Finite Group Databasebeta

← #414468

Subgroups — #414468

Order 512.

1240 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2083C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1942C211yesmaximal, characteristic
5#15145C211yesmaximal, characteristic
6#15143C211yesmaximal, characteristic
7#11324C212yesmaximal
8#11324C212yesmaximal
9#10862C212yesmaximal
10#10841C212yesmaximal
11C8×#95C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2212yes—
15C22×#107C2212yes—
16C22×#107C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C22×C4×C8C2212yes—
19C2×#398—22no—
20C2×#395—22no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2212yes—
25C2×#384C2212yes—
26#2083—24no—
27#2057—24no—
28#1990—24no—
29#1942—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1444—22no—
39#1444—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2312yes—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—24no—
70C2×#107—24no—
71C2×#107—22no—
72C2×#107—24no—
73C2×#107—24no—
74C2×#107—42no—
75C2×#107—42no—
76C2×#107—42no—
77C2×C4×C8—42no—
78C2×C4×C8—42no—
79C2×C4×C8—42no—
80C2×C4×C8D414yes—
81C2×C4×C8Q818yescomplemented
82#398—48no—
83#395—48no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#384—24no—
89#384—24no—
90#384—22no—
91#384—22no—
92#384—42no—
93#384—42no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95D412yes—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—24no—
106#336D414yes—
107#336Q818yescomplemented
108C22×Q8—44no—
109C22×Q8—42no—
110C22×Q8—44no—
111C22×Q8—82no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D414yes—
121C23×C4C2411yescharacteristic
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8C2×D414yes—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8C2×Q814yes—
136C22×C8C2×Q814yes—
137C22×C8C2×Q814yes—
138C22×C8C2×Q814yes—
139C4⋊Q8—28no—
140C4⋊Q8—48no—
141C4⋊Q8—44no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—48no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—84no—
185C2×C4⋊C4—84no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4#4114yes—
188C2×C4⋊C4#4112yes—
189C2×C4⋊C4#4112yes—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×Q814yes—
199C2×C42C2×Q814yes—
200#107—28no—
201#107—48no—
202#107—48no—
203#107—44no—
204C4×C8—44no—
205C4×C8—48no—
206#95—42no—
207#95—42no—
208#95—48no—
209#95—48no—
210#95D818yes—
211#95Q1618yes—
212C24C22×D411yescharacteristic
213C2×Q8—48no—
214C2×Q8—816no—
215C2×Q8—84no—
216C2×Q8—84no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4#12712yes—
254C22×C4#12712yes—
255C22×C4C4⋊Q818yes—
256C22×C4C2×D814yes—
257C22×C4C2×D818yes—
258C22×C4C2×Q1614yes—
259C22×C4C2×Q1618yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×Q812yes—
263C22×C4C22×Q812yes—
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4112yes—
266C22×C4C2×#4111yescharacteristic
267C22×C4H(5,𝔽2)12yes—
268C22×C4H(5,𝔽2)11yescharacteristic
269C22×C42−1+411yescharacteristic
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8C4⋊Q814yes—
285C2×C8C4⋊Q814yes—
286C2×C8C4⋊Q818yes—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—48no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—816no—
311C4⋊C4—816no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C42—22no—
317C42—24no—
318C42—22no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—24no—
323C23C2×#12711yescharacteristic
324C23C2×C4⋊Q812yes—
325C23#54612yes—
326C23#54611yescharacteristic
327C23#54811yescharacteristic
328C23C22×D812yes—
329C23C22×Q1612yes—
330C23#57712yes—
331C23#57712yes—
332Q8—816no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—28no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4#49314yes—
374C2×C4#49414yes—
375C2×C4#500116yes—
376C2×C4#52511yescharacteristic
377C2×C4#52511yescharacteristic
378C2×C4C2×C4⋊Q811yescharacteristic
379C2×C4C2×C4⋊Q811yescharacteristic
380C2×C4C2×C4⋊Q812yes—
381C2×C4#53312yes—
382C2×C4#53312yes—
383C2×C4#53411yescharacteristic
384C2×C4#53711yescharacteristic
385C2×C4#54312yes—
386C2×C4#54412yes—
387C2×C4#54611yescharacteristic
388C2×C4#54612yes—
389C2×C4#54811yescharacteristic
390C2×C4#55412yes—
391C2×C4#55412yes—
392C2×C4#55412yes—
393C2×C4#55712yes—
394C8—24no—
395C8—24no—
396C8—28no—
397C22#253911yescharacteristic
398C22#254011yescharacteristic
399C22#254912yes—
400C22#255012yes—
401C22#286811yescharacteristic
402C22C2×#49311yescharacteristic
403C22C2×#49411yescharacteristic
404C22#315411yescharacteristic
405C22#315411yescharacteristic
406C22C2×#50014yes—
407C22#316314yes—
408C22#329412yes—
409C22#329414yes—
410C22#329912yes—
411C22#329914yes—
412C22#335812yes—
413C22#336312yes—
414C4—216no—
415C4—24no—
416C4—24no—
417C4—84no—
418C4—82no—
419C4—82no—
420C4—161no—
421C4—161no—
422C4—162no—
423C4#239911yescharacteristic
424C4#254912yes—
425C4#255012yes—
426C4#258311yescharacteristic
427C4#258411yescharacteristic
428C4#265911yescharacteristic
429C2#1517311yescharacteristic
430C2#2832911yescharacteristic
431C2#2833311yescharacteristic
432C2#2839711yescharacteristic
433C2#2840111yescharacteristic
434C2#2876914yes—
435C2#2878914yes—
436C2#3331211yescharacteristic
437C2#3331811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
81. C2×C4×C8Q8semidirect
107. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20832
C2×#20572
C2×#19902
C2×#19422
#151452
#151432
#113242
#108622
#108412
C8×#952