Finite Group Databasebeta

← #230402

Subgroups — #230402

Order 512.

520 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19159C211yesmaximal, characteristic
2#18089C211yesmaximal, characteristic, complemented
3#18025C211yesmaximal, characteristic
4#18016C211yesmaximal, characteristic, complemented
5#17839C211yesmaximal, characteristic, complemented
6#9854C211yesmaximal, characteristic, complemented
7#9636C211yesmaximal, characteristic, complemented
8#9587C211yesmaximal, characteristic, complemented
9#9586C211yesmaximal, characteristic, complemented
10#8972C211yesmaximal, characteristic, complemented
11#8969C211yesmaximal, characteristic
12#8936C211yesmaximal, characteristic, complemented
13#8934C211yesmaximal, characteristic, complemented
14#8270C211yesmaximal, characteristic, complemented
15#8195C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3107C2211yescharacteristic
18#3106C2211yescharacteristic
19#2991C2211yescharacteristic
20#2990C2211yescharacteristic
21#2986C2211yescharacteristic
22#2979C2211yescharacteristic
23#2965C2211yescharacteristic
24#1994C2211yescharacteristic
25#1962C2211yescharacteristic
26C4×#104C2211yescharacteristic
27#1744C2211yescharacteristic
28#1704C411yescharacteristic, complemented
29#1703C411yescharacteristic, complemented
30#1702C411yescharacteristic, complemented
31#1701C411yescharacteristic, complemented
32#1692C411yescharacteristic, complemented
33#1691C411yescharacteristic, complemented
34#1686C411yescharacteristic, complemented
35#1685C411yescharacteristic, complemented
36#1594—21no—
37#1592—21no—
38#1591—21no—
39#1587—21no—
40#1585—21no—
41#1583—21no—
42#1583—21no—
43#1582—21no—
44#1581—21no—
45SD16⋊C8—21no—
46SD16⋊C8—21no—
47C8×SD16—21no—
48#1574C2211yescharacteristic
49#1574C2211yescharacteristic
50#1574C2211yescharacteristic
51#1574C2211yescharacteristic
52#1572—21no—
53#1571—21no—
54#1569—21no—
55#1567C2211yescharacteristic
56#1565—21no—
57#1564C2211yescharacteristic
58#1564C2211yescharacteristic
59#1544C2211yescharacteristic
60#1543C2211yescharacteristic
61#1493C2211yescharacteristic
62#1492C2211yescharacteristic
63#1485C2211yescharacteristic
64#1484C2211yescharacteristic
65#1476C2211yescharacteristic
66#1452C2211yescharacteristic
67#1451C2211yescharacteristic
68#1450C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1278C2211yescharacteristic
72#1276C2211yescharacteristic
73#1276C2211yescharacteristic
74#1276C2211yescharacteristic
75#533C2311yescharacteristic
76#532C2311yescharacteristic
77C4×#41C2311yescharacteristic
78#498C2×C411yescharacteristic
79#492C2×C411yescharacteristic
80#487C2×C411yescharacteristic
81#486C2×C411yescharacteristic
82#485—21no—
83#482—21no—
84#477C2×C411yescharacteristic
85#476C2×C411yescharacteristic
86#475C2×C411yescharacteristic
87#474C2×C411yescharacteristic
88#465—21no—
89#465—21no—
90#462C2×C411yescharacteristic
91#461C2×C411yescharacteristic
92#460C2×C411yescharacteristic
93#459C2×C411yescharacteristic
94#452—21no—
95#449—21no—
96#446—21no—
97#446—21no—
98#446—21no—
99#446C2×C411yescharacteristic
100#446C2×C411yescharacteristic
101C8×Q8—21no—
102C8×Q8—21no—
103C8×Q8—21no—
104C4×Q16—21no—
105C4×SD16—21no—
106C4×SD16—21no—
107C4×SD16—21no—
108C4×SD16—21no—
109C4×D8—21no—
110#436—21no—
111#436—21no—
112#436—21no—
113#436—41no—
114#436C2×C411yescharacteristic
115#436C2×C411yescharacteristic
116#435—21no—
117#435—21no—
118#435—41no—
119C8×D4—21no—
120C8×D4—21no—
121C8×D4—21no—
122#433—41no—
123#432—21no—
124#431—21no—
125#431—21no—
126#431C2311yescharacteristic
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#423—21no—
133#423C2311yescharacteristic
134#423C2311yescharacteristic
135C2×C4⋊C8—21no—
136C2×#104—21no—
137C2×#104—41no—
138#408—41no—
139C4×M16—21no—
140C4×M16C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C43C2311yescharacteristic
144#339—21no—
145#339—41no—
146#336—21no—
147#336—21no—
148#335—21no—
149#335—21no—
150#335—21no—
151#335—21no—
152#335—21no—
153#335—21no—
154#335C2×C411yescharacteristic
155#335C2×C411yescharacteristic
156#335D411yescharacteristic
157#335D411yescharacteristic
158#334—21no—
159#334—21no—
160#334—21no—
161#334—21no—
162#334C2×C411yescharacteristic
163#334C2×C411yescharacteristic
164#334D411yescharacteristic
165#334D411yescharacteristic
166#333C2×C411yescharacteristic
167#333C2×C411yescharacteristic
168#332C2×C411yescharacteristic
169#332C2×C411yescharacteristic
170#332C2×C411yescharacteristic
171#332C2×C411yescharacteristic
172#331C2×C411yescharacteristic
173#331C2×C411yescharacteristic
174Q8⋊C8—21no—
175Q8⋊C8—21no—
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8D411yescharacteristic
179Q8⋊C8D411yescharacteristic
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8D411yescharacteristic
185D4⋊C8D411yescharacteristic
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191C82—21no—
192C2×#41—21no—
193C2×#41#4111yescharacteristic
194C2×Q16—21no—
195C2×SD16—21no—
196C2×SD16—21no—
197C2×SD16—21no—
198C2×SD16—21no—
199C2×D8—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—41no—
206C2×M16—41no—
207C22×C8—21no—
208C22×C8—41no—
209C22×C8—41no—
210C22×C8#4111yescharacteristic
211C22×C8#4111yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213C4⋊Q8C22×C411yescharacteristic
214C4⋊Q8C22×C411yescharacteristic
215#127C22×C411yescharacteristic
216#125—21no—
217#124—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220#121—21no—
221#121—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8C22×C411yescharacteristic
225C4×Q8C2×D411yescharacteristic
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4C22×C411yescharacteristic
231C4×D4C2×D411yescharacteristic
232#117—21no—
233#117—21no—
234#117#4111yescharacteristic
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C2411yescharacteristic
240#107—21no—
241#107—21no—
242#106—21no—
243#106—21no—
244#106—21no—
245#106—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8C22×C411yescharacteristic
257C4⋊C8C22×C411yescharacteristic
258C4⋊C8C22×C411yescharacteristic
259C4⋊C8C22×C411yescharacteristic
260C4⋊C8C22×C411yescharacteristic
261C4⋊C8C22×C411yescharacteristic
262C4⋊C8C2×D411yescharacteristic
263C4⋊C8C2×D411yescharacteristic
264C4⋊C8C2×D411yescharacteristic
265C4⋊C8C2×D411yescharacteristic
266C4⋊C8C2×D411yescharacteristic
267C4⋊C8C2×D411yescharacteristic
268#104—42no—
269#104—82no—
270#103—21no—
271#103—21no—
272#103—21no—
273#103—21no—
274#103—21no—
275#103—21no—
276#102—21no—
277#102—21no—
278#102—21no—
279#102—21no—
280#102—21no—
281#102—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—41no—
291C22⋊C8—41no—
292C22⋊C8—41no—
293C22⋊C8—41no—
294#97—21no—
295#97—21no—
296#97—21no—
297#97—41no—
298#97—41no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—41no—
308C4×C8—41no—
309C4×C8C22×C411yescharacteristic
310C4×C8C22×C411yescharacteristic
311C4×C8C2×D411yescharacteristic
312C4×C8C2×D411yescharacteristic
313C4×C8C2×D411yescharacteristic
314C4×C8C2×D411yescharacteristic
315#41—41no—
316#41—41no—
317#41—81no—
318#41—81no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8C4×D411yescharacteristic
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—41no—
328C2×D4C4×D411yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337Q16—81no—
338SD16—42no—
339SD16—42no—
340SD16—42no—
341SD16—81no—
342D8—81no—
343M16—42no—
344M16—41no—
345M16—41no—
346M16—81no—
347M16—81no—
348M16—81no—
349M16—161no—
350M16—161no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—161no—
373C2×C8—161no—
374C2×C8C4×D411yescharacteristic
375C2×C8C4×D411yescharacteristic
376C2×C8C4×D411yescharacteristic
377C2×C8C4×D411yescharacteristic
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4C4×D411yescharacteristic
389C4⋊C4C4×D411yescharacteristic
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—22no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42C23×C411yescharacteristic
407C42C22×D411yescharacteristic
408C42C22×D411yescharacteristic
409C422−1+411yescharacteristic
410C23—21no—
411C23—41no—
412C23#55311yescharacteristic
413Q8—21no—
414Q8—21no—
415Q8—41no—
416Q8—41no—
417Q8—41no—
418Q8—41no—
419Q8—81no—
420D4—21no—
421D4—21no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—81no—
427D4—81no—
428D4—161no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4#52011yescharacteristic
454C2×C4D4211yescharacteristic
455C2×C4#54711yescharacteristic
456C2×C4#54811yescharacteristic
457C8—21no—
458C8—21no—
459C8—21no—
460C8—21no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—81no—
476C8—81no—
477C8—81no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22#236111yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4#295511yescharacteristic
499C4#295511yescharacteristic
500C4#295611yescharacteristic
501C4#295711yescharacteristic
502C4#328811yescharacteristic
503C4#328911yescharacteristic
504C2—41no—
505C2—81no—
506C2—161no—
507C2#2301711yescharacteristic
508C2#2302111yescharacteristic
509C2#2304911yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#3106C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #18089C2semidirect
4. #18016C2semidirect
5. #17839C2semidirect
6. #9854C2semidirect
7. #9636C2semidirect
8. #9587C2semidirect
9. #9586C2semidirect
10. #8972C2semidirect
12. #8936C2semidirect
13. #8934C2semidirect
14. #8270C2semidirect
15. #8195C2semidirect
28. #1704C4semidirect
29. #1703C4semidirect
30. #1702C4semidirect
31. #1701C4semidirect
32. #1692C4semidirect
33. #1691C4semidirect
34. #1686C4semidirect
35. #1685C4semidirect

Maximal subgroups

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

subgroupindex
#191592
#180892
#180252
#180162
#178392
#98542
#96362
#95872
#95862
#89722
#89692
#89362
#89342
#82702
#81952