Finite Group Databasebeta

← #364299

Subgroups — #364299

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1767C211yesmaximal, characteristic
2#14688C211yesmaximal, characteristic
3#13755C211yesmaximal, characteristic
4#11232C212yesmaximal
5#11231C212yesmaximal
6#10908C212yesmaximal
7#10907C212yesmaximal
8#10863C212yesmaximal
9#10821C212yesmaximal
10C22×#107C2212yes—
11C22×#106C2212yes—
12C22×C4⋊C8C2212yes—
13C22×#97C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C2×#383—21no—
16C2×C4×C4⋊C4C2212yes—
17C2×#376—21no—
18C2×C43C2211yescharacteristic
19#1767C418yescomplemented
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C4⋊#95C2212yes—
28C4⋊#95C2212yes—
29C4⋊#95C2211yescharacteristic
30C4⋊#95C2211yescharacteristic
31C4⋊#95C2212yes—
32C4⋊#95C2211yescharacteristic
33C4⋊#95C2211yescharacteristic
34#1434—21no—
35#1434—21no—
36#1434—21no—
37#1434—21no—
38#1434—21no—
39C4×#95—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C42—22no—
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×#107—24no—
55C2×#107—24no—
56C2×#107—42no—
57C2×#106—24no—
58C2×#106—24no—
59C2×#106—42no—
60C2×C4⋊C8—24no—
61C2×C4⋊C8—24no—
62C2×C4⋊C8C2×C418yes—
63C2×#97—41no—
64C2×#97—41no—
65C2×#97—41no—
66C2×#97—41no—
67C2×#97C2×C414yes—
68C2×#97C2×C414yes—
69#383—44no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—24no—
72#376—44no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—22no—
88C43C2×C414yes—
89C43Q814yes—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#336C2×C418yes—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2411yescharacteristic
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8C22×C414yes—
119C22×C8C22×C414yes—
120C22×C8#4114yes—
121C22×C8#4114yes—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
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#4114yes—
145C2×C4⋊C4#4114yes—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—28no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42C4⋊C414yes—
164C2×C42C4⋊C414yes—
165C2×C42C4⋊C414yes—
166C2×C42C4⋊C414yes—
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C2×Q812yes—
171C2×C42C2×Q812yes—
172C2×C42C2×Q812yes—
173#107—48no—
174#106—48no—
175C4⋊C8—216no—
176#97—44no—
177#97—44no—
178#95—28no—
179#95—44no—
180#95—44no—
181#95—44no—
182#95—42no—
183#95—42no—
184#95—44no—
185#95—42no—
186#95—42no—
187#95—44no—
188#95—42no—
189#95—42no—
190#95—44no—
191#95—42no—
192#95—42no—
193C24C22×D411yescharacteristic
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—42no—
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—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C2×C4⋊C412yes—
245C22×C4C2×C4⋊C412yes—
246C22×C4C2×C4⋊C412yes—
247C22×C4C2×C4⋊C412yes—
248C22×C4C2×C4⋊C412yes—
249C22×C4C2×C4⋊C412yes—
250C22×C4AGL(1,ℤ/8)12yes—
251C22×C4AGL(1,ℤ/8)12yes—
252C22×C4#13712yes—
253C22×C4#13712yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×Q811yescharacteristic
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C22×C4C2×#4112yes—
259C22×C4C2×#4112yes—
260C22×C4H(5,𝔽2)11yescharacteristic
261C22×C4H(5,𝔽2)11yescharacteristic
262C22×C42−1+411yescharacteristic
263C22×C42−1+411yescharacteristic
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8#11718yes—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C42—24no—
295C42—24no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—28no—
305C42—28no—
306C42—44no—
307C42—44no—
308C42—416no—
309C23C22×C4⋊C411yescharacteristic
310C23#52512yes—
311C23#53212yes—
312C23#53711yescharacteristic
313C23#53711yescharacteristic
314C23C2×AGL(1,ℤ/8)11yescharacteristic
315C23C2×AGL(1,ℤ/8)11yescharacteristic
316C23C2×#13711yescharacteristic
317C23C2×#13711yescharacteristic
318C23#57512yes—
319C23#57512yes—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4#42814yes—
379C2×C4#42814yes—
380C2×C4C2×#11712yes—
381C2×C4C4×#4112yes—
382C2×C4#51811yescharacteristic
383C2×C4#51911yescharacteristic
384C2×C4#53311yescharacteristic
385C2×C4#53311yescharacteristic
386C2×C4#53311yescharacteristic
387C2×C4#53311yescharacteristic
388C2×C4#54411yescharacteristic
389C2×C4#54411yescharacteristic
390C2×C4#55112yes—
391C2×C4#55112yes—
392C2×C4#55212yes—
393C2×C4#55212yes—
394C2×C4#55312yes—
395C2×C4#55312yes—
396C2×C4#55612yes—
397C2×C4#55612yes—
398C8—28no—
399C8—44no—
400C22#228511yescharacteristic
401C22#229611yescharacteristic
402C22#231611yescharacteristic
403C22#260112yes—
404C22#260112yes—
405C22C2×#42811yescharacteristic
406C22C2×#42811yescharacteristic
407C22#291111yescharacteristic
408C22#291111yescharacteristic
409C22#311412yes—
410C22#311512yes—
411C22#311612yes—
412C22#311712yes—
413C22#317112yes—
414C22#317212yes—
415C22#318012yes—
416C22#318112yes—
417C22#325311yescharacteristic
418C22#325411yescharacteristic
419C22#325411yescharacteristic
420C22#325511yescharacteristic
421C22#325612yes—
422C22#325612yes—
423C4—24no—
424C4—24no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—84no—
430C4—84no—
431C4—162no—
432C4—162no—
433C4#235312yes—
434C4#235612yes—
435C4#260511yescharacteristic
436C4#260511yescharacteristic
437C4#260611yescharacteristic
438C4#260611yescharacteristic
439C2#1396411yescharacteristic
440C2#2081911yescharacteristic
441C2#2082311yescharacteristic
442C2#2136311yescharacteristic
443C2#2136411yescharacteristic
444C2#2196411yescharacteristic
445C2#2196511yescharacteristic
446C2#2944512yes—
447C2#2944712yes—
448C2#2944912yes—
449C2#2945112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1767C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17672
#146882
#137552
#112322
#112312
#109082
#109072
#108632
#108212