Finite Group Databasebeta

← #196447

Subgroups — #196447

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2975C211yesmaximal, characteristic
2#35172C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic, complemented
4C2×#1503C211yesmaximal, characteristic, complemented
5#13941C211yesmaximal, characteristic
6#9115C218yesmaximal, complemented
7#8026C211yesmaximal, characteristic, complemented
8#8014C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2975—24no—
12#2975—24no—
13C2×#436—22no—
14C2×#436—22no—
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#407C2214yes—
18#2848C2214yes—
19#2626—22no—
20#2580—22no—
21#2566—22no—
22#2563C412yescomplemented
23#2542C412yescomplemented
24#2538C414yescomplemented
25#2317—21no—
26#2309—21no—
27C2×#390—21no—
28#2285C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1518—24no—
32#1503—24no—
33C2×#328C2214yes—
34C2×#328C2214yes—
35#1439—22no—
36C4⋊#95—21no—
37C22⋊#95C2214yes—
38#1294C2214yes—
39#1294C2214yes—
40C23×D4D411yescharacteristic
41C24×C4C2312yes—
42C22×M16—24no—
43#533—22no—
44#533—22no—
45#533—22no—
46#533—22no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2×C418yes—
52C2×C22⋊Q8C2312yes—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C412yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C412yes—
61C2×C4×D4D412yes—
62C2×C4×D4D412yes—
63C2×#117—21no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4—24no—
69C22×C22⋊C4D412yes—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#436—216no—
73#436—216no—
74#431—22no—
75#431—22no—
76#431D414yes—
77#431D414yes—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80#407—28no—
81#407—28no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#399—24no—
88#396—28no—
89#393—24no—
90#390—22no—
91#390—22no—
92#390—24no—
93#390—44no—
94#389—24no—
95#385—28no—
96#385—24no—
97#384—22no—
98#384C2×C412yes—
99#382—22no—
100#382—21no—
101#382—22no—
102#382—22no—
103#380—21no—
104#380C2×C418yes—
105#380C2312yes—
106#379—24no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111C2×#95—21no—
112C2×#95—24no—
113C2×#95C2312yes—
114C2×#95C2312yes—
115#328—28no—
116#328—28no—
117C25C2×D412yes—
118C22×Q8—21no—
119C22×Q8C22×C412yes—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C22⋊C412yes—
127C22×D4C22⋊C412yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—84no—
135C23×C4C22×C414yes—
136C23×C4C2×D412yes—
137C23×C4C2×D414yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—216no—
142C2×M16—88no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149#125—42no—
150#125—42no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C22⋊Q8—88no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170#117—42no—
171#117—42no—
172#117—42no—
173#117—42no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C414yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—48no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—84no—
202C2×C22⋊C4C22⋊C414yes—
203C2×C22⋊C4C22⋊C414yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4112yes—
216C2×C42#4112yes—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C22⋊C8—216no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8C2×D414yes—
223C22⋊C8C2×D414yes—
224C4×C8—24no—
225C4×C8—24no—
226#95—22no—
227#95—28no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—42no—
234#95—84no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—44no—
242C24—44no—
243C24C2×C22⋊C414yes—
244C24C22×D411yescharacteristic
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—44no—
248C2×Q8—42no—
249C2×Q8—82no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—48no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—84no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4C2×C22⋊C414yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4AGL(1,ℤ/8)12yes—
300C22×C4#13712yes—
301C22×C4#13712yes—
302C22×C4#13712yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309M16—816no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—44no—
313C2×C8—44no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—44no—
319C4⋊C4—48no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—82no—
328C4⋊C4—88no—
329C4⋊C4—84no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—88no—
337C22⋊C4—88no—
338C22⋊C4—816no—
339C42—44no—
340C42—44no—
341C42—84no—
342C42C4×D412yes—
343C42C4×D412yes—
344C42C4×D412yes—
345C42C4×D412yes—
346C42#12012yes—
347C42#12012yes—
348C42#12312yes—
349C42#12312yes—
350C23—22no—
351C23—22no—
352C23—28no—
353C23—24no—
354C23—24no—
355C23—28no—
356C23—24no—
357C23—48no—
358C23—48no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—84no—
368C23—84no—
369C23C22×C22⋊C411yescharacteristic
370C23C2×#12111yescharacteristic
371C23C2×#12111yescharacteristic
372C23C2×AGL(1,ℤ/8)11yescharacteristic
373C23C2×#13711yescharacteristic
374C23C2×#13711yescharacteristic
375C23C2×#13711yescharacteristic
376Q8—82no—
377Q8—82no—
378Q8—84no—
379D4—48no—
380D4—48no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—28no—
386C2×C4—24no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—84no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4#38612yes—
419C2×C4#38612yes—
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4#41712yes—
423C2×C4#41712yes—
424C2×C4#41912yes—
425C2×C4#41912yes—
426C2×C4#46814yes—
427C2×C4#46814yes—
428C2×C4#47014yes—
429C2×C4#47014yes—
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×#12011yescharacteristic
433C2×C4C2×#12311yescharacteristic
434C8—44no—
435C8—44no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—21no—
443C22—21no—
444C22—44no—
445C22—48no—
446C22—48no—
447C22—44no—
448C22—44no—
449C22—84no—
450C22—84no—
451C22—84no—
452C22—84no—
453C22C2×#38611yescharacteristic
454C22#288312yes—
455C22C2×#41911yescharacteristic
456C22C2×#41711yescharacteristic
457C22#290112yes—
458C22C2×#46811yescharacteristic
459C22C2×#46811yescharacteristic
460C22C2×#47011yescharacteristic
461C22C2×#47011yescharacteristic
462C4—22no—
463C4—22no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—44no—
469C4—84no—
470C4—82no—
471C4—84no—
472C4—82no—
473C4—162no—
474C4#189414yes—
475C4#189414yes—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—84no—
481C2#1762712yes—
482C2#1763012yes—
483C2#2090311yescharacteristic
484C2C2×#189411yescharacteristic
485C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1518C2semidirect
4. C2×#1503C2semidirect
6. #9115C2semidirect
7. #8026C2semidirect
8. #8014C2semidirect
22. #2563C4semidirect
23. #2542C4semidirect
24. #2538C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29752
#351722
C2×#15182
C2×#15032
#139412
#91152
#80262
#80142