Finite Group Databasebeta

← #229259

Subgroups — #229259

Order 512.

812 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23185C211yesmaximal, characteristic, complemented
2#23093C212yesmaximal, complemented
3#22658C211yesmaximal, characteristic, complemented
4C2×#1667C211yesmaximal, characteristic
5#9824C211yesmaximal, characteristic, complemented
6#9614C211yesmaximal, characteristic, complemented
7#9564C212yesmaximal, complemented
8#8980C212yesmaximal, complemented
9#8921C212yesmaximal, complemented
10C8×#107C211yesmaximal, characteristic, complemented
11C8×#102C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#459C2212yes—
14C2×C4×D8—21no—
15C2×C8×D4C2211yescharacteristic
16C2×#427—21no—
17#2366C2212yes—
18#2354C2212yes—
19#1982C2212yes—
20#1970—21no—
21#1970—22no—
22#1957C2212yes—
23#1957C2212yes—
24#1948C2212yes—
25#1947C2212yes—
26#1925—22no—
27#1879—22no—
28#1879—22no—
29#1854—22no—
30#1845C2211yescharacteristic
31#1808—21no—
32C4×#107C2211yescharacteristic
33C8×C4⋊C4C2211yescharacteristic
34C8×C4⋊C4C2212yes—
35C4×#102—22no—
36C4×#102C2211yescharacteristic
37#1667C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#331C2212yes—
40C2×D4⋊C8C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2212yes—
45#1277C2211yescharacteristic
46C22×D8—22no—
47C22×D8—21no—
48C22×D8—21no—
49C23×C8—21no—
50C2×#127C2312yes—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4C2311yescharacteristic
54C2×#117—22no—
55#493—22no—
56#493C2×C414yes—
57#459—24no—
58#459C2×C418yescomplemented
59C4×D8—44no—
60C8×D4—22no—
61C8×D4C2×C414yescomplemented
62#427—44no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—22no—
66C2×#107D411yescharacteristic
67C2×#107D411yescharacteristic
68C2×#106—22no—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2312yes—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—22no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—42no—
78C2×#102D411yescharacteristic
79C2×#102D411yescharacteristic
80C2×C22⋊C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—22no—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#390—22no—
87#390—22no—
88#390—22no—
89#388—42no—
90#386—42no—
91#384C2312yes—
92#380—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2312yes—
95C4×C4⋊C4C2311yescharacteristic
96#340—22no—
97#340—22no—
98#340D412yes—
99#340D411yescharacteristic
100#340D411yescharacteristic
101#336—21no—
102#336—22no—
103#335C2×C414yescomplemented
104#331C2×C418yescomplemented
105D4⋊C8—22no—
106C82—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4C2×D411yescharacteristic
111C23×C4—21no—
112C23×C4—42no—
113C2×D8—28no—
114C2×D8—24no—
115C2×D8—24no—
116C2×D8—42no—
117C2×D8—42no—
118C2×D8—81no—
119C2×D8—81no—
120C2×D8—82no—
121C2×D8—82no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—24no—
126C22×C8—21no—
127C22×C8—42no—
128C22×C8—42no—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133#127—22no—
134#127—22no—
135#127—22no—
136#127C22×C414yes—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—44no—
140C4×D4—44no—
141C4×D4—82no—
142C4×D4—82no—
143C4×D4C22×C412yes—
144C4×D4#4112yes—
145#117—44no—
146#117—82no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C22⋊C4—21no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C42—21no—
162C2×C42—22no—
163C2×C42—42no—
164C2×C42C2411yescharacteristic
165#107—22no—
166#107—42no—
167#107—82no—
168#107—81no—
169#107D814yescomplemented
170#106—44no—
171C4⋊C8—21no—
172C4⋊C8—22no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8C22×C412yes—
176C4⋊C8C22×C414yes—
177#102—22no—
178#102—28no—
179#102—22no—
180#102—44no—
181#102—44no—
182#102—82no—
183#102—82no—
184#102—84no—
185C22⋊C8—24no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8—21no—
189C4×C8—21no—
190C4×C8—42no—
191C4×C8—82no—
192C4×C8C22×C412yes—
193C4×C8#4112yes—
194#95—21no—
195#95—22no—
196#95—42no—
197C24—21no—
198C24—42no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4C4×D412yes—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—84no—
234C22×C4—84no—
235C22×C4—162no—
236C22×C4C22×D411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4H(5,𝔽2)11yescharacteristic
239D8—48no—
240D8—84no—
241D8—88no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—48no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8C4×D412yes—
263C2×C8C4×D412yes—
264C2×C8C2×D812yes—
265C2×C8#13512yes—
266C2×C8#13512yes—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—82no—
279C4⋊C4C4×D412yes—
280C4⋊C4C2×D814yes—
281C4⋊C4#13512yes—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—84no—
285C22⋊C4—84no—
286C22⋊C4—84no—
287C22⋊C4—162no—
288C22⋊C4—162no—
289C42—41no—
290C42—42no—
291C42—42no—
292C42—41no—
293C42—84no—
294C42C23×C411yescharacteristic
295C42C2×#4111yescharacteristic
296C42C2×#4111yescharacteristic
297C422−1+411yescharacteristic
298C23—22no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—44no—
306C23—82no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23D4211yescharacteristic
313D4—22no—
314D4—22no—
315D4—41no—
316D4—41no—
317D4—44no—
318D4—44no—
319D4—44no—
320D4—44no—
321D4—84no—
322D4—82no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—162no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—168no—
352C2×C4C2×C4×D411yescharacteristic
353C2×C4C2×C4×D411yescharacteristic
354C2×C4#52011yescharacteristic
355C2×C4#54711yescharacteristic
356C2×C4#54811yescharacteristic
357C2×C4#55311yescharacteristic
358C2×C4C22×D811yescharacteristic
359C2×C4C2×#13511yescharacteristic
360C2×C4C2×#13511yescharacteristic
361C2×C4C2×#13511yescharacteristic
362C2×C4#57611yescharacteristic
363C2×C4#57611yescharacteristic
364C2×C4#57611yescharacteristic
365C2×C4#57711yescharacteristic
366C8—22no—
367C8—22no—
368C8—42no—
369C8—41no—
370C8—42no—
371C8—44no—
372C8—81no—
373C8C4×D814yes—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—84no—
382C22—84no—
383C22—84no—
384C22—162no—
385C22—162no—
386C22—162no—
387C22—162no—
388C22#236111yescharacteristic
389C22D4×D811yescharacteristic
390C22#328011yescharacteristic
391C22#328411yescharacteristic
392C22#328711yescharacteristic
393C22#329011yescharacteristic
394C22#329211yescharacteristic
395C4—21no—
396C4—21no—
397C4—44no—
398C4—42no—
399C4—42no—
400C4—81no—
401C4—82no—
402C4—162no—
403C4C2×C4×D811yescharacteristic
404C4C4×#13511yescharacteristic
405C4#294611yescharacteristic
406C4#294811yescharacteristic
407C4#333911yescharacteristic
408C4#334111yescharacteristic
409C4#335811yescharacteristic
410C4#336111yescharacteristic
411C2—82no—
412C2—164no—
413C2#1996811yescharacteristic
414C2#1997511yescharacteristic
415C2#1999411yescharacteristic
416C2#2001811yescharacteristic
417C2#2299511yescharacteristic
418C2#2302711yescharacteristic
419C2#2304611yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#1667C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23185C2semidirect
2. #23093C2semidirect
3. #22658C2semidirect
5. #9824C2semidirect
6. #9614C2semidirect
7. #9564C2semidirect
8. #8980C2semidirect
9. #8921C2semidirect
10. C8×#107C2semidirect
11. C8×#102C2semidirect
37. #1667C4semidirect
58. #459C2×C4semidirect
61. C8×D4C2×C4semidirect
103. #335C2×C4semidirect
104. #331C2×C4semidirect
169. #107D8semidirect

Maximal subgroups

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

subgroupindex
#231852
#230932
#226582
C2×#16672
#98242
#96142
#95642
#89802
#89212
C8×#1072
C8×#1022