Finite Group Databasebeta

← #496204

Subgroups — #496204

Order 512.

1732 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1732 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#35676C211yesmaximal, characteristic
2#20587C212yesmaximal, complemented
3#13089C218yesmaximal, complemented
4#12857C212yesmaximal, complemented
5#12697C212yesmaximal, complemented
6C22×2−1+4C2211yescharacteristic
7C2×#535C411yescharacteristic, complemented
8C2×#535C2211yescharacteristic
9C22×#124C411yescharacteristic, complemented
10C2×#410—22no—
11C2×#410C2214yescomplemented
12#2436—22no—
13#2436C414yescomplemented
14#2431—22no—
15#2431C2214yes—
16#2404—22no—
17#2404—22no—
18#2310—22no—
19#2310C411yescharacteristic, complemented
20#2297C411yescharacteristic, complemented
21#2297C2211yescharacteristic
22C2×#353C2218yes—
23#1782—22no—
24#1782—22no—
25#1782C2214yes—
26#1782C2214yescomplemented
27#1393C2218yescomplemented
28C2×2−1+4—44no—
29C2×2−1+4—44no—
30C2×2−1+4—44no—
31C22×#41C2×C411yescharacteristic
32C22×#41D412yes—
33C22×#41D412yes—
34C22×#41D412yes—
35C22×#41C2312yes—
36C22×#41C2311yescharacteristic
37C23×Q8C2×C411yescharacteristic
38C23×Q8C2×C412yes—
39C23×Q8D412yes—
40C23×D4C2×C411yescharacteristic
41#535—44no—
42#535—44no—
43C2×#127—21no—
44C2×#127C2312yes—
45C2×#124—22no—
46C2×#124—21no—
47C2×#124—44no—
48C2×#124C2×C414yes—
49C2×#124C2×C412yes—
50C2×#121—22no—
51C2×#121C2×C414yes—
52C2×#120—22no—
53C2×#120—22no—
54C2×#117—24no—
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C42C2×C411yescharacteristic
60#410—48no—
61#410—48no—
62C2×#99—24no—
63C2×#99—24no—
64C2×#99—24no—
65C2×#99C2318yes—
66#393—24no—
67#393—24no—
68#392—24no—
69#392—24no—
70#391—22no—
71#391—21no—
72#391—48no—
73#391C2×C414yes—
74#391C2×C412yes—
75#390—21no—
76#390C2312yes—
77#386—22no—
78#386—22no—
79#386—24no—
80#386—24no—
81#386—24no—
82#386—24no—
83#381—21no—
84#381—21no—
85#381—24no—
86#381—24no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4C2×C414yes—
89#353—48no—
90C25—21no—
912−1+4—416no—
922−1+4—432no—
932−1+4—416no—
94C2×#41—24no—
95C2×#41—24no—
96C2×#41—28no—
97C2×#41—28no—
98C2×#41—42no—
99C2×#41—44no—
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—
109C2×#41—44no—
110C2×#41—44no—
111C2×#41—44no—
112C22×Q8—28no—
113C22×Q8—24no—
114C22×Q8—44no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—44no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8C22⋊C412yes—
122C22×Q8C22⋊C412yes—
123C22×Q8C22⋊C412yes—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C412yes—
126C22×Q8C22×C411yescharacteristic
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—21no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4C22⋊C412yes—
134C22×D4C22⋊C411yescharacteristic
135C22×D4C22×C412yes—
136C22×D4C2×D412yes—
137C22×D4C2×D412yes—
138C22×D4C2×D412yes—
139C22×D4C2411yescharacteristic
140C23×C4—21no—
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150#127—28no—
151#127—44no—
152#124—44no—
153#124—48no—
154#124—44no—
155#124—48no—
156#121—48no—
157#121—48no—
158#120—48no—
159#120—48no—
160#117—42no—
161#117—42no—
162#117—48no—
163#117—48no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—42no—
181C2×C42—44no—
182C2×C42C22×C412yes—
183C2×C42C22×C412yes—
184#99—216no—
185#99—48no—
186#99—48no—
187#99—48no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—48no—
193#95—48no—
194C24—22no—
195C24—42no—
196C24—42no—
197C24—42no—
198C24—84no—
199C24C2×C22⋊C412yes—
200C24C2×C22⋊C411yescharacteristic
201C24C22×D412yes—
202#41—416no—
203#41—416no—
204#41—416no—
205#41—416no—
206#41—48no—
207#41—48no—
208#41—48no—
209#41—48no—
210#41—432no—
211#41—432no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—41no—
229C2×Q8—416no—
230C2×Q8—416no—
231C2×Q8—48no—
232C2×Q8—48no—
233C2×Q8—432no—
234C2×Q8#9914yes—
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—22no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—42no—
260C2×D4—84no—
261C2×D4—84no—
262C2×D4—88no—
263C2×D4—88no—
264C2×D4H(5,𝔽2)14yes—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—48no—
296C22×C4—82no—
297C22×C4#9914yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C23×C411yescharacteristic
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—88no—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—84no—
316C22⋊C4—84no—
317C42—24no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—88no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—44no—
331C23—44no—
332C23—48no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—84no—
337C23—88no—
338C23—164no—
339C23—164no—
340C23C2×#9912yes—
341C23C22×C22⋊C411yescharacteristic
342C23#53414yes—
343Q8—44no—
344Q8—48no—
345Q8—48no—
346Q8—48no—
347Q8—44no—
348Q8—48no—
349D4—44no—
350D4—48no—
351D4—48no—
352D4—44no—
353D4—48no—
354D4—48no—
355D4—84no—
356D4—84no—
357D4—168no—
358D4—168no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—48no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—84no—
392C2×C4—88no—
393C2×C4—816no—
394C2×C4C2×#9914yes—
395C2×C4C2×#9912yes—
396C2×C4C2×#9914yes—
397C2×C4#51812yes—
398C22—22no—
399C22—24no—
400C22—22no—
401C22—42no—
402C22—44no—
403C22—42no—
404C22—44no—
405C22—42no—
406C22—44no—
407C22—42no—
408C22—82no—
409C22—82no—
410C22—82no—
411C22—84no—
412C22—84no—
413C22—84no—
414C22—82no—
415C22—168no—
416C22#230811yescharacteristic
417C22C22×#9911yescharacteristic
418C22C22×#9911yescharacteristic
419C4—22no—
420C4—22no—
421C4—24no—
422C4—24no—
423C4—42no—
424C4—42no—
425C4—82no—
426C4—82no—
427C4—82no—
428C4—168no—
429C2—21no—
430C2—21no—
431C2—44no—
432C2—42no—
433C2—82no—
434C2—162no—
435C2#2174611yescharacteristic
436C2#3336212yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radical#35676C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C22×2−1+4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20587C2semidirect
3. #13089C2semidirect
4. #12857C2semidirect
5. #12697C2semidirect
7. C2×#535C4semidirect
9. C22×#124C4semidirect
11. C2×#410C22semidirect
13. #2436C4semidirect
19. #2310C4semidirect
20. #2297C4semidirect
26. #1782C22semidirect
27. #1393C22semidirect

Maximal subgroups

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

subgroupindex
#356762
#205872
#130892
#128572
#126972