Finite Group Databasebeta

← #232142

Subgroups — #232142

Order 512.

545 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21693C211yesmaximal, characteristic
2#9094C211yesmaximal, characteristic, complemented
3#9093C211yesmaximal, characteristic
4#9092C211yesmaximal, characteristic, complemented
5#9091C211yesmaximal, characteristic, complemented
6#9090C211yesmaximal, characteristic, complemented
7#9089C211yesmaximal, characteristic
8#9088C211yesmaximal, characteristic, complemented
9#9087C211yesmaximal, characteristic, complemented
10#8964C211yesmaximal, characteristic, complemented
11#8961C211yesmaximal, characteristic, complemented
12#8960C211yesmaximal, characteristic, complemented
13#8957C211yesmaximal, characteristic, complemented
14#8801C211yesmaximal, characteristic, complemented
15#8801C211yesmaximal, characteristic, complemented
16C2×#495C411yescharacteristic
17#2983C2211yescharacteristic
18#2982C2211yescharacteristic
19#2975C2211yescharacteristic
20#2972C2211yescharacteristic
21C2×#442C2211yescharacteristic
22C2×#441C2211yescharacteristic
23C2×#440C2211yescharacteristic
24C2×#440C2211yescharacteristic
25C4×#124C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#1973C411yescharacteristic
28#1973C411yescharacteristic
29#1972C411yescharacteristic
30#1971C411yescharacteristic
31#1958C411yescharacteristic
32#1958C411yescharacteristic
33#1846C411yescharacteristic
34C4×#97C2211yescharacteristic
35#1479C2211yescharacteristic
36#1479C2211yescharacteristic
37#1479C2211yescharacteristic
38#1479C2211yescharacteristic
39#1478C2211yescharacteristic
40#1478C2211yescharacteristic
41#1478C2211yescharacteristic
42#1478C2211yescharacteristic
43#1461C2211yescharacteristic
44#1461C2211yescharacteristic
45#1460C2211yescharacteristic
46#1460C2211yescharacteristic
47#1281C2211yescharacteristic
48#1281C2211yescharacteristic
49#1281C2211yescharacteristic
50#1281C2211yescharacteristic
51#1276C2211yescharacteristic
52#1276C2211yescharacteristic
53#1276C2211yescharacteristic
54#1276C2211yescharacteristic
55#1275C2211yescharacteristic
56#1275C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59C22×Q16C2×C411yescharacteristic
60C22×SD16C2×C411yescharacteristic
61C22×SD16C2×C411yescharacteristic
62C22×D8C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#124C2×C411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69#495—24no—
70#446—41no—
71#446—41no—
72C8×Q8—41no—
73C8×Q8—41no—
74#442—41no—
75#442—41no—
76#441—41no—
77#441—41no—
78#440—41no—
79#440—41no—
80#440—41no—
81#440—41no—
82#436—41no—
83#436—41no—
84C8×D4—41no—
85C8×D4—41no—
86#432—41no—
87#431—41no—
88#431C2311yescharacteristic
89#431C2311yescharacteristic
90#431C2311yescharacteristic
91#431C2311yescharacteristic
92#431C2311yescharacteristic
93#431C2311yescharacteristic
94#431C2311yescharacteristic
95#431C2311yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#107C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×#106C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108#407—41no—
109C2×C22⋊C8—41no—
110C2×#97C2×C411yescharacteristic
111C2×#97C2×C411yescharacteristic
112C2×#97C2311yescharacteristic
113C2×#97C2311yescharacteristic
114C2×C4×C8C2×C411yescharacteristic
115C2×C4×C8C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#390C2×C411yescharacteristic
120#388—41no—
121#388—41no—
122#382C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124C4×C22⋊C4—41no—
125C4×C22⋊C4—41no—
126C43C2311yescharacteristic
127Q8⋊C8—41no—
128Q8⋊C8—41no—
129Q8⋊C8—41no—
130Q8⋊C8—41no—
131D4⋊C8—41no—
132D4⋊C8—41no—
133D4⋊C8—41no—
134D4⋊C8—41no—
135#324—41no—
136#324—41no—
137#323—41no—
138#323—41no—
139C22×Q8C22×C411yescharacteristic
140C22×Q8C22×C411yescharacteristic
141C22×D4C22×C411yescharacteristic
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C2×Q16—22no—
146C2×Q16—22no—
147C2×Q16—41no—
148C2×SD16—22no—
149C2×SD16—22no—
150C2×SD16—22no—
151C2×SD16—22no—
152C2×SD16—41no—
153C2×SD16—41no—
154C2×D8—22no—
155C2×D8—22no—
156C2×D8—41no—
157C2×M16—42no—
158C22×C8—42no—
159C22×C8C22×C411yescharacteristic
160C22×C8C22×C411yescharacteristic
161C22×C8C22×C411yescharacteristic
162C22×C8C22×C411yescharacteristic
163#124—22no—
164#124—41no—
165#124—41no—
166#124M1614yescomplemented
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C2411yescharacteristic
202#107—41no—
203#107—41no—
204#106—41no—
205#106—41no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—42no—
215C4⋊C8—42no—
216#103—41no—
217#103—41no—
218#103—41no—
219#103—41no—
220#102—41no—
221#102—41no—
222#102—41no—
223#102—41no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—42no—
233C22⋊C8—42no—
234#97—22no—
235#97—22no—
236#97—21no—
237#97—21no—
238#97—21no—
239#97—21no—
240#97—42no—
241C4×C8—22no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—41no—
251C4×C8—42no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24—21no—
257C24—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8C2×M1612yes—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4C2×M1612yes—
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—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4C23×C411yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C4H(5,𝔽2)11yescharacteristic
299Q16—44no—
300SD16—44no—
301SD16—44no—
302D8—44no—
303M16—162no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—162no—
323C2×C8#13112yes—
324C2×C8#13112yes—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—81no—
353C42—81no—
354C42C2×M1612yes—
355C42H(5,𝔽2)11yescharacteristic
356C42H(5,𝔽2)11yescharacteristic
357C42H(5,𝔽2)11yescharacteristic
358C42H(5,𝔽2)11yescharacteristic
359C23—22no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23#51811yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374Q8—41no—
375Q8—41no—
376Q8—41no—
377Q8—41no—
378D4—22no—
379D4—22no—
380D4—41no—
381D4—41no—
382D4—41no—
383D4—41no—
384D4—41no—
385D4—41no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—161no—
411C2×C4—161no—
412C2×C4#51811yescharacteristic
413C2×C4#51811yescharacteristic
414C2×C4#56011yescharacteristic
415C2×C4#56011yescharacteristic
416C2×C4#56011yescharacteristic
417C2×C4#56011yescharacteristic
418C2×C4C22×M1611yescharacteristic
419C2×C4C2×#13111yescharacteristic
420C2×C4C2×#13111yescharacteristic
421C2×C4#56811yescharacteristic
422C8—22no—
423C8—22no—
424C8—41no—
425C8—41no—
426C8—81no—
427C8—81no—
428C8—81no—
429C8—81no—
430C8—81no—
431C8—81no—
432C8—81no—
433C8—81no—
434C22—21no—
435C22—21no—
436C22—42no—
437C22—82no—
438C22—82no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—161no—
445C22—161no—
446C22#238011yescharacteristic
447C22#297311yescharacteristic
448C22#297911yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—42no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4#297411yescharacteristic
462C4#297411yescharacteristic
463C4#298111yescharacteristic
464C4#298111yescharacteristic
465C2—21no—
466C2—21no—
467C2—82no—
468C2—161no—
469C2#2335611yescharacteristic
470C2#2336211yescharacteristic
471C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
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. #9094C2semidirect
4. #9092C2semidirect
5. #9091C2semidirect
6. #9090C2semidirect
8. #9088C2semidirect
9. #9087C2semidirect
10. #8964C2semidirect
11. #8961C2semidirect
12. #8960C2semidirect
13. #8957C2semidirect
14. #8801C2semidirect
15. #8801C2semidirect
166. #124M16semidirect

Maximal subgroups

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

subgroupindex
#216932
#90942
#90932
#90922
#90912
#90902
#90892
#90882
#90872
#89642
#89612
#89602
#89572
#88012