Finite Group Databasebeta

← #106920

Subgroups — #106920

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2356C211yesmaximal, characteristic
2#20647C211yesmaximal, characteristic
3C2×#1479C211yesmaximal, characteristic
4C2×#1303C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1286C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#419—21no—
13#2356—22no—
14#2356—22no—
15#2356—22no—
16#2356C414yescomplemented
17C2×#391—21no—
18C2×#387—22no—
19#2285C2211yescharacteristic
20C2×#383—21no—
21C2×#382C2211yescharacteristic
22C2×C4×C4⋊C4—21no—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×#376—21no—
25#1805—22no—
26#1803C414yescomplemented
27#1736—21no—
28#1479—22no—
29#1479C414yescomplemented
30C2×Q8⋊C8—22no—
31#1303—22no—
32#1294—21no—
33#1294—21no—
34#1294—22no—
35#1286—22no—
36C23×Q8D411yescharacteristic
37C22×M16—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8D412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8Q812yes—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C42—21no—
56C22×C42—22no—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66#431C2×C412yes—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2×C412yes—
75#419—42no—
76#419—42no—
77C2×#103—24no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#391—24no—
83#391—24no—
84#387—28no—
85#387—28no—
86#384C2×C412yes—
87#383—24no—
88#383—24no—
89#382—21no—
90#382—21no—
91#382—21no—
92#382C2×C412yes—
93#382D412yes—
94#380—21no—
95#380—21no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4D412yes—
104#376—24no—
105#376—24no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112C2×#95D412yes—
113#340—24no—
114Q8⋊C8—28no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8C22⋊C412yes—
127C22×Q8C22⋊C412yes—
128C22×Q8C4⋊C412yes—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—41no—
138C2×M16—41no—
139C2×M16—41no—
140C2×M16—41no—
141C2×M16—41no—
142C2×M16—41no—
143C22×C8—22no—
144C22×C8—41no—
145C22×C8—41no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8C4214yes—
159C4×Q8C22⋊C414yes—
160C4×Q8C4⋊C414yes—
161C4×Q8C4⋊C414yes—
162#117—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C4⋊C412yes—
190C2×C22⋊C4—41no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—28no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—48no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q811yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226#103—48no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C4×C8—22no—
232C4×C8—22no—
233C4×C8—41no—
234C4×C8—41no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—41no—
244#95—41no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C414yes—
248C24#12011yescharacteristic
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8#9514yes—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C4⋊C411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4C22⋊Q811yescharacteristic
321C22×C4C22⋊Q811yescharacteristic
322C22×C4#12311yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4#13511yescharacteristic
325C22×C4#13511yescharacteristic
326C22×C4#13511yescharacteristic
327C22×C4#13511yescharacteristic
328C22×C4AGL(1,ℤ/8)11yescharacteristic
329C22×C4AGL(1,ℤ/8)11yescharacteristic
330C22×C4#13711yescharacteristic
331C22×C4#13711yescharacteristic
332M16—81no—
333M16—81no—
334M16—81no—
335M16—81no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—82no—
342C2×C8—82no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4#9514yes—
361C22⋊C4—81no—
362C22⋊C4—81no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—416no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—81no—
373C42—81no—
374C42C2×C4211yescharacteristic
375C42C2×C22⋊C411yescharacteristic
376C42C2×C4⋊C411yescharacteristic
377C42C2×C4⋊C411yescharacteristic
378C42#11711yescharacteristic
379C42#11711yescharacteristic
380C42C4×D411yescharacteristic
381C42C4×D411yescharacteristic
382C42C4×D411yescharacteristic
383C42C4×D411yescharacteristic
384C42#12111yescharacteristic
385C42#12111yescharacteristic
386C42C22⋊Q811yescharacteristic
387C42C22⋊Q811yescharacteristic
388C42#12311yescharacteristic
389C42#12311yescharacteristic
390C23—22no—
391C23—22no—
392C23#37911yescharacteristic
393C23#38511yescharacteristic
394C23#38611yescharacteristic
395C23#44911yescharacteristic
396C23#44911yescharacteristic
397C23#45211yescharacteristic
398C23#45211yescharacteristic
399Q8—24no—
400Q8—24no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4C2×#9511yescharacteristic
449C2×C4C4×C22⋊C411yescharacteristic
450C2×C4C4×C22⋊C411yescharacteristic
451C2×C4#38011yescharacteristic
452C2×C4#38011yescharacteristic
453C2×C4#38111yescharacteristic
454C2×C4#38511yescharacteristic
455C2×C4#38511yescharacteristic
456C2×C4#38511yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#41611yescharacteristic
461C2×C4#41611yescharacteristic
462C2×C4#41611yescharacteristic
463C2×C4#41611yescharacteristic
464C2×C4#41711yescharacteristic
465C2×C4#41711yescharacteristic
466C2×C4#41811yescharacteristic
467C2×C4#41911yescharacteristic
468C2×C4#42711yescharacteristic
469C2×C4#42711yescharacteristic
470C2×C4#42811yescharacteristic
471C2×C4#42811yescharacteristic
472C2×C4#44911yescharacteristic
473C2×C4#44911yescharacteristic
474C2×C4#45211yescharacteristic
475C2×C4#45211yescharacteristic
476C2×C4#48211yescharacteristic
477C2×C4#48211yescharacteristic
478C2×C4#48211yescharacteristic
479C2×C4#48211yescharacteristic
480C2×C4#48511yescharacteristic
481C2×C4#48511yescharacteristic
482C2×C4#48511yescharacteristic
483C2×C4#48511yescharacteristic
484C8—82no—
485C8—82no—
486C8—82no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22C22⋊#9511yescharacteristic
496C22#178811yescharacteristic
497C22#178811yescharacteristic
498C22#185411yescharacteristic
499C22#185411yescharacteristic
500C22#187311yescharacteristic
501C22#187511yescharacteristic
502C22#190512yes—
503C22#190612yes—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—44no—
513C4—44no—
514C4—44no—
515C4—82no—
516C4—84no—
517C4—162no—
518C4#173511yescharacteristic
519C4#173611yescharacteristic
520C4#182511yescharacteristic
521C4#182511yescharacteristic
522C4#185411yescharacteristic
523C4#185411yescharacteristic
524C4#187311yescharacteristic
525C4#187511yescharacteristic
526C2—21no—
527C2—21no—
528C2—21no—
529C2—21no—
530C2#917812yes—
531C2#918212yes—
532C2#1073111yescharacteristic
533C2#1077211yescharacteristic
534C2#1078611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2356C4semidirect
26. #1803C4semidirect
29. #1479C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23562
#206472
C2×#14792
C2×#13032
#80262
#80142
C2×#12862