Finite Group Databasebeta

← #468737

Subgroups — #468737

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2058C212yesmaximal
2#15151C211yesmaximal, characteristic
3#14161C211yesmaximal, characteristic
4#11323C212yesmaximal
5#11316C212yesmaximal
6#11257C212yesmaximal
7#11231C212yesmaximal
8#11202C211yesmaximal, characteristic
9#10884C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#106C2212yes—
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C2×#398—21no—
15C2×#398—21no—
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#2058—28no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30#1446—21no—
31#1446—21no—
32#1445—21no—
33#1445—21no—
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C23×Q8—21no—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8D414yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—42no—
66C2×#106—42no—
67C2×#106—42no—
68C2×#106—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—41no—
71#398—44no—
72#398—44no—
73#389—24no—
74#389—24no—
75#384—41no—
76#384—41no—
77C4×C4⋊C4—41no—
78C4×C4⋊C4—41no—
79C4×C4⋊C4—41no—
80C4×C4⋊C4—41no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D411yescharacteristic
92C2×#95D411yescharacteristic
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#340Q818yescomplemented
99#336—24no—
100#336—24no—
101#336—41no—
102C22×Q8—24no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8—82no—
106C22×Q8—82no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
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×C8C2×Q814yes—
131C22×C8#4114yes—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
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—24no—
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—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4SD1614yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×Q814yes—
177C2×C4⋊C4C2×Q814yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42C2×D412yes—
194C2×C42#4112yes—
195#106—48no—
196#106—48no—
197C4⋊C8—44no—
198#95—41no—
199#95—41no—
200#95—42no—
201#95—42no—
202#95—44no—
203#95—44no—
204#95—42no—
205#95—44no—
206#95—82no—
207#95—82no—
208#95—82no—
209#95—82no—
210C24C22×D411yescharacteristic
211C2×Q8—48no—
212C2×Q8—88no—
213C2×Q8—84no—
214C2×Q8—84no—
215C2×Q8—88no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
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#12112yes—
270C22×C4#12112yes—
271C22×C4C2×SD1612yes—
272C22×C4C2×SD1612yes—
273C22×C4C2×SD1612yes—
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4#13712yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×Q812yes—
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C42−1+412yes—
285C22×C42−1+412yes—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—48no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—88no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—42no—
327C42—42no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—84no—
333C23C2×#12111yescharacteristic
334C23#53411yescharacteristic
335C23#53511yescharacteristic
336C23#54712yes—
337C23D4×Q812yes—
338C23C22×SD1611yescharacteristic
339C23C2×AGL(1,ℤ/8)11yescharacteristic
340C23C2×#13711yescharacteristic
341C23#57511yescharacteristic
342C23#57712yes—
343C23#57712yes—
344Q8—816no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
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#46014yes—
405C2×C4#53312yes—
406C2×C4#53411yescharacteristic
407C2×C4#53411yescharacteristic
408C2×C4#53811yescharacteristic
409C2×C4#53811yescharacteristic
410C2×C4#53912yes—
411C2×C4#54112yes—
412C2×C4#54212yes—
413C2×C4#54412yes—
414C2×C4#54412yes—
415C2×C4#54812yes—
416C2×C4D4×Q812yes—
417C2×C4#55111yescharacteristic
418C2×C4#55412yes—
419C2×C4Q8211yescharacteristic
420C2×C4#56311yescharacteristic
421C2×C4#56312yes—
422C2×C4#56311yescharacteristic
423C8—44no—
424C8—44no—
425C22#242912yes—
426C22#261711yescharacteristic
427C22#271711yescharacteristic
428C22#272012yes—
429C22#273411yescharacteristic
430C22C2×#46011yescharacteristic
431C22#303711yescharacteristic
432C22#304212yes—
433C22#318711yescharacteristic
434C22#319311yescharacteristic
435C22#320011yescharacteristic
436C22#320011yescharacteristic
437C22#320611yescharacteristic
438C22#321311yescharacteristic
439C22#322212yes—
440C22#333512yes—
441C22#334412yes—
442C22#334812yes—
443C22#335412yes—
444C22Q8×SD1612yes—
445C22#338412yes—
446C22#338712yes—
447C22#338812yes—
448C4—24no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—81no—
454C4—84no—
455C4—84no—
456C4—84no—
457C4—162no—
458C4#258611yescharacteristic
459C4#266612yes—
460C4#272811yescharacteristic
461C4#272912yes—
462C4#273411yescharacteristic
463C4#283311yescharacteristic
464C2#1662911yescharacteristic
465C2#2545112yes—
466C2#2546412yes—
467C2#2971311yescharacteristic
468C2#2972311yescharacteristic
469C2#3225911yescharacteristic
470C2#3227011yescharacteristic
471C2#3234612yes—
472C2#3237112yes—
473C2#3268311yescharacteristic
474C2#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
98. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
#151512
#141612
#113232
#113162
#112572
#112312
#112022
#108842