Finite Group Databasebeta

← #6433832

Subgroups — #6433832

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34634C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic
3#30302C214yesmaximal, complemented
4#30300C214yesmaximal, complemented
5#30299C218yesmaximal, complemented
6#23659C212yesmaximal, complemented
7#23657C212yesmaximal, complemented
8#23554C212yesmaximal, complemented
9#23552C212yesmaximal, complemented
10#20661C214yesmaximal
11#20335C211yesmaximal, characteristic
12C23×M16C2212yes—
13C2×#533C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#428C2214yes—
17C2×#428C2214yes—
18#2629C2214yes—
19#2628—22no—
20#2628—22no—
21#2628C2214yes—
22#2628C2214yes—
23#2628C2212yes—
24#2628C2212yes—
25#2627—21no—
26#2627—21no—
27#2626—21no—
28#2626—21no—
29#2396—22no—
30#2396—22no—
31C2×#400—21no—
32C2×#399—21no—
33C2×#395—21no—
34C2×#393—21no—
35#2296C2212yes—
36C2×#389C2212yes—
37#2285C2211yescharacteristic
38C2×#382C2211yescharacteristic
39#2280C2212yes—
40C2×#380C2212yes—
41#2085C2218yescomplemented
42#2085C2218yescomplemented
43#2085C2218yescomplemented
44#2084C2214yescomplemented
45#2083C2214yescomplemented
46#2059C2218yescomplemented
47#2058C2218yescomplemented
48#2058C2214yescomplemented
49#2057C2218yescomplemented
50#2057C2214yescomplemented
51#1809C2218yes—
52#1809C2218yes—
53#1808C2218yes—
54#1807C2218yes—
55#1736C2214yes—
56#1736C2214yes—
57#1729C2214yes—
58#1729C2214yes—
59#1728C2212yes—
60#1728C2212yes—
61C23×Q8—21no—
62C24×C4C2311yescharacteristic
63C22×M16—44no—
64C22×M16—44no—
65C22×M16Q8116yescomplemented
66C22×M16C2314yes—
67C22×M16C2314yes—
68C23×C8C2314yes—
69#533—44no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—44no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2314yes—
74C2×C4⋊Q8C2311yescharacteristic
75C2×#125C2312yes—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×#117—44no—
81C2×#117D412yes—
82C2×#117D412yes—
83C2×#117C2312yes—
84C2×#117C2312yes—
85C2×#117C2312yes—
86C2×#117C2312yes—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—22no—
94C22×C22⋊C4D411yescharacteristic
95C22×C22⋊C4D411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#428—48no—
99#428—48no—
100C2×#107C2318yes—
101C2×#107C2318yes—
102C2×#106C2318yes—
103C2×#106C2318yes—
104#400—22no—
105#400—22no—
106#400—24no—
107#400—24no—
108#399—22no—
109#399—22no—
110#399—24no—
111#399—24no—
112#398—24no—
113#398—24no—
114#396—24no—
115#396—24no—
116#395—22no—
117#395—22no—
118#395—24no—
119#395—24no—
120#393—22no—
121#393—22no—
122#393—24no—
123#393—24no—
124#389—22no—
125#389—22no—
126#389C2314yes—
127#389C2314yes—
128#389C2314yes—
129#389C2314yes—
130#385—22no—
131#385—22no—
132#384C2312yes—
133#382—22no—
134#382C2314yes—
135#382C2312yes—
136#381—22no—
137#380—21no—
138#380—21no—
139#380—24no—
140#380C2314yes—
141#380C2318yes—
142#380C2314yes—
143#379—22no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—24no—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—21no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#336C2318yes—
157#336C2314yes—
158#336C2314yes—
159C25C2×D411yescharacteristic
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—24no—
163C22×Q8—44no—
164C22×Q8—84no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×Q818yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C2×M16—28no—
180C2×M16—28no—
181C2×M16—44no—
182C2×M16—44no—
183C2×M16—432no—
184C2×M16C2×Q8116yes—
185C2×M16C2×Q8116yescomplemented
186C22×C8—44no—
187C22×C8—44no—
188C22×C8C2×Q8116yescomplemented
189C22×C8C2418yes—
190C4⋊Q8—44no—
191C4⋊Q8—44no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4⋊Q8—48no—
195#125—44no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C4×Q8—44no—
199C4×Q8—44no—
200#117—28no—
201#117—24no—
202#117—24no—
203#117—44no—
204#117—44no—
205#117—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—84no—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2414yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2412yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2414yes—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—84no—
246C2×C22⋊C4C2×D412yes—
247C2×C22⋊C4C2×D412yes—
248C2×C22⋊C4C2×D412yes—
249C2×C22⋊C4C2×D412yes—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D412yes—
257C2×C42C2411yescharacteristic
258C2×C42C2411yescharacteristic
259C2×C42C2412yes—
260#107—48no—
261#107—48no—
262#106—48no—
263#106—48no—
264#95—28no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—24no—
274#95—24no—
275C24—22no—
276C24—22no—
277C24—42no—
278C24—42no—
279C24C4⋊Q814yes—
280C24C22×D412yes—
281C24C22×D411yescharacteristic
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—48no—
285C2×Q8—82no—
286C2×Q8—82no—
287C2×Q8—88no—
288C2×Q8—816no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—216no—
294C22×C4—216no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4C4⋊Q814yes—
319C22×C4C4⋊Q818yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D412yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D412yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×Q814yes—
329C22×C4C22×Q814yes—
330C22×C4C22×Q814yes—
331C22×C4H(5,𝔽2)14yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C42−1+412yes—
334C22×C4C2511yescharacteristic
335M16—232no—
336M16—416no—
337C2×C8—28no—
338C2×C8—28no—
339C2×C8—416no—
340C2×C8C22×Q8116yes—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—48no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—82no—
360C4⋊C4—84no—
361C4⋊C4—82no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—84no—
372C22⋊C4—84no—
373C22⋊C4—88no—
374C42—22no—
375C42—22no—
376C42—24no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—44no—
382C42—44no—
383C23—22no—
384C23—22no—
385C23—24no—
386C23—24no—
387C23—22no—
388C23—22no—
389C23—28no—
390C23—48no—
391C23—82no—
392C23—82no—
393C23—84no—
394C23C2×C4⋊Q814yes—
395C23C2×C4⋊Q812yes—
396C23#53412yes—
397C23#53712yes—
398C23C23×D411yescharacteristic
399Q8—84no—
400Q8—84no—
401Q8—816no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—216no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—88no—
440C2×C4—88no—
441C2×C4C2×C4⋊Q814yes—
442C2×C4C2×C4⋊Q812yes—
443C2×C4#53412yes—
444C2×C4#53712yes—
445C2×C4#54318yes—
446C2×C4#54418yes—
447C2×C4C23×Q812yes—
448C2×C4C2×H(5,𝔽2)11yescharacteristic
449C2×C4C2×H(5,𝔽2)12yes—
450C2×C4C2×2−1+411yescharacteristic
451C8—216no—
452C22—22no—
453C22—22no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—28no—
458C22—42no—
459C22—42no—
460C22—44no—
461C22—88no—
462C22#265914yes—
463C22C22×C4⋊Q811yescharacteristic
464C22C2×#53411yescharacteristic
465C22C2×#53711yescharacteristic
466C22#358512yes—
467C22#358612yes—
468C4—24no—
469C4—42no—
470C4—41no—
471C4—41no—
472C4—42no—
473C4—41no—
474C4—41no—
475C4—84no—
476C4—82no—
477C4—82no—
478C4—84no—
479C4—84no—
480C4#265914yes—
481C4C2×#54312yes—
482C4C2×#54412yes—
483C2—24no—
484C2—44no—
485C2C2×#265911yescharacteristic
486C2#6066412yes—
487C2#6075311yescharacteristic
488C2#6075411yescharacteristic
489C2#6091912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30302C2semidirect
4. #30300C2semidirect
5. #30299C2semidirect
6. #23659C2semidirect
7. #23657C2semidirect
8. #23554C2semidirect
9. #23552C2semidirect
41. #2085C22semidirect
42. #2085C22semidirect
43. #2085C22semidirect
44. #2084C22semidirect
45. #2083C22semidirect
46. #2059C22semidirect
47. #2058C22semidirect
48. #2058C22semidirect
49. #2057C22semidirect
50. #2057C22semidirect
65. C22×M16Q8semidirect
185. C2×M16C2×Q8semidirect
188. C22×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#346342
C2×#26282
#303022
#303002
#302992
#236592
#236572
#235542
#235522
#206612
#203352