Finite Group Databasebeta

← #232171

Subgroups — #232171

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23804C211yesmaximal, characteristic
2#9094C211yesmaximal, characteristic, complemented
3#9092C211yesmaximal, characteristic, complemented
4#9090C211yesmaximal, characteristic, complemented
5#9089C211yesmaximal, characteristic
6#9089C211yesmaximal, characteristic
7#9088C211yesmaximal, characteristic, complemented
8#9087C211yesmaximal, characteristic, complemented
9#9087C211yesmaximal, characteristic, complemented
10#8964C211yesmaximal, characteristic, complemented
11#8960C211yesmaximal, characteristic, complemented
12#8957C211yesmaximal, characteristic, complemented
13#8957C211yesmaximal, characteristic, complemented
14#8802C211yesmaximal, characteristic, complemented
15#8802C211yesmaximal, characteristic, complemented
16#2983C2211yescharacteristic
17#2982C2211yescharacteristic
18#2972C2211yescharacteristic
19#2972C2211yescharacteristic
20C2×#442C2211yescharacteristic
21C2×#441C2211yescharacteristic
22C2×#440C2211yescharacteristic
23C2×#440C2211yescharacteristic
24C4×#124C2211yescharacteristic
25C4×#124C2211yescharacteristic
26#2097C411yescharacteristic
27#2097C411yescharacteristic
28#2092C411yescharacteristic
29#2092C411yescharacteristic
30#2015C411yescharacteristic
31#2015C411yescharacteristic
32#2012C411yescharacteristic
33#2012C411yescharacteristic
34#1745C2211yescharacteristic
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#446—41no—
70#446—41no—
71C8×Q8—41no—
72C8×Q8—41no—
73#442—41no—
74#442—41no—
75#441—41no—
76#441—41no—
77#440—41no—
78#440—41no—
79#440—41no—
80#440—41no—
81C8×D4—41no—
82C8×D4—41no—
83C8×D4—41no—
84C8×D4—41no—
85#432—41no—
86#431—41no—
87#431C2311yescharacteristic
88#431C2311yescharacteristic
89#431C2311yescharacteristic
90#431C2311yescharacteristic
91#431C2311yescharacteristic
92#431C2311yescharacteristic
93#431C2311yescharacteristic
94#431C2311yescharacteristic
95C2×#107C2×C411yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×C22⋊C8—41no—
108C2×C22⋊C8—41no—
109C2×#97C2311yescharacteristic
110C2×#97C2311yescharacteristic
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#388—41no—
116#388—41no—
117#382C2×C411yescharacteristic
118#382C2×C411yescharacteristic
119C4×C22⋊C4—41no—
120C4×C22⋊C4—41no—
121C43C2311yescharacteristic
122#336C2×C411yescharacteristic
123#336C2×C411yescharacteristic
124#336C2×C411yescharacteristic
125#336C2×C411yescharacteristic
126Q8⋊C8—41no—
127Q8⋊C8—41no—
128Q8⋊C8—41no—
129Q8⋊C8—41no—
130D4⋊C8—41no—
131D4⋊C8—41no—
132D4⋊C8—41no—
133D4⋊C8—41no—
134#324—41no—
135#324—41no—
136#323—41no—
137#323—41no—
138C22×Q8C22×C411yescharacteristic
139C22×Q8C22×C411yescharacteristic
140C22×D4C22×C411yescharacteristic
141C22×D4C22×C411yescharacteristic
142C23×C4—21no—
143C23×C4—21no—
144C2×Q16—22no—
145C2×Q16—41no—
146C2×Q16—41no—
147C2×SD16—22no—
148C2×SD16—22no—
149C2×SD16—22no—
150C2×SD16—41no—
151C2×SD16—41no—
152C2×SD16—41no—
153C2×D8—22no—
154C2×D8—22no—
155C2×D8—41no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8C22×C411yescharacteristic
159C22×C8C22×C411yescharacteristic
160C22×C8C22×C411yescharacteristic
161C22×C8C22×C411yescharacteristic
162#124—22no—
163#124—22no—
164#124—22no—
165#124—41no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2411yescharacteristic
201#107—41no—
202#107—41no—
203#106—41no—
204#106—41no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—42no—
214C4⋊C8—42no—
215#103—41no—
216#103—41no—
217#103—41no—
218#103—41no—
219#102—41no—
220#102—41no—
221#102—41no—
222#102—41no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—42no—
232C22⋊C8—42no—
233#97—21no—
234#97—21no—
235#97—21no—
236#97—21no—
237#97—42no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—42no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—21no—
252C24—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8#13112yes—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4#13112yes—
276C2×D4#13112yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C23×C411yescharacteristic
292C22×C4H(5,𝔽2)11yescharacteristic
293C22×C4H(5,𝔽2)11yescharacteristic
294Q16—44no—
295SD16—44no—
296SD16—44no—
297D8—44no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—162no—
319C2×C8—162no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—81no—
349C42—81no—
350C42H(5,𝔽2)11yescharacteristic
351C42H(5,𝔽2)11yescharacteristic
352C42H(5,𝔽2)11yescharacteristic
353C42H(5,𝔽2)11yescharacteristic
354C23—22no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23#51811yescharacteristic
365Q8—22no—
366Q8—22no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370Q8—41no—
371Q8—41no—
372Q8—41no—
373D4—22no—
374D4—22no—
375D4—22no—
376D4—22no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—41no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
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—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—161no—
406C2×C4—161no—
407C2×C4#51811yescharacteristic
408C2×C4#51811yescharacteristic
409C2×C4#56011yescharacteristic
410C2×C4#56011yescharacteristic
411C2×C4#56011yescharacteristic
412C2×C4#56011yescharacteristic
413C2×C4C2×#13111yescharacteristic
414C2×C4C2×#13111yescharacteristic
415C2×C4C2×#13111yescharacteristic
416C2×C4#56811yescharacteristic
417C8—41no—
418C8—41no—
419C8—41no—
420C8—41no—
421C8—81no—
422C8—81no—
423C8—81no—
424C8—81no—
425C8—81no—
426C8—81no—
427C8—81no—
428C8—81no—
429C22—21no—
430C22—21no—
431C22—42no—
432C22—42no—
433C22—82no—
434C22—82no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—161no—
440C22—161no—
441C22#238011yescharacteristic
442C22#297611yescharacteristic
443C22#297711yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4#297611yescharacteristic
457C4#297611yescharacteristic
458C4#297711yescharacteristic
459C4#297711yescharacteristic
460C2—21no—
461C2—21no—
462C2—82no—
463C2—82no—
464C2#2335811yescharacteristic
465C2#2336211yescharacteristic
466C2#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
3. #9092C2semidirect
4. #9090C2semidirect
7. #9088C2semidirect
8. #9087C2semidirect
9. #9087C2semidirect
10. #8964C2semidirect
11. #8960C2semidirect
12. #8957C2semidirect
13. #8957C2semidirect
14. #8802C2semidirect
15. #8802C2semidirect

Maximal subgroups

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

subgroupindex
#238042
#90942
#90922
#90902
#90892
#90882
#90872
#89642
#89602
#89572
#88022