Finite Group Databasebeta

← #232136

Subgroups — #232136

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21701C211yesmaximal, characteristic
2#9094C211yesmaximal, characteristic, complemented
3#9094C211yesmaximal, characteristic, complemented
4#9093C211yesmaximal, characteristic, complemented
5#9091C211yesmaximal, characteristic, complemented
6#9089C211yesmaximal, characteristic, complemented
7#9089C211yesmaximal, characteristic, complemented
8#9089C211yesmaximal, characteristic, complemented
9#9087C211yesmaximal, characteristic, complemented
10#8964C211yesmaximal, characteristic, complemented
11#8963C211yesmaximal, characteristic, complemented
12#8959C211yesmaximal, characteristic, complemented
13#8957C211yesmaximal, characteristic, complemented
14#8801C211yesmaximal, characteristic, complemented
15#8800C211yesmaximal, characteristic, complemented
16C2×#496C411yescharacteristic
17#2983C2211yescharacteristic
18#2975C2211yescharacteristic
19#2972C2211yescharacteristic
20#2972C2211yescharacteristic
21C2×#442C2211yescharacteristic
22C2×#442C2211yescharacteristic
23C2×#440C2211yescharacteristic
24C2×#440C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#1970C411yescharacteristic
28#1970C411yescharacteristic
29#1968C411yescharacteristic
30#1968C411yescharacteristic
31#1961C411yescharacteristic
32#1959C411yescharacteristic
33#1847C411yescharacteristic
34C4×#97C2211yescharacteristic
35#1479C2211yescharacteristic
36#1479C2211yescharacteristic
37#1478C2211yescharacteristic
38#1478C2211yescharacteristic
39#1478C2211yescharacteristic
40#1478C2211yescharacteristic
41#1478C2211yescharacteristic
42#1478C2211yescharacteristic
43#1461C2211yescharacteristic
44#1460C2211yescharacteristic
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×SD16C2×C411yescharacteristic
60C22×SD16C2×C411yescharacteristic
61C22×D8C2×C411yescharacteristic
62C22×D8C2×C411yescharacteristic
63C2×#127C2×C411yescharacteristic
64C2×#124C2×C411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—41no—
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70#496—24no—
71#446—41no—
72#446—41no—
73#442—41no—
74#442—41no—
75#442—41no—
76#442—41no—
77#440—41no—
78#440—41no—
79#440—41no—
80#440—41no—
81#436—41no—
82#436—41no—
83C8×D4—41no—
84C8×D4—41no—
85C8×D4—41no—
86C8×D4—41no—
87#432—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×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108#407—41no—
109C2×C22⋊C8—41no—
110C2×C22⋊C8—41no—
111C2×#97C2×C411yescharacteristic
112C2×#97C2×C411yescharacteristic
113C2×#97C2311yescharacteristic
114C2×#97C2311yescharacteristic
115C2×C4×C8C2×C411yescharacteristic
116C2×C4×C8C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#390—41no—
119#390C2×C411yescharacteristic
120#390C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#388—41no—
123#384C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125C4×C22⋊C4—41no—
126C43C2311yescharacteristic
127Q8⋊C8—41no—
128Q8⋊C8—41no—
129D4⋊C8—41no—
130D4⋊C8—41no—
131D4⋊C8—41no—
132D4⋊C8—41no—
133D4⋊C8—41no—
134D4⋊C8—41no—
135#324—41no—
136#323—41no—
137#323—41no—
138#323—41no—
139C22×Q8C22×C411yescharacteristic
140C22×D4—41no—
141C22×D4C22×C411yescharacteristic
142C22×D4C22×C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
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—
156C2×D8—41no—
157C2×D8—41no—
158C2×D8M1614yescomplemented
159C2×M16—42no—
160C22×C8—42no—
161C22×C8—42no—
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166#127—22no—
167#127—22no—
168#124—22no—
169#124—22no—
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—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—84no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2411yescharacteristic
206#107—41no—
207#107—41no—
208#106—41no—
209#106—41no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—42no—
219#103—41no—
220#103—41no—
221#102—41no—
222#102—41no—
223#102—41no—
224#102—41no—
225#102—41no—
226#102—41no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235C22⋊C8—42no—
236C22⋊C8—42no—
237C22⋊C8—42no—
238#97—22no—
239#97—22no—
240#97—21no—
241#97—21no—
242#97—21no—
243#97—21no—
244#97—42no—
245C4×C8—22no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—41no—
255#95—41no—
256#95—41no—
257C24—21no—
258C24—21no—
259C24—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—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—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—44no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4C2×M1612yes—
286C2×D4C2×M1612yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4C23×C411yescharacteristic
304C22×C4H(5,𝔽2)11yescharacteristic
305C22×C4H(5,𝔽2)11yescharacteristic
306SD16—44no—
307SD16—44no—
308D8—44no—
309D8—44no—
310M16—162no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—162no—
330C2×C8—162no—
331C2×C8C2×M1612yes—
332C2×C8#13112yes—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—82no—
344C22⋊C4—44no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—42no—
358C42—41no—
359C42—41no—
360C42—81no—
361C42#13112yes—
362C42H(5,𝔽2)11yescharacteristic
363C42H(5,𝔽2)11yescharacteristic
364C42H(5,𝔽2)11yescharacteristic
365C42H(5,𝔽2)11yescharacteristic
366C23—22no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23#51811yescharacteristic
382Q8—41no—
383Q8—41no—
384Q8—41no—
385Q8—41no—
386D4—22no—
387D4—22no—
388D4—22no—
389D4—22no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—88no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—161no—
422C2×C4—161no—
423C2×C4—161no—
424C2×C4#51811yescharacteristic
425C2×C4#51811yescharacteristic
426C2×C4#56011yescharacteristic
427C2×C4#56011yescharacteristic
428C2×C4#56011yescharacteristic
429C2×C4#56011yescharacteristic
430C2×C4C22×M1611yescharacteristic
431C2×C4C2×#13111yescharacteristic
432C2×C4C2×#13111yescharacteristic
433C2×C4#56811yescharacteristic
434C8—22no—
435C8—22no—
436C8—41no—
437C8—41no—
438C8—81no—
439C8—81no—
440C8—81no—
441C8—81no—
442C8—81no—
443C8—81no—
444C8—81no—
445C8—81no—
446C22—21no—
447C22—21no—
448C22—42no—
449C22—42no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22#238011yescharacteristic
464C22#297411yescharacteristic
465C22#298111yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—42no—
471C4—42no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4#297311yescharacteristic
478C4#297411yescharacteristic
479C4#297911yescharacteristic
480C4#298111yescharacteristic
481C2—21no—
482C2—21no—
483C2—82no—
484C2—82no—
485C2—161no—
486C2#2335611yescharacteristic
487C2#2336111yescharacteristic
488C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
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. #9094C2semidirect
3. #9094C2semidirect
4. #9093C2semidirect
5. #9091C2semidirect
6. #9089C2semidirect
7. #9089C2semidirect
8. #9089C2semidirect
9. #9087C2semidirect
10. #8964C2semidirect
11. #8963C2semidirect
12. #8959C2semidirect
13. #8957C2semidirect
14. #8801C2semidirect
15. #8800C2semidirect
158. C2×D8M16semidirect

Maximal subgroups

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

subgroupindex
#217012
#90942
#90932
#90912
#90892
#90872
#89642
#89632
#89592
#89572
#88012
#88002