Finite Group Databasebeta

← #6493650

Subgroups — #6493650

Order 512.

1717 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
1#35316C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#33117C214yesmaximal, complemented
4#29930C214yesmaximal, complemented
5#27972C212yesmaximal, complemented
6#27908C212yesmaximal, complemented
7#27892C212yesmaximal, complemented
8#27889C212yesmaximal, complemented
9#27592C214yesmaximal, complemented
10#24016C214yesmaximal
11#21984C212yesmaximal, complemented
12#21804C212yesmaximal, complemented
13#20718C211yesmaximal, characteristic
14C2×#536C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17C2×#485C2218yes—
18C2×#431C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2621C2212yes—
22#2621C2212yes—
23#2547—22no—
24#2546—22no—
25#2486—22no—
26#2486—22no—
27#2355—22no—
28#2354—22no—
29#2352—22no—
30#2352—22no—
31#2314C2212yes—
32#2314C2212yes—
33#2311C2212yes—
34#2310C2212yes—
35#2285C2212yes—
36#2285C2212yes—
37#2285C2211yescharacteristic
38#2285C2211yescharacteristic
39C2×#384C2211yescharacteristic
40#2274C2212yes—
41#2097C2218yes—
42#2072C2218yes—
43#2036C2218yes—
44#2035C2218yes—
45#1981C2218yes—
46#1956C2212yes—
47#1956C2212yes—
48#1956C2212yes—
49#1956C2212yes—
50#1949C2218yes—
51#1928C2218yes—
52#1927C2218yes—
53#1872C2218yes—
54#1854C2218yes—
55#1845C2212yes—
56#1844C2212yes—
57#1834C2212yes—
58#1834C2212yes—
59#1818C2212yes—
60#1798C2212yes—
61#1798C2212yes—
62#1787C2218yes—
63#1780C2212yes—
64C4×C22⋊C8C2212yes—
65C4×C22⋊C8C2211yescharacteristic
66C4×C22⋊C8C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C23×Q8C2312yes—
69#536—48no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8C2312yes—
74C2×C4⋊Q8C2312yes—
75C2×#124—22no—
76C2×#123—22no—
77C2×#123—22no—
78C2×C22⋊Q8C2314yes—
79C2×C22⋊Q8C2314yes—
80C4×#41—28no—
81C2×C4×Q8D412yes—
82C2×C4×D4D412yes—
83C2×C4×D4D412yes—
84C2×C4×D4D412yes—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117C2312yes—
88C2×#117C2312yes—
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C22×C4⋊C4—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4C2312yes—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#485—416no—
99#431—28no—
100C2×#107C2318yes—
101C2×#106C2318yes—
102C2×C4⋊C8C2314yes—
103C2×#103C2318yes—
104C2×#103C2318yes—
105C2×#103C2318yes—
106C2×#103C2318yes—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2312yes—
110C2×C22⋊C8C2312yes—
111C2×C4×C8C2312yes—
112C2×C4×C8C2312yes—
113#398—24no—
114#398—24no—
115#397—24no—
116#397—24no—
117#391—22no—
118#391—24no—
119#391C2314yes—
120#391C2314yes—
121#390—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#388—24no—
127#388—24no—
128#387—24no—
129#387—24no—
130#386—22no—
131#386—22no—
132#384—22no—
133#384C2312yes—
134#384C2312yes—
135#384C2312yes—
136#384C2312yes—
137#382—22no—
138#382—22no—
139#382C2312yes—
140#382C2312yes—
141#381—24no—
142#380—22no—
143#380—22no—
144#380—22no—
145#380—22no—
146#380—22no—
147#380—22no—
148#380—22no—
149#380—22no—
150#380C2314yes—
151#380C2314yes—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4C2314yes—
158C4×C4⋊C4C2314yes—
159C4×C22⋊C4—24no—
160#340C2318yes—
161#340C2318yes—
162#336C2314yes—
163#336C2312yes—
164#336C2312yes—
165C2×#41—24no—
166C2×#41—24no—
167C2×#41—42no—
168C22×Q8—22no—
169C22×Q8—28no—
170C22×Q8—44no—
171C22×Q8—44no—
172C22×Q8C2×D411yescharacteristic
173C22×Q8C2414yes—
174C22×D4C2×D411yescharacteristic
175C22×D4C2×D411yescharacteristic
176C22×D4C2×D411yescharacteristic
177C23×C4—22no—
178C23×C4—22no—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8C2414yes—
186C22×C8C2414yes—
187C4⋊Q8—28no—
188C4⋊Q8—28no—
189C4⋊Q8—48no—
190C4⋊Q8—48no—
191#124—48no—
192#123—48no—
193#123—48no—
194C22⋊Q8—216no—
195C22⋊Q8—48no—
196C4×Q8—28no—
197C4×D4—28no—
198C4×D4—28no—
199C4×D4—28no—
200#117—24no—
201#117—24no—
202#117—24no—
203#117—28no—
204#117—44no—
205#117—84no—
206#117—84no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—84no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2414yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4C2×D412yes—
236C2×C22⋊C4C2×D412yes—
237C2×C22⋊C4C2×D412yes—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42#4118yes—
248C2×C42C2411yescharacteristic
249C2×C42C2411yescharacteristic
250C2×C42C2411yescharacteristic
251C2×C42C2411yescharacteristic
252#107—48no—
253#106—48no—
254C4⋊C8—28no—
255#103—48no—
256#103—48no—
257#103—48no—
258#103—48no—
259C22⋊C8—28no—
260C22⋊C8—28no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263C4×C8—24no—
264C4×C8—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275C24C22×D411yescharacteristic
276C24C22×D411yescharacteristic
277C24C22×D411yescharacteristic
278#41—416no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—28no—
282C2×Q8—28no—
283C2×Q8—28no—
284C2×Q8—41no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—816no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—84no—
322C22×C4#12318yes—
323C22×C4C22×D412yes—
324C22×C4C22×D412yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D412yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C2×#4114yes—
331C22×C4H(5,𝔽2)14yes—
332C22×C4C2511yescharacteristic
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—28no—
344C4⋊C4—28no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—48no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—84no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—44no—
362C22⋊C4—48no—
363C22⋊C4—44no—
364C22⋊C4—48no—
365C22⋊C4—84no—
366C22⋊C4—84no—
367C42—24no—
368C42—24no—
369C42—22no—
370C42—22no—
371C42—44no—
372C42—44no—
373C42—84no—
374C42—84no—
375C42#12318yes—
376C42C2×#4114yes—
377C42C2×#4114yes—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—22no—
390C23—22no—
391C23C2×#12312yes—
392C23#53512yes—
393C23C23×D411yescharacteristic
394Q8—44no—
395Q8—416no—
396Q8—88no—
397D4—44no—
398D4—44no—
399D4—44no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—216no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—48no—
419C2×C4—48no—
420C2×C4—44no—
421C2×C4—44no—
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—44no—
433C2×C4—88no—
434C2×C4—164no—
435C2×C4C2×#12312yes—
436C2×C4C2×#12314yes—
437C2×C4C2×#12314yes—
438C2×C4#53512yes—
439C2×C4#57714yes—
440C2×C4#57814yes—
441C2×C4C22×#4112yes—
442C2×C4C2×H(5,𝔽2)12yes—
443C2×C4#58512yes—
444C8—44no—
445C8—44no—
446C22—24no—
447C22—24no—
448C22—22no—
449C22—22no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—42no—
459C22—42no—
460C22—41no—
461C22C22×#12311yescharacteristic
462C22C2×#53511yescharacteristic
463C22#344911yescharacteristic
464C22C2×#57712yes—
465C22C2×#57812yes—
466C4—24no—
467C4—22no—
468C4—22no—
469C4—42no—
470C4—42no—
471C4—41no—
472C4—84no—
473C4—84no—
474C4—84no—
475C4—84no—
476C4#309718yes—
477C4#349414yes—
478C2—42no—
479C2—42no—
480C2—81no—
481C2#3526211yescharacteristic
482C2C2×#309712yes—
483C2#6082412yes—
484C2#6101211yescharacteristic
485C2#6101311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
3. #33117C2semidirect
4. #29930C2semidirect
5. #27972C2semidirect
6. #27908C2semidirect
7. #27892C2semidirect
8. #27889C2semidirect
9. #27592C2semidirect
11. #21984C2semidirect
12. #21804C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#341882
#331172
#299302
#279722
#279082
#278922
#278892
#275922
#240162
#219842
#218042
#207182