Finite Group Databasebeta

← #196914

Subgroups — #196914

Order 512.

1231 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35111C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic, complemented
3#21822C211yesmaximal, characteristic, complemented
4C2×#1512C211yesmaximal, characteristic
5C2×#1509C211yesmaximal, characteristic
6#9157C212yesmaximal
7#9153C212yesmaximal, complemented
8#9149C212yesmaximal
9#9145C212yesmaximal, complemented
10#8021C211yesmaximal, characteristic, complemented
11#8017C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16#2855C2212yes—
17#2854C2212yes—
18#2623—21no—
19#2614—21no—
20#2604—22no—
21#2599—22no—
22#2592C412yescomplemented
23#2584C412yescomplemented
24#2581C412yescomplemented
25#2580C412yescomplemented
26#2356—24no—
27#2356—24no—
28#2322—22no—
29C2×#391—21no—
30C2×#387—22no—
31#2283C2211yescharacteristic
32C2×#383C2212yes—
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376C2212yes—
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—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8C2×C412yes—
72C2×C4×Q8D412yes—
73C2×C4×Q8D412yes—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D412yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C42—22no—
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85#432—22no—
86#432—22no—
87#432—22no—
88#432—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2311yescharacteristic
93C2×C22⋊C8C2312yes—
94C2×#97—21no—
95C2×#97—21no—
96#400—22no—
97#399—22no—
98#398—22no—
99#397—22no—
100#395—22no—
101#395—22no—
102#393—22no—
103#393—22no—
104#391—28no—
105#391C2×C414yes—
106#390C2×C414yes—
107#389—24no—
108#389C2×C414yes—
109#389C2×C414yes—
110#388—24no—
111#388—24no—
112#387—216no—
113#387—24no—
114#387—24no—
115#386—22no—
116#385—22no—
117#385—24no—
118#383—21no—
119#383—24no—
120#383—24no—
121#382—22no—
122#382C2×C412yes—
123#382D414yes—
124#381—22no—
125#381—21no—
126#380C2312yes—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4D414yes—
131C4×C22⋊C4—21no—
132#376—21no—
133#376—24no—
134#376—24no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142#336—24no—
143#336—24no—
144#328—22no—
145#328—22no—
146#328—24no—
147#327—22no—
148#327—22no—
149#327—24no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8C22⋊C412yes—
155C22×Q8C22⋊C412yes—
156C22×Q8C22×C411yescharacteristic
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171#126—42no—
172#126—42no—
173#125—22no—
174#125—22no—
175#124—22no—
176#124—22no—
177#123—44no—
178#123—44no—
179#123—42no—
180#123—42no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185#121—21no—
186#121—21no—
187#121—21no—
188#121—21no—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—42no—
202C4×D4—42no—
203#117—42no—
204#117—42no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C22⋊C414yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C414yes—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—28no—
239C2×C42—21no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42—48no—
244C2×C42C22×C414yes—
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D414yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4112yes—
253C2×C42#4112yes—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—22no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—22no—
262#97—42no—
263#97—42no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—21no—
276#95—22no—
277#95—21no—
278#95—22no—
279#95C2×D412yes—
280#95C2×D414yes—
281#95C2×D412yes—
282C24—21no—
283C24C22×D411yescharacteristic
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—48no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—42no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—82no—
332C22×C4C2×C22⋊C414yes—
333C22×C4C2×C22⋊C414yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4#12012yes—
339C22×C4#12012yes—
340C22×C4#12312yes—
341C22×C4#12312yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C2×C8—42no—
349C2×C8—44no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—41no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—84no—
372C22⋊C4—48no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—84no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—24no—
387C42—48no—
388C42—416no—
389C42—42no—
390C42—82no—
391C42C4×D412yes—
392C42C4×D412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42#12112yes—
396C42#12112yes—
397C42#12112yes—
398C42#12112yes—
399C23—22no—
400C23—22no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—41no—
405C23C22×C22⋊C411yescharacteristic
406C23C2×#12011yescharacteristic
407C23C2×#12311yescharacteristic
408C23#57512yes—
409C23#57512yes—
410Q8—48no—
411Q8—48no—
412Q8—81no—
413Q8—81no—
414D4—81no—
415D4—81no—
416D4—82no—
417D4—81no—
418D4—81no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—48no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—88no—
468C2×C4#38612yes—
469C2×C4#38612yes—
470C2×C4#38612yes—
471C2×C4#38612yes—
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×#12111yescharacteristic
476C8—82no—
477C8—81no—
478C8—81no—
479C22—24no—
480C22—24no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—81no—
486C22—84no—
487C22—81no—
488C22—81no—
489C22C2×#38611yescharacteristic
490C22#288412yes—
491C22#289612yes—
492C22#290112yes—
493C22#300511yescharacteristic
494C22#300511yescharacteristic
495C22#309312yes—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—44no—
505C4—44no—
506C4—44no—
507C4—82no—
508C4—84no—
509C4—81no—
510C4—84no—
511C4—81no—
512C4—81no—
513C2—21no—
514C2—21no—
515C2—22no—
516C2—161no—
517C2#1762414yes—
518C2#2091011yescharacteristic
519C2#2098011yescharacteristic
520C2#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×#2356C2semidirect
3. #21822C2semidirect
7. #9153C2semidirect
9. #9145C2semidirect
10. #8021C2semidirect
11. #8017C2semidirect
22. #2592C4semidirect
23. #2584C4semidirect
24. #2581C4semidirect
25. #2580C4semidirect

Maximal subgroups

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

subgroupindex
#351112
C2×#23562
#218222
C2×#15122
C2×#15092
#91572
#91532
#91492
#91452
#80212
#80172