Finite Group Databasebeta

← #496205

Subgroups — #496205

Order 512.

1822 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1822 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35616C211yesmaximal, characteristic
2#20585C212yesmaximal
3#13090C218yesmaximal, complemented
4#12857C212yesmaximal, complemented
5#12698C212yesmaximal, complemented
6C22×H(5,𝔽2)C2211yescharacteristic
7C2×#536C411yescharacteristic, complemented
8C2×#536C2211yescharacteristic
9C22×#124C411yescharacteristic, complemented
10C22×#99—22no—
11C22×#99C2214yes—
12#2435—22no—
13#2435C2214yes—
14#2432—22no—
15#2432C414yescomplemented
16#2405—22no—
17#2405—22no—
18#2309C411yescharacteristic, complemented
19#2308—22no—
20#2297C411yescharacteristic, complemented
21#2297C2211yescharacteristic
22C2×#354C2218yescomplemented
23#1782—22no—
24#1782C2214yes—
25#1781—22no—
26#1781C2214yes—
27#1393C2218yescomplemented
28C2×H(5,𝔽2)—44no—
29C2×H(5,𝔽2)—44no—
30C2×H(5,𝔽2)—44no—
31C22×#41C2×C412yes—
32C22×#41C2×C411yescharacteristic
33C22×#41D412yes—
34C22×#41C2311yescharacteristic
35C23×Q8C2×C411yescharacteristic, complemented
36C23×D4C2×C411yescharacteristic
37C23×D4D412yes—
38C23×D4D412yes—
39C23×D4D412yes—
40C23×D4C2312yes—
41#536—44no—
42#536—44no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yescomplemented
45C2×#124—22no—
46C2×#124—21no—
47C2×#124—44no—
48C2×#124C2×C414yescomplemented
49C2×#124C2312yes—
50C2×#123—22no—
51C2×#123—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2×C414yescomplemented
54C2×#117C2×C411yescharacteristic, complemented
55C2×#117C2×C411yescharacteristic, complemented
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—24no—
59C22×C42C2×C411yescharacteristic, complemented
60C2×#99—24no—
61C2×#99—24no—
62C2×#99—24no—
63C2×#99—48no—
64C2×#99—48no—
65C2×#99C2318yes—
66#394—24no—
67#394—24no—
68#394—24no—
69#394—24no—
70#391—21no—
71#391C2×C412yescomplemented
72#390—22no—
73#390—21no—
74#390C2×C414yescomplemented
75#390C2312yes—
76#386—22no—
77#386—22no—
78#386—48no—
79#385—24no—
80#385—24no—
81#381—24no—
82#379—21no—
83#379—21no—
84#379—24no—
85#379—24no—
86#379—24no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4C2×C414yescomplemented
89#354—48no—
90C25C22⋊C411yescharacteristic
91C25C22⋊C411yescharacteristic
92C25C2×D412yes—
93C25C2×D412yes—
94H(5,𝔽2)—416no—
95H(5,𝔽2)—432no—
96H(5,𝔽2)—416no—
97C2×#41—24no—
98C2×#41—24no—
99C2×#41—42no—
100C2×#41—42no—
101C2×#41—42no—
102C2×#41—44no—
103C2×#41—44no—
104C2×#41—44no—
105C2×#41—44no—
106C2×#41—42no—
107C2×#41—44no—
108C2×#41—44no—
109C22×Q8—21no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8C22⋊C411yescharacteristic
114C22×Q8C22×C412yes—
115C22×Q8C22×C411yescharacteristic
116C22×D4—24no—
117C22×D4—28no—
118C22×D4—28no—
119C22×D4—28no—
120C22×D4—42no—
121C22×D4—42no—
122C22×D4—44no—
123C22×D4—44no—
124C22×D4—44no—
125C22×D4—44no—
126C22×D4—44no—
127C22×D4—44no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4C22⋊C412yes—
132C22×D4C22⋊C412yes—
133C22×D4C22⋊C412yes—
134C22×D4C22⋊C412yes—
135C22×D4C22⋊C411yescharacteristic
136C22×D4C22×C412yes—
137C22×D4C2×D412yes—
138C22×D4C2×D412yes—
139C22×D4C2×D412yes—
140C22×D4C2411yescharacteristic
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C4⋊Q8—44no—
150C4⋊Q8—44no—
151#124—28no—
152#124—48no—
153#124—44no—
154#124—48no—
155#123—48no—
156#123—48no—
157C22⋊Q8—48no—
158C22⋊Q8—48no—
159#117—42no—
160#117—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4C22×C412yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—48no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—42no—
185C2×C42C22×C412yes—
186C2×C42C22×C412yes—
187#99—216no—
188#99—48no—
189#99—48no—
190#99—48no—
191#95—24no—
192#95—24no—
193#95—48no—
194C24—24no—
195C24—24no—
196C24—24no—
197C24—22no—
198C24—24no—
199C24—24no—
200C24—41no—
201C24—41no—
202C24—41no—
203C24—41no—
204C24—42no—
205C24—82no—
206C24—82no—
207C24—84no—
208C24—84no—
209C24#9914yes—
210C24C2×C22⋊C412yes—
211C24C2×C22⋊C412yes—
212C24C2×C22⋊C412yes—
213C24C2×C22⋊C411yescharacteristic
214C24C22×D412yes—
215#41—48no—
216#41—48no—
217#41—48no—
218#41—48no—
219#41—432no—
220#41—432no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—24no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—84no—
234C2×Q8—84no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—48no—
264C2×D4—48no—
265C2×D4—416no—
266C2×D4—416no—
267C2×D4—416no—
268C2×D4—416no—
269C2×D4—416no—
270C2×D4—416no—
271C2×D4—432no—
272C2×D4#9914yes—
273C2×D4H(5,𝔽2)14yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—48no—
298C22×C4—82no—
299C22×C4—88no—
300C22×C4—168no—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C23×C411yescharacteristic
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—48no—
317C22⋊C4—84no—
318C22⋊C4—84no—
319C22⋊C4—84no—
320C22⋊C4—84no—
321C22⋊C4—88no—
322C22⋊C4—168no—
323C42—24no—
324C42—42no—
325C42—42no—
326C42—42no—
327C23—28no—
328C23—28no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—44no—
341C23—44no—
342C23—48no—
343C23—42no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—88no—
349C23—88no—
350C23—84no—
351C23—84no—
352C23—162no—
353C23—162no—
354C23—164no—
355C23—164no—
356C23—164no—
357C23—162no—
358C23—162no—
359C23—164no—
360C23C2×#9914yes—
361C23C2×#9914yes—
362C23C2×#9912yes—
363C23C22×C22⋊C411yescharacteristic
364C23#53414yes—
365Q8—44no—
366Q8—44no—
367Q8—84no—
368Q8—84no—
369D4—44no—
370D4—44no—
371D4—48no—
372D4—48no—
373D4—48no—
374D4—48no—
375D4—48no—
376D4—48no—
377D4—48no—
378D4—48no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—88no—
408C2×C4—816no—
409C2×C4—1616no—
410C2×C4C2×#9912yes—
411C2×C4#51812yes—
412C22—22no—
413C22—28no—
414C22—22no—
415C22—24no—
416C22—24no—
417C22—24no—
418C22—42no—
419C22—44no—
420C22—44no—
421C22—42no—
422C22—44no—
423C22—44no—
424C22—44no—
425C22—42no—
426C22—44no—
427C22—42no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—88no—
434C22—88no—
435C22—164no—
436C22—168no—
437C22—164no—
438C22#230811yescharacteristic
439C22C22×#9911yescharacteristic
440C22C22×#9911yescharacteristic
441C4—22no—
442C4—22no—
443C4—42no—
444C4—42no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—82no—
449C4—168no—
450C2—21no—
451C2—21no—
452C2—44no—
453C2—44no—
454C2—42no—
455C2—44no—
456C2—82no—
457C2#2174611yescharacteristic
458C2#3336112yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13090C2semidirect
4. #12857C2semidirect
5. #12698C2semidirect
7. C2×#536C4semidirect
9. C22×#124C4semidirect
15. #2432C4semidirect
18. #2309C4semidirect
20. #2297C4semidirect
22. C2×#354C22semidirect
27. #1393C22semidirect
35. C23×Q8C2×C4semidirect
44. C2×C4⋊Q8C2×C4semidirect
48. C2×#124C2×C4semidirect
53. C2×C22⋊Q8C2×C4semidirect
54. C2×#117C2×C4semidirect
55. C2×#117C2×C4semidirect
59. C22×C42C2×C4semidirect
71. #391C2×C4semidirect
74. #390C2×C4semidirect
88. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#356162
#205852
#130902
#128572
#126982