Finite Group Databasebeta

← #193265

Subgroups — #193265

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8923C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7903C211yesmaximal, characteristic
11#7896C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2369—21no—
17#2369—21no—
18#2365C2212yes—
19#2365C2212yes—
20#2345—22no—
21Q8×C22⋊C4—22no—
22#2331—22no—
23#2316—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2274—21no—
29C4×#117C2211yescharacteristic
30#1993—22no—
31#1990C2212yes—
32#1990C2212yes—
33#1948—21no—
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2212yes—
37#1947—21no—
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2212yes—
41#1854—21no—
42#1854—21no—
43#1809—22no—
44C4×#107—21no—
45C4×#106—21no—
46#1736—22no—
47#1546C418yescomplemented
48C2×#332C2212yes—
49C2×#332C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C23×Q8—21no—
55C22×M16—21no—
56C22×M16—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8Q814yes—
64C2×C4⋊Q8C2312yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C42C2311yescharacteristic
74#431—22no—
75#431C2×C414yes—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—22no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—22no—
82#423—22no—
83#423C2×C414yes—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C22⋊C8—21no—
89C4×M16—22no—
90C4×M16—22no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8C2312yes—
94#391—24no—
95#391—24no—
96#391—22no—
97#391—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#387—24no—
105#387—24no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#382—21no—
111#382—21no—
112#380—24no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380D412yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134#376—21no—
135C43—21no—
136#340—22no—
137#340—22no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—24no—
143#340—24no—
144#332C2×C418yes—
145#332C2×C418yes—
146C22×Q8—24no—
147C22×Q8—22no—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—81no—
151C22×Q8—81no—
152C22×Q8—81no—
153C22×Q8—81no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—24no—
159C2×M16—24no—
160C2×M16—42no—
161C2×M16—42no—
162C2×M16—81no—
163C2×M16—81no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8C4⋊C418yes—
176C4⋊Q8C22×C414yes—
177C4⋊Q8C2×Q814yes—
178C4×Q8—82no—
179C4×Q8—82no—
180C4×Q8—82no—
181C4×Q8—82no—
182#117—22no—
183#117—22no—
184#117—41no—
185#117—41no—
186#117—42no—
187#117—42no—
188#117—44no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
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—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4C4⋊C418yes—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—41no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—41no—
229C2×C42C22×C412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×Q812yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42C2411yescharacteristic
237#107—42no—
238#107—42no—
239#107—82no—
240#106—42no—
241#106—42no—
242#106—82no—
243C4⋊C8—24no—
244C4⋊C8—22no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8C22×C414yes—
248C4⋊C8C22×C412yes—
249C4⋊C8C22×C412yes—
250C22⋊C8—24no—
251#97—42no—
252C4×C8—24no—
253C4×C8—22no—
254C4×C8—22no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—21no—
260#95—21no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—41no—
266#95—41no—
267C24#12011yescharacteristic
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—28no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4C2×C4⋊C412yes—
304C22×C4C4×D412yes—
305C22×C4C4×D412yes—
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4C22⋊Q812yes—
310C22×C4C22⋊Q812yes—
311C22×C4#12312yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315M16—48no—
316M16—84no—
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
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—84no—
359C4⋊C4—84no—
360C4⋊C4C2×C4⋊C418yes—
361C4⋊C4#13712yes—
362C4⋊C4#13712yes—
363C22⋊C4—24no—
364C22⋊C4—42no—
365C22⋊C4—44no—
366C22⋊C4—82no—
367C42—24no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—42no—
376C42—42no—
377C42—84no—
378C42—84no—
379C42—82no—
380C42C2×C4⋊C412yes—
381C42C4×D412yes—
382C42C4×D412yes—
383C42C22⋊Q812yes—
384C42C22⋊Q812yes—
385C42#12312yes—
386C42C23×C411yescharacteristic
387C42C22×Q811yescharacteristic
388C42C2×#4111yescharacteristic
389C42C2×#4111yescharacteristic
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23#38512yes—
397C23C2×#12011yescharacteristic
398Q8—44no—
399Q8—44no—
400Q8—44no—
401Q8—44no—
402Q8—88no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—28no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—88no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4#38512yes—
448C2×C4#38512yes—
449C2×C4#38512yes—
450C2×C4#45412yes—
451C2×C4#45612yes—
452C2×C4C22×C4⋊C411yescharacteristic
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×C22⋊Q811yescharacteristic
456C2×C4C2×C22⋊Q811yescharacteristic
457C2×C4C2×#12311yescharacteristic
458C2×C4C2×#13711yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57611yescharacteristic
463C2×C4#57611yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57711yescharacteristic
466C8—44no—
467C8—42no—
468C8—42no—
469C8—44no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22C2×#38511yescharacteristic
478C22#300411yescharacteristic
479C22#300511yescharacteristic
480C22#301211yescharacteristic
481C22#301211yescharacteristic
482C22C2×#45411yescharacteristic
483C22C2×#45611yescharacteristic
484C4—21no—
485C4—21no—
486C4—22no—
487C4—22no—
488C4—48no—
489C4—42no—
490C4—42no—
491C4—81no—
492C4—82no—
493C4—81no—
494C4—82no—
495C4—161no—
496C4—161no—
497C4#185714yes—
498C4C4×#13711yescharacteristic
499C4C4×#13711yescharacteristic
500C4#294811yescharacteristic
501C4#294911yescharacteristic
502C4#307811yescharacteristic
503C4#307811yescharacteristic
504C4#308111yescharacteristic
505C4#308111yescharacteristic
506C2—42no—
507C2#1858311yescharacteristic
508C2#1858511yescharacteristic
509C2#1859111yescharacteristic
510C2#1859311yescharacteristic
511C2C2×#185711yescharacteristic
512C2#2087411yescharacteristic
513C2#2087911yescharacteristic
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. #8926C2semidirect
7. #8925C2semidirect
8. #8923C2semidirect
9. #8922C2semidirect
47. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#342162
#219642
#213622
#213242
C2×#15462
#89262
#89252
#89232
#89222
#79032
#78962