Finite Group Databasebeta

← #6431970

Subgroups — #6431970

Order 512.

4625 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#578C211yesmaximal, characteristic
2C2×#2906C211yesmaximal, characteristic, complemented
3C2×#2898C212yesmaximal, complemented
4#34259C212yesmaximal, complemented
5#23547C216yesmaximal, complemented
6#23545C216yesmaximal, complemented
7#23543C213yesmaximal, complemented
8#23542C213yesmaximal, complemented
9#23539C213yesmaximal, complemented
10C2×#2056C213yesmaximal, complemented
11#20348C211yesmaximal, characteristic, complemented
12C22×2−1+4C2212yes—
13C2×#578—216no—
14C2×#578—28no—
15C22×#137C2216yes—
16C22×#135C2213yes—
17C23×Q16C2213yes—
18C22×#131C2211yescharacteristic
19C22×#131C2211yescharacteristic
20C2×#520C2212yes—
21C2×C4×#41C2211yescharacteristic
22#2906—28no—
23C2×#424C2213yes—
24C2×#424C2213yes—
25C2×#423C2213yes—
26C22×C4⋊C8C2213yes—
27#2898—216no—
28C2×#417C2216yes—
29C2×#419C2216yes—
30C2×#416C2216yes—
31C22×#103C2216yes—
32Q8×C22⋊C4—22no—
33Q8×C22⋊C4—22no—
34#2339—26no—
35#2339—26no—
36#2311C2216yes—
37#2310C2216yes—
38#2297C2216yes—
39C2×#391C2216yes—
40#2056C2213yes—
41#2056C2216yes—
42#2056C2216yes—
43#2056C2216yes—
44#2056C2213yes—
45#2056C22112yes—
46#2055C2216yes—
47#2055C2216yes—
48#2055C22112yes—
49#2054C2211yescharacteristic
50#2054C2213yes—
51#1736C2216yes—
52#1735C2213yes—
53C2×#340C2213yes—
54C2×2−1+4—28no—
55C2×2−1+4—28no—
56C2×2−1+4—28no—
57C2×2−1+4—28no—
58C2×2−1+4—28no—
59C2×2−1+4D4116yes—
60C22×#41—26no—
61C22×#41C2316yes—
62C22×#41C2311yescharacteristic
63C23×Q8—22no—
64C23×Q8C2316yes—
65#578—2128no—
66C2×#137—224no—
67C2×#137—224no—
68C2×#137—224no—
69C2×#135—224no—
70C2×#135—212no—
71C22×Q16—224no—
72C22×Q16—212no—
73C22×Q16C2313yes—
74C22×Q16C2316yes—
75C22×SD16C2316yes—
76C22×D8C2311yescharacteristic
77C2×#131—28no—
78C2×#131—28no—
79C2×#131—24no—
80C2×#131D418yes—
81C22×M16C2313yes—
82C22×M16C2313yes—
83C23×C8C2313yes—
84C23×C8C2313yes—
85#520—28no—
86#520—28no—
87C4×#41—24no—
88C4×#41—24no—
89C2×C4×Q8—22no—
90C2×C4×Q8D411yescharacteristic
91C2×C4×Q8D411yescharacteristic
92C2×C4×D4—26no—
93C2×C4×D4D413yes—
94C2×C4×D4D413yes—
95C2×#117C2316yes—
96C2×#117C2313yes—
97C22×C4⋊C4C2316yes—
98C22×C42C2313yes—
99#424—212no—
100#424—212no—
101#423—212no—
102C2×C4⋊C8—212no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2313yes—
105C2×C4⋊C8C2313yes—
106C2×C4⋊C8C2313yes—
107C2×C4⋊C8C2316yes—
108#419—224no—
109#417—224no—
110#416—224no—
111C2×#103—224no—
112C2×#103C2316yes—
113C2×#103C23112yes—
114C2×#103C2316yes—
115C2×#102C2312yes—
116C2×#102C2316yes—
117#391—26no—
118#391—26no—
119#391—26no—
120#391—26no—
121#391C2316yes—
122#391C23112yes—
123#391C2316yes—
124#390C2316yes—
125#390C2312yes—
126#386—26no—
127#386—26no—
128#386—212no—
129#381—212no—
130#380—26no—
131#380—26no—
132C4×C22⋊C4—26no—
133C4×C22⋊C4—26no—
134C2×#95—26no—
135C2×#95—26no—
136#340C2313yes—
137#340C2311yescharacteristic
138#340C2316yes—
139#340C2316yes—
1402−1+4—264no—
1412−1+4—264no—
1422−1+4—432no—
1432−1+4—432no—
144C2×#41—248no—
145C2×#41—212no—
146C2×#41—212no—
147C2×#41—212no—
148C2×#41—212no—
149C2×#41—212no—
150C2×#41—424no—
151C2×#41—424no—
152C2×#41C2×D4124yes—
153C2×#41C2×D418yes—
154C2×#41C2×D414yes—
155C2×#41#4114yes—
156C2×#41#4118yes—
157C22×Q8—216no—
158C22×Q8—26no—
159C22×Q8—212no—
160C22×Q8—212no—
161C22×Q8—212no—
162C22×Q8—212no—
163C22×Q8—212no—
164C22×Q8—48no—
165C22×Q8—48no—
166C22×Q8C2×D4124yes—
167C22×Q8C2×D411yescharacteristic
168C22×Q8C2416yes—
169C22×D4—26no—
170C22×D4C2×D413yes—
171C22×D4C2412yes—
172C23×C4—26no—
173C23×C4—26no—
174C23×C4—26no—
175C23×C4C2×D413yes—
176C23×C4C2×D413yes—
177C23×C4C2413yes—
178#137—2192no—
179#135—296no—
180C2×Q16—296no—
181C2×Q16—212no—
182C2×Q16—224no—
183C2×Q16—26no—
184C2×Q16—212no—
185C2×SD16—212no—
186C2×SD16—212no—
187C2×SD16—212no—
188C2×D8—24no—
189C2×D8—22no—
190#131—232no—
191#131—416no—
192C2×M16—212no—
193C2×M16—212no—
194C2×M16—26no—
195C2×M16C2×D4112yes—
196C22×C8—212no—
197C22×C8—212no—
198C22×C8—26no—
199C22×C8C2×D4112yes—
200C22×C8C2411yescharacteristic
201C22×C8C2413yes—
202C22×C8C2411yescharacteristic
203C22×C8C2413yes—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—48no—
207C4×D4—212no—
208C4×D4—212no—
209C4×D4—424no—
210#117—26no—
211#117—26no—
212#117—212no—
213#117—212no—
214C2×C4⋊C4—212no—
215C2×C4⋊C4—212no—
216C2×C4⋊C4—26no—
217C2×C4⋊C4C2×D413yes—
218C2×C4⋊C4C2×D413yes—
219C2×C4⋊C4C2416yes—
220C2×C4⋊C4C2412yes—
221C2×C22⋊C4—26no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—26no—
225C2×C22⋊C4—26no—
226C2×C22⋊C4C2×D416yes—
227C2×C42—26no—
228C2×C42—26no—
229C2×C42—26no—
230C2×C42C2×D413yes—
231C2×C42C2×D413yes—
232C2×C42C2413yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—22no—
235C4⋊C8—26no—
236C4⋊C8—26no—
237C4⋊C8—26no—
238C4⋊C8—212no—
239#103—212no—
240#103—224no—
241#103—212no—
242#102—24no—
243#102—212no—
244#95—26no—
245#95—26no—
246#95—212no—
247#95—212no—
248#95—26no—
249#95—26no—
250C24—22no—
251C24C22×D413yes—
252#41—248no—
253#41—248no—
254#41—4192no—
255#41—424no—
256#41—424no—
257#41#120116yes—
258#41#123132yes—
259#41#124116yes—
260C2×Q8—224no—
261C2×Q8—26no—
262C2×Q8—248no—
263C2×Q8—248no—
264C2×Q8—26no—
265C2×Q8—26no—
266C2×Q8—26no—
267C2×Q8—26no—
268C2×Q8—464no—
269C2×Q8—412no—
270C2×Q8—412no—
271C2×Q8—424no—
272C2×Q8—424no—
273C2×Q8#12114yes—
274C2×Q8#12118yes—
275C2×Q8C22×D4112yes—
276C2×D4—224no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—412no—
283C2×D4—412no—
284C2×D4#121112yes—
285C2×D4#121124yes—
286C2×D4C22×D414yes—
287C22×C4—26no—
288C22×C4—224no—
289C22×C4—26no—
290C22×C4—26no—
291C22×C4—224no—
292C22×C4—26no—
293C22×C4—22no—
294C22×C4—26no—
295C22×C4—43no—
296C22×C4—46no—
297C22×C4—412no—
298C22×C4—412no—
299C22×C4—412no—
300C22×C4—412no—
301C22×C4—412no—
302C22×C4—86no—
303C22×C4—86no—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D413yes—
306C22×C4C22×D413yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D4112yes—
309C22×C4C22×D416yes—
310C22×C4C22×D413yes—
311C22×C4C2×#4116yes—
312C22×C4C2×#41112yes—
313C22×C4C2511yescharacteristic
314Q16—224no—
315Q16—248no—
316SD16—248no—
317D8—28no—
318M16—224no—
319M16—412no—
320C2×C8—224no—
321C2×C8—22no—
322C2×C8—26no—
323C2×C8—22no—
324C2×C8—26no—
325C2×C8—21no—
326C2×C8—23no—
327C2×C8—412no—
328C2×C8C22×D412yes—
329C2×C8C22×D416yes—
330C4⋊C4—26no—
331C4⋊C4—26no—
332C4⋊C4—26no—
333C4⋊C4—22no—
334C4⋊C4—26no—
335C4⋊C4—22no—
336C4⋊C4—412no—
337C22⋊C4—212no—
338C22⋊C4—44no—
339C22⋊C4—412no—
340C22⋊C4—44no—
341C22⋊C4—86no—
342C22⋊C4—86no—
343C42—26no—
344C42—26no—
345C42—21no—
346C42—23no—
347C42—21no—
348C42—23no—
349C42—412no—
350C23—23no—
351C23—26no—
352C23—23no—
353C23—24no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23C2×#12116yes—
361C23C2×#121112yes—
362C23C23×D411yescharacteristic
363Q8—212no—
364Q8—212no—
365Q8—448no—
366Q8—46no—
367Q8—46no—
368Q8#394116yes—
369D4—24no—
370D4—24no—
371D4—448no—
372D4—42no—
373D4—42no—
374D4#394148yes—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—23no—
378C2×C4—22no—
379C2×C4—26no—
380C2×C4—21no—
381C2×C4—23no—
382C2×C4—23no—
383C2×C4—212no—
384C2×C4—23no—
385C2×C4—412no—
386C2×C4—448no—
387C2×C4—46no—
388C2×C4—46no—
389C2×C4—448no—
390C2×C4—46no—
391C2×C4—42no—
392C2×C4—46no—
393C2×C4—42no—
394C2×C4—46no—
395C2×C4—42no—
396C2×C4—86no—
397C2×C4—1612no—
398C2×C4C2×#120112yes—
399C2×C4C2×#12116yes—
400C2×C4C2×#121112yes—
401C2×C4C2×#123124yes—
402C2×C4C2×#124112yes—
403C2×C4#57812yescomplemented
404C2×C4C23×D412yes—
405C2×C4C23×D411yescharacteristic
406C2×C4C22×#4111yescharacteristic
407C2×C4C22×#4112yes—
408C8—22no—
409C8—26no—
410C8—41no—
411C8—43no—
412C22—212no—
413C22—212no—
414C22—26no—
415C22—26no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22C2×#394112yes—
424C22C22×#12112yes—
425C22C22×#12111yescharacteristic
426C22C2×#57811yescharacteristic
427C22#358411yescharacteristic
428C22#358612yes—
429C4—22no—
430C4—43no—
431C4—43no—
432C4—41no—
433C4—412no—
434C4—82no—
435C4—86no—
436C4C2×#394112yes—
437C4C22×#12011yescharacteristic
438C4C22×#12312yes—
439C4C22×#12411yescharacteristic
440C2—212no—
441C2—84no—
442C2C22×#39411yescharacteristic
443C2#6043212yes—
444C2#6043312yes—
445C2#6045911yescharacteristic
446C2#6046211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#578C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2906C2semidirect
3. C2×#2898C2semidirect
4. #34259C2semidirect
5. #23547C2semidirect
6. #23545C2semidirect
7. #23543C2semidirect
8. #23542C2semidirect
9. #23539C2semidirect
10. C2×#2056C2semidirect
11. #20348C2semidirect
403. C2×C4#578semidirect

Maximal subgroups

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

subgroupindex
C22×#5782
C2×#29062
C2×#28982
#342592
#235472
#235452
#235432
#235422
#235392
C2×#20562
#203482