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8C2×D814yes—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—82no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—82no—
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12112yes—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12312yes—
314C22×C4#12312yes—
315C22×C4#12412yes—
316C22×C4C2×D812yes—
317C22×C4#13512yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4#13712yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323M16—84no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—48no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—44no—
334C2×C8—84no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—44no—
340C4⋊C4—416no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C22⋊C4—416no—
356C22⋊C4—44no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—88no—
361C22⋊C4—82no—
362C42—42no—
363C42—42no—
364C42—44no—
365C42—44no—
366C42—82no—
367C42#12112yes—
368C42#12112yes—
369C42#12112yes—
370C42#12312yes—
371C42#12312yes—
372C42#12412yes—
373C42C22×D411yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C42C2×#4111yescharacteristic
377C23—22no—
378C23—22no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23#39412yes—
385C23#44712yes—
386C23#45212yes—
387C23C2×#12011yescharacteristic
388Q8—24no—
389Q8—24no—
390Q8—42no—
391Q8—42no—
392Q8—48no—
393Q8—84no—
394D4—44no—
395D4—42no—
396D4—42no—
397D4—84no—
398D4—88no—
399D4—84no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#39412yes—
429C2×C4#39412yes—
430C2×C4#39412yes—
431C2×C4#44712yes—
432C2×C4#45212yes—
433C2×C4#46614yes—
434C2×C4#46814yes—
435C2×C4#48014yes—
436C2×C4#48214yes—
437C2×C4C2×#12111yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12111yescharacteristic
440C2×C4C2×#12311yescharacteristic
441C2×C4C2×#12311yescharacteristic
442C2×C4C2×#12411yescharacteristic
443C2×C4C22×D811yescharacteristic
444C2×C4C2×#13511yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4#57712yes—
448C2×C4#57812yes—
449C8—24no—
450C8—42no—
451C8—82no—
452C8—82no—
453C22—22no—
454C22—22no—
455C22—24no—
456C22—24no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—84no—
461C22—84no—
462C22#203314yes—
463C22C2×#39411yescharacteristic
464C22C2×#44711yescharacteristic
465C22C2×#45211yescharacteristic
466C22#301312yes—
467C22#301812yes—
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—44no—
473C4—82no—
474C4—82no—
475C4—84no—
476C4—162no—
477C4#203314yes—
478C4C2×#46611yescharacteristic
479C4C2×#46811yescharacteristic
480C4#305711yescharacteristic
481C4#305911yescharacteristic
482C4C2×#48011yescharacteristic
483C4C2×#48211yescharacteristic
484C4#309712yes—
485C2—24no—
486C2—162no—
487C2#1866912yes—
488C2#1867412yes—
489C2C2×#203311yescharacteristic
490C2#2348711yescharacteristic
491C2#2349811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20817C2semidirect
4. #20647C2semidirect
6. C22×Q8⋊C8C2semidirect
7. #10434C2semidirect
8. #10433C2semidirect
9. #8045C2semidirect

Maximal subgroups

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

subgroupindex
#351782
#218842
#208172
#206472
C2×#14922
C22×Q8⋊C82
#104342
#104332
#80452