Finite Group Databasebeta

← #496307

Subgroups — #496307

Order 512.

1450 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1450 conjugacy classes of subgroups — too many to draw. Only its 202 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
1C2×#2544C211yesmaximal, characteristic
2#20722C212yesmaximal, complemented
3#13225C218yesmaximal, complemented
4#12694C212yesmaximal, complemented
5#12693C212yesmaximal
6C22×#127C2211yescharacteristic
7C22×#124C411yescharacteristic, complemented
8C22×#124C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10#2544—22no—
11#2544—22no—
12#2544C414yescomplemented
13#2544C2214yes—
14C2×#394—22no—
15C2×#391C411yescharacteristic, complemented
16C2×#390C411yescharacteristic, complemented
17C2×#386—22no—
18#2280—22no—
19C2×C4×C4⋊C4C411yescharacteristic, complemented
20C2×#354C2218yescomplemented
21C2×#353C2218yes—
22#1819—22no—
23#1799—22no—
24#1799C2214yescomplemented
25#1781C2214yes—
26C4×#99—22no—
27C4×#99C2214yescomplemented
28C23×Q8C2×C411yescharacteristic
29C23×D4—22no—
30C23×D4C2×C411yescharacteristic
31C23×D4C2311yescharacteristic
32C24×C4D412yes—
33C2×#127—22no—
34C2×#127—44no—
35C2×#127C2×C412yes—
36C2×#127C2312yes—
37C2×#124—22no—
38C2×#124—22no—
39C2×#124—42no—
40C2×#124—41no—
41C2×#124—41no—
42C2×#124—44no—
43C2×#124C2×C414yes—
44C2×#124C2×C412yes—
45C2×#124C2312yes—
46C2×C4×D4—42no—
47C2×C4×D4—41no—
48C2×C4×D4—41no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4D412yes—
51C2×C4×D4D412yes—
52C2×C4×D4C2312yes—
53C2×#117—42no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D412yes—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C2×#99—24no—
65C2×#99C2318yes—
66#394—24no—
67#394—24no—
68#394—24no—
69#394—24no—
70#391—21no—
71#391—21no—
72#391C2×C414yes—
73#390—21no—
74#390—21no—
75#390C2×C414yes—
76#386—24no—
77#386—24no—
78#386—24no—
79#386—24no—
80#381—42no—
81#380—24no—
82#380—42no—
83#379—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2×C414yes—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—42no—
88C2×#95—22no—
89C2×#95—21no—
90#354—48no—
91#353—48no—
92#342—24no—
93#342—24no—
94C25—21no—
95C25C2×D412yes—
96C22×Q8—21no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8C22×C412yes—
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—24no—
105C22×D4—21no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4—41no—
109C22×D4—41no—
110C22×D4—41no—
111C22×D4—41no—
112C22×D4—48no—
113C22×D4—42no—
114C22×D4—48no—
115C22×D4—84no—
116C22×D4C22×C412yes—
117C22×D4C22×C411yescharacteristic
118C22×D4C2×D412yes—
119C22×D4C2411yescharacteristic
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—84no—
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136#127—28no—
137#127—44no—
138#127—48no—
139#124—28no—
140#124—44no—
141#124—44no—
142#124—44no—
143#124—44no—
144#124—44no—
145#124—48no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154#117—88no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22×C412yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—84no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C2×D412yes—
184C2×C22⋊C4C2×D412yes—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—44no—
193C2×C42C22⋊C412yes—
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196C2×C42C22×C412yes—
197#99—216no—
198#99—48no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—44no—
203#95—44no—
204C24—24no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—42no—
209C24—44no—
210C24—44no—
211C24—44no—
212C24—42no—
213C24—42no—
214C24—44no—
215C24—41no—
216C24—41no—
217C24C2×C22⋊C411yescharacteristic
218C24C2×C22⋊C412yes—
219C24C22×D412yes—
220C2×Q8—24no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—84no—
227C2×Q8—84no—
228C2×D4—24no—
229C2×D4—22no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—48no—
238C2×D4—48no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—48no—
244C2×D4—48no—
245C2×D4—48no—
246C2×D4—48no—
247C2×D4—42no—
248C2×D4—84no—
249C2×D4—84no—
250C2×D4—84no—
251C2×D4—84no—
252C2×D4—832no—
253C2×D4H(5,𝔽2)14yes—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—48no—
288C22×C4—84no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4#9912yes—
296C22×C4#9911yescharacteristic
297C22×C4#9911yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C23×C411yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—48no—
308C4⋊C4—88no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—84no—
321C22⋊C4—84no—
322C22⋊C4—88no—
323C22⋊C4—88no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—24no—
329C42—24no—
330C42—42no—
331C42—42no—
332C42—88no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—44no—
343C23—44no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—416no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—82no—
357C23—82no—
358C23—88no—
359C23—88no—
360C23—82no—
361C23—82no—
362C23—88no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23C2×#9911yescharacteristic
368C23C2×#9911yescharacteristic
369C23C22×C22⋊C411yescharacteristic
370C23#53514yes—
371Q8—84no—
372Q8—84no—
373D4—44no—
374D4—48no—
375D4—44no—
376D4—84no—
377D4—84no—
378D4—816no—
379D4—816no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—84no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—88no—
422C2×C4—88no—
423C2×C4—88no—
424C2×C4C2×#9912yes—
425C2×C4C2×#9912yes—
426C2×C4#41012yes—
427C2×C4#51812yes—
428C22—24no—
429C22—24no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—24no—
437C22—44no—
438C22—44no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—88no—
446C22—88no—
447C22—82no—
448C22—84no—
449C22—84no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—164no—
454C22—164no—
455C22#212612yes—
456C22#212712yes—
457C22#230911yescharacteristic
458C22C22×#9911yescharacteristic
459C22C2×#41011yescharacteristic
460C4—22no—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—44no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—82no—
469C4—82no—
470C4—168no—
471C2—21no—
472C2—21no—
473C2—22no—
474C2—22no—
475C2—44no—
476C2—82no—
477C2—162no—
478C2#2179311yescharacteristic
479C2C2×#212611yescharacteristic
480C2C2×#212711yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20722C2semidirect
3. #13225C2semidirect
4. #12694C2semidirect
7. C22×#124C4semidirect
12. #2544C4semidirect
15. C2×#391C4semidirect
16. C2×#390C4semidirect
19. C2×C4×C4⋊C4C4semidirect
20. C2×#354C22semidirect
24. #1799C22semidirect
27. C4×#99C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25442
#207222
#132252
#126942
#126932