Finite Group Databasebeta

← #219653

Subgroups — #219653

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19669C211yesmaximal, characteristic
2#17972C211yesmaximal, characteristic, complemented
3#17971C211yesmaximal, characteristic
4#9763C211yesmaximal, characteristic, complemented
5#9761C211yesmaximal, characteristic
6#9286C211yesmaximal, characteristic, complemented
7#9286C211yesmaximal, characteristic, complemented
8#9285C211yesmaximal, characteristic
9#9284C211yesmaximal, characteristic
10#9282C211yesmaximal, characteristic
11#9280C211yesmaximal, characteristic, complemented
12#9280C211yesmaximal, characteristic, complemented
13#9279C211yesmaximal, characteristic
14#8797C211yesmaximal, characteristic, complemented
15#8138C211yesmaximal, characteristic, complemented
16#3231C2211yescharacteristic
17#3231C2211yescharacteristic
18#3228C2211yescharacteristic
19#3210C2211yescharacteristic
20#3170C2211yescharacteristic
21#3167C2211yescharacteristic
22#2985C2211yescharacteristic
23#2985C2211yescharacteristic
24#2985C2211yescharacteristic
25#2981C2211yescharacteristic
26#1726C2211yescharacteristic
27#1723C411yescharacteristic, complemented
28#1722C411yescharacteristic, complemented
29#1716C411yescharacteristic, complemented
30#1714C411yescharacteristic, complemented
31#1711C411yescharacteristic, complemented
32#1710C411yescharacteristic, complemented
33#1708C411yescharacteristic, complemented
34#1707C411yescharacteristic, complemented
35#1593—21no—
36#1592—21no—
37#1591—21no—
38#1590—21no—
39#1579—21no—
40SD16⋊C8—21no—
41C8×Q16—21no—
42C8×SD16—21no—
43#1568C2211yescharacteristic
44#1568C2211yescharacteristic
45#1531C2211yescharacteristic
46#1531C2211yescharacteristic
47#1531C2211yescharacteristic
48#1530C2211yescharacteristic
49#1506C2211yescharacteristic
50#1506C2211yescharacteristic
51#1506C2211yescharacteristic
52#1505C2211yescharacteristic
53#1497C2211yescharacteristic
54#1497C2211yescharacteristic
55#1497C2211yescharacteristic
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#533C2311yescharacteristic
70#532C2311yescharacteristic
71#500C2×C411yescharacteristic
72#500C2×C411yescharacteristic
73#500C2×C411yescharacteristic
74#500C2×C411yescharacteristic
75#498C2×C411yescharacteristic
76#498C2×C411yescharacteristic
77#498C2×C411yescharacteristic
78#498C2×C411yescharacteristic
79#494C2×C411yescharacteristic
80#494C2×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#487C2×C411yescharacteristic
90#487C2×C411yescharacteristic
91#486C2×C411yescharacteristic
92#486C2×C411yescharacteristic
93#486C2×C411yescharacteristic
94#486C2×C411yescharacteristic
95#485—41no—
96#485—41no—
97#482—41no—
98#479—41no—
99#479—41no—
100#478—41no—
101#467—21no—
102#467—21no—
103#465—21no—
104#465—21no—
105#464—41no—
106#452—41no—
107#446—21no—
108#446—21no—
109#446—21no—
110#446—41no—
111#446—41no—
112#446—41no—
113C8×Q8—21no—
114C8×Q8—21no—
115C8×Q8—21no—
116C4×Q16—21no—
117C4×Q16—21no—
118C4×SD16—21no—
119C4×SD16—21no—
120#436—21no—
121#435—41no—
122C8×D4—21no—
123#433—41no—
124#433—41no—
125#433—41no—
126#432C2311yescharacteristic
127#432C2311yescharacteristic
128#432C2311yescharacteristic
129#432C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#431C2311yescharacteristic
133#431C2311yescharacteristic
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#330—41no—
147#329—41no—
148#328—41no—
149#328—41no—
150#328—41no—
151#327—41no—
152Q8⋊C8—21no—
153Q8⋊C8—21no—
154Q8⋊C8—21no—
155Q8⋊C8—21no—
156Q8⋊C8—21no—
157Q8⋊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×Q16—21no—
170C2×Q16—21no—
171C2×Q16—41no—
172C2×SD16—21no—
173C2×SD16—21no—
174C2×SD16—41no—
175C2×M16—41no—
176C22×C8—21no—
177C22×C8—41no—
178C22×C8#4111yescharacteristic
179C22×C8#4111yescharacteristic
180C4⋊Q8C22×C411yescharacteristic
181C4⋊Q8C22×C411yescharacteristic
182C4⋊Q8C22×C411yescharacteristic
183C4⋊Q8C22×C411yescharacteristic
184C4⋊Q8C22×C411yescharacteristic
185C4⋊Q8C22×C411yescharacteristic
186C4⋊Q8C22×C411yescharacteristic
187#127C22×C411yescharacteristic
188#125—21no—
189#125—21no—
190#125—21no—
191#124—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198#121—21no—
199#121—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×D4—21no—
207C4×D4—21no—
208#117—21no—
209#117—21no—
210#117—21no—
211C2×C42—21no—
212C2×C42C2411yescharacteristic
213#107—21no—
214#107—21no—
215#107—41no—
216#107—41no—
217#107—41no—
218#106—21no—
219#106—21no—
220#106—41no—
221#106—41no—
222#106—41no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—41no—
232C4⋊C8—41no—
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—21no—
242#103—21no—
243#103—41no—
244#103—41no—
245#103—41no—
246#103—41no—
247#103—41no—
248#102—21no—
249#102—21no—
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—
263#97—24no—
264#97—21no—
265#97—21no—
266#97—21no—
267#97—21no—
268#97—21no—
269#97—21no—
270#97—41no—
271#97—41no—
272#97—41no—
273C4×C8—42no—
274C4×C8—41no—
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—
292C2×Q8—21no—
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—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4C2×#4111yescharacteristic
306C22×C4H(5,𝔽2)11yescharacteristic
307C22×C4H(5,𝔽2)11yescharacteristic
308Q16—42no—
309Q16—42no—
310Q16—161no—
311SD16—42no—
312SD16—42no—
313SD16—161no—
314M16—161no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—161no—
333C2×C8C4×D411yescharacteristic
334C2×C8C4×D411yescharacteristic
335C2×C8C4×D411yescharacteristic
336C2×C8C4×D411yescharacteristic
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C42—22no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42C23×C411yescharacteristic
368C42C22×D411yescharacteristic
369C42H(5,𝔽2)11yescharacteristic
370C42H(5,𝔽2)11yescharacteristic
371C23—41no—
372C23#54011yescharacteristic
373Q8—41no—
374Q8—41no—
375Q8—41no—
376Q8—41no—
377Q8—41no—
378Q8—41no—
379Q8—81no—
380Q8—81no—
381Q8—81no—
382Q8—81no—
383D4—41no—
384D4—41no—
385D4—81no—
386D4—81no—
387D4—161no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4C2×C4×D411yescharacteristic
405C2×C4#51811yescharacteristic
406C2×C4#51811yescharacteristic
407C2×C4#53511yescharacteristic
408C2×C4#53511yescharacteristic
409C2×C4#54011yescharacteristic
410C8—21no—
411C8—21no—
412C8—41no—
413C8—41no—
414C8—41no—
415C8—41no—
416C8—42no—
417C8—41no—
418C8—41no—
419C8—41no—
420C8—41no—
421C8—81no—
422C8—81no—
423C8—81no—
424C8—81no—
425C8#44312yes—
426C8#44412yes—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22#232811yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—82no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4C2×#44311yescharacteristic
448C4C2×#44411yescharacteristic
449C2—41no—
450C2—161no—
451C2#2228011yescharacteristic
452C2#2228411yescharacteristic
453C2#2228511yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#532C23
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. #17972C2semidirect
4. #9763C2semidirect
6. #9286C2semidirect
7. #9286C2semidirect
11. #9280C2semidirect
12. #9280C2semidirect
14. #8797C2semidirect
15. #8138C2semidirect
27. #1723C4semidirect
28. #1722C4semidirect
29. #1716C4semidirect
30. #1714C4semidirect
31. #1711C4semidirect
32. #1710C4semidirect
33. #1708C4semidirect
34. #1707C4semidirect

Maximal subgroups

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

subgroupindex
#196692
#179722
#179712
#97632
#97612
#92862
#92852
#92842
#92822
#92802
#92792
#87972
#81382