Finite Group Databasebeta

← #238701

Subgroups — #238701

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic, complemented
7#10434C214yesmaximal, complemented
8#10433C214yesmaximal, complemented
9#8039C211yesmaximal, characteristic, complemented
10C2×#536—21no—
11C2×#532C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103—21no—
17#2854—22no—
18#2618—22no—
19#2548—21no—
20#2546—21no—
21#2545—22no—
22#2538C2212yes—
23#2538C2212yes—
24#2536—22no—
25#2311—21no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#1990—21no—
29#1957—21no—
30#1947—22no—
31#1874—22no—
32#1852—22no—
33#1845C2212yes—
34#1834C2212yes—
35#1803—22no—
36#1803C2214yes—
37#1802C2214yes—
38C2×#340—21no—
39#1480—24no—
40#1479—24no—
41C2×Q8⋊C8C2214yes—
42C2×D4⋊C8C2214yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×#41—21no—
46C23×Q8D411yescharacteristic
47C23×C8—21no—
48#536—44no—
49#536—44no—
50#532—22no—
51#532—22no—
52#532—22no—
53#532—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8D411yescharacteristic
56C2×#127D411yescharacteristic
57C2×#124—22no—
58C2×#124—21no—
59C2×#123—24no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×#121—21no—
63C2×#121D412yes—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8D412yes—
71C2×C4×Q8C2312yes—
72C2×C4×D4—21no—
73C2×C4×D4C2312yes—
74C2×#117—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82#431—22no—
83#431—22no—
84#431—22no—
85#431D414yes—
86C2×#107—24no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×#103—24no—
94C2×#103—44no—
95C2×#102—24no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2312yes—
100#398—24no—
101#396—24no—
102#396—24no—
103#394—24no—
104#391—22no—
105#390—22no—
106#390—22no—
107#390—22no—
108#389—22no—
109#389—22no—
110#388—24no—
111#388—24no—
112#386—24no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384C2312yes—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#380—21no—
121#380—22no—
122#380—22no—
123#380D412yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4D412yes—
126#340—24no—
127#340—24no—
128#340—24no—
129Q8⋊C8—28no—
130D4⋊C8—28no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—42no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8D814yes—
145C22×Q8C2×D412yes—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C2×D412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—42no—
161C22×C8—42no—
162C22×C8—42no—
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—44no—
173C22⋊Q8—44no—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176#121—24no—
177#121—24no—
178#121—42no—
179#121—42no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—42no—
195C4×D4—42no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42#4112yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42C2411yescharacteristic
246#107—84no—
247C4⋊C8—24no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—42no—
253#103—48no—
254#102—84no—
255C22⋊C8—24no—
256C22⋊C8—42no—
257C4×C8—22no—
258C4×C8—42no—
259C4×C8C2×D414yes—
260#95—24no—
261#95—24no—
262#95—22no—
263#95—22no—
264C24—22no—
265C24#12011yescharacteristic
266#41—84no—
267#41—84no—
268#41—84no—
269#41—84no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—28no—
277C2×Q8—28no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8C2×D814yes—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—81no—
298C2×D4—81no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12312yes—
331C22×C4#12312yes—
332C22×C4#12412yes—
333C22×C4C2×D812yes—
334C22×C4#13512yes—
335C22×C4AGL(1,ℤ/8)12yes—
336C22×C4#13712yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—84no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C22⋊C4—48no—
377C22⋊C4—48no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—44no—
382C22⋊C4—84no—
383C22⋊C4—84no—
384C42—42no—
385C42—42no—
386C42—44no—
387C42—44no—
388C42#12112yes—
389C42#12112yes—
390C42#12112yes—
391C42#12312yes—
392C42#12312yes—
393C42#12412yes—
394C42C22×D411yescharacteristic
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23#39412yes—
410C23#44712yes—
411C23#45212yes—
412C23C2×#12011yescharacteristic
413Q8—24no—
414Q8—24no—
415Q8—42no—
416Q8—42no—
417Q8—48no—
418Q8—84no—
419D4—44no—
420D4—44no—
421D4—84no—
422D4—88no—
423D4—84no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—48no—
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—84no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—164no—
459C2×C4#39412yes—
460C2×C4#39412yes—
461C2×C4#39412yes—
462C2×C4#44712yes—
463C2×C4#45212yes—
464C2×C4#46614yes—
465C2×C4#46814yes—
466C2×C4#48014yes—
467C2×C4#48214yes—
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×#12111yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4C2×#12411yescharacteristic
474C2×C4C22×D811yescharacteristic
475C2×C4C2×#13511yescharacteristic
476C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C2×C4#57712yes—
479C2×C4#57812yes—
480C8—44no—
481C8—82no—
482C8—82no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22#203314yes—
497C22C2×#39411yescharacteristic
498C22C2×#44711yescharacteristic
499C22C2×#45211yescharacteristic
500C22#301011yescharacteristic
501C22#301111yescharacteristic
502C22#302111yescharacteristic
503C22#302211yescharacteristic
504C4—22no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—44no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—162no—
514C4#203314yes—
515C4C2×#46611yescharacteristic
516C4C2×#46811yescharacteristic
517C4#306312yes—
518C4C2×#48011yescharacteristic
519C4C2×#48211yescharacteristic
520C4#309712yes—
521C2—24no—
522C2—162no—
523C2#1889911yescharacteristic
524C2#1890911yescharacteristic
525C2#1891211yescharacteristic
526C2#1892211yescharacteristic
527C2C2×#203311yescharacteristic
528C2#2348411yescharacteristic
529C2#2350011yescharacteristic
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
2. #21982C2semidirect
4. C2×#1803C2semidirect
6. C2×#1479C2semidirect
7. #10434C2semidirect
8. #10433C2semidirect
9. #8039C2semidirect

Maximal subgroups

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

subgroupindex
#351782
#219822
#218592
C2×#18032
C2×#14802
C2×#14792
#104342
#104332
#80392