Finite Group Databasebeta

← #219648

Subgroups — #219648

Order 512.

485 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#19665C211yesmaximal, characteristic
2#17971C211yesmaximal, characteristic, complemented
3#17971C211yesmaximal, characteristic, complemented
4#9761C211yesmaximal, characteristic, complemented
5#9761C211yesmaximal, characteristic, complemented
6#9286C211yesmaximal, characteristic, complemented
7#9285C211yesmaximal, characteristic, complemented
8#9284C211yesmaximal, characteristic, complemented
9#9283C211yesmaximal, characteristic
10#9282C211yesmaximal, characteristic, complemented
11#9281C211yesmaximal, characteristic
12#9280C211yesmaximal, characteristic, complemented
13#9279C211yesmaximal, characteristic, complemented
14#8796C211yesmaximal, characteristic, complemented
15#8137C211yesmaximal, characteristic, complemented
16#3231C2211yescharacteristic
17#3228C2211yescharacteristic
18#3210C2211yescharacteristic
19#3207C2211yescharacteristic
20#3167C2211yescharacteristic
21#3167C2211yescharacteristic
22#2985C2211yescharacteristic
23#2985C2211yescharacteristic
24#2981C2211yescharacteristic
25#2981C2211yescharacteristic
26#1726C2211yescharacteristic
27#1720C411yescharacteristic, complemented
28#1719C411yescharacteristic, complemented
29#1718C411yescharacteristic, complemented
30#1717C411yescharacteristic, complemented
31#1708C411yescharacteristic, complemented
32#1707C411yescharacteristic, complemented
33#1706C411yescharacteristic, complemented
34#1705C411yescharacteristic, complemented
35#1592—21no—
36#1592—21no—
37#1590—21no—
38#1590—21no—
39SD16⋊C8—21no—
40SD16⋊C8—21no—
41C8×SD16—21no—
42C8×SD16—21no—
43#1568C2211yescharacteristic
44#1568C2211yescharacteristic
45#1531C2211yescharacteristic
46#1531C2211yescharacteristic
47#1530C2211yescharacteristic
48#1530C2211yescharacteristic
49#1506C2211yescharacteristic
50#1506C2211yescharacteristic
51#1505C2211yescharacteristic
52#1505C2211yescharacteristic
53#1497C2211yescharacteristic
54#1497C2211yescharacteristic
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#533C2311yescharacteristic
69#532C2311yescharacteristic
70#532C2311yescharacteristic
71#500C2×C411yescharacteristic
72#500C2×C411yescharacteristic
73#498C2×C411yescharacteristic
74#498C2×C411yescharacteristic
75#498C2×C411yescharacteristic
76#498C2×C411yescharacteristic
77#494C2×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#487C2×C411yescharacteristic
88#487C2×C411yescharacteristic
89#486C2×C411yescharacteristic
90#486C2×C411yescharacteristic
91#486C2×C411yescharacteristic
92#486C2×C411yescharacteristic
93#486C2×C411yescharacteristic
94#486C2×C411yescharacteristic
95#485—41no—
96#482—41no—
97#479—41no—
98#478—41no—
99#465—21no—
100#465—21no—
101#465—21no—
102#465—21no—
103#464—41no—
104#463—41no—
105#452—41no—
106#449—41no—
107#446—21no—
108#446—21no—
109#446—41no—
110#446—41no—
111C8×Q8—21no—
112C8×Q8—21no—
113C4×SD16—21no—
114C4×SD16—21no—
115C4×SD16—21no—
116C4×SD16—21no—
117#436—21no—
118#436—21no—
119#435—41no—
120#435—41no—
121C8×D4—21no—
122C8×D4—21no—
123#433—41no—
124#433—41no—
125#432C2311yescharacteristic
126#432C2311yescharacteristic
127#432C2311yescharacteristic
128#432C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#431C2311yescharacteristic
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#330—41no—
146#329—41no—
147#329—41no—
148#328—41no—
149#328—41no—
150#327—41no—
151#327—41no—
152Q8⋊C8—21no—
153Q8⋊C8—21no—
154Q8⋊C8—21no—
155Q8⋊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×Q16—41no—
171C2×SD16—21no—
172C2×SD16—21no—
173C2×SD16—21no—
174C2×SD16—21no—
175C2×SD16—41no—
176C2×SD16—41no—
177C2×D8—41no—
178C2×M16—41no—
179C2×M16—41no—
180C22×C8—21no—
181C22×C8—41no—
182C22×C8—41no—
183C22×C8#4111yescharacteristic
184C22×C8#4111yescharacteristic
185C4⋊Q8C22×C411yescharacteristic
186C4⋊Q8C22×C411yescharacteristic
187C4⋊Q8C22×C411yescharacteristic
188C4⋊Q8C22×C411yescharacteristic
189C4⋊Q8C22×C411yescharacteristic
190C4⋊Q8C22×C411yescharacteristic
191#127C22×C411yescharacteristic
192#127C22×C411yescharacteristic
193#125—21no—
194#125—21no—
195#124—21no—
196#124—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204#121—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213#117—21no—
214#117—21no—
215C2×C42—21no—
216C2×C42C2411yescharacteristic
217#107—41no—
218#107—41no—
219#106—21no—
220#106—21no—
221#106—21no—
222#106—21no—
223#106—41no—
224#106—41no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—41no—
234C4⋊C8—41no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237#103—21no—
238#103—21no—
239#103—21no—
240#103—21no—
241#103—41no—
242#103—41no—
243#103—41no—
244#103—41no—
245#102—21no—
246#102—21no—
247#102—21no—
248#102—21no—
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—
265#97—21no—
266#97—21no—
267#97—21no—
268#97—21no—
269#97—21no—
270#97—21no—
271#97—42no—
272#97—41no—
273#97—41no—
274C4×C8—24no—
275C4×C8—41no—
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—
293C2×Q8—21no—
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—41no—
306C2×D4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4C2×#4111yescharacteristic
311C22×C4H(5,𝔽2)11yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313Q16—161no—
314SD16—42no—
315SD16—42no—
316SD16—42no—
317SD16—42no—
318SD16—161no—
319SD16—161no—
320D8—161no—
321M16—161no—
322M16—161no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—161no—
341C2×C8—161no—
342C2×C8C4×D411yescharacteristic
343C2×C8C4×D411yescharacteristic
344C2×C8C4×D411yescharacteristic
345C2×C8C4×D411yescharacteristic
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C42—22no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42C23×C411yescharacteristic
373C42C22×D411yescharacteristic
374C42H(5,𝔽2)11yescharacteristic
375C42H(5,𝔽2)11yescharacteristic
376C23—41no—
377C23—41no—
378C23#54011yescharacteristic
379Q8—41no—
380Q8—41no—
381Q8—41no—
382Q8—41no—
383Q8—81no—
384Q8—81no—
385Q8—81no—
386Q8—81no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—81no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—161no—
396D4—161no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4C2×C4×D411yescharacteristic
415C2×C4#51811yescharacteristic
416C2×C4#51811yescharacteristic
417C2×C4#53511yescharacteristic
418C2×C4#53511yescharacteristic
419C2×C4#54011yescharacteristic
420C8—21no—
421C8—21no—
422C8—42no—
423C8—41no—
424C8—41no—
425C8—41no—
426C8—41no—
427C8—41no—
428C8—41no—
429C8—41no—
430C8—41no—
431C8—81no—
432C8—81no—
433C8—81no—
434C8—81no—
435C8#44312yes—
436C8#44412yes—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22#232811yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—82no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4C2×#44311yescharacteristic
460C4C2×#44411yescharacteristic
461C2—41no—
462C2—161no—
463C2—161no—
464C2#2228011yescharacteristic
465C2#2228411yescharacteristic
466C2#2228411yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3207C22
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. #17971C2semidirect
4. #9761C2semidirect
5. #9761C2semidirect
6. #9286C2semidirect
7. #9285C2semidirect
8. #9284C2semidirect
10. #9282C2semidirect
12. #9280C2semidirect
13. #9279C2semidirect
14. #8796C2semidirect
15. #8137C2semidirect
27. #1720C4semidirect
28. #1719C4semidirect
29. #1718C4semidirect
30. #1717C4semidirect
31. #1708C4semidirect
32. #1707C4semidirect
33. #1706C4semidirect
34. #1705C4semidirect

Maximal subgroups

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

subgroupindex
#196652
#179712
#97612
#92862
#92852
#92842
#92832
#92822
#92812
#92802
#92792
#87962
#81372