Finite Group Databasebeta

← #414183

Subgroups — #414183

Order 512.

1200 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2086C211yesmaximal, characteristic
2C2×#2056C211yesmaximal, characteristic
3C2×#1969C211yesmaximal, characteristic
4C2×#1933C212yesmaximal
5C2×C8×C4⋊C4C211yesmaximal, characteristic
6#15237C212yesmaximal
7#11379C211yesmaximal, characteristic
8#11052C212yesmaximal
9#11024C211yesmaximal, characteristic
10#10926C212yesmaximal
11#10842C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#382C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2086—24no—
28#2056—24no—
29#1969—24no—
30#1933—28no—
31C8×C4⋊C4—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1443—24no—
40#1441—24no—
41C4⋊#95C2212yes—
42C23×Q8C2312yes—
43C22×Q16—24no—
44C22×Q16—22no—
45C22×Q16—22no—
46C22×Q16—22no—
47C22×Q16—42no—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8D414yes—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—42no—
75C2×C4×C8—22no—
76C2×C4×C8—22no—
77C2×C4×C8—22no—
78C2×C4×C8—42no—
79C2×C4×C8—42no—
80C2×C4×C8D414yes—
81C2×C4×C8D414yes—
82#391—24no—
83#391—24no—
84#391—24no—
85#391—24no—
86#391—24no—
87#391—42no—
88#389—24no—
89#389—24no—
90#382—24no—
91#382—42no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C2×#95—24no—
95C2×#95—22no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D412yes—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—24no—
106#340—24no—
107#336—22no—
108#336—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8#4114yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C2×Q16—28no—
133C2×Q16—48no—
134C2×Q16—48no—
135C2×Q16—44no—
136C2×Q16—44no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8#4112yes—
156C22×C8#4112yes—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4#4114yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42#4112yes—
194#107—44no—
195C4⋊C8—28no—
196#103—216no—
197#103—48no—
198#103—48no—
199C4×C8—28no—
200C4×C8—44no—
201C4×C8—44no—
202#95—24no—
203#95—48no—
204#95—48no—
205#95—48no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—48no—
210C24C22×D411yescharacteristic
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
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—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4#12114yes—
276C22×C4#12712yes—
277C22×C4#12712yes—
278C22×C4#13514yes—
279C22×C4#13514yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4112yes—
286C22×C4C2×#4112yes—
287C22×C4C2×#4111yescharacteristic
288C22×C4H(5,𝔽2)12yes—
289C22×C42−1+412yes—
290Q16—416no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8#12114yes—
310C2×C8#12114yes—
311C4⋊C4—216no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—88no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—41no—
333C42—41no—
334C42—42no—
335C23C2×#12111yescharacteristic
336C23C2×#12711yescharacteristic
337C23#53211yescharacteristic
338C23#54612yes—
339C23#54812yes—
340C23C2×#13512yes—
341C23C2×#13512yes—
342C23#57814yes—
343Q8—44no—
344Q8—44no—
345Q8—44no—
346Q8—44no—
347Q8—82no—
348Q8—82no—
349Q8—82no—
350Q8—82no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—48no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—82no—
375C2×C4—84no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—84no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4#49514yes—
392C2×C4C2×#12111yescharacteristic
393C2×C4C2×#12111yescharacteristic
394C2×C4#52512yes—
395C2×C4#52511yescharacteristic
396C2×C4#52912yes—
397C2×C4#53211yescharacteristic
398C2×C4#53211yescharacteristic
399C2×C4#53211yescharacteristic
400C2×C4#53411yescharacteristic
401C2×C4#53611yescharacteristic
402C2×C4#53712yes—
403C2×C4#53912yes—
404C2×C4#54612yes—
405C2×C4#54812yes—
406C2×C4#55112yes—
407C2×C4#55212yes—
408C2×C4#55312yes—
409C2×C4#55612yes—
410C8—24no—
411C8—24no—
412C8—42no—
413C8—42no—
414C22#253611yescharacteristic
415C22#254012yes—
416C22#254311yescharacteristic
417C22#256012yes—
418C22#285611yescharacteristic
419C22#305012yes—
420C22#305912yes—
421C22C2×#49511yescharacteristic
422C22#314711yescharacteristic
423C22#315612yes—
424C22#316812yes—
425C22#317712yes—
426C22#330014yes—
427C22#330214yes—
428C22#336414yes—
429C22#336414yes—
430C4—24no—
431C4—44no—
432C4—44no—
433C4—42no—
434C4—84no—
435C4—84no—
436C4—162no—
437C4—162no—
438C4#239211yescharacteristic
439C4#253411yescharacteristic
440C4#254511yescharacteristic
441C4#255812yes—
442C4#260112yes—
443C4#261811yescharacteristic
444C2#1516311yescharacteristic
445C2#2819012yes—
446C2#2840312yes—
447C2#2840712yes—
448C2#2855412yes—
449C2#2907414yes—
450C2#3317311yescharacteristic
451C2#3318011yescharacteristic
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

Maximal subgroups

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

subgroupindex
C2×#20862
C2×#20562
C2×#19692
C2×#19332
C2×C8×C4⋊C42
#152372
#113792
#110522
#110242
#109262
#108422