Finite Group Databasebeta

← #465217

Subgroups — #465217

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2084C212yesmaximal
2C2×#1948C211yesmaximal, characteristic
3C2×#1919C211yesmaximal, characteristic
4#15145C211yesmaximal, characteristic
5#13933C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11316C212yesmaximal
8#11267C212yesmaximal
9#11255C211yesmaximal, characteristic
10#11227C211yesmaximal, characteristic
11#11220C212yesmaximal
12C22×#106C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#398—21no—
16C2×#398—21no—
17C2×#389C2211yescharacteristic
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×#384C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22#2084—28no—
23#1948—24no—
24#1919—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1444—22no—
37#1442—22no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43C4×#95C2211yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42—21no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—24no—
67C2×#106D414yes—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—22no—
70#398—44no—
71#398—44no—
72#389—24no—
73#389—22no—
74#389—22no—
75#389—42no—
76#384—24no—
77#384—22no—
78#384—22no—
79#384—42no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4D414yes—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95D411yescharacteristic
92C2×#95D412yes—
93C2×#95D411yescharacteristic
94#340—24no—
95#340—24no—
96#340—24no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—42no—
102#336—24no—
103#336—24no—
104#336—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8C2×D412yes—
132C22×C8#4114yes—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
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—21no—
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—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—81no—
176C2×C4⋊C4—81no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4#4112yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—82no—
193C2×C42C2×D412yes—
194C2×C42#4112yes—
195#106—28no—
196#106—48no—
197C4⋊C8—28no—
198#95—24no—
199#95—22no—
200#95—24no—
201#95—44no—
202#95—44no—
203#95—42no—
204#95—44no—
205#95—42no—
206#95—82no—
207#95—81no—
208#95—82no—
209#95SD1614yes—
210C24C22×D411yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
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#12112yes—
266C22×C4#12112yes—
267C22×C4C2×SD1614yes—
268C22×C4C2×SD1612yes—
269C22×C4AGL(1,ℤ/8)12yes—
270C22×C4#13714yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4C2×#4111yescharacteristic
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C42−1+412yes—
278C22×C42−1+412yes—
279C22×C42−1+412yes—
280C22×C42−1+411yescharacteristic
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
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—88no—
320C4⋊C4—88no—
321C4⋊C4—84no—
322C4⋊C4—88no—
323C4⋊C4—84no—
324C4⋊C4—84no—
325C4⋊C4#12114yes—
326C42—22no—
327C42—22no—
328C42—24no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—84no—
334C23C2×#12111yescharacteristic
335C23#53612yes—
336C23D4211yescharacteristic
337C23#54712yes—
338C23#54811yescharacteristic
339C23C22×SD1611yescharacteristic
340C23C2×AGL(1,ℤ/8)11yescharacteristic
341C23C2×#13712yes—
342C23#57611yescharacteristic
343C23#57712yes—
344C23#57711yescharacteristic
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—81no—
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—81no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4#46014yes—
405C2×C4C2×#12111yescharacteristic
406C2×C4#52511yescharacteristic
407C2×C4#53611yescharacteristic
408C2×C4#53611yescharacteristic
409C2×C4#53611yescharacteristic
410C2×C4#53612yes—
411C2×C4#53711yescharacteristic
412C2×C4#54112yes—
413C2×C4#54112yes—
414C2×C4#54112yes—
415C2×C4#54212yes—
416C2×C4#54711yescharacteristic
417C2×C4#54812yes—
418C2×C4#55212yes—
419C2×C4#55212yes—
420C2×C4#55311yescharacteristic
421C2×C4#56312yes—
422C2×C4#56412yes—
423C8—42no—
424C8—44no—
425C8—81no—
426C22#241311yescharacteristic
427C22#241611yescharacteristic
428C22#262111yescharacteristic
429C22#267412yes—
430C22#271912yes—
431C22C2×#46011yescharacteristic
432C22#303911yescharacteristic
433C22#304211yescharacteristic
434C22#304511yescharacteristic
435C22#322912yes—
436C22#323912yes—
437C22#324112yes—
438C22#324412yes—
439C22#326612yes—
440C22#327411yescharacteristic
441C22D4×SD1611yescharacteristic
442C22#333512yes—
443C22#334512yes—
444C22#334612yes—
445C22#335612yes—
446C22#336211yescharacteristic
447C22#336811yescharacteristic
448C22#337212yes—
449C4—24no—
450C4—42no—
451C4—44no—
452C4—44no—
453C4—44no—
454C4—82no—
455C4—82no—
456C4—161no—
457C4—162no—
458C4—162no—
459C4#253611yescharacteristic
460C4#262111yescharacteristic
461C4#263711yescharacteristic
462C4#271912yes—
463C4#276612yes—
464C4#282211yescharacteristic
465C2#1651711yescharacteristic
466C2#2457811yescharacteristic
467C2#2459611yescharacteristic
468C2#2472811yescharacteristic
469C2#2474811yescharacteristic
470C2#2986611yescharacteristic
471C2#2987411yescharacteristic
472C2#3077712yes—
473C2#3079912yes—
474C2#3231212yes—
475C2#3233012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20842
C2×#19482
C2×#19192
#151452
#139332
#113732
#113162
#112672
#112552
#112272
#112202