Finite Group Databasebeta

← #239820

Subgroups — #239820

Order 512.

1142 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#21859C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#21171C211yesmaximal, characteristic, complemented
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1492C211yesmaximal, characteristic
7#10417C214yesmaximal, complemented
8#10416C214yesmaximal, complemented
9#7890C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2862—21no—
17#2858—21no—
18#2640—22no—
19#2563—21no—
20#2563—21no—
21#2547C2212yes—
22#2547C2212yes—
23#2504—22no—
24#2436—22no—
25#2311—21no—
26#2311—21no—
27C2×#384C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1990—22no—
30#1990—21no—
31#1990C2214yes—
32#1957—21no—
33#1948—21no—
34#1948C2212yes—
35#1947—21no—
36#1947C2212yes—
37#1874—22no—
38#1802C2214yes—
39#1736—21no—
40C2×#340—22no—
41#1549—24no—
42C2×#332C2214yes—
43#1492—24no—
44C2×D4⋊C8C2214yes—
45#1279C2212yes—
46#1279C2212yes—
47C22×#41—21no—
48C23×Q8—21no—
49C22×M16—21no—
50C23×C8—21no—
51#532—22no—
52#532—22no—
53#532—22no—
54#532—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8C2312yes—
64C2×#127D411yescharacteristic
65C2×#124—22no—
66C2×#124—22no—
67C2×#124—21no—
68C2×#121—21no—
69C2×#121D412yes—
70C2×C4×D4—21no—
71C2×C4×D4C2312yes—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D412yes—
79C22×C42C2311yescharacteristic
80#431—22no—
81#431—22no—
82#428—44no—
83C2×#107—22no—
84C2×#106—22no—
85#423—22no—
86#423D414yes—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×#102—24no—
91C2×C22⋊C8—21no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#397—24no—
98#397—24no—
99#393—24no—
100#393—24no—
101#391—22no—
102#391—22no—
103#391—22no—
104#391—22no—
105#390—22no—
106#390—22no—
107#390—22no—
108#390—22no—
109#386—24no—
110#386—24no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384C2312yes—
115#382—22no—
116#382—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120#380—22no—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127#376—21no—
128C43—21no—
129#340—24no—
130#340—22no—
131#340—22no—
132#340—28no—
133#332—28no—
134D4⋊C8—28no—
135C2×#41—42no—
136C2×#41—42no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—82no—
147C22×Q8—82no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C2×D412yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—24no—
159C2×M16—42no—
160C2×M16—42no—
161C22×C8—24no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—42no—
166C22×C8—42no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8—24no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179#127—22no—
180#127—22no—
181#124—42no—
182#124—42no—
183#124—84no—
184#124—84no—
185#121—24no—
186#121—24no—
187#121—42no—
188#121—42no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—42no—
194C4×D4—42no—
195#117—22no—
196#117—22no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D412yes—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244C2×C42C2411yescharacteristic
245#107—44no—
246#106—44no—
247C4⋊C8—24no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8C2×D414yes—
252#102—84no—
253C22⋊C8—42no—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—42no—
257C4×C8—42no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266C24—22no—
267C24#12011yescharacteristic
268#41—84no—
269#41—84no—
270#41—84no—
271#41—84no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—42no—
277C2×Q8—44no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×Q8—88no—
283C2×Q8—88no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12112yes—
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4#12312yes—
327C22×C4#12312yes—
328C22×C4#12412yes—
329C22×C4#13512yes—
330C22×C4#13512yes—
331C22×C4AGL(1,ℤ/8)12yes—
332C22×C4#13712yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336M16—48no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—42no—
340C2×C8—48no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C22⋊C4—84no—
377C22⋊C4—84no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42#12112yes—
387C42#12112yes—
388C42#12112yes—
389C42#12312yes—
390C42#12312yes—
391C42#12412yes—
392C42C22×D411yescharacteristic
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23#39412yes—
408C23#44912yes—
409C23#45212yes—
410C23C2×#12011yescharacteristic
411Q8—88no—
412Q8—84no—
413Q8—84no—
414Q8—84no—
415D4—44no—
416D4—44no—
417D4—88no—
418D4—84no—
419D4—84no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—88no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—164no—
455C2×C4#39412yes—
456C2×C4#39412yes—
457C2×C4#39412yes—
458C2×C4#44912yes—
459C2×C4#45212yes—
460C2×C4#45412yes—
461C2×C4#45612yes—
462C2×C4#46314yes—
463C2×C4#46414yes—
464C2×C4#48814yes—
465C2×C4#48914yes—
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×#12311yescharacteristic
470C2×C4C2×#12311yescharacteristic
471C2×C4C2×#12411yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
475C2×C4C2×#13711yescharacteristic
476C2×C4#57612yes—
477C2×C4#57712yes—
478C8—44no—
479C8—44no—
480C8—82no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22#204214yes—
495C22C2×#39411yescharacteristic
496C22C2×#44911yescharacteristic
497C22C2×#45211yescharacteristic
498C22#300811yescharacteristic
499C22#300911yescharacteristic
500C22C2×#45411yescharacteristic
501C22C2×#45611yescharacteristic
502C4—22no—
503C4—21no—
504C4—21no—
505C4—84no—
506C4—82no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—162no—
512C4#203814yes—
513C4#204214yes—
514C4C2×#46311yescharacteristic
515C4C2×#46411yescharacteristic
516C4#306212yes—
517C4#309612yes—
518C4C2×#48811yescharacteristic
519C4C2×#48911yescharacteristic
520C2—24no—
521C2—162no—
522C2#1864611yescharacteristic
523C2#1864711yescharacteristic
524C2#1865611yescharacteristic
525C2#1865711yescharacteristic
526C2#2348311yescharacteristic
527C2C2×#203811yescharacteristic
528C2C2×#204211yescharacteristic
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. C2×#1990C2semidirect
4. #21171C2semidirect
5. C2×#1549C2semidirect
7. #10417C2semidirect
8. #10416C2semidirect
9. #7890C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#218592
C2×#19902
#211712
C2×#15492
C2×#14922
#104172
#104162
#78902