Finite Group Databasebeta

← #192703

Subgroups — #192703

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2#21894C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic
6#8888C212yesmaximal, complemented
7#8887C212yesmaximal, complemented
8#8884C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8043C211yesmaximal, characteristic
11#7903C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—22no—
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2360—22no—
18#2360—22no—
19#2355—21no—
20#2355C2212yes—
21Q8×C4⋊C4—21no—
22Q8×C4⋊C4C2212yes—
23#2330—22no—
24#2317—21no—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8—21no—
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#2274—21no—
30C4×#117C2211yescharacteristic
31#1988—21no—
32#1981—21no—
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1894—22no—
38#1854—21no—
39#1854—21no—
40#1847—22no—
41#1845C2212yes—
42#1844C2212yes—
43#1803C2212yes—
44#1803C2212yes—
45C4×#107—21no—
46C4×#106—21no—
47#1485C418yescomplemented
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×M16—21no—
55C22×M16—21no—
56#533—22no—
57#533—22no—
58#533C2×C414yes—
59#533Q814yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C2×#117—21no—
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C42C2311yescharacteristic
77#431—22no—
78#431C2×C414yes—
79#428—48no—
80C2×#107—24no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#106—24no—
84C2×#106—21no—
85C2×#106—21no—
86#423—22no—
87#423—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×#103—22no—
93C2×#103—22no—
94C2×C22⋊C8—21no—
95C4×M16—22no—
96C4×M16C2×C414yes—
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2312yes—
100#391—22no—
101#391—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—24no—
106#389—22no—
107#389—24no—
108#387—24no—
109#387—24no—
110#385—24no—
111#384—22no—
112#384—21no—
113#384—21no—
114#384D412yes—
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2312yes—
118#382—22no—
119#382—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—24no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138C43—21no—
139#340—22no—
140#340—22no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145Q8⋊C8C2×C418yescomplemented
146Q8⋊C8C2×C418yescomplemented
147C22×Q8—21no—
148C22×Q8C2×D412yes—
149C23×C4—22no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×M16—24no—
154C2×M16—24no—
155C2×M16—24no—
156C2×M16—42no—
157C2×M16—81no—
158C2×M16—81no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8C4⋊C414yes—
166C4⋊Q8C4⋊C414yes—
167#125—24no—
168#125—24no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8C4⋊C418yes—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—82no—
178C4×Q8—82no—
179C4×Q8—84no—
180C4×Q8C22×C414yes—
181C4×Q8C2×Q814yes—
182#117—24no—
183#117—24no—
184#117—22no—
185#117—22no—
186#117—41no—
187#117—41no—
188#117—44no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D414yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—41no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—41no—
232C2×C42C22×C412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×Q812yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42C2411yescharacteristic
240#107—42no—
241#107—48no—
242#107—42no—
243#106—42no—
244#106—48no—
245#106—42no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—42no—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253#103—82no—
254#103—82no—
255C22⋊C8—24no—
256#97—24no—
257C4×C8—24no—
258C4×C8—22no—
259C4×C8C22×C414yes—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—41no—
269#95—41no—
270C24#12011yescharacteristic
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8C2×C4⋊C414yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—44no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4C2×C4⋊C412yes—
303C22×C4C4×D412yes—
304C22×C4C4×D412yes—
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4C22⋊Q812yes—
309C22×C4C22⋊Q812yes—
310C22×C4#12312yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—216no—
315M16—84no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—82no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4C2×C4⋊C414yes—
362C4⋊C4#13712yes—
363C4⋊C4#13712yes—
364C22⋊C4—24no—
365C22⋊C4—28no—
366C22⋊C4—42no—
367C22⋊C4—82no—
368C42—24no—
369C42—24no—
370C42—24no—
371C42—42no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—42no—
377C42—42no—
378C42—84no—
379C42—84no—
380C42—82no—
381C42C2×C4⋊C412yes—
382C42C4×D412yes—
383C42C4×D412yes—
384C42C22⋊Q812yes—
385C42C22⋊Q812yes—
386C42#12312yes—
387C42C23×C411yescharacteristic
388C42C22×Q811yescharacteristic
389C42C2×#4111yescharacteristic
390C42C2×#4111yescharacteristic
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23#38512yes—
398C23C2×#12011yescharacteristic
399Q8—24no—
400Q8—24no—
401Q8—42no—
402Q8—42no—
403Q8—48no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—164no—
448C2×C4#38512yes—
449C2×C4#38512yes—
450C2×C4#38512yes—
451C2×C4#45312yes—
452C2×C4#45612yes—
453C2×C4C22×C4⋊C411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C22⋊Q811yescharacteristic
457C2×C4C2×C22⋊Q811yescharacteristic
458C2×C4C2×#12311yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4C2×#13711yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57511yescharacteristic
463C2×C4#57611yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57811yescharacteristic
467C8—28no—
468C8—42no—
469C8—42no—
470C8—82no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22C2×#38511yescharacteristic
479C22#300411yescharacteristic
480C22#300511yescharacteristic
481C22#301211yescharacteristic
482C22#301311yescharacteristic
483C22C2×#45311yescharacteristic
484C22C2×#45611yescharacteristic
485C4—22no—
486C4—21no—
487C4—21no—
488C4—22no—
489C4—44no—
490C4—44no—
491C4—42no—
492C4—42no—
493C4—81no—
494C4—84no—
495C4—81no—
496C4—161no—
497C4—161no—
498C4#185914yes—
499C4#291211yescharacteristic
500C4#291311yescharacteristic
501C4C4×#13711yescharacteristic
502C4C4×#13711yescharacteristic
503C4#307811yescharacteristic
504C4#307811yescharacteristic
505C4#309611yescharacteristic
506C4#309711yescharacteristic
507C2—42no—
508C2#1856711yescharacteristic
509C2#1856811yescharacteristic
510C2#1859111yescharacteristic
511C2#1859211yescharacteristic
512C2#2084911yescharacteristic
513C2C2×#185911yescharacteristic
514C2#2087411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ 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
6. #8888C2semidirect
7. #8887C2semidirect
8. #8884C2semidirect
9. #8883C2semidirect
47. #1485C4semidirect
145. Q8⋊C8C2×C4semidirect
146. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342142
#218942
#213622
#208002
C2×#14852
#88882
#88872
#88842
#88832
#80432
#79032