Finite Group Databasebeta

← #6479793

Subgroups — #6479793

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C212yesmaximal, complemented
2#33207C211yesmaximal, characteristic, complemented
3#33171C211yesmaximal, characteristic
4#33048C212yesmaximal, complemented
5#29776C214yesmaximal, complemented
6#29437C218yesmaximal, complemented
7#29190C212yesmaximal, complemented
8#29178C212yesmaximal, complemented
9#28654C212yesmaximal, complemented
10#28615C212yesmaximal, complemented
11#22193C214yesmaximal, complemented
12C4×#424C211yesmaximal, characteristic, complemented
13C2×#498C2211yescharacteristic
14C2×#493C2211yescharacteristic
15C2×#486C2212yes—
16C2×#466C2214yes—
17C2×#424C2212yes—
18C2×C4×M16C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2859C2212yes—
21#2858C2212yes—
22#2857C2212yes—
23#2856C2212yes—
24#2644C2214yes—
25#2624C2214yes—
26#2513—22no—
27#2490—22no—
28#2454—22no—
29#2448—22no—
30#2447—22no—
31#2427—22no—
32#2426—22no—
33#2420—22no—
34#2324C2214yes—
35#2319C2214yes—
36C4×C4⋊Q8C2212yes—
37C4×#127C2212yes—
38C4×#117C2211yescharacteristic
39#2086C2218yes—
40#2065C2218yes—
41#2065C2218yes—
42#2054C2218yes—
43#1990C2212yes—
44#1990C2212yes—
45#1970C2214yes—
46#1957C2212yes—
47#1957C2212yes—
48#1941C2214yes—
49#1936C2218yes—
50#1926C2218yes—
51#1918C2218yes—
52#1895C2218yes—
53#1879C2218yes—
54#1864C2218yes—
55#1845C2211yescharacteristic
56#1844C2211yescharacteristic
57#1834C2212yes—
58#1807C2214yes—
59#1768C2214yes—
60C4×C4⋊C8C2212yes—
61C4×C4⋊C8C2212yes—
62#1729C2212yes—
63C22×D8C2314yes—
64C22×M16C2312yes—
65C23×C8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#127C2312yes—
68C2×#125—22no—
69C2×#124—22no—
70C2×#124—22no—
71C2×C22⋊Q8—22no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121C2314yes—
75C2×C4×Q8—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4C2314yes—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#498—42no—
83#493—42no—
84#486—44no—
85#466—216no—
86C2×#107C2314yes—
87C2×#106C2314yes—
88C2×#106C2314yes—
89#424—28no—
90C2×C4⋊C8C2314yes—
91C2×C4⋊C8C2314yes—
92C2×#102C2318yes—
93C2×#102C2318yes—
94C2×#102C2318yes—
95C4×M16—42no—
96C2×#97C2312yes—
97C2×C4×C8—42no—
98C2×C4×C8C2312yes—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2312yes—
102#399—24no—
103#397—24no—
104#397—24no—
105#397—24no—
106#397—24no—
107#396—24no—
108#394—24no—
109#394—24no—
110#391—22no—
111#391—22no—
112#390—24no—
113#390—22no—
114#390—22no—
115#390—22no—
116#390—22no—
117#390—22no—
118#390—22no—
119#390C2314yes—
120#390C2314yes—
121#390C2314yes—
122#390C2314yes—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389C2314yes—
128#389C2314yes—
129#389C2314yes—
130#389C2314yes—
131#388—24no—
132#388—24no—
133#388—24no—
134#386—24no—
135#385—24no—
136#384C2312yes—
137#384C2312yes—
138#382—22no—
139#381—24no—
140#380—22no—
141#380—24no—
142#380C2314yes—
143C4×C4⋊C4—22no—
144C4×C4⋊C4D412yes—
145C4×C4⋊C4D412yes—
146C4×C4⋊C4C2314yes—
147C4×C22⋊C4D412yes—
148C4×C22⋊C4D412yes—
149#376C2312yes—
150C43C2312yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C22×Q8—22no—
157C22×D4—24no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4C2414yes—
161C23×C4—24no—
162C23×C4C2×D412yes—
163C23×C4C2411yescharacteristic
164C2×D8—28no—
165C2×D8—44no—
166C2×D8—44no—
167C2×M16—24no—
168C2×M16—42no—
169C2×M16—42no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8#4118yes—
173C22×C8C2414yes—
174C22×C8C2412yes—
175C22×C8C2412yes—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178#127—24no—
179#127—24no—
180#125—84no—
181#124—84no—
182#124—84no—
183C22⋊Q8—84no—
184#121—216no—
185#121—84no—
186#121—84no—
187C4×Q8—84no—
188C4×D4—48no—
189C4×D4—84no—
190C4×D4—84no—
191#117—24no—
192#117—42no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2414yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4C2×D412yes—
220C2×C22⋊C4C2×D411yescharacteristic
221C2×C22⋊C4C2×D411yescharacteristic
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2412yes—
231C2×C42C2412yes—
232C2×C42C2411yescharacteristic
233C2×C42C2411yescharacteristic
234#107—28no—
235#106—44no—
236#106—44no—
237C4⋊C8—28no—
238C4⋊C8—28no—
239#102—216no—
240#102—48no—
241#102—48no—
242#97—42no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—24no—
246C4×C8—42no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95C2×D412yes—
254#95C2×D412yes—
255#95C2×D412yes—
256#95C2×D412yes—
257C24—24no—
258C24C22×D411yescharacteristic
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×D4—28no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—28no—
266C2×D4—48no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—84no—
274C2×D4—84no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—48no—
285C22×C4—84no—
286C22×C4—84no—
287C22×C4C22×D412yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D412yes—
294C22×C4C2×#4114yes—
295C22×C4C2511yescharacteristic
296D8—88no—
297M16—48no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—48no—
306C2×C8C2×#4114yes—
307C2×C8C2×#4114yes—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C22⋊C4—24no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—84no—
330C22⋊C4—84no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C22⋊C4—84no—
335C22⋊C4—84no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—21no—
345C42—21no—
346C42—24no—
347C42—84no—
348C42H(5,𝔽2)12yes—
349C422−1+412yes—
350C23—21no—
351C23—21no—
352C23—22no—
353C23—22no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23#53212yes—
362C23C23×D411yescharacteristic
363Q8—88no—
364D4—48no—
365D4—48no—
366D4—88no—
367D4—88no—
368D4—168no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—168no—
399C2×C4#53212yes—
400C2×C4#53714yes—
401C2×C4#57614yes—
402C2×C4#57714yes—
403C2×C4C22×#4112yes—
404C2×C4C2×H(5,𝔽2)11yescharacteristic
405C2×C4C2×2−1+411yescharacteristic
406C2×C4#58512yes—
407C8—24no—
408C8—24no—
409C8—44no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—42no—
414C22—42no—
415C22—84no—
416C22—84no—
417C22—84no—
418C22—84no—
419C22—84no—
420C22—84no—
421C22—84no—
422C22C2×#53211yescharacteristic
423C22C2×#53711yescharacteristic
424C22#344911yescharacteristic
425C22C2×#57612yes—
426C22C2×#57712yes—
427C4—22no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—84no—
434C4—84no—
435C4—84no—
436C4#325718yes—
437C4#350512yes—
438C4#350712yes—
439C2—42no—
440C2—164no—
441C2#3510311yescharacteristic
442C2#6071411yescharacteristic
443C2#6071611yescharacteristic
444C2C2×#325712yes—
445C2#6100612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#33171C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35288C2semidirect
2. #33207C2semidirect
4. #33048C2semidirect
5. #29776C2semidirect
6. #29437C2semidirect
7. #29190C2semidirect
8. #29178C2semidirect
9. #28654C2semidirect
10. #28615C2semidirect
11. #22193C2semidirect
12. C4×#424C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#332072
#331712
#330482
#297762
#294372
#291902
#291782
#286542
#286152
#221932
C4×#4242