Finite Group Databasebeta

← #219547

Subgroups — #219547

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23812C211yesmaximal, characteristic
2#17880C211yesmaximal, characteristic, complemented
3#17877C211yesmaximal, characteristic
4#9669C211yesmaximal, characteristic, complemented
5#9668C211yesmaximal, characteristic
6#9106C211yesmaximal, characteristic, complemented
7#9103C211yesmaximal, characteristic, complemented
8#9102C211yesmaximal, characteristic, complemented
9#9099C211yesmaximal, characteristic, complemented
10#9094C211yesmaximal, characteristic, complemented
11#9091C211yesmaximal, characteristic, complemented
12#9090C211yesmaximal, characteristic, complemented
13#9087C211yesmaximal, characteristic, complemented
14#8783C211yesmaximal, characteristic, complemented
15#8141C211yesmaximal, characteristic, complemented
16#2983C2211yescharacteristic
17#2982C2211yescharacteristic
18#2975C2211yescharacteristic
19#2972C2211yescharacteristic
20C2×#440C2211yescharacteristic
21C2×#440C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24#2326C2211yescharacteristic
25#2326C2211yescharacteristic
26#2095C411yescharacteristic
27#2095C411yescharacteristic
28#2093C411yescharacteristic
29#2093C411yescharacteristic
30#2015C411yescharacteristic
31#2015C411yescharacteristic
32#2012C411yescharacteristic
33#2012C411yescharacteristic
34#1746C2211yescharacteristic
35#1587—22no—
36#1586—22no—
37#1584—22no—
38#1581—22no—
39#1564C2211yescharacteristic
40#1564C2211yescharacteristic
41#1479C2211yescharacteristic
42#1479C2211yescharacteristic
43#1479C2211yescharacteristic
44#1479C2211yescharacteristic
45#1478C2211yescharacteristic
46#1478C2211yescharacteristic
47#1478C2211yescharacteristic
48#1478C2211yescharacteristic
49#1461C2211yescharacteristic
50#1461C2211yescharacteristic
51#1460C2211yescharacteristic
52#1460C2211yescharacteristic
53#1452C2211yescharacteristic
54#1450C2211yescharacteristic
55#1281C2211yescharacteristic
56#1281C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59#1274C2211yescharacteristic
60#1274C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C22×Q16C2×C411yescharacteristic
64C22×SD16C2×C411yescharacteristic
65C22×SD16C2×C411yescharacteristic
66C22×D8C2×C411yescharacteristic
67C2×#124C2×C411yescharacteristic
68C2×#124C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#446—22no—
74#446—41no—
75C8×Q8—22no—
76C8×Q8—41no—
77#440—41no—
78#440—41no—
79#440—41no—
80#440—41no—
81C4×Q16—21no—
82C4×Q16—21no—
83C4×Q16—21no—
84C4×Q16—21no—
85C4×D8—21no—
86C4×D8—21no—
87C4×D8—21no—
88C4×D8—21no—
89#436—22no—
90#436—41no—
91C8×D4—22no—
92C8×D4—41no—
93#432—41no—
94#431—41no—
95#431C2311yescharacteristic
96#431C2311yescharacteristic
97#431C2311yescharacteristic
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103C2×#107C2×C411yescharacteristic
104C2×#107C2×C411yescharacteristic
105C2×#107C2×C411yescharacteristic
106C2×#107C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115#407—41no—
116C2×C22⋊C8—41no—
117C2×#97C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#388—41no—
124#388—41no—
125#385—41no—
126#385—41no—
127#384C2×C411yescharacteristic
128#384C2×C411yescharacteristic
129#376C2311yescharacteristic
130#336C2×C411yescharacteristic
131#336C2×C411yescharacteristic
132#336C2×C411yescharacteristic
133#336C2×C411yescharacteristic
134#335—21no—
135#335—21no—
136#334—21no—
137#334—21no—
138Q8⋊C8—22no—
139Q8⋊C8—22no—
140Q8⋊C8—41no—
141Q8⋊C8—41no—
142D4⋊C8—22no—
143D4⋊C8—22no—
144D4⋊C8—41no—
145D4⋊C8—41no—
146#324—41no—
147#324—41no—
148#323—41no—
149#323—41no—
150#322—21no—
151#322D412yes—
152#322D412yes—
153C82—21no—
154C22×Q8C22×C411yescharacteristic
155C22×Q8C22×C411yescharacteristic
156C22×D4C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C2×Q16—22no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—41no—
164C2×SD16—22no—
165C2×SD16—22no—
166C2×SD16—41no—
167C2×SD16—41no—
168C2×SD16—41no—
169C2×SD16—41no—
170C2×D8—22no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—41no—
174C2×M16—42no—
175C22×C8—42no—
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180#124—22no—
181#124—22no—
182#124—41no—
183#124—41no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C2411yescharacteristic
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—41no—
224#107—41no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235#103—21no—
236#103—21no—
237#103—21no—
238#103—21no—
239#103—41no—
240#103—41no—
241#102—21no—
242#102—21no—
243#102—21no—
244#102—21no—
245#102—41no—
246#102—41no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—42no—
256C22⋊C8—42no—
257#97—21no—
258#97—21no—
259#97—42no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—42no—
267C4×C8C2×D412yes—
268C4×C8C2×D412yes—
269C4×C8C2×D411yescharacteristic
270C4×C8C2×D411yescharacteristic
271C4×C8#4111yescharacteristic
272C4×C8#4111yescharacteristic
273#95—21no—
274#95—21no—
275#95—41no—
276#95—41no—
277C24—21no—
278C24—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8#13112yes—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4#13112yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—161no—
316C22×C4—161no—
317C22×C4C23×C411yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320Q16—42no—
321Q16—42no—
322SD16—44no—
323SD16—44no—
324D8—42no—
325D8—42no—
326M16—162no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—162no—
346C2×C8C4×D411yescharacteristic
347C2×C8C4×D411yescharacteristic
348C2×C8C4×D411yescharacteristic
349C2×C8C4×D411yescharacteristic
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—161no—
363C4⋊C4—161no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—161no—
371C22⋊C4—161no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42C22×D411yescharacteristic
381C42C2×#4111yescharacteristic
382C42H(5,𝔽2)11yescharacteristic
383C42H(5,𝔽2)11yescharacteristic
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23#51811yescharacteristic
395Q8—22no—
396Q8—22no—
397Q8—41no—
398Q8—41no—
399Q8—41no—
400Q8—41no—
401Q8—41no—
402Q8—41no—
403D4—22no—
404D4—22no—
405D4—41no—
406D4—41no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
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—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—161no—
436C2×C4—161no—
437C2×C4C2×C4×D411yescharacteristic
438C2×C4#51811yescharacteristic
439C2×C4#53511yescharacteristic
440C2×C4#53511yescharacteristic
441C2×C4#54011yescharacteristic
442C2×C4#54011yescharacteristic
443C2×C4C2×#13111yescharacteristic
444C2×C4C2×#13111yescharacteristic
445C2×C4#56811yescharacteristic
446C2×C4#56811yescharacteristic
447C8—21no—
448C8—21no—
449C8—21no—
450C8—21no—
451C8—42no—
452C8—42no—
453C8—41no—
454C8—41no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C8—81no—
461C22—21no—
462C22—21no—
463C22—42no—
464C22—82no—
465C22—82no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—161no—
472C22—161no—
473C22#232811yescharacteristic
474C22#297611yescharacteristic
475C22#297611yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4#293311yescharacteristic
489C4#293311yescharacteristic
490C4#297711yescharacteristic
491C4#297811yescharacteristic
492C2—21no—
493C2—21no—
494C2—82no—
495C2—161no—
496C2#2226711yescharacteristic
497C2#2228511yescharacteristic
498C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17880C2semidirect
4. #9669C2semidirect
6. #9106C2semidirect
7. #9103C2semidirect
8. #9102C2semidirect
9. #9099C2semidirect
10. #9094C2semidirect
11. #9091C2semidirect
12. #9090C2semidirect
13. #9087C2semidirect
14. #8783C2semidirect
15. #8141C2semidirect

Maximal subgroups

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

subgroupindex
#238122
#178802
#178772
#96692
#96682
#91062
#91032
#91022
#90992
#90942
#90912
#90902
#90872
#87832
#81412