Finite Group Databasebeta

← #415589

Subgroups — #415589

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2#21171C212yesmaximal
3#13910C211yesmaximal, characteristic
4#11392C212yesmaximal
5#11342C212yesmaximal
6#11340C214yesmaximal
7#10852C212yesmaximal
8#10847C211yesmaximal, characteristic
9C2×#544C2211yescharacteristic
10C2×#428C2214yes—
11C2×#423C2212yes—
12#2331—22no—
13#2330—22no—
14#2329—22no—
15#2317C2212yes—
16#2316C2211yescharacteristic
17C4×C4⋊Q8—22no—
18C4×#125—22no—
19C4×C22⋊Q8—22no—
20C2×#389—22no—
21C2×#384C2211yescharacteristic
22C2×#384C2212yes—
23C4×#117C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1948—22no—
27#1948—22no—
28#1947—22no—
29#1947—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32#1736C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1444—22no—
36C4⋊#95C2212yes—
37C22×M16C2314yes—
38#544—24no—
39#544—22no—
40#544Q814yes—
41C2×C4⋊Q8—22no—
42C2×#125—22no—
43C2×C22⋊Q8—22no—
44C2×C4×Q8—22no—
45C2×#117C2312yes—
46C2×#117C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42D412yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57#428—44no—
58#428—44no—
59C2×#107—24no—
60C2×#106—24no—
61#423—24no—
62#423Q818yescomplemented
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65#391—24no—
66#391—22no—
67#389—24no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—48no—
74#387—24no—
75#387—24no—
76#387—24no—
77#385—24no—
78#384—22no—
79#384—22no—
80#384—22no—
81#384—44no—
82#382—22no—
83#380—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4D412yes—
92C4×C4⋊C4D412yes—
93C4×C4⋊C4Q814yescomplemented
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96#376—22no—
97C2×#95—21no—
98C2×#95—24no—
99C2×#95D412yes—
100C43—22no—
101#343—24no—
102#343—42no—
103#340—24no—
104#340—24no—
105#339—28no—
106#339—44no—
107C22×Q8—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16C2×Q818yes—
121C22×C8—24no—
122C4⋊Q8—28no—
123C4⋊Q8—28no—
124#125—28no—
125#125—28no—
126C22⋊Q8—28no—
127C22⋊Q8—28no—
128C4×Q8—84no—
129#117—22no—
130#117—22no—
131#117—22no—
132#117—22no—
133#117—22no—
134#117—22no—
135#117C2×Q814yes—
136#117#4114yes—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—84no—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×Q812yes—
169C2×C4⋊C4#4112yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—44no—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×Q814yes—
187C2×C42C2×Q814yes—
188C2×C42C2×Q812yes—
189C2×C42#4112yes—
190#107—48no—
191#106—48no—
192C4⋊C8—28no—
193C4⋊C8—28no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—24no—
203#95—48no—
204C24C22×D411yescharacteristic
205C2×Q8—28no—
206C2×Q8—82no—
207C2×Q8—82no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4#12011yescharacteristic
238C22×C4#12012yes—
239C22×C4#12011yescharacteristic
240C22×C4C22⋊Q812yes—
241C22×C4C22⋊Q814yes—
242C22×C4C22⋊Q812yes—
243C22×C4C4⋊Q812yes—
244C22×C4C4⋊Q812yes—
245C22×C4C22×D412yes—
246C22×C4C22×Q811yescharacteristic
247C22×C4C22×Q812yes—
248C22×C4C22×Q811yescharacteristic
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4112yes—
251C22×C4C2×#4111yescharacteristic
252C22×C4H(5,𝔽2)11yescharacteristic
253C22×C4H(5,𝔽2)12yes—
254C22×C42−1+411yescharacteristic
255M16—28no—
256M16—28no—
257M16—44no—
258M16—44no—
259C2×C8—28no—
260C2×C8—44no—
261C2×C8—44no—
262C4⋊C4—216no—
263C4⋊C4—28no—
264C4⋊C4—216no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—84no—
283C4⋊C4—816no—
284C4⋊C4—82no—
285C4⋊C4C22⋊Q814yes—
286C4⋊C4C22⋊Q814yes—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C42—24no—
292C42—24no—
293C42—28no—
294C42—42no—
295C42—42no—
296C42—44no—
297C42—44no—
298C42—44no—
299C42—44no—
300C42—82no—
301C42—82no—
302C42—82no—
303C42C22⋊Q814yes—
304C42C22⋊Q814yes—
305C42C4⋊Q814yes—
306C42C4⋊Q814yes—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23C2×#12011yescharacteristic
312C23C2×C22⋊Q811yescharacteristic
313C23C2×C22⋊Q811yescharacteristic
314C23#52511yescharacteristic
315C23C2×C4⋊Q811yescharacteristic
316C23#53511yescharacteristic
317C23#53711yescharacteristic
318Q8—88no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4#39512yes—
362C2×C4#39514yes—
363C2×C4#39512yes—
364C2×C4#45312yes—
365C2×C4#45612yes—
366C2×C4C2×C22⋊Q812yes—
367C2×C4C2×C22⋊Q812yes—
368C2×C4#52512yes—
369C2×C4C2×C4⋊Q812yes—
370C2×C4#53512yes—
371C2×C4#53712yes—
372C2×C4#55412yes—
373C2×C4#55412yes—
374C2×C4#55412yes—
375C2×C4#55711yescharacteristic
376C2×C4#55712yes—
377C2×C4#55711yescharacteristic
378C2×C4Q8211yescharacteristic
379C2×C4Q8212yes—
380C2×C4#55911yescharacteristic
381C2×C4#55912yes—
382C8—48no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22C2×#39511yescharacteristic
396C22#256711yescharacteristic
397C22#258011yescharacteristic
398C22#258211yescharacteristic
399C22#258411yescharacteristic
400C22#261511yescharacteristic
401C22#284811yescharacteristic
402C22C2×#45311yescharacteristic
403C22C2×#45611yescharacteristic
404C22#302111yescharacteristic
405C22#302211yescharacteristic
406C4—22no—
407C4—24no—
408C4—48no—
409C4—44no—
410C4—44no—
411C4—44no—
412C4—82no—
413C4—82no—
414C4—82no—
415C4—162no—
416C4#206014yes—
417C4#256712yes—
418C4#258012yes—
419C4#258212yes—
420C4#258412yes—
421C2—21no—
422C2—21no—
423C2—21no—
424C2—21no—
425C2#1520211yescharacteristic
426C2C2×#206011yescharacteristic
427C2#2356211yescharacteristic
428C2#2967811yescharacteristic
429C2#2968111yescharacteristic
430C2#3296911yescharacteristic
431C2#3297211yescharacteristic
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
93. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5442
#211712
#139102
#113922
#113422
#113402
#108522
#108472