Finite Group Databasebeta

← #415590

Subgroups — #415590

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2#21171C212yesmaximal
3#13911C211yesmaximal, characteristic
4#11392C212yesmaximal
5#11342C212yesmaximal
6#11339C214yesmaximal
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×#384C2211yescharacteristic
21C2×#382C2212yes—
22C4×#117C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1948—22no—
26#1948—22no—
27#1947—22no—
28#1947—22no—
29C2×#343C2212yes—
30C2×#343C2212yes—
31#1736C2214yes—
32C2×#339C2214yes—
33C2×#339C2214yes—
34#1442—22no—
35C4⋊#95—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—21no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C42D412yes—
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56#428—44no—
57#428—44no—
58C2×#107—24no—
59C2×#106—24no—
60#423—24no—
61#423Q818yescomplemented
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64#391—24no—
65#391—22no—
66#389—24no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#387—24no—
73#387—24no—
74#387—24no—
75#385—24no—
76#384—22no—
77#384—22no—
78#384—22no—
79#382—22no—
80#382—44no—
81#380—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4D412yes—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4Q814yescomplemented
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94#376—22no—
95C2×#95—21no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95D412yes—
99C43—22no—
100#343—24no—
101#343—42no—
102#340—24no—
103#340—24no—
104#339—28no—
105#339—44no—
106C22×Q8—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
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—28no—
122C4⋊Q8—28no—
123#125—28no—
124#125—28no—
125C22⋊Q8—28no—
126C22⋊Q8—28no—
127C4×Q8—84no—
128#117—22no—
129#117—22no—
130#117—22no—
131#117—22no—
132#117—22no—
133#117—22no—
134#117C2×Q814yes—
135#117#4114yes—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×Q812yes—
165C2×C4⋊C4#4112yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C42—22no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—44no—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×Q814yes—
183C2×C42C2×Q814yes—
184C2×C42C2×Q812yes—
185C2×C42#4112yes—
186#107—48no—
187#106—48no—
188C4⋊C8—28no—
189C4⋊C8—28no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—24no—
199#95—48no—
200#95—84no—
201C24C22×D411yescharacteristic
202C2×Q8—28no—
203C2×Q8—82no—
204C2×Q8—82no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4#12011yescharacteristic
235C22×C4#12012yes—
236C22×C4#12011yescharacteristic
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q814yes—
239C22×C4C22⋊Q812yes—
240C22×C4C4⋊Q812yes—
241C22×C4C4⋊Q812yes—
242C22×C4C22×D412yes—
243C22×C4C22×Q811yescharacteristic
244C22×C4C22×Q812yes—
245C22×C4C22×Q811yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4112yes—
248C22×C4C2×#4111yescharacteristic
249C22×C4H(5,𝔽2)11yescharacteristic
250C22×C4H(5,𝔽2)12yes—
251C22×C42−1+411yescharacteristic
252M16—28no—
253M16—28no—
254M16—44no—
255M16—44no—
256C2×C8—28no—
257C2×C8—44no—
258C2×C8—44no—
259C4⋊C4—28no—
260C4⋊C4—216no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—48no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C4⋊C4—82no—
277C4⋊C4—84no—
278C4⋊C4—82no—
279C4⋊C4—84no—
280C4⋊C4—82no—
281C4⋊C4C22⋊Q814yes—
282C4⋊C4C22⋊Q814yes—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C42—24no—
288C42—24no—
289C42—28no—
290C42—42no—
291C42—42no—
292C42—44no—
293C42—44no—
294C42—44no—
295C42—44no—
296C42—82no—
297C42—82no—
298C42—82no—
299C42C22⋊Q814yes—
300C42C22⋊Q814yes—
301C42C4⋊Q814yes—
302C42C4⋊Q814yes—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23C2×#12011yescharacteristic
308C23C2×C22⋊Q811yescharacteristic
309C23C2×C22⋊Q811yescharacteristic
310C23#52511yescharacteristic
311C23C2×C4⋊Q811yescharacteristic
312C23#53511yescharacteristic
313C23#53711yescharacteristic
314Q8—88no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
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—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#39512yes—
358C2×C4#39514yes—
359C2×C4#39512yes—
360C2×C4#45412yes—
361C2×C4#45512yes—
362C2×C4C2×C22⋊Q812yes—
363C2×C4C2×C22⋊Q812yes—
364C2×C4#52512yes—
365C2×C4C2×C4⋊Q812yes—
366C2×C4#53512yes—
367C2×C4#53712yes—
368C2×C4#55412yes—
369C2×C4#55412yes—
370C2×C4#55412yes—
371C2×C4#55711yescharacteristic
372C2×C4#55712yes—
373C2×C4#55711yescharacteristic
374C2×C4Q8211yescharacteristic
375C2×C4Q8212yes—
376C2×C4#55911yescharacteristic
377C2×C4#55912yes—
378C8—48no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22C2×#39511yescharacteristic
392C22#256711yescharacteristic
393C22#258011yescharacteristic
394C22#258211yescharacteristic
395C22#258411yescharacteristic
396C22#261511yescharacteristic
397C22#284811yescharacteristic
398C22C2×#45511yescharacteristic
399C22C2×#45411yescharacteristic
400C22#302111yescharacteristic
401C22#302211yescharacteristic
402C4—22no—
403C4—24no—
404C4—44no—
405C4—44no—
406C4—44no—
407C4—84no—
408C4—82no—
409C4—82no—
410C4—82no—
411C4—162no—
412C4#206114yes—
413C4#256712yes—
414C4#258012yes—
415C4#258212yes—
416C4#258412yes—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2#1520211yescharacteristic
422C2C2×#206111yescharacteristic
423C2#2356211yescharacteristic
424C2#2967911yescharacteristic
425C2#2968011yescharacteristic
426C2#3297011yescharacteristic
427C2#3297111yescharacteristic
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
61. #423Q8semidirect
91. 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
#139112
#113922
#113422
#113392
#108522
#108472