Finite Group Databasebeta

← #438779

Subgroups — #438779

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2058C212yesmaximal
2C2×#1943C212yesmaximal
3#15128C211yesmaximal, characteristic
4#13910C211yesmaximal, characteristic
5#11327C212yesmaximal
6#11323C212yesmaximal
7#11219C212yesmaximal
8#10907C212yesmaximal
9#10875C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#107C2212yes—
12C22×#106C2212yes—
13C22×#106C2212yes—
14C22×#97C2211yescharacteristic
15C2×#395—21no—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#384—21no—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#2058—28no—
24#1943—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1446—21no—
34#1444—21no—
35#1444—21no—
36#1444—21no—
37#1441—21no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×Q8—21no—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—42no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#107—24no—
65C2×#107—24no—
66C2×#107Q818yescomplemented
67C2×#106—24no—
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—42no—
71C2×#106—42no—
72C2×#106—42no—
73C2×#97—24no—
74C2×#97—41no—
75#395—44no—
76#389—24no—
77#389—24no—
78#389—24no—
79#389—42no—
80#389—42no—
81#389—42no—
82#384—41no—
83#384—41no—
84#384—44no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—41no—
87C4×C4⋊C4—41no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98#340—24no—
99#340—24no—
100#340—24no—
101#340—42no—
102#340D418yes—
103#336—24no—
104#336—41no—
105#336Q818yescomplemented
106C22×Q8—44no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—82no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8C2×D414yes—
133C22×C8C2×Q814yes—
134C22×C8C2×Q814yes—
135C22×C8C2×Q814yes—
136C4⋊Q8—44no—
137C4⋊Q8—48no—
138C4⋊Q8—44no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2×Q814yes—
183C2×C4⋊C4C2×Q814yes—
184C2×C4⋊C4#4114yes—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C2×Q814yes—
200#107—216no—
201#106—48no—
202#106—48no—
203#97—44no—
204#95—24no—
205#95—41no—
206#95—41no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—44no—
213#95—82no—
214C24C22×D411yescharacteristic
215C2×Q8—48no—
216C2×Q8—88no—
217C2×Q8—84no—
218C2×Q8—88no—
219C2×Q8—84no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4C4⋊Q812yes—
275C22×C4C4⋊Q812yes—
276C22×C4AGL(1,ℤ/8)12yes—
277C22×C4AGL(1,ℤ/8)12yes—
278C22×C4#13712yes—
279C22×C4#13712yes—
280C22×C4C22×D412yes—
281C22×C4C22×Q812yes—
282C22×C4C22×Q812yes—
283C22×C4C2×#4112yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)12yes—
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C42−1+412yes—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8C4⋊Q818yes—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—48no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—88no—
321C4⋊C4—84no—
322C4⋊C4—88no—
323C4⋊C4—88no—
324C4⋊C4—84no—
325C4⋊C4C22⋊Q818yes—
326C42—24no—
327C42—24no—
328C42—48no—
329C42—42no—
330C42—42no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C23#52511yescharacteristic
336C23C2×C4⋊Q811yescharacteristic
337C23#53411yescharacteristic
338C23D4212yes—
339C23D4×Q812yes—
340C23C2×AGL(1,ℤ/8)11yescharacteristic
341C23C2×AGL(1,ℤ/8)11yescharacteristic
342C23C2×#13711yescharacteristic
343C23C2×#13711yescharacteristic
344C23#57712yes—
345C23#57712yes—
346Q8—816no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#50114yes—
407C2×C4C2×C22⋊Q812yes—
408C2×C4C2×C4⋊Q812yes—
409C2×C4#53512yes—
410C2×C4#53912yes—
411C2×C4#54412yes—
412C2×C4#54412yes—
413C2×C4D4211yescharacteristic
414C2×C4#54712yes—
415C2×C4D4×Q812yes—
416C2×C4#55412yes—
417C2×C4#55412yes—
418C2×C4#55412yes—
419C2×C4Q8211yescharacteristic
420C2×C4Q8212yes—
421C2×C4#56011yescharacteristic
422C2×C4#56311yescharacteristic
423C8—28no—
424C8—44no—
425C22#244912yes—
426C22#255112yes—
427C22#258311yescharacteristic
428C22#266911yescharacteristic
429C22#273411yescharacteristic
430C22#310911yescharacteristic
431C22#311311yescharacteristic
432C22#312512yes—
433C22C2×#50111yescharacteristic
434C22#316312yes—
435C22#316511yescharacteristic
436C22#319311yescharacteristic
437C22#319311yescharacteristic
438C22#320011yescharacteristic
439C22#320011yescharacteristic
440C22#326512yes—
441C22#326612yes—
442C22#327212yes—
443C22#327812yes—
444C22#338412yes—
445C22#338412yes—
446C22#338712yes—
447C22#338712yes—
448C4—24no—
449C4—48no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—84no—
454C4—81no—
455C4—84no—
456C4—81no—
457C4—162no—
458C4#244012yes—
459C4#256712yes—
460C4#272012yes—
461C4#279212yes—
462C2#1562511yescharacteristic
463C2#2638912yes—
464C2#2639212yes—
465C2#2881712yes—
466C2#2881912yes—
467C2#2930111yescharacteristic
468C2#2931111yescharacteristic
469C2#3051011yescharacteristic
470C2#3051611yescharacteristic
471C2#3269411yescharacteristic
472C2#3269411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
66. C2×#107Q8semidirect
105. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
C2×#19432
#151282
#139102
#113272
#113232
#112192
#109072
#108752