Finite Group Databasebeta

← #225608

Subgroups — #225608

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22766C212yesmaximal
2#22658C211yesmaximal, characteristic, complemented
3D4×#107C211yesmaximal, characteristic, complemented
4C2×C8×D8C211yesmaximal, characteristic
5#9616C212yesmaximal, complemented
6#8897C212yesmaximal, complemented
7#8894C212yesmaximal, complemented
8#8816C211yesmaximal, characteristic, complemented
9#8644C211yesmaximal, characteristic, complemented
10#8644C211yesmaximal, characteristic, complemented
11C8×#107C211yesmaximal, characteristic, complemented
12C2×C4×D8C2211yescharacteristic
13C2×C8×D4C2211yescharacteristic
14C2×C8×D4C2211yescharacteristic
15C2×#427—21no—
16C22×#107—21no—
17#2354C2212yes—
18D4×C4⋊C4C2212yes—
19#1948C2212yes—
20#1947C2212yes—
21#1934—22no—
22#1918—22no—
23#1879—22no—
24#1854—22no—
25#1852—22no—
26#1845C2211yescharacteristic
27#1845C2211yescharacteristic
28#1834C2212yes—
29#1834C2212yes—
30#1808—21no—
31#1808—21no—
32#1802C2212yes—
33#1802C2212yes—
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#102—22no—
37C8×D8C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#335C2211yescharacteristic
40C2×D4⋊C8C2211yescharacteristic
41C2×D4⋊C8C2211yescharacteristic
42C2×C82C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1279C2212yes—
46C22×D8D411yescharacteristic
47C23×C8—21no—
48C23×C8—21no—
49C2×C4×D4—22no—
50C2×C4×D4C2311yescharacteristic
51C2×C4×D4C2311yescharacteristic
52C2×#117—22no—
53C22×C4⋊C4—22no—
54C4×D8C2×C414yes—
55C4×D8Q814yes—
56C8×D4—22no—
57C8×D4—22no—
58C8×D4C2×C414yes—
59C8×D4C2×C414yescomplemented
60#427—44no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—22no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—44no—
67C2×#107D411yescharacteristic
68C2×#107D411yescharacteristic
69C2×#107D411yescharacteristic
70C2×#106—22no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×#102—22no—
75C2×#102—22no—
76C2×#102D411yescharacteristic
77C2×#102D411yescharacteristic
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C4×C8—22no—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84#390—22no—
85#389—22no—
86#388—42no—
87#385—42no—
88#384C2312yes—
89#384C2312yes—
90#380—22no—
91#380—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2312yes—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—42no—
101#340D412yes—
102#336—22no—
103#335—22no—
104#335C2×C414yescomplemented
105D4⋊C8C2×C414yes—
106D4⋊C8C2×C414yescomplemented
107C82C2×C414yescomplemented
108C22×D4C2×D411yescharacteristic
109C22×D4C2×D411yescharacteristic
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—42no—
113C2×D8—22no—
114C2×D8—22no—
115C2×D8C4⋊C414yes—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—24no—
120C22×C8—24no—
121C22×C8—22no—
122C22×C8—42no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4—44no—
135C4×D4—82no—
136C4×D4C22×C412yes—
137C4×D4C22×C412yes—
138C4×D4C2×Q812yes—
139C4×D4C2×Q812yes—
140#117—44no—
141#117—82no—
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—44no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D412yes—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C42—22no—
161C2×C42—42no—
162C2×C42C2411yescharacteristic
163#107—22no—
164#107—42no—
165#107—42no—
166#107—44no—
167#107—81no—
168#107—81no—
169#107C4⋊C414yes—
170#107D814yescomplemented
171#106—44no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8C22×C412yes—
177C4⋊C8C22×C412yes—
178#102—44no—
179#102—82no—
180#102C4⋊C414yes—
181#102C4⋊C414yes—
182C22⋊C8—24no—
183C22⋊C8—24no—
184C4×C8—21no—
185C4×C8—82no—
186C4×C8C22×C412yes—
187C4×C8C22×C412yes—
188C4×C8C22×C412yes—
189C4×C8C2×Q812yes—
190C4×C8#4112yes—
191#95—22no—
192#95—42no—
193#95—42no—
194C24—21no—
195C24—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4C2×C4⋊C412yes—
205C2×D4C2×C4⋊C412yes—
206C2×D4#13512yes—
207C2×D4#13512yes—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—84no—
227C22×C4—84no—
228C22×C4—162no—
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4H(5,𝔽2)11yescharacteristic
232D8—28no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—48no—
249C2×C8—48no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8C2×C4⋊C412yes—
254C2×C8C2×C4⋊C412yes—
255C2×C8C4×D412yes—
256C2×C8C4×D412yes—
257C2×C8C2×D812yes—
258C2×C8#13512yes—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—84no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4C2×C4⋊C412yes—
276C4⋊C4C2×C4⋊C412yes—
277C4⋊C4C2×D814yes—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—84no—
283C22⋊C4—84no—
284C22⋊C4—162no—
285C42—42no—
286C42—42no—
287C42—84no—
288C42C23×C411yescharacteristic
289C42C22×Q811yescharacteristic
290C42C2×#4111yescharacteristic
291C422−1+411yescharacteristic
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23D4211yescharacteristic
307D4—22no—
308D4—22no—
309D4—22no—
310D4—22no—
311D4—22no—
312D4—41no—
313D4—41no—
314D4#42714yescomplemented
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—168no—
344C2×C4C22×C4⋊C411yescharacteristic
345C2×C4C2×C4×D411yescharacteristic
346C2×C4#52011yescharacteristic
347C2×C4#54711yescharacteristic
348C2×C4D4×Q811yescharacteristic
349C2×C4#55411yescharacteristic
350C2×C4C22×D811yescharacteristic
351C2×C4C2×#13511yescharacteristic
352C2×C4C2×#13511yescharacteristic
353C2×C4C2×#13511yescharacteristic
354C2×C4#57611yescharacteristic
355C2×C4#57611yescharacteristic
356C2×C4#57811yescharacteristic
357C2×C4#57811yescharacteristic
358C8—22no—
359C8—22no—
360C8—22no—
361C8—42no—
362C8—42no—
363C8—42no—
364C8—82no—
365C8C4×D814yes—
366C22—24no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22D4×C4⋊C411yescharacteristic
381C22D4×D811yescharacteristic
382C22#328011yescharacteristic
383C22#328311yescharacteristic
384C22#328511yescharacteristic
385C22#329011yescharacteristic
386C22#329011yescharacteristic
387C4—21no—
388C4—21no—
389C4—44no—
390C4—42no—
391C4—42no—
392C4—82no—
393C4—162no—
394C4—162no—
395C4C2×#42711yescharacteristic
396C4#291311yescharacteristic
397C4C2×C4×D811yescharacteristic
398C4#295111yescharacteristic
399C4#333611yescharacteristic
400C4#333911yescharacteristic
401C4Q8×D811yescharacteristic
402C4#338011yescharacteristic
403C2—44no—
404C2—82no—
405C2#1996811yescharacteristic
406C2#1997311yescharacteristic
407C2#1998911yescharacteristic
408C2#2001111yescharacteristic
409C2#2265611yescharacteristic
410C2C4⋊C4×D811yescharacteristic
411C2#2269111yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
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
2. #22658C2semidirect
3. D4×#107C2semidirect
5. #9616C2semidirect
6. #8897C2semidirect
7. #8894C2semidirect
8. #8816C2semidirect
9. #8644C2semidirect
10. #8644C2semidirect
11. C8×#107C2semidirect
37. C8×D8C4semidirect
59. C8×D4C2×C4semidirect
104. #335C2×C4semidirect
106. D4⋊C8C2×C4semidirect
107. C82C2×C4semidirect
170. #107D8semidirect
314. D4#427semidirect

Maximal subgroups

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

subgroupindex
#227662
#226582
D4×#1072
C2×C8×D82
#96162
#88972
#88942
#88162
#86442
C8×#1072