Finite Group Databasebeta

← #193275

Subgroups — #193275

Order 512.

1086 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34209C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8923C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7902C211yesmaximal, characteristic
11#7900C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#519—21no—
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2365—21no—
18#2365—21no—
19#2365C2212yes—
20#2365C2212yes—
21#2344—22no—
22#2344—22no—
23#2327—22no—
24#2310—21no—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8—21no—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1990C2212yes—
31#1990C2212yes—
32#1987—22no—
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2212yes—
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2212yes—
39#1854—22no—
40#1854—21no—
41#1854—21no—
42#1847—21no—
43#1836—21no—
44#1824—22no—
45#1776—21no—
46#1776—21no—
47#1556C418yescomplemented
48C2×#332C2212yes—
49C2×#332C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C23×Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8Q814yes—
62C2×C4⋊Q8C2312yes—
63#519—44no—
64#519—44no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×#117—24no—
70C2×#117—21no—
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432—22no—
79#432C2×C414yes—
80#432C2×C414yes—
81#431—22no—
82#431—22no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—22no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2311yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8C2312yes—
99#391—22no—
100#391—22no—
101#391—22no—
102#391—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#387—24no—
108#387—24no—
109#387—24no—
110#387—24no—
111#384D411yescharacteristic
112#384D411yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#382—21no—
117#381—24no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380D412yes—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136#376—21no—
137C43—21no—
138#340—24no—
139#340—24no—
140#340—22no—
141#340—22no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#332C2×C418yes—
147#332C2×C418yes—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—42no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8C4⋊C418yes—
172C4⋊Q8C22×C414yes—
173C4⋊Q8C2×Q814yes—
174C4×Q8—48no—
175C4×Q8—48no—
176C4×Q8—44no—
177C4×Q8—44no—
178C4×Q8—44no—
179C4×Q8—44no—
180#117—22no—
181#117—22no—
182#117—416no—
183#117—41no—
184#117—41no—
185#117—42no—
186#117—42no—
187#117—82no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
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—21no—
206C2×C4⋊C4C4⋊C418yes—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—41no—
218C2×C42—21no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—41no—
228C2×C42C22×C412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×Q812yes—
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236#107—42no—
237#107—42no—
238#107—82no—
239#106—42no—
240#106—42no—
241#106—82no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—22no—
245C4⋊C8—42no—
246C4⋊C8C22×C412yes—
247C4⋊C8C22×C412yes—
248C4⋊C8C22×C414yes—
249C22⋊C8—24no—
250C22⋊C8—24no—
251C22⋊C8—42no—
252#97—24no—
253#97—24no—
254C4×C8—22no—
255C4×C8—22no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—22no—
266#95—41no—
267#95—41no—
268C24#12011yescharacteristic
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—28no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8—48no—
277C2×Q8—48no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—28no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—41no—
301C22×C4—41no—
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
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4C2×C4⋊C418yes—
355C4⋊C4AGL(1,ℤ/8)12yes—
356C4⋊C4AGL(1,ℤ/8)12yes—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—82no—
361C42—24no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—48no—
366C42—48no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—82no—
372C42C2×C4⋊C412yes—
373C42C4×D412yes—
374C42C4×D412yes—
375C42C22⋊Q812yes—
376C42C22⋊Q812yes—
377C42#12312yes—
378C42C23×C411yescharacteristic
379C42C22×Q811yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23#38512yes—
389C23C2×#12011yescharacteristic
390Q8—416no—
391Q8—44no—
392Q8—44no—
393Q8—44no—
394Q8—44no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—28no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—416no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#38512yes—
440C2×C4#38512yes—
441C2×C4#38512yes—
442C2×C4C22×C4⋊C411yescharacteristic
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×#12311yescharacteristic
448C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
449C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
450C2×C4#57511yescharacteristic
451C2×C4#57511yescharacteristic
452C2×C4#57712yes—
453C2×C4#57812yes—
454C8—42no—
455C8—42no—
456C8—44no—
457C8—82no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22C2×#38511yescharacteristic
466C22#300311yescharacteristic
467C22#300511yescharacteristic
468C22#301011yescharacteristic
469C22#301111yescharacteristic
470C22#302011yescharacteristic
471C22#302011yescharacteristic
472C4—21no—
473C4—21no—
474C4—22no—
475C4—22no—
476C4—22no—
477C4—48no—
478C4—42no—
479C4—42no—
480C4—82no—
481C4—82no—
482C4—81no—
483C4—81no—
484C4—161no—
485C4—161no—
486C4C4×AGL(1,ℤ/8)11yescharacteristic
487C4C4×AGL(1,ℤ/8)11yescharacteristic
488C4#294712yes—
489C4#307711yescharacteristic
490C4#307711yescharacteristic
491C4#308212yes—
492C2—42no—
493C2#1882611yescharacteristic
494C2#1882711yescharacteristic
495C2#1883011yescharacteristic
496C2#1883111yescharacteristic
497C2#2085711yescharacteristic
498C2#2087311yescharacteristic
499C2#2087711yescharacteristic
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. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#342092
#219882
#218342
#213642
C2×#15562
#89262
#89252
#89232
#89222
#79022
#79002