Finite Group Databasebeta

← #219616

Subgroups — #219616

Order 512.

580 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22282C211yesmaximal, characteristic, complemented
2#22230C211yesmaximal, characteristic, complemented
3#19701C211yesmaximal, characteristic
4#9769C211yesmaximal, characteristic, complemented
5#9756C211yesmaximal, characteristic, complemented
6#9302C211yesmaximal, characteristic, complemented
7#9301C211yesmaximal, characteristic, complemented
8#9299C211yesmaximal, characteristic, complemented
9#9299C211yesmaximal, characteristic, complemented
10#9293C211yesmaximal, characteristic, complemented
11#9293C211yesmaximal, characteristic, complemented
12#9289C211yesmaximal, characteristic, complemented
13#9287C211yesmaximal, characteristic, complemented
14#8795C211yesmaximal, characteristic, complemented
15#8141C211yesmaximal, characteristic, complemented
16#3227C2211yescharacteristic
17#3206C2211yescharacteristic
18#3205C2211yescharacteristic
19#3205C2211yescharacteristic
20#3147C2211yescharacteristic
21C2×#486C2211yescharacteristic
22#2932C2211yescharacteristic
23#2932C2211yescharacteristic
24#2331C2211yescharacteristic
25#2328C2211yescharacteristic
26#1993C2211yescharacteristic
27#1963C2211yescharacteristic
28#1963C2211yescharacteristic
29#1961C2211yescharacteristic
30#1926—21no—
31#1925—21no—
32#1876—21no—
33#1876—21no—
34#1876—21no—
35#1876—21no—
36#1864—21no—
37#1862—21no—
38#1846C2211yescharacteristic
39#1837C2211yescharacteristic
40#1746C2211yescharacteristic
41#1721C412yescomplemented
42#1720C412yescomplemented
43#1717C412yescomplemented
44#1715C412yescomplemented
45#1549C2211yescharacteristic
46#1548C2211yescharacteristic
47#1545C2211yescharacteristic
48#1545C2211yescharacteristic
49#1517C2211yescharacteristic
50#1517C2211yescharacteristic
51#1517C2211yescharacteristic
52#1517C2211yescharacteristic
53#1452C2211yescharacteristic
54#1450C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1286C2211yescharacteristic
58#1286C2211yescharacteristic
59#1275C2211yescharacteristic
60#1275C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C2×#135—21no—
64C2×#135—21no—
65C2×#131—21no—
66C2×#131—21no—
67#532C2311yescharacteristic
68#532C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×C4×D4—21no—
72#498C2×C412yescomplemented
73#495—21no—
74#495—21no—
75#494C2×C411yescharacteristic, complemented
76#493C2×C412yes—
77#493C2×C412yes—
78#493C2×C411yescharacteristic, complemented
79#492C2×C411yescharacteristic, complemented
80#492C2×C411yescharacteristic, complemented
81#492C2×C412yescomplemented
82#487C2×C412yescomplemented
83#486—21no—
84#486—21no—
85#486C2×C412yes—
86#486C2×C412yes—
87#486C2×C412yes—
88#486C2×C412yes—
89#486C2×C412yescomplemented
90#481—41no—
91#476—41no—
92#461—41no—
93#459—41no—
94#459—41no—
95#448—41no—
96#447—41no—
97#447—41no—
98#435—41no—
99#435—41no—
100#435—41no—
101#435—41no—
102#431C2311yescharacteristic
103#431C2311yescharacteristic
104#431C2311yescharacteristic
105#431C2311yescharacteristic
106C2×#107—21no—
107C2×#106—21no—
108#423C2311yescharacteristic
109#423C2311yescharacteristic
110C2×#104—21no—
111C2×#104—21no—
112C2×#104—21no—
113C2×#104—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118#408—21no—
119#408—21no—
120C2×#97C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—41no—
123#390—21no—
124#389—21no—
125#387—41no—
126#386—41no—
127#386—41no—
128#382C2311yescharacteristic
129#382C2311yescharacteristic
130C4×C4⋊C4—21no—
131#376C2311yescharacteristic
132#341—21no—
133#341—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138#336D411yescharacteristic
139#336D411yescharacteristic
140#333—41no—
141#332—41no—
142#331—41no—
143#331—41no—
144#327—41no—
145#327—41no—
146#327—41no—
147#327—41no—
148#322—21no—
149#322C2×C412yescomplemented
150#322C2×C412yescomplemented
151C82—21no—
152C2×#41—21no—
153C2×#41—21no—
154C22×Q8—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C23×C4—41no—
159#135—81no—
160#135—81no—
161#135—81no—
162#135—81no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×SD16—21no—
166C2×SD16—21no—
167C2×SD16—21no—
168C2×SD16—21no—
169C2×SD16—42no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—42no—
173C2×D8—42no—
174#131—81no—
175#131—81no—
176#131—81no—
177#131—81no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—41no—
181C2×M16—41no—
182C22×C8—41no—
183C22×C8—41no—
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8C22×C412yes—
192C4⋊Q8C22×C411yescharacteristic
193C4⋊Q8C22×C411yescharacteristic
194#127—21no—
195#127—21no—
196#127—21no—
197#127C22×C412yes—
198#127C22×C411yescharacteristic
199#127C22×C411yescharacteristic
200#124—21no—
201#124—21no—
202#121—21no—
203#121—21no—
204#121—21no—
205#121—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—81no—
211C4×D4—81no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—41no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224#107—81no—
225#106—42no—
226#106—81no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235#104—81no—
236#104—81no—
237#104—81no—
238#104—81no—
239#103—42no—
240#102—22no—
241#102—22no—
242#102—22no—
243#102—22no—
244#102—42no—
245#102—42no—
246#102—42no—
247#102—81no—
248#102—81no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—42no—
254C22⋊C8—42no—
255#97—21no—
256#97—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8C22×C412yes—
260C4×C8C22×C412yes—
261C4×C8C22×C411yescharacteristic
262C4×C8C22×C411yescharacteristic
263C4×C8#4111yescharacteristic
264C4×C8#4111yescharacteristic
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—41no—
272#95—41no—
273C24—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279#41—41no—
280#41—41no—
281#41—41no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—161no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—161no—
306C2×D4—161no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4—161no—
316C22×C4—161no—
317C22×C4C22×D411yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320Q16—81no—
321Q16—81no—
322SD16—81no—
323SD16—81no—
324SD16—81no—
325SD16—81no—
326SD16—162no—
327D8—81no—
328D8—81no—
329D8—162no—
330D8—162no—
331M16—41no—
332M16—41no—
333M16—41no—
334M16—41no—
335M16—82no—
336M16—82no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—82no—
354C2×C8—82no—
355C2×C8C4×D411yescharacteristic
356C2×C8C4×D411yescharacteristic
357C2×C8C4×D411yescharacteristic
358C2×C8C4×D411yescharacteristic
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—161no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—82no—
377C22⋊C4—161no—
378C22⋊C4—161no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—161no—
384C42C23×C411yescharacteristic
385C42C2×#4111yescharacteristic
386C42H(5,𝔽2)11yescharacteristic
387C42H(5,𝔽2)11yescharacteristic
388C23—41no—
389C23—41no—
390C23—42no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23#53511yescharacteristic
398Q8—41no—
399Q8—41no—
400Q8—41no—
401Q8—41no—
402Q8—82no—
403Q8—161no—
404D4—41no—
405D4—41no—
406D4—41no—
407D4—41no—
408D4—42no—
409D4—42no—
410D4—81no—
411D4—81no—
412D4—82no—
413D4—82no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—161no—
423D4—161no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—164no—
448C2×C4C2×C4×D411yescharacteristic
449C2×C4#51811yescharacteristic
450C2×C4#51811yescharacteristic
451C2×C4#53511yescharacteristic
452C2×C4#54011yescharacteristic
453C2×C4#54011yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C8—21no—
459C8—21no—
460C8—21no—
461C8—21no—
462C8—42no—
463C8—42no—
464C8—41no—
465C8—41no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—81no—
470C22—21no—
471C22—21no—
472C22—81no—
473C22—81no—
474C22—82no—
475C22—82no—
476C22—81no—
477C22—81no—
478C22—82no—
479C22—81no—
480C22—81no—
481C22—161no—
482C22—161no—
483C22—161no—
484C22—161no—
485C22#232811yescharacteristic
486C22#322011yescharacteristic
487C22#322111yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—82no—
497C4—161no—
498C4—161no—
499C4#294611yescharacteristic
500C4#294611yescharacteristic
501C4#322411yescharacteristic
502C4#322411yescharacteristic
503C2—21no—
504C2—21no—
505C2—161no—
506C2—161no—
507C2—162no—
508C2#2227911yescharacteristic
509C2#2228411yescharacteristic
510C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#19701C2
LCSG ▷ C42 ▷ C22 ▷ 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
1. #22282C2semidirect
2. #22230C2semidirect
4. #9769C2semidirect
5. #9756C2semidirect
6. #9302C2semidirect
7. #9301C2semidirect
8. #9299C2semidirect
9. #9299C2semidirect
10. #9293C2semidirect
11. #9293C2semidirect
12. #9289C2semidirect
13. #9287C2semidirect
14. #8795C2semidirect
15. #8141C2semidirect
41. #1721C4semidirect
42. #1720C4semidirect
43. #1717C4semidirect
44. #1715C4semidirect
72. #498C2×C4semidirect
75. #494C2×C4semidirect
78. #493C2×C4semidirect
79. #492C2×C4semidirect
80. #492C2×C4semidirect
81. #492C2×C4semidirect
82. #487C2×C4semidirect
89. #486C2×C4semidirect
149. #322C2×C4semidirect
150. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222822
#222302
#197012
#97692
#97562
#93022
#93012
#92992
#92932
#92892
#92872
#87952
#81412