Finite Group Databasebeta

← #219708

Subgroups — #219708

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23808C211yesmaximal, characteristic
2#17879C211yesmaximal, characteristic, complemented
3#17877C211yesmaximal, characteristic, complemented
4#9668C211yesmaximal, characteristic, complemented
5#9667C211yesmaximal, characteristic, complemented
6#9106C211yesmaximal, characteristic, complemented
7#9103C211yesmaximal, characteristic, complemented
8#9101C211yesmaximal, characteristic, complemented
9#9101C211yesmaximal, characteristic, complemented
10#9094C211yesmaximal, characteristic, complemented
11#9093C211yesmaximal, characteristic, complemented
12#9089C211yesmaximal, characteristic, complemented
13#9087C211yesmaximal, characteristic, complemented
14#8782C211yesmaximal, characteristic, complemented
15#8141C211yesmaximal, characteristic, complemented
16#2983C2211yescharacteristic
17#2975C2211yescharacteristic
18#2972C2211yescharacteristic
19#2972C2211yescharacteristic
20C2×#442C2211yescharacteristic
21C2×#440C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24#2328C2211yescharacteristic
25#2326C2211yescharacteristic
26#2097C411yescharacteristic
27#2094C411yescharacteristic
28#2093C411yescharacteristic
29#2092C411yescharacteristic
30#2014C411yescharacteristic
31#2012C411yescharacteristic
32#2012C411yescharacteristic
33#2011C411yescharacteristic
34#1746C2211yescharacteristic
35#1586—22no—
36#1585—22no—
37#1582—22no—
38#1581—22no—
39#1564C2211yescharacteristic
40#1564C2211yescharacteristic
41#1479C2211yescharacteristic
42#1479C2211yescharacteristic
43#1478C2211yescharacteristic
44#1478C2211yescharacteristic
45#1478C2211yescharacteristic
46#1478C2211yescharacteristic
47#1478C2211yescharacteristic
48#1478C2211yescharacteristic
49#1461C2211yescharacteristic
50#1460C2211yescharacteristic
51#1460C2211yescharacteristic
52#1460C2211yescharacteristic
53#1452C2211yescharacteristic
54#1450C2211yescharacteristic
55#1281C2211yescharacteristic
56#1281C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59#1274C2211yescharacteristic
60#1274C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C22×SD16C2×C411yescharacteristic
64C22×SD16C2×C411yescharacteristic
65C22×D8C2×C411yescharacteristic
66C22×D8C2×C411yescharacteristic
67C2×#127C2×C411yescharacteristic
68C2×#124C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#446—22no—
74#446—41no—
75#442—41no—
76#442—41no—
77#440—41no—
78#440—41no—
79C4×SD16—21no—
80C4×SD16—21no—
81C4×SD16—21no—
82C4×SD16—21no—
83C4×D8—21no—
84C4×D8—21no—
85C4×D8—21no—
86C4×D8—21no—
87#436—22no—
88#436—41no—
89C8×D4—22no—
90C8×D4—22no—
91C8×D4—41no—
92C8×D4—41no—
93#432—41no—
94#431C2311yescharacteristic
95#431C2311yescharacteristic
96#431C2311yescharacteristic
97#431C2311yescharacteristic
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102C2×#107C2×C411yescharacteristic
103C2×#107C2×C411yescharacteristic
104C2×#106C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114#407—41no—
115C2×C22⋊C8—41no—
116C2×C22⋊C8—41no—
117C2×#97C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391C2×C411yescharacteristic
120#390C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#388—41no—
124#386—41no—
125#386—41no—
126#385—41no—
127#384C2×C411yescharacteristic
128#382C2×C411yescharacteristic
129#376C2311yescharacteristic
130#336C2×C411yescharacteristic
131#336C2×C411yescharacteristic
132#336C2×C411yescharacteristic
133#336C2×C411yescharacteristic
134#335—21no—
135#335—21no—
136#334—21no—
137#334—21no—
138Q8⋊C8—22no—
139Q8⋊C8—41no—
140D4⋊C8—22no—
141D4⋊C8—22no—
142D4⋊C8—22no—
143D4⋊C8—41no—
144D4⋊C8—41no—
145D4⋊C8—41no—
146#324—41no—
147#323—41no—
148#323—41no—
149#323—41no—
150#322—21no—
151#322D412yescomplemented
152#322D412yescomplemented
153C82—21no—
154C22×Q8C22×C411yescharacteristic
155C22×D4—41no—
156C22×D4C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C2×SD16—22no—
163C2×SD16—22no—
164C2×SD16—21no—
165C2×SD16—21no—
166C2×SD16—41no—
167C2×SD16—41no—
168C2×SD16—41no—
169C2×D8—22no—
170C2×D8—22no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—41no—
174C2×D8—41no—
175C2×D8—41no—
176C2×M16—42no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183#127—22no—
184#127—22no—
185#124—41no—
186#124—41no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2411yescharacteristic
220#107—21no—
221#107—21no—
222#107—41no—
223#106—21no—
224#106—21no—
225#106—41no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—42no—
235#103—21no—
236#103—21no—
237#103—41no—
238#102—21no—
239#102—21no—
240#102—21no—
241#102—21no—
242#102—21no—
243#102—21no—
244#102—41no—
245#102—41no—
246#102—41no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—42no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258#97—21no—
259#97—21no—
260#97—42no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8C2×D412yes—
268C4×C8C2×D412yes—
269C4×C8C2×D411yescharacteristic
270C4×C8C2×D411yescharacteristic
271C4×C8#4111yescharacteristic
272C4×C8#4111yescharacteristic
273#95—21no—
274#95—21no—
275#95—41no—
276#95—41no—
277C24—21no—
278C24—21no—
279C24—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—44no—
303C2×D4—161no—
304C2×D4—161no—
305C2×D4#13112yes—
306C2×D4#13112yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—161no—
321C22×C4—161no—
322C22×C4—161no—
323C22×C4C23×C411yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C4H(5,𝔽2)11yescharacteristic
326SD16—42no—
327SD16—42no—
328SD16—44no—
329D8—42no—
330D8—42no—
331D8—44no—
332M16—162no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—162no—
352C2×C8—162no—
353C2×C8C4×D411yescharacteristic
354C2×C8C4×D411yescharacteristic
355C2×C8C4×D411yescharacteristic
356C2×C8C4×D411yescharacteristic
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—161no—
368C22⋊C4—44no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—161no—
376C22⋊C4—161no—
377C22⋊C4—161no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—41no—
383C42—41no—
384C42C22×D411yescharacteristic
385C42C2×#4111yescharacteristic
386C42H(5,𝔽2)11yescharacteristic
387C42H(5,𝔽2)11yescharacteristic
388C23—22no—
389C23—22no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23#51811yescharacteristic
404Q8—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408D4—22no—
409D4—22no—
410D4—22no—
411D4—22no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—164no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
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—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—161no—
444C2×C4—161no—
445C2×C4—161no—
446C2×C4C2×C4×D411yescharacteristic
447C2×C4#51811yescharacteristic
448C2×C4#53511yescharacteristic
449C2×C4#53511yescharacteristic
450C2×C4#54011yescharacteristic
451C2×C4#54011yescharacteristic
452C2×C4C2×#13111yescharacteristic
453C2×C4C2×#13111yescharacteristic
454C2×C4#56811yescharacteristic
455C2×C4#56811yescharacteristic
456C8—21no—
457C8—21no—
458C8—21no—
459C8—21no—
460C8—42no—
461C8—42no—
462C8—41no—
463C8—41no—
464C8—81no—
465C8—81no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—81no—
470C22—21no—
471C22—21no—
472C22—42no—
473C22—42no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—161no—
485C22—161no—
486C22—161no—
487C22#232811yescharacteristic
488C22#297711yescharacteristic
489C22#297811yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4#293311yescharacteristic
502C4#293311yescharacteristic
503C4#297611yescharacteristic
504C4#297611yescharacteristic
505C2—21no—
506C2—21no—
507C2—82no—
508C2—82no—
509C2—161no—
510C2#2226711yescharacteristic
511C2#2228411yescharacteristic
512C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2011C4
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. #17879C2semidirect
3. #17877C2semidirect
4. #9668C2semidirect
5. #9667C2semidirect
6. #9106C2semidirect
7. #9103C2semidirect
8. #9101C2semidirect
9. #9101C2semidirect
10. #9094C2semidirect
11. #9093C2semidirect
12. #9089C2semidirect
13. #9087C2semidirect
14. #8782C2semidirect
15. #8141C2semidirect
151. #322D4semidirect
152. #322D4semidirect

Maximal subgroups

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

subgroupindex
#238082
#178792
#178772
#96682
#96672
#91062
#91032
#91012
#90942
#90932
#90892
#90872
#87822
#81412