Finite Group Databasebeta

← #202253

Subgroups — #202253

Order 512.

676 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#21974C211yesmaximal, characteristic
2#20535C211yesmaximal, characteristic
3C4×#428C211yesmaximal, characteristic
4#9540C211yesmaximal, characteristic, complemented
5#9539C211yesmaximal, characteristic, complemented
6#9538C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#9536C211yesmaximal, characteristic, complemented
9#9535C211yesmaximal, characteristic, complemented
10#9534C211yesmaximal, characteristic, complemented
11#9533C211yesmaximal, characteristic, complemented
12#7902C211yesmaximal, characteristic
13#7901C211yesmaximal, characteristic
14#7876C211yesmaximal, characteristic
15#7850C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#428C411yescharacteristic
20C2×#428C411yescharacteristic
21C2×#423C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C4×#117C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1851—21no—
30#1851—21no—
31#1842—21no—
32#1831—21no—
33#1776C411yescharacteristic
34#1776C411yescharacteristic
35#1776C411yescharacteristic
36#1776C411yescharacteristic
37C4×#107C2211yescharacteristic
38C4×#107C2211yescharacteristic
39C4×#106C2211yescharacteristic
40C4×#106C2211yescharacteristic
41#1771—21no—
42#1771—21no—
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1749—21no—
48C4×C22⋊C8—21no—
49#1736C411yescharacteristic
50#1736C411yescharacteristic
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53#1279C2211yescharacteristic
54#1279C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C22×M16C2×C411yescharacteristic
69C22×M16C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#432—22no—
77#432Q814yes—
78#431—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82#428—24no—
83#428—24no—
84C2×#107C2×C411yescharacteristic
85C2×#107C2×C411yescharacteristic
86C2×#107C2×C411yescharacteristic
87C2×#107C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×#106C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92#423—22no—
93#423—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8D411yescharacteristic
109C2×C22⋊C8D411yescharacteristic
110C4×M16—22no—
111C4×M16—22no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97C2×C411yescharacteristic
116C2×#97C2×C411yescharacteristic
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138C2×#95—21no—
139C43—21no—
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—41no—
166C2×M16—41no—
167C2×M16—42no—
168C2×M16—81no—
169C2×M16—81no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184#117—22no—
185#117—22no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—81no—
215C2×C42—81no—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4111yescharacteristic
223C2×C42C2411yescharacteristic
224#107—22no—
225#107—22no—
226#107—22no—
227#107M1614yescomplemented
228#106—22no—
229#106—22no—
230#106—22no—
231#106M1614yescomplemented
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244C4⋊C8—81no—
245C4⋊C8—81no—
246C4⋊C8C2×Q812yes—
247C4⋊C8C2×Q812yes—
248C22⋊C8—24no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—42no—
252#97—22no—
253#97—22no—
254#97—24no—
255#97—81no—
256#97—81no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—81no—
270C4×C8—81no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279C24C4×D411yescharacteristic
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4#11712yes—
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×Q812yes—
309C22×C4C22⋊Q812yes—
310C22×C4#12312yes—
311C22×C4#12512yes—
312C22×C4#12612yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316M16—28no—
317M16—44no—
318M16—84no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8C2×M1612yes—
360C2×C8C2×M1612yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4C2×M1612yes—
376C4⋊C4C2×M1612yes—
377C4⋊C4C2×M1612yes—
378C4⋊C4C2×M1612yes—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—81no—
394C42—81no—
395C42#11712yes—
396C42C4×Q812yes—
397C42C22⋊Q812yes—
398C42#12312yes—
399C42#12512yes—
400C42#12612yes—
401C42C22×Q811yescharacteristic
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23#38712yes—
412C23C2×C4×D411yescharacteristic
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4#38712yes—
458C2×C4#38712yes—
459C2×C4#38712yes—
460C2×C4C2×#11711yescharacteristic
461C2×C4C2×C4×Q811yescharacteristic
462C2×C4C2×C22⋊Q811yescharacteristic
463C2×C4C2×#12311yescharacteristic
464C2×C4C2×#12511yescharacteristic
465C2×C4C2×#12611yescharacteristic
466C2×C4C22×M1611yescharacteristic
467C2×C4C22×M1611yescharacteristic
468C2×C4#56811yescharacteristic
469C2×C4#56811yescharacteristic
470C2×C4#57611yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C2×C4#57811yescharacteristic
474C8—22no—
475C8—22no—
476C8—42no—
477C8—42no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—81no—
485C8—81no—
486C8—81no—
487C8—81no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22C2×#38711yescharacteristic
496C22D4×M1611yescharacteristic
497C22D4×M1611yescharacteristic
498C22#294611yescharacteristic
499C22#294711yescharacteristic
500C22#295511yescharacteristic
501C22#295511yescharacteristic
502C4—22no—
503C4—22no—
504C4—21no—
505C4—21no—
506C4—22no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4#292011yescharacteristic
516C4#292111yescharacteristic
517C4Q8×M1611yescharacteristic
518C4Q8×M1611yescharacteristic
519C4#308311yescharacteristic
520C4#308411yescharacteristic
521C4#309611yescharacteristic
522C4#309711yescharacteristic
523C2—42no—
524C2#1805011yescharacteristic
525C2#1805111yescharacteristic
526C2#1805211yescharacteristic
527C2#1805311yescharacteristic
528C2#2102311yescharacteristic
529C2#2103511yescharacteristic
530C2#2104111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9540C2semidirect
5. #9539C2semidirect
6. #9538C2semidirect
7. #9537C2semidirect
8. #9536C2semidirect
9. #9535C2semidirect
10. #9534C2semidirect
11. #9533C2semidirect
227. #107M16semidirect
231. #106M16semidirect

Maximal subgroups

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

subgroupindex
#219742
#205352
C4×#4282
#95402
#95392
#95382
#95372
#95362
#95352
#95342
#95332
#79022
#79012
#78762
#78502