Finite Group Databasebeta

← #196912

Subgroups — #196912

Order 512.

1383 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35111C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic, complemented
3#21822C211yesmaximal, characteristic, complemented
4C2×#1512C211yesmaximal, characteristic
5C2×#1509C211yesmaximal, characteristic
6#9156C212yesmaximal
7#9152C212yesmaximal, complemented
8#9148C212yesmaximal
9#9144C212yesmaximal, complemented
10#8021C211yesmaximal, characteristic, complemented
11#8017C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#432C2211yescharacteristic
17#2855C2212yes—
18#2854C2212yes—
19#2623—21no—
20#2614—21no—
21#2604—22no—
22#2599—22no—
23#2592C412yescomplemented
24#2584C412yescomplemented
25#2581C412yescomplemented
26#2580C412yescomplemented
27#2351—24no—
28#2351—24no—
29#2322—22no—
30C2×#389—22no—
31C2×#389—21no—
32#2283C2211yescharacteristic
33C2×#382C2212yes—
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1985—22no—
38#1985—22no—
39#1817C2212yes—
40#1817C2212yes—
41#1512—24no—
42#1509—24no—
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#328C2212yes—
46C2×#327C2211yescharacteristic
47C2×#327C2211yescharacteristic
48C2×#327C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#529—22no—
55#529—22no—
56#529—22no—
57#529—22no—
58C2×#126—21no—
59C2×#125C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#123—22no—
62C2×#123—21no—
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121C2311yescharacteristic
65C2×C4×Q8—28no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—48no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C412yes—
73C2×C4×Q8C2×C412yes—
74C2×C4×Q8D412yes—
75C2×C4×Q8D412yes—
76C2×C4×D4—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×#117—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D412yes—
84C22×C4⋊C4D411yescharacteristic
85C22×C42—22no—
86C22×C42—21no—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89#432—22no—
90#432—22no—
91#432—22no—
92#432—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C22⋊C8C2311yescharacteristic
96C2×C22⋊C8C2311yescharacteristic
97C2×C22⋊C8C2312yes—
98C2×#97—21no—
99C2×#97—21no—
100#400—22no—
101#399—22no—
102#398—22no—
103#397—22no—
104#395—22no—
105#395—22no—
106#393—22no—
107#393—22no—
108#391C2×C414yes—
109#390C2×C414yes—
110#389—28no—
111#389—216no—
112#389—24no—
113#389C2×C414yes—
114#389C2×C414yes—
115#388—24no—
116#388—24no—
117#387—24no—
118#387—24no—
119#386—22no—
120#385—22no—
121#385—24no—
122#383—21no—
123#382—22no—
124#382—24no—
125#382—24no—
126#382C2×C412yes—
127#382D414yes—
128#381—22no—
129#381—21no—
130#380C2312yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4D414yes—
136C4×C22⋊C4—21no—
137#376—21no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143#336—24no—
144#336—24no—
145#328—22no—
146#328—22no—
147#328—24no—
148#327—22no—
149#327—22no—
150#327—24no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8C22⋊C412yes—
156C22×Q8C22⋊C412yes—
157C22×Q8C22×C411yescharacteristic
158C22×D4—21no—
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172#126—42no—
173#126—42no—
174#125—22no—
175#125—22no—
176#124—22no—
177#124—22no—
178#123—42no—
179#123—42no—
180#123—44no—
181#123—44no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186#121—21no—
187#121—21no—
188#121—21no—
189#121—21no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—432no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—42no—
204C4×D4—42no—
205#117—42no—
206#117—42no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22⋊C414yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C414yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—28no—
246C2×C42—21no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42C22×C414yes—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D414yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4112yes—
261C2×C42#4112yes—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—22no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—22no—
270#97—42no—
271#97—42no—
272#95—28no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—22no—
285#95C2×D412yes—
286#95C2×D414yes—
287#95C2×D412yes—
288C24—21no—
289C24C22×D411yescharacteristic
290C2×Q8—28no—
291C2×Q8—216no—
292C2×Q8—28no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—28no—
296C2×Q8—28no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—82no—
337C22×C4C2×C22⋊C414yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C414yes—
343C22×C4#12012yes—
344C22×C4#12012yes—
345C22×C4#12312yes—
346C22×C4#12312yes—
347C22×C4AGL(1,ℤ/8)14yes—
348C22×C4#13714yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C2×C8—42no—
356C2×C8—44no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—41no—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—416no—
372C4⋊C4—48no—
373C4⋊C4—416no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—48no—
380C4⋊C4—84no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—48no—
391C22⋊C4—84no—
392C42—24no—
393C42—24no—
394C42—24no—
395C42—24no—
396C42—48no—
397C42—416no—
398C42—42no—
399C42—82no—
400C42C4×D412yes—
401C42C4×D412yes—
402C42C4×D412yes—
403C42C4×D412yes—
404C42#12112yes—
405C42#12112yes—
406C42#12112yes—
407C42#12112yes—
408C23—22no—
409C23—22no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—41no—
414C23C22×C22⋊C411yescharacteristic
415C23C2×#12011yescharacteristic
416C23C2×#12311yescharacteristic
417C23C2×AGL(1,ℤ/8)12yes—
418C23C2×#13712yes—
419Q8—216no—
420Q8—216no—
421Q8—81no—
422Q8—81no—
423D4—81no—
424D4—81no—
425D4—82no—
426D4—81no—
427D4—81no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4#38612yes—
478C2×C4#38612yes—
479C2×C4#38612yes—
480C2×C4#38612yes—
481C2×C4#41718yes—
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C8—82no—
487C8—81no—
488C8—81no—
489C22—24no—
490C22—24no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—81no—
496C22—84no—
497C22—81no—
498C22—81no—
499C22C2×#38611yescharacteristic
500C22#288412yes—
501C22C2×#41712yes—
502C22#290112yes—
503C22#300311yescharacteristic
504C22#300411yescharacteristic
505C22#309212yes—
506C4—22no—
507C4—22no—
508C4—28no—
509C4—22no—
510C4—22no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—44no—
516C4—44no—
517C4—82no—
518C4—84no—
519C4—81no—
520C4—84no—
521C4—81no—
522C4—81no—
523C2—21no—
524C2—22no—
525C2—21no—
526C2—161no—
527C2#1762814yes—
528C2#2090911yescharacteristic
529C2#2098011yescharacteristic
530C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2351C2semidirect
3. #21822C2semidirect
7. #9152C2semidirect
9. #9144C2semidirect
10. #8021C2semidirect
11. #8017C2semidirect
23. #2592C4semidirect
24. #2584C4semidirect
25. #2581C4semidirect
26. #2580C4semidirect

Maximal subgroups

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

subgroupindex
#351112
C2×#23512
#218222
C2×#15122
C2×#15092
#91562
#91522
#91482
#91442
#80212
#80172