Finite Group Databasebeta

← #219618

Subgroups — #219618

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22275C211yesmaximal, characteristic, complemented
2#22236C211yesmaximal, characteristic, complemented
3#19697C211yesmaximal, characteristic
4#9768C211yesmaximal, characteristic, complemented
5#9757C211yesmaximal, characteristic, complemented
6#9300C211yesmaximal, characteristic, complemented
7#9299C211yesmaximal, characteristic
8#9296C211yesmaximal, characteristic, complemented
9#9295C211yesmaximal, characteristic, complemented
10#9294C211yesmaximal, characteristic, complemented
11#9293C211yesmaximal, characteristic
12#9292C211yesmaximal, characteristic, complemented
13#9291C211yesmaximal, characteristic, complemented
14#8794C211yesmaximal, characteristic, complemented
15#8142C211yesmaximal, characteristic, complemented
16#3229C2211yescharacteristic
17#3226C2211yescharacteristic
18#3208C2211yescharacteristic
19#3205C2211yescharacteristic
20C2×#492C2211yescharacteristic
21#3131C2211yescharacteristic
22#2961C2211yescharacteristic
23#2932C2211yescharacteristic
24#2328C2211yescharacteristic
25#2327C2211yescharacteristic
26#1995C2211yescharacteristic
27#1963C2211yescharacteristic
28#1961C2211yescharacteristic
29#1960C2211yescharacteristic
30#1929—21no—
31#1929—21no—
32#1880—21no—
33#1880—21no—
34#1877—21no—
35#1877—21no—
36#1866—21no—
37#1866—21no—
38#1848C2211yescharacteristic
39#1835C2211yescharacteristic
40#1746C2211yescharacteristic
41#1719C412yescomplemented
42#1717C412yescomplemented
43#1715C412yescomplemented
44#1714C412yescomplemented
45#1546C2211yescharacteristic
46#1546C2211yescharacteristic
47#1545C2211yescharacteristic
48#1545C2211yescharacteristic
49#1518C2211yescharacteristic
50#1518C2211yescharacteristic
51#1517C2211yescharacteristic
52#1517C2211yescharacteristic
53#1452C2211yescharacteristic
54#1450C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1286C2211yescharacteristic
58#1286C2211yescharacteristic
59#1275C2211yescharacteristic
60#1275C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C22×SD16—21no—
64C22×SD16—21no—
65C2×#131—21no—
66#533C2311yescharacteristic
67#532C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72#500C2×C412yescomplemented
73#498C2×C412yescomplemented
74#493C2×C412yes—
75#492—21no—
76#492—21no—
77#492C2×C412yes—
78#492C2×C412yes—
79#492C2×C412yes—
80#488—21no—
81#488—21no—
82#487C2×C411yescharacteristic, complemented
83#487C2×C411yescharacteristic, complemented
84#487C2×C412yescomplemented
85#486C2×C412yes—
86#486C2×C412yes—
87#486C2×C411yescharacteristic, complemented
88#486C2×C411yescharacteristic, complemented
89#486C2×C412yescomplemented
90#483—41no—
91#480—41no—
92#475—41no—
93#474—41no—
94#460—41no—
95#459—41no—
96#450—41no—
97#447—41no—
98#446—41no—
99#446—41no—
100#435—41no—
101#435—41no—
102#431C2311yescharacteristic
103#431C2311yescharacteristic
104#431C2311yescharacteristic
105#431C2311yescharacteristic
106C2×#108—21no—
107C2×#108—21no—
108#424—21no—
109#423C2311yescharacteristic
110#423C2311yescharacteristic
111C2×#104—21no—
112C2×#104—21no—
113#416—21no—
114#416—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#102—21no—
118C2×#102—21no—
119#408—21no—
120#405—21no—
121C2×#97C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#390—21no—
125#387—41no—
126#387—41no—
127#386—41no—
128#386—41no—
129#382C2311yescharacteristic
130#382C2311yescharacteristic
131#376C2311yescharacteristic
132#340—21no—
133#340—21no—
134#339—21no—
135#339—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138#336D411yescharacteristic
139#336D411yescharacteristic
140#332—41no—
141#332—41no—
142#331—41no—
143#331—41no—
144#328—41no—
145#328—41no—
146#327—41no—
147#327—41no—
148#322—21no—
149#322C2×C412yescomplemented
150#322C2×C412yescomplemented
151C82—21no—
152C2×#41—21no—
153C22×Q8—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C23×C4—41no—
158C2×SD16—22no—
159C2×SD16—22no—
160C2×SD16—22no—
161C2×SD16—22no—
162C2×SD16—42no—
163C2×SD16—81no—
164C2×SD16—81no—
165C2×SD16—81no—
166C2×SD16—81no—
167C2×D8—42no—
168#131—81no—
169#131—81no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—41no—
173C22×C8—41no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C411yescharacteristic
183C4⋊Q8C22×C411yescharacteristic
184C4⋊Q8C22×C411yescharacteristic
185#127—21no—
186#127—21no—
187#127—21no—
188#127C22×C412yes—
189#127C22×C411yescharacteristic
190#125—21no—
191#124—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194#121—21no—
195#121—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—81no—
199C4×Q8—81no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—81no—
203C4×D4—81no—
204#117—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C42—41no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2411yescharacteristic
217#108—81no—
218#108—81no—
219#107—42no—
220#106—42no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—42no—
230#104—81no—
231#104—81no—
232#103—21no—
233#103—21no—
234#103—21no—
235#103—21no—
236#103—42no—
237#103—81no—
238#103—81no—
239#102—21no—
240#102—21no—
241#102—21no—
242#102—21no—
243#102—42no—
244#102—42no—
245#102—42no—
246#102—81no—
247#102—81no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—42no—
253#97—21no—
254#97—21no—
255#97—41no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—41no—
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—42no—
272C24—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—161no—
301C2×D4—161no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—82no—
309C22×C4—161no—
310C22×C4—161no—
311C22×C4C22×D411yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314SD16—82no—
315SD16—82no—
316SD16—82no—
317SD16—82no—
318SD16—162no—
319D8—162no—
320M16—41no—
321M16—41no—
322M16—41no—
323M16—41no—
324M16—82no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—82no—
342C2×C8C4×D411yescharacteristic
343C2×C8C4×D411yescharacteristic
344C2×C8C4×D411yescharacteristic
345C2×C8C4×D411yescharacteristic
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—82no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—82no—
367C22⋊C4—161no—
368C22⋊C4—161no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—41no—
374C42—82no—
375C42C23×C411yescharacteristic
376C42C2×#4111yescharacteristic
377C42H(5,𝔽2)11yescharacteristic
378C42H(5,𝔽2)11yescharacteristic
379C23—41no—
380C23—42no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23#53511yescharacteristic
388Q8—42no—
389Q8—42no—
390Q8—41no—
391Q8—41no—
392Q8—81no—
393Q8—81no—
394Q8—81no—
395Q8—81no—
396D4—42no—
397D4—42no—
398D4—41no—
399D4—41no—
400D4—81no—
401D4—82no—
402D4—82no—
403D4—81no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—81no—
408D4—161no—
409D4—161no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—164no—
433C2×C4C2×C4×D411yescharacteristic
434C2×C4#51811yescharacteristic
435C2×C4#51811yescharacteristic
436C2×C4#53511yescharacteristic
437C2×C4#54011yescharacteristic
438C2×C4#54011yescharacteristic
439C2×C4#57611yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57711yescharacteristic
443C8—21no—
444C8—21no—
445C8—21no—
446C8—21no—
447C8—42no—
448C8—42no—
449C8—41no—
450C8—41no—
451C8—81no—
452C8—81no—
453C8—81no—
454C8—81no—
455C22—21no—
456C22—21no—
457C22—81no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—161no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22#232811yescharacteristic
468C22#322411yescharacteristic
469C22#322411yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—82no—
478C4—81no—
479C4—81no—
480C4—161no—
481C4—161no—
482C4#294611yescharacteristic
483C4#294611yescharacteristic
484C4#322011yescharacteristic
485C4#322111yescharacteristic
486C2—21no—
487C2—21no—
488C2—161no—
489C2—162no—
490C2#2227911yescharacteristic
491C2#2228411yescharacteristic
492C2#2228411yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3205C22
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. #22275C2semidirect
2. #22236C2semidirect
4. #9768C2semidirect
5. #9757C2semidirect
6. #9300C2semidirect
8. #9296C2semidirect
9. #9295C2semidirect
10. #9294C2semidirect
12. #9292C2semidirect
13. #9291C2semidirect
14. #8794C2semidirect
15. #8142C2semidirect
41. #1719C4semidirect
42. #1717C4semidirect
43. #1715C4semidirect
44. #1714C4semidirect
72. #500C2×C4semidirect
73. #498C2×C4semidirect
82. #487C2×C4semidirect
83. #487C2×C4semidirect
84. #487C2×C4semidirect
87. #486C2×C4semidirect
88. #486C2×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
#222752
#222362
#196972
#97682
#97572
#93002
#92992
#92962
#92952
#92942
#92932
#92922
#92912
#87942
#81422