Finite Group Databasebeta

← #360681

Subgroups — #360681

Order 512.

1226 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34189C211yesmaximal, characteristic
2#20800C211yesmaximal, characteristic, complemented
3#20648C211yesmaximal, characteristic
4C2×#1759C211yesmaximal, characteristic
5#14348C211yesmaximal, characteristic, complemented
6#11113C214yesmaximal
7#10899C214yesmaximal, complemented
8#10858C211yesmaximal, characteristic, complemented
9#10840C211yesmaximal, characteristic, complemented
10C2×C4×#41C2211yescharacteristic
11C2×#428C2214yes—
12C2×#423C2211yescharacteristic
13C22×#104C2212yes—
14C2×#417C2212yes—
15C2×C4×M16C2211yescharacteristic
16#2357—22no—
17#2356—22no—
18#2353—24no—
19#2314—21no—
20#2314—21no—
21#2311C2212yes—
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2283—22no—
25C2×#382C2211yescharacteristic
26#2274C2212yes—
27C2×#376C2211yescharacteristic
28#1847—22no—
29#1845—21no—
30#1844—21no—
31#1803—22no—
32#1802—22no—
33#1759C418yescomplemented
34C2×#343C2214yes—
35#1736C2212yes—
36C2×#339C2214yes—
37C2×#339C2212yes—
38#1442—24no—
39C4⋊#95C2214yes—
40C22×#41C2311yescharacteristic
41C22×M16C2312yes—
42C22×M16C2312yes—
43C4×#41—22no—
44C4×#41—22no—
45C4×#41C2×C414yes—
46C4×#41Q814yes—
47C2×C4×Q8D412yes—
48C2×C4×D4—22no—
49C2×C4×D4D412yes—
50C2×#117—21no—
51C2×#117C2312yes—
52C2×#117C2311yescharacteristic
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42—21no—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59#428—28no—
60#428—44no—
61C2×#107—24no—
62C2×#106—24no—
63#423—22no—
64#423—22no—
65C2×C4⋊C8—22no—
66C2×#104—24no—
67C2×#104—24no—
68C2×#104C2×C418yes—
69#417—44no—
70C2×#103—22no—
71C2×#102—22no—
72C4×M16—22no—
73C4×M16C2×C414yes—
74C2×#97—21no—
75C2×C4×C8—21no—
76#391—22no—
77#390—22no—
78#388—24no—
79#388—24no—
80#387—24no—
81#387—24no—
82#386—24no—
83#384—22no—
84#384—21no—
85#384—21no—
86#384D411yescharacteristic
87#384D411yescharacteristic
88#383—24no—
89#382—21no—
90#382—22no—
91#382—21no—
92#382D412yes—
93#381—24no—
94#380—22no—
95#380—22no—
96#380—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104#376—24no—
105#376—22no—
106#376C2×C414yes—
107C2×#95—22no—
108C2×#95—28no—
109C2×#95D414yes—
110#343—28no—
111#340—24no—
112#339—44no—
113#339C2×C418yes—
114C2×#41—21no—
115C2×#41—21no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41C22×C412yes—
121C2×#41C2×Q812yes—
122C22×Q8C2×D411yescharacteristic
123C22×D4—21no—
124C22×D4C2×D411yescharacteristic
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4C2×D414yes—
130C23×C4C2×D412yes—
131C23×C4C2411yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—24no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—24no—
139C2×M16C22×C414yes—
140C2×M16#4114yes—
141C2×M16#4114yes—
142C22×C8—22no—
143C22×C8—22no—
144C4×Q8—24no—
145C4×Q8C4⋊C418yes—
146C4×D4—28no—
147C4×D4—28no—
148C4×D4—24no—
149C4×D4C4⋊C418yes—
150#117—21no—
151#117—21no—
152#117—24no—
153#117—24no—
154#117—24no—
155#117—24no—
156#117C22×C412yes—
157#117C2×Q812yes—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—84no—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42C22×C414yes—
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×Q812yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200#107—48no—
201#106—48no—
202C4⋊C8—44no—
203#104—216no—
204#103—44no—
205#102—44no—
206#97—24no—
207C4×C8—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—28no—
213#95—24no—
214#95—24no—
215#95—22no—
216#95—22no—
217#95—88no—
218C24—21no—
219C24C22×D411yescharacteristic
220#41—24no—
221#41—24no—
222#41—24no—
223#41—24no—
224#41—42no—
225#41—42no—
226#41—42no—
227#41—42no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8C2×C4⋊C412yes—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4C2×C4⋊C412yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—28no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4C4×D414yes—
270C22×C4#12012yes—
271C22×C4#12012yes—
272C22×C4C22⋊Q814yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×Q811yescharacteristic
277C22×C4C2×#4112yes—
278C22×C4C2×#4112yes—
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+412yes—
283M16—24no—
284M16—24no—
285M16—24no—
286M16—44no—
287M16—44no—
288M16#11718yes—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—22no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C4⋊C4C2×C4⋊C414yes—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—28no—
313C22⋊C4—44no—
314C22⋊C4—84no—
315C22⋊C4—84no—
316C42—22no—
317C42—22no—
318C42—28no—
319C42—24no—
320C42—24no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—84no—
326C42—84no—
327C42C2×C4⋊C414yes—
328C42C4×D412yes—
329C42C4×D412yes—
330C42C22⋊Q812yes—
331C42C22⋊Q812yes—
332C42#12314yes—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23C2×C4×D411yescharacteristic
341C23C2×#12011yescharacteristic
342C23C2×C22⋊Q811yescharacteristic
343C23#54712yes—
344C23#55012yes—
345Q8—22no—
346Q8—22no—
347Q8—41no—
348Q8—41no—
349D4—22no—
350D4—22no—
351D4—44no—
352D4—44no—
353D4—41no—
354D4—41no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—88no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—88no—
385C2×C4—88no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4#38514yes—
389C2×C4#38514yes—
390C2×C4C22×C4⋊C411yescharacteristic
391C2×C4C2×#11712yes—
392C2×C4C2×C4×D411yescharacteristic
393C2×C4#52012yes—
394C2×C4C2×C22⋊Q811yescharacteristic
395C2×C4C2×#12311yescharacteristic
396C2×C4#53211yescharacteristic
397C2×C4#53211yescharacteristic
398C2×C4#53312yes—
399C2×C4#53511yescharacteristic
400C2×C4#53611yescharacteristic
401C2×C4#54412yes—
402C2×C4#54612yes—
403C2×C4#54812yes—
404C2×C4#55212yes—
405C2×C4#55312yes—
406C2×C4#55512yes—
407C2×C4#55612yes—
408C8—28no—
409C8—82no—
410C22—22no—
411C22—22no—
412C22—24no—
413C22—22no—
414C22—22no—
415C22—42no—
416C22—44no—
417C22—81no—
418C22—81no—
419C22—82no—
420C22C2×#38511yescharacteristic
421C22#235412yes—
422C22#243111yescharacteristic
423C22#243211yescharacteristic
424C22#244112yes—
425C22#302112yes—
426C22#302212yes—
427C4—24no—
428C4—24no—
429C4—42no—
430C4—44no—
431C4—84no—
432C4—82no—
433C4—82no—
434C4—84no—
435C4—162no—
436C4#228511yescharacteristic
437C4#231711yescharacteristic
438C4#235212yes—
439C4#243912yes—
440C4#249911yescharacteristic
441C4#250011yescharacteristic
442C2—22no—
443C2—22no—
444C2—82no—
445C2#1387911yescharacteristic
446C2#2086512yes—
447C2#2561911yescharacteristic
448C2#2562011yescharacteristic
449C2#2569212yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20800C2semidirect
5. #14348C2semidirect
7. #10899C2semidirect
8. #10858C2semidirect
9. #10840C2semidirect
33. #1759C4semidirect

Maximal subgroups

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

subgroupindex
#341892
#208002
#206482
C2×#17592
#143482
#111132
#108992
#108582
#108402