Finite Group Databasebeta

← #219652

Subgroups — #219652

Order 512.

498 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19670C211yesmaximal, characteristic
2#17971C211yesmaximal, characteristic, complemented
3#17970C211yesmaximal, characteristic, complemented
4#9762C211yesmaximal, characteristic, complemented
5#9761C211yesmaximal, characteristic, complemented
6#9285C211yesmaximal, characteristic, complemented
7#9284C211yesmaximal, characteristic, complemented
8#9283C211yesmaximal, characteristic, complemented
9#9283C211yesmaximal, characteristic, complemented
10#9282C211yesmaximal, characteristic, complemented
11#9281C211yesmaximal, characteristic, complemented
12#9281C211yesmaximal, characteristic, complemented
13#9279C211yesmaximal, characteristic, complemented
14#8797C211yesmaximal, characteristic, complemented
15#8138C211yesmaximal, characteristic, complemented
16#3228C2211yescharacteristic
17#3210C2211yescharacteristic
18#3207C2211yescharacteristic
19#3207C2211yescharacteristic
20#3169C2211yescharacteristic
21#3167C2211yescharacteristic
22#2985C2211yescharacteristic
23#2981C2211yescharacteristic
24#2981C2211yescharacteristic
25#2981C2211yescharacteristic
26#1726C2211yescharacteristic
27#1723C411yescharacteristic, complemented
28#1721C411yescharacteristic, complemented
29#1715C411yescharacteristic, complemented
30#1714C411yescharacteristic, complemented
31#1711C411yescharacteristic, complemented
32#1709C411yescharacteristic, complemented
33#1708C411yescharacteristic, complemented
34#1706C411yescharacteristic, complemented
35#1594—21no—
36#1592—21no—
37#1590—21no—
38#1589—21no—
39#1580—21no—
40SD16⋊C8—21no—
41C8×SD16—21no—
42C8×D8—21no—
43#1568C2211yescharacteristic
44#1568C2211yescharacteristic
45#1531C2211yescharacteristic
46#1530C2211yescharacteristic
47#1530C2211yescharacteristic
48#1530C2211yescharacteristic
49#1506C2211yescharacteristic
50#1505C2211yescharacteristic
51#1505C2211yescharacteristic
52#1505C2211yescharacteristic
53#1497C2211yescharacteristic
54#1496C2211yescharacteristic
55#1496C2211yescharacteristic
56#1496C2211yescharacteristic
57#1450C2211yescharacteristic
58#1450C2211yescharacteristic
59#1287C2211yescharacteristic
60#1287C2211yescharacteristic
61#1287C2211yescharacteristic
62#1287C2211yescharacteristic
63#1275C2211yescharacteristic
64#1275C2211yescharacteristic
65#1274C2211yescharacteristic
66#1274C2211yescharacteristic
67#533C2311yescharacteristic
68#532C2311yescharacteristic
69#532C2311yescharacteristic
70#532C2311yescharacteristic
71#500C2×C411yescharacteristic
72#500C2×C411yescharacteristic
73#498C2×C411yescharacteristic
74#498C2×C411yescharacteristic
75#494C2×C411yescharacteristic
76#493C2×C411yescharacteristic
77#493C2×C411yescharacteristic
78#493C2×C411yescharacteristic
79#492C2×C411yescharacteristic
80#492C2×C411yescharacteristic
81#492C2×C411yescharacteristic
82#492C2×C411yescharacteristic
83#487C2×C411yescharacteristic
84#487C2×C411yescharacteristic
85#487C2×C411yescharacteristic
86#487C2×C411yescharacteristic
87#486C2×C411yescharacteristic
88#486C2×C411yescharacteristic
89#486C2×C411yescharacteristic
90#486C2×C411yescharacteristic
91#486C2×C411yescharacteristic
92#486C2×C411yescharacteristic
93#486C2×C411yescharacteristic
94#486C2×C411yescharacteristic
95#482—41no—
96#478—41no—
97#466—21no—
98#466—21no—
99#465—21no—
100#465—21no—
101#464—41no—
102#463—41no—
103#463—41no—
104#452—41no—
105#449—41no—
106#449—41no—
107#446—21no—
108#446—41no—
109C8×Q8—21no—
110C4×SD16—21no—
111C4×SD16—21no—
112C4×D8—21no—
113C4×D8—21no—
114#436—21no—
115#436—21no—
116#436—21no—
117#435—41no—
118#435—41no—
119#435—41no—
120C8×D4—21no—
121C8×D4—21no—
122C8×D4—21no—
123#433—41no—
124#432C2311yescharacteristic
125#432C2311yescharacteristic
126#432C2311yescharacteristic
127#432C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#408—41no—
133#408—41no—
134#408—41no—
135C2×#97—21no—
136C2×#97C2311yescharacteristic
137C2×C4×C8—21no—
138C2×C4×C8C2311yescharacteristic
139#376C2311yescharacteristic
140#335—21no—
141#335—21no—
142#334—21no—
143#334—21no—
144#330—41no—
145#329—41no—
146#329—41no—
147#329—41no—
148#328—41no—
149#327—41no—
150#327—41no—
151#327—41no—
152Q8⋊C8—21no—
153Q8⋊C8—21no—
154D4⋊C8—21no—
155D4⋊C8—21no—
156D4⋊C8—21no—
157D4⋊C8—21no—
158D4⋊C8—21no—
159D4⋊C8—21no—
160#322C2×C411yescharacteristic
161#322C2×C411yescharacteristic
162#322D411yescharacteristic
163#322D411yescharacteristic
164C82C2×C411yescharacteristic
165C82C2×C411yescharacteristic
166C82D411yescharacteristic
167C82D411yescharacteristic
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×Q16—41no—
172C2×SD16—21no—
173C2×SD16—21no—
174C2×SD16—41no—
175C2×SD16—41no—
176C2×SD16—41no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—41no—
180C2×D8—41no—
181C2×M16—41no—
182C2×M16—41no—
183C2×M16—41no—
184C22×C8—21no—
185C22×C8—41no—
186C22×C8—41no—
187C22×C8—41no—
188C22×C8#4111yescharacteristic
189C22×C8#4111yescharacteristic
190C4⋊Q8C22×C411yescharacteristic
191C4⋊Q8C22×C411yescharacteristic
192C4⋊Q8C22×C411yescharacteristic
193C4⋊Q8C22×C411yescharacteristic
194C4⋊Q8C22×C411yescharacteristic
195#127C22×C411yescharacteristic
196#127C22×C411yescharacteristic
197#127C22×C411yescharacteristic
198#125—21no—
199#124—21no—
200#124—21no—
201#124—21no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204#121—21no—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209#121—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218#117—21no—
219C2×C42—21no—
220C2×C42C2411yescharacteristic
221#107—21no—
222#107—21no—
223#107—41no—
224#106—21no—
225#106—21no—
226#106—41no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237#103—21no—
238#103—21no—
239#103—41no—
240#103—41no—
241#103—41no—
242#102—21no—
243#102—21no—
244#102—21no—
245#102—21no—
246#102—21no—
247#102—21no—
248#102—41no—
249#102—41no—
250#102—41no—
251#102—41no—
252#102—41no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267#97—24no—
268#97—21no—
269#97—21no—
270#97—21no—
271#97—21no—
272#97—21no—
273#97—21no—
274#97—41no—
275C4×C8—42no—
276C4×C8—41no—
277C4×C8C22×C411yescharacteristic
278C4×C8C22×C411yescharacteristic
279C4×C8C22×C411yescharacteristic
280C4×C8C22×C411yescharacteristic
281C4×C8C22×C411yescharacteristic
282C4×C8C22×C411yescharacteristic
283C4×C8C2×D411yescharacteristic
284C4×C8C2×D411yescharacteristic
285C4×C8C2×D411yescharacteristic
286C4×C8C2×D411yescharacteristic
287C4×C8C2×D411yescharacteristic
288C4×C8C2×D411yescharacteristic
289#95—21no—
290#95—21no—
291#41—161no—
292#41—161no—
293#41—161no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4C2×#4111yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C4H(5,𝔽2)11yescharacteristic
318Q16—161no—
319SD16—42no—
320SD16—42no—
321SD16—161no—
322SD16—161no—
323SD16—161no—
324D8—42no—
325D8—42no—
326D8—161no—
327D8—161no—
328M16—161no—
329M16—161no—
330M16—161no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—161no—
349C2×C8—161no—
350C2×C8—161no—
351C2×C8C4×D411yescharacteristic
352C2×C8C4×D411yescharacteristic
353C2×C8C4×D411yescharacteristic
354C2×C8C4×D411yescharacteristic
355C4⋊C4—21no—
356C4⋊C4—21no—
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—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C42—22no—
375C42—41no—
376C42—41no—
377C42C23×C411yescharacteristic
378C42C22×D411yescharacteristic
379C42H(5,𝔽2)11yescharacteristic
380C42H(5,𝔽2)11yescharacteristic
381C23—41no—
382C23—41no—
383C23—41no—
384C23#54011yescharacteristic
385Q8—41no—
386Q8—41no—
387Q8—81no—
388Q8—81no—
389Q8—81no—
390Q8—81no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—81no—
403D4—161no—
404D4—161no—
405D4—161no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4C2×C4×D411yescharacteristic
425C2×C4#51811yescharacteristic
426C2×C4#51811yescharacteristic
427C2×C4#53511yescharacteristic
428C2×C4#53511yescharacteristic
429C2×C4#54011yescharacteristic
430C8—21no—
431C8—21no—
432C8—42no—
433C8—41no—
434C8—41no—
435C8—41no—
436C8—41no—
437C8—41no—
438C8—41no—
439C8—41no—
440C8—41no—
441C8—81no—
442C8—81no—
443C8—81no—
444C8—81no—
445C8#44312yes—
446C8#44412yes—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22#232811yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—82no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4C2×#44311yescharacteristic
472C4C2×#44411yescharacteristic
473C2—41no—
474C2—161no—
475C2—161no—
476C2—161no—
477C2#2228011yescharacteristic
478C2#2228411yescharacteristic
479C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#19670C2
LCSG ▷ C42 ▷ C22 ▷ 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. #17971C2semidirect
3. #17970C2semidirect
4. #9762C2semidirect
5. #9761C2semidirect
6. #9285C2semidirect
7. #9284C2semidirect
8. #9283C2semidirect
9. #9283C2semidirect
10. #9282C2semidirect
11. #9281C2semidirect
12. #9281C2semidirect
13. #9279C2semidirect
14. #8797C2semidirect
15. #8138C2semidirect
27. #1723C4semidirect
28. #1721C4semidirect
29. #1715C4semidirect
30. #1714C4semidirect
31. #1711C4semidirect
32. #1709C4semidirect
33. #1708C4semidirect
34. #1706C4semidirect

Maximal subgroups

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

subgroupindex
#196702
#179712
#179702
#97622
#97612
#92852
#92842
#92832
#92822
#92812
#92792
#87972
#81382