Finite Group Databasebeta

← #6342583

Subgroups — #6342583

Order 512.

3823 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×2−1+4C211yesmaximal, characteristic
2#60248C212yesmaximal, complemented
3#17372C216yesmaximal, complemented
4#17371C216yesmaximal, complemented
5#17366C216yesmaximal, complemented
6#17361C216yesmaximal, complemented
7#17353C212yesmaximal, complemented
8#17352C212yesmaximal, complemented
9C22×2−1+4C411yescharacteristic
10C4×2−1+4—28no—
11C4×2−1+4—28no—
12C2×#520C413yes—
13C2×#520C413yes—
14C2×#520C2211yescharacteristic
15C2×#520C2213yes—
16C2×#519C411yescharacteristic
17C2×#519C2211yescharacteristic
18C2×#519C2213yes—
19C2×C4×#41C413yes—
20C2×C4×#41C411yescharacteristic
21C2×C4×#41C2213yes—
22C2×C4×#41C2213yes—
23C22×C4×Q8C413yes—
24C22×C4×Q8C411yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26#2984—28no—
27#2983C22112yes—
28#2982C2214yes—
29Q8×M16—28no—
30#2920C22112yes—
31#1467C22112yes—
32#1466C22112yes—
33#1465C2214yes—
34#1465C22112yes—
35#1464C22124yes—
36#1461C22112yes—
37#1460C22112yes—
38#1458C22112yes—
39#1458C22112yes—
40C2×2−1+4—28no—
41C2×2−1+4—28no—
42C2×2−1+4—28no—
43C22×#41C2×C413yes—
44C22×#41C2×C413yes—
45C22×#41C2×C413yes—
46C22×#41C2×C411yescharacteristic
47C23×Q8C2×C413yes—
48C23×Q8C2×C411yescharacteristic
49C23×Q8C2×C411yescharacteristic
50#520—28no—
51#520—224no—
52#520—224no—
53#520—224no—
54#519—28no—
55#519—28no—
56#519—224no—
57C4×#41—212no—
58C4×#41—212no—
59C4×#41—24no—
60C4×#41—24no—
61C4×#41—212no—
62C4×#41—212no—
63C4×#41—212no—
64C4×#41—212no—
65C2×C4×Q8—212no—
66C2×C4×Q8—212no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8C2×C413yes—
70C2×C4×Q8C2×C413yes—
71C2×C4×Q8C2×C413yes—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C416yes—
74C2×C4×Q8C2×C413yes—
75C2×C4×Q8C2×C413yes—
76C2×C4×Q8C2×C413yes—
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8C2×C413yes—
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×Q8D418yes—
81C2×C4×Q8D418yes—
82C2×C4×Q8C2313yes—
83C2×C4×Q8C2313yes—
84C2×C4×Q8C2311yescharacteristic
85C2×C4×Q8C2313yes—
86C2×C4×D4C2×C413yes—
87C2×C4×D4C2×C413yes—
88C2×C4×D4C2×C416yes—
89C2×C4×D4C2×C413yes—
90C2×C4×D4C2×C413yes—
91C2×C4×D4C2×C413yes—
92C2×C4×D4C2×C413yes—
93C2×C4×D4C2×C416yes—
94C2×C4×D4C2311yescharacteristic
95C2×C4×D4C2313yes—
96C2×C4×D4C2313yes—
97C2×C4×D4C2313yes—
98C2×#117C2×C416yes—
99C2×#117C2×C413yes—
100C2×#117C2×C413yes—
101C2×#117C2×C413yes—
102C2×#117C2×C413yes—
103C2×#117C2313yes—
104C2×#117C2313yes—
105C2×#117C2316yes—
106C22×C4⋊C4C2×C416yes—
107C22×C4⋊C4C2×C413yes—
108C22×C4⋊C4C2×C413yes—
109C22×C4⋊C4C2×C413yes—
110C22×C42C2×C416yes—
111C22×C42C2×C413yes—
112C22×C42C2×C413yes—
113C22×C42C2313yes—
114#446—248no—
115C8×Q8—216no—
116#432—212no—
117#432C23124yes—
118#431—212no—
119#431C23124yes—
120#423—212no—
121#407C2318yes—
122C4×M16—212no—
123#324C2318yes—
124#324C23124yes—
125#323C2318yes—
126#323C23124yes—
1272−1+4—2128no—
128C2×#41—212no—
129C2×#41—24no—
130C2×#41—212no—
131C2×#41—24no—
132C2×#41—212no—
133C2×#41—212no—
134C2×#41—212no—
135C2×#41—212no—
136C2×#41—212no—
137C2×#41—212no—
138C2×#41—212no—
139C2×#41M1618yescomplemented
140C22×Q8—212no—
141C22×Q8—24no—
142C22×Q8—212no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—212no—
147C22×Q8C22⋊C418yes—
148C22×Q8M1618yescomplemented
149C22×Q8C22×C413yes—
150C22×Q8C22×C413yes—
151C22×Q8C22×C413yes—
152C22×Q8C22×C411yescharacteristic
153C22×D4C22×C413yes—
154C22×D4C22×C413yes—
155C22×D4C22×C411yescharacteristic
156C22×D4C22×C413yes—
157C23×C4C22×C416yes—
158C23×C4C22×C413yes—
159C23×C4C22×C413yes—
160C23×C4C22×C413yes—
161C23×C4C22×C413yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C416yes—
164C23×C4C22×C413yes—
165C23×C4C22×C413yes—
166C23×C4C22×C413yes—
167C23×C4C2411yescharacteristic
168C2×M16—28no—
169C4×Q8—212no—
170C4×Q8—24no—
171C4×Q8—212no—
172C4×Q8—24no—
173C4×Q8—224no—
174C4×Q8—212no—
175C4×Q8—212no—
176C4×Q8—26no—
177C4×Q8—26no—
178C4×Q8—26no—
179C4×Q8—26no—
180C4×Q8—212no—
181C4×Q8—212no—
182C4×Q8C22⋊C4112yes—
183C4×Q8C22⋊C4112yes—
184C4×Q8C22⋊C414yes—
185C4×Q8C22⋊C414yes—
186C4×Q8C2×D418yes—
187C4×Q8C2×D4124yes—
188C4×D4—212no—
189C4×D4—212no—
190C4×D4—224no—
191C4×D4—22no—
192C4×D4—26no—
193C4×D4—22no—
194C4×D4—26no—
195C4×D4—224no—
196C4×D4—212no—
197C4×D4—212no—
198C4×D4—26no—
199C4×D4—26no—
200C4×D4—26no—
201C4×D4—26no—
202C4×D4—212no—
203C4×D4—212no—
204#117—212no—
205#117—212no—
206#117—224no—
207#117—212no—
208#117—212no—
209#117—212no—
210#117—212no—
211#117—224no—
212C2×C4⋊C4—224no—
213C2×C4⋊C4—212no—
214C2×C4⋊C4—212no—
215C2×C4⋊C4C22⋊C4124yes—
216C2×C4⋊C4C22×C413yes—
217C2×C4⋊C4C22×C413yes—
218C2×C4⋊C4C22×C413yes—
219C2×C4⋊C4C22×C413yes—
220C2×C4⋊C4C22×C416yes—
221C2×C4⋊C4C22×C413yes—
222C2×C4⋊C4C22×C413yes—
223C2×C4⋊C4C22×C416yes—
224C2×C4⋊C4C22×C413yes—
225C2×C4⋊C4C22×C413yes—
226C2×C4⋊C4C2411yescharacteristic
227C2×C4⋊C4C2413yes—
228C2×C22⋊C4C22×C411yescharacteristic
229C2×C22⋊C4C22×C413yes—
230C2×C22⋊C4C22×C413yes—
231C2×C22⋊C4C22×C413yes—
232C2×C22⋊C4C22×C416yes—
233C2×C22⋊C4C2411yescharacteristic
234C2×C22⋊C4C2413yes—
235C2×C42—212no—
236C2×C42—26no—
237C2×C42—26no—
238C2×C42—212no—
239C2×C42—26no—
240C2×C42—26no—
241C2×C42C22×C416yes—
242C2×C42C22×C413yes—
243C2×C42C22×C413yes—
244C2×C42C22×C416yes—
245C2×C42C22×C413yes—
246C2×C42C22×C413yes—
247C2×C42C22×C413yes—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C413yes—
250C2×C42C22×C413yes—
251C2×C42C2×D4112yes—
252C2×C42C2×D4112yes—
253C2×C42C2413yes—
254C2×C42C2413yes—
255C4⋊C8—224no—
256C4⋊C8—224no—
257C22⋊C8—28no—
258C22⋊C8C24116yes—
259#97—224no—
260C4×C8—224no—
261C24C23×C411yescharacteristic
262C24C23×C413yes—
263C24C23×C411yescharacteristic
264#41—296no—
265#41—296no—
266#41—296no—
267#41—232no—
268C2×Q8—296no—
269C2×Q8—232no—
270C2×Q8—232no—
271C2×Q8—26no—
272C2×Q8—26no—
273C2×Q8—26no—
274C2×Q8—26no—
275C2×Q8—26no—
276C2×Q8—26no—
277C2×Q8—26no—
278C2×Q8C2×C22⋊C4112yes—
279C2×Q8C2×C22⋊C414yes—
280C2×Q8C2×M16112yes—
281C2×Q8C2×M1614yes—
282C2×Q8C2×M1614yes—
283C2×D4—26no—
284C2×D4—22no—
285C2×D4—26no—
286C2×D4—26no—
287C2×D4—26no—
288C2×D4—22no—
289C2×D4—26no—
290C2×D4—26no—
291C2×D4—26no—
292C2×D4—22no—
293C2×D4—26no—
294C2×D4C2×M16112yes—
295C22×C4—212no—
296C22×C4—212no—
297C22×C4—26no—
298C22×C4—26no—
299C22×C4—212no—
300C22×C4—26no—
301C22×C4—26no—
302C22×C4—26no—
303C22×C4—26no—
304C22×C4—23no—
305C22×C4—21no—
306C22×C4—23no—
307C22×C4—21no—
308C22×C4—212no—
309C22×C4—26no—
310C22×C4—26no—
311C22×C4—42no—
312C22×C4—46no—
313C22×C4—42no—
314C22×C4—83no—
315C22×C4—86no—
316C22×C4—83no—
317C22×C4—83no—
318C22×C4C2×C22⋊C4112yes—
319C22×C4C2×C22⋊C4112yes—
320C22×C4C2×M16112yes—
321C22×C4C2×M16112yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C23×C413yes—
324C22×C4C23×C413yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C23×C413yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C23×C413yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C23×C413yes—
331C22×C4C23×C413yes—
332C22×C4C23×C413yes—
333C22×C4C22×D412yes—
334C22×C4C22×D412yes—
335C22×C4C2511yescharacteristic
336M16—416no—
337C2×C8—216no—
338C4⋊C4—26no—
339C4⋊C4—26no—
340C4⋊C4—212no—
341C4⋊C4—26no—
342C4⋊C4—26no—
343C4⋊C4—26no—
344C4⋊C4—26no—
345C4⋊C4—22no—
346C4⋊C4—26no—
347C4⋊C4C2×C22⋊C4124yes—
348C4⋊C4C2×C22⋊C4112yes—
349C4⋊C4C2×C22⋊C4112yes—
350C22⋊C4—26no—
351C22⋊C4—22no—
352C22⋊C4—26no—
353C22⋊C4—26no—
354C22⋊C4—212no—
355C22⋊C4—22no—
356C22⋊C4—26no—
357C42—23no—
358C42—21no—
359C42—23no—
360C42—21no—
361C42—212no—
362C42—26no—
363C42—26no—
364C42—212no—
365C42—26no—
366C42—26no—
367C42C2×C22⋊C416yes—
368C42C2×C22⋊C416yes—
369C42C2×C22⋊C416yes—
370C42C2×C22⋊C416yes—
371C42C22×D4112yes—
372C42C22×D4112yes—
373C23—23no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—23no—
379C23—41no—
380C23—41no—
381C23—43no—
382C23—41no—
383C23—43no—
384C23—41no—
385C23C22×C22⋊C412yes—
386C23C22×M1616yes—
387C23C22×M1612yes—
388C23C24×C411yescharacteristic
389Q8—224no—
390Q8—224no—
391Q8—224no—
392Q8#407116yes—
393D4—224no—
394D4—224no—
395D4—28no—
396D4—224no—
397C2×C4—248no—
398C2×C4—23no—
399C2×C4—21no—
400C2×C4—23no—
401C2×C4—23no—
402C2×C4—23no—
403C2×C4—21no—
404C2×C4—224no—
405C2×C4—224no—
406C2×C4—224no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—23no—
410C2×C4—21no—
411C2×C4—23no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—43no—
416C2×C4—43no—
417C2×C4—412no—
418C2×C4—46no—
419C2×C4—42no—
420C2×C4—46no—
421C2×C4—42no—
422C2×C4—43no—
423C2×C4—41no—
424C2×C4—43no—
425C2×C4—41no—
426C2×C4—812no—
427C2×C4—86no—
428C2×C4—86no—
429C2×C4C22×C22⋊C411yescharacteristic
430C2×C4C22×C22⋊C416yes—
431C2×C4C22×C22⋊C416yes—
432C2×C4C22×C22⋊C416yes—
433C2×C4C22×C22⋊C416yes—
434C2×C4C22×C22⋊C411yescharacteristic
435C2×C4C22×M1616yes—
436C2×C4C22×M1612yes—
437C2×C4C22×M1616yes—
438C2×C4C22×M1616yes—
439C2×C4C23×D412yes—
440C8—416no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—26no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—46no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—46no—
454C22—43no—
455C22—41no—
456C22—43no—
457C22—41no—
458C22C23×C22⋊C411yescharacteristic
459C22C23×M1611yescharacteristic
460C22C23×M1611yescharacteristic
461C4—26no—
462C4—26no—
463C4—22no—
464C4—22no—
465C4—21no—
466C4—26no—
467C4—26no—
468C4—43no—
469C4—41no—
470C4—43no—
471C4—41no—
472C4C2×#407112yes—
473C4#288412yes—
474C2—21no—
475C2—21no—
476C2—42no—
477C2—46no—
478C2—42no—
479C2C22×#40711yescharacteristic
480C2C2×#288411yescharacteristic
481C2#6007611yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60248C2semidirect
3. #17372C2semidirect
4. #17371C2semidirect
5. #17366C2semidirect
6. #17361C2semidirect
7. #17353C2semidirect
8. #17352C2semidirect
139. C2×#41M16semidirect
148. C22×Q8M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×2−1+42
#602482
#173722
#173712
#173662
#173612
#173532
#173522