Finite Group Databasebeta

← #219703

Subgroups — #219703

Order 512.

606 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21704C211yesmaximal, characteristic
2#17878C211yesmaximal, characteristic, complemented
3#17877C211yesmaximal, characteristic, complemented
4#9672C211yesmaximal, characteristic, complemented
5#9670C211yesmaximal, characteristic, complemented
6#9105C211yesmaximal, characteristic, complemented
7#9105C211yesmaximal, characteristic, complemented
8#9102C211yesmaximal, characteristic, complemented
9#9099C211yesmaximal, characteristic, complemented
10#9093C211yesmaximal, characteristic, complemented
11#9091C211yesmaximal, characteristic, complemented
12#9090C211yesmaximal, characteristic, complemented
13#9089C211yesmaximal, characteristic, complemented
14#8780C211yesmaximal, characteristic, complemented
15#8140C211yesmaximal, characteristic, complemented
16C2×#496C411yescharacteristic
17#2982C2211yescharacteristic
18#2975C2211yescharacteristic
19#2975C2211yescharacteristic
20#2972C2211yescharacteristic
21C2×#442C2211yescharacteristic
22C2×#440C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C2×C4×D8C2211yescharacteristic
25#2328C2211yescharacteristic
26#2326C2211yescharacteristic
27#1973C411yescharacteristic
28#1972C411yescharacteristic
29#1970C411yescharacteristic
30#1968C411yescharacteristic
31#1958C411yescharacteristic
32#1957C411yescharacteristic
33#1847C411yescharacteristic
34#1726C2211yescharacteristic
35#1588—22no—
36#1586—22no—
37#1583—22no—
38#1581—22no—
39#1564C2211yescharacteristic
40#1564C2211yescharacteristic
41#1479C2211yescharacteristic
42#1479C2211yescharacteristic
43#1478C2211yescharacteristic
44#1478C2211yescharacteristic
45#1478C2211yescharacteristic
46#1478C2211yescharacteristic
47#1478C2211yescharacteristic
48#1478C2211yescharacteristic
49#1461C2211yescharacteristic
50#1460C2211yescharacteristic
51#1460C2211yescharacteristic
52#1460C2211yescharacteristic
53#1452C2211yescharacteristic
54#1452C2211yescharacteristic
55#1281C2211yescharacteristic
56#1281C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59#1274C2211yescharacteristic
60#1274C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C22×SD16C2×C411yescharacteristic
64C22×SD16C2×C411yescharacteristic
65C22×D8C2×C411yescharacteristic
66C22×D8C2×C411yescharacteristic
67C2×#127C2×C411yescharacteristic
68C2×#124C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#496—24no—
74C8×Q8—22no—
75C8×Q8—41no—
76#442—41no—
77#442—41no—
78#440—41no—
79#440—41no—
80C4×SD16—21no—
81C4×SD16—21no—
82C4×SD16—21no—
83C4×SD16—21no—
84C4×D8—21no—
85C4×D8—21no—
86C4×D8—21no—
87C4×D8—21no—
88#436—22no—
89#436—22no—
90#436—41no—
91#436—41no—
92C8×D4—22no—
93C8×D4—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×#106C2×C411yescharacteristic
106C2×#106C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115#407—41no—
116#407—41no—
117C2×C22⋊C8—41no—
118C2×#97C2×C411yescharacteristic
119C2×#97C2×C411yescharacteristic
120C2×#97C2311yescharacteristic
121C2×C4×C8C2×C411yescharacteristic
122C2×C4×C8C2×C411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#388—41no—
129#386—41no—
130#386—41no—
131#385—41no—
132#384C2×C411yescharacteristic
133#382C2×C411yescharacteristic
134#376C2311yescharacteristic
135#335—21no—
136#335—21no—
137#334—21no—
138#334—21no—
139Q8⋊C8—22no—
140Q8⋊C8—41no—
141D4⋊C8—22no—
142D4⋊C8—22no—
143D4⋊C8—22no—
144D4⋊C8—41no—
145D4⋊C8—41no—
146D4⋊C8—41no—
147#324—41no—
148#323—41no—
149#323—41no—
150#323—41no—
151#322—21no—
152#322—21no—
153#322D412yescomplemented
154#322D412yescomplemented
155C22×Q8C22×C411yescharacteristic
156C22×D4—41no—
157C22×D4C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C2×SD16—22no—
164C2×SD16—22no—
165C2×SD16—21no—
166C2×SD16—21no—
167C2×SD16—22no—
168C2×SD16—41no—
169C2×SD16—41no—
170C2×D8—22no—
171C2×D8—22no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—22no—
175C2×D8—41no—
176C2×D8—41no—
177C2×M16—42no—
178C2×M16—42no—
179C22×C8—42no—
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184#127—22no—
185#127—41no—
186#124—41no—
187#124M1614yescomplemented
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—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—21no—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C2411yescharacteristic
221#107—21no—
222#107—21no—
223#107—41no—
224#106—21no—
225#106—21no—
226#106—41no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—42no—
236#103—21no—
237#103—21no—
238#103—41no—
239#102—21no—
240#102—21no—
241#102—21no—
242#102—21no—
243#102—21no—
244#102—21no—
245#102—41no—
246#102—41no—
247#102—41no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259#97—22no—
260#97—21no—
261#97—21no—
262#97—41no—
263C4×C8—22no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—41no—
271C4×C8—42no—
272C4×C8C2×D412yes—
273C4×C8C2×D412yes—
274C4×C8C2×D411yescharacteristic
275C4×C8C2×D411yescharacteristic
276C4×C8#4111yescharacteristic
277C4×C8#4111yescharacteristic
278#95—21no—
279#95—21no—
280#95—41no—
281#95—41no—
282C24—21no—
283C24—21no—
284C24—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8C2×M1612yes—
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—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—44no—
309C2×D4—161no—
310C2×D4—161no—
311C2×D4C2×M1612yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—161no—
326C22×C4—161no—
327C22×C4—161no—
328C22×C4C23×C411yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331SD16—42no—
332SD16—42no—
333SD16—44no—
334D8—42no—
335D8—42no—
336D8—44no—
337M16—162no—
338M16—162no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—162no—
357C2×C8C4×D411yescharacteristic
358C2×C8C4×D411yescharacteristic
359C2×C8C4×D411yescharacteristic
360C2×C8C4×D411yescharacteristic
361C2×C8#13112yes—
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—41no—
371C4⋊C4—41no—
372C4⋊C4—161no—
373C22⋊C4—44no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—161no—
381C22⋊C4—161no—
382C22⋊C4—161no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—41no—
387C42—41no—
388C42C2×M1612yes—
389C42C22×D411yescharacteristic
390C42C2×#4111yescharacteristic
391C42H(5,𝔽2)11yescharacteristic
392C42H(5,𝔽2)11yescharacteristic
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23#51811yescharacteristic
409Q8—22no—
410Q8—22no—
411Q8—41no—
412Q8—41no—
413D4—22no—
414D4—22no—
415D4—41no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—164no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—161no—
449C2×C4—161no—
450C2×C4—161no—
451C2×C4C2×C4×D411yescharacteristic
452C2×C4#51811yescharacteristic
453C2×C4#53511yescharacteristic
454C2×C4#53511yescharacteristic
455C2×C4#54011yescharacteristic
456C2×C4#54011yescharacteristic
457C2×C4C22×M1611yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4#56811yescharacteristic
460C2×C4#56811yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—21no—
465C8—22no—
466C8—42no—
467C8—42no—
468C8—41no—
469C8—81no—
470C8—81no—
471C8—81no—
472C8—81no—
473C8—81no—
474C8—81no—
475C22—21no—
476C22—21no—
477C22—42no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—161no—
490C22—161no—
491C22—161no—
492C22#232811yescharacteristic
493C22#297311yescharacteristic
494C22#298111yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—42no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4D4×M1611yescharacteristic
507C4#293511yescharacteristic
508C4#297411yescharacteristic
509C4#298011yescharacteristic
510C2—21no—
511C2—21no—
512C2—82no—
513C2—161no—
514C2—161no—
515C2#2226611yescharacteristic
516C2#2228411yescharacteristic
517C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#496C4
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. #17878C2semidirect
3. #17877C2semidirect
4. #9672C2semidirect
5. #9670C2semidirect
6. #9105C2semidirect
7. #9105C2semidirect
8. #9102C2semidirect
9. #9099C2semidirect
10. #9093C2semidirect
11. #9091C2semidirect
12. #9090C2semidirect
13. #9089C2semidirect
14. #8780C2semidirect
15. #8140C2semidirect
153. #322D4semidirect
154. #322D4semidirect
187. #124M16semidirect

Maximal subgroups

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

subgroupindex
#217042
#178782
#178772
#96722
#96702
#91052
#91022
#90992
#90932
#90912
#90902
#90892
#87802
#81402