Finite Group Databasebeta

← #412229

Subgroups — #412229

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2060C212yesmaximal
3#21170C212yesmaximal
4#13779C211yesmaximal, characteristic
5#11392C212yesmaximal
6#11342C212yesmaximal
7#11211C212yesmaximal
8#10847C211yesmaximal, characteristic
9#10840C212yesmaximal
10C2×#544C2212yes—
11C2×#428C2214yes—
12C2×#423C2212yes—
13#2857—22no—
14#2659—22no—
15#2552—22no—
16#2551—22no—
17#2550—24no—
18#2316C2212yes—
19#2285C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#2060—28no—
26#1949—24no—
27#1948—22no—
28#1947—22no—
29C2×#343C2212yes—
30C2×#343C2212yes—
31#1736C2214yes—
32C2×#339C2214yes—
33C2×#339C2214yes—
34#1436—22no—
35#1436—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C22×M16C2314yes—
39#544—24no—
40#544—24no—
41#544—44no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—24no—
44C2×#125—24no—
45C2×C22⋊Q8—24no—
46C2×#117C2314yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C42—21no—
53C22×C42D412yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#428—28no—
58#428—44no—
59C2×#107—24no—
60C2×#106—24no—
61#423—24no—
62#423Q818yescomplemented
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65#398—28no—
66#395—28no—
67#391—24no—
68#389—24no—
69#389—24no—
70#389—24no—
71#389—24no—
72#384—22no—
73#384—22no—
74#384D414yes—
75#384D412yes—
76#384D412yes—
77#382—22no—
78#382D414yes—
79#382Q814yescomplemented
80#380—24no—
81#380—22no—
82#380—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89#376—22no—
90#376Q814yescomplemented
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98#343—24no—
99#343—24no—
100#340—24no—
101#340—24no—
102#339—28no—
103#339—44no—
104C22×Q8—24no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2411yescharacteristic
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16C2×Q818yes—
120C22×C8—24no—
121C4⋊Q8—48no—
122C4⋊Q8—48no—
123C4⋊Q8—84no—
124#125—48no—
125#125—48no—
126C22⋊Q8—48no—
127C22⋊Q8—48no—
128#117—24no—
129#117—24no—
130#117—24no—
131#117—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D414yes—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×Q812yes—
158C2×C4⋊C4C2×Q812yes—
159C2×C4⋊C4#4112yes—
160C2×C4⋊C4#4112yes—
161C2×C22⋊C4—24no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—24no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—42no—
172C2×C42—82no—
173C2×C42—82no—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×Q812yes—
178C2×C42C2×Q812yes—
179C2×C42C2×Q812yes—
180C2×C42C2×Q814yes—
181C2×C42C2×Q812yes—
182C2×C42#4112yes—
183#107—48no—
184#106—48no—
185C4⋊C8—28no—
186C4⋊C8—28no—
187#95—28no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—28no—
194#95—41no—
195#95—41no—
196#95—82no—
197#95—82no—
198C24C22×D411yescharacteristic
199C2×Q8—48no—
200C2×Q8—84no—
201C2×Q8—84no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4#12011yescharacteristic
241C22×C4#12012yes—
242C22×C4#12011yescharacteristic
243C22×C4#12114yes—
244C22×C4C22⋊Q814yes—
245C22×C4C22⋊Q812yes—
246C22×C4C22⋊Q812yes—
247C22×C4C4⋊Q812yes—
248C22×C4C4⋊Q812yes—
249C22×C4C22×D412yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C22×Q812yes—
252C22×C4C22×Q811yescharacteristic
253C22×C4C22×Q811yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4H(5,𝔽2)12yes—
258C22×C42−1+412yes—
259M16—28no—
260M16—28no—
261M16—44no—
262M16—44no—
263C2×C8—28no—
264C2×C8—44no—
265C2×C8—44no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—416no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C4⋊C4—88no—
286C4⋊C4—84no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—24no—
293C42—44no—
294C42—42no—
295C42—44no—
296C42—44no—
297C42—82no—
298C42—84no—
299C42C22⋊Q814yes—
300C42C22⋊Q814yes—
301C42C4⋊Q814yes—
302C42C4⋊Q814yes—
303C42C4⋊Q814yescomplemented
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C2×#12011yescharacteristic
309C23C2×#12111yescharacteristic
310C23C2×C22⋊Q811yescharacteristic
311C23C2×C22⋊Q811yescharacteristic
312C23C2×C4⋊Q811yescharacteristic
313C23#53511yescharacteristic
314C23#53611yescharacteristic
315Q8—168no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4#39512yes—
367C2×C4#39512yes—
368C2×C4#39514yes—
369C2×C4C2×C22⋊Q812yes—
370C2×C4C2×C4⋊Q812yes—
371C2×C4C2×C4⋊Q811yescharacteristic
372C2×C4#53211yescharacteristic
373C2×C4#53311yescharacteristic
374C2×C4#53311yescharacteristic
375C2×C4#54212yes—
376C2×C4#54412yes—
377C2×C4D4212yes—
378C2×C4#54612yes—
379C2×C4#54812yes—
380C2×C4D4×Q812yes—
381C2×C4D4×Q812yes—
382C2×C4#55412yes—
383C2×C4#55712yes—
384C2×C4Q8212yes—
385C8—48no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22C2×#39511yescharacteristic
399C22#240811yescharacteristic
400C22#240911yescharacteristic
401C22#254611yescharacteristic
402C22#256711yescharacteristic
403C22#262811yescharacteristic
404C22#285711yescharacteristic
405C22#301911yescharacteristic
406C22#302011yescharacteristic
407C22#302111yescharacteristic
408C22#302211yescharacteristic
409C4—22no—
410C4—22no—
411C4—24no—
412C4—44no—
413C4—44no—
414C4—44no—
415C4—82no—
416C4—82no—
417C4—84no—
418C4—84no—
419C4—82no—
420C4#245112yes—
421C4#245212yes—
422C4#255012yes—
423C4#255112yes—
424C2—21no—
425C2—21no—
426C2—21no—
427C2—21no—
428C2#1512811yescharacteristic
429C2#2356111yescharacteristic
430C2#2356211yescharacteristic
431C2#2422611yescharacteristic
432C2#2422911yescharacteristic
433C2#2431711yescharacteristic
434C2#2431811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
62. #423Q8semidirect
79. #382Q8semidirect
90. #376Q8semidirect
303. C42C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#20602
#211702
#137792
#113922
#113422
#112112
#108472
#108402