Finite Group Databasebeta

← #498454

Subgroups — #498454

Order 512.

561 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33579C211yesmaximal, characteristic
2#23179C211yesmaximal, characteristic, complemented
3#22802C211yesmaximal, characteristic, complemented
4#13242C212yesmaximal, complemented
5#13208C211yesmaximal, characteristic, complemented
6#13179C211yesmaximal, characteristic, complemented
7#13134C212yesmaximal, complemented
8#13011C212yesmaximal, complemented
9#12662C211yesmaximal, characteristic, complemented
10#12642C212yesmaximal, complemented
11C4×#359C211yesmaximal, characteristic, complemented
12C2×#508C2212yes—
13#3147C2211yescharacteristic
14C2×#443C2212yes—
15C4×#135C2211yescharacteristic
16#2935—21no—
17#2925—21no—
18#2211C412yescomplemented
19#2210C412yescomplemented
20#2209C411yescharacteristic, complemented
21#2208C411yescharacteristic, complemented
22#2207C411yescharacteristic, complemented
23#2206C411yescharacteristic, complemented
24C2×#368C2212yes—
25C2×C4⋊C16C2211yescharacteristic
26C2×#359C2211yescharacteristic
27C2×#359C2211yescharacteristic
28C2×#359C2212yes—
29C2×#359C2212yes—
30#2138C2212yes—
31C2×#346C2212yes—
32C2×C4×C16C2211yescharacteristic
33#1980—21no—
34#1974C2212yes—
35#1974C2212yes—
36#1929—22no—
37#1876—22no—
38#1848C2211yescharacteristic
39#1806—21no—
40C4×#108C2211yescharacteristic
41#1772C2212yes—
42C8×C4⋊C4C2211yescharacteristic
43#1382C2211yescharacteristic
44#1382C2211yescharacteristic
45#1381C2212yes—
46#1374C2212yes—
47C2×#135—21no—
48C2×#135C2312yes—
49C2×#135C2311yescharacteristic
50C2×#131—22no—
51#532—21no—
52C4×#41—21no—
53#508—42no—
54#508—42no—
55C2×Q32C2×C412yes—
56C2×SD32C2×C412yes—
57C2×SD32C2×C412yes—
58C2×D16C2×C412yes—
59C22×C16C2311yescharacteristic
60C22×C16C2312yes—
61C22×C16C2311yescharacteristic
62#495—21no—
63#495C2×C412yes—
64#494C2×C411yescharacteristic
65#493C2×C411yescharacteristic
66#492—21no—
67#443—24no—
68#443—42no—
69C4×Q16C2×C411yescharacteristic
70C4×Q16C2×C411yescharacteristic
71C4×SD16—21no—
72C4×SD16—21no—
73C4×D8C2×C411yescharacteristic
74C4×D8C2×C411yescharacteristic
75#436—41no—
76#435—42no—
77C8×D4—41no—
78#433—42no—
79#432—41no—
80#431—41no—
81C2×#108—21no—
82C2×#108C2312yes—
83C2×#108C2311yescharacteristic
84C2×#108C2311yescharacteristic
85#427C2311yescharacteristic
86#423—21no—
87C2×C4⋊C8D411yescharacteristic
88C2×#104—22no—
89C2×#104—22no—
90C2×#104—22no—
91#416—21no—
92#408—22no—
93#408—22no—
94#405—22no—
95C4×M16—21no—
96C2×C4×C8C2312yes—
97C2×C4×C8C2311yescharacteristic
98C4×C4⋊C4D411yescharacteristic
99#368—42no—
100C4⋊C16C2×C411yescharacteristic
101C4⋊C16C2×C411yescharacteristic
102C4⋊C16C2×C412yes—
103#359—22no—
104#359—24no—
105#359—22no—
106#359—42no—
107#358C2×C412yes—
108#358C2×C412yes—
109#357C2×C412yes—
110#357C2×C412yes—
111#346—42no—
112C4×C16—21no—
113C4×C16—21no—
114#341—21no—
115#339—22no—
116#336D412yes—
117C2×#41—21no—
118C2×#41—22no—
119#135—22no—
120#135—21no—
121#135—21no—
122#135—22no—
123#135—22no—
124#135—22no—
125#135—21no—
126#135—21no—
127#135—81no—
128#135—81no—
129C2×Q16—21no—
130C2×Q16C22×C412yes—
131C2×Q16C22×C411yescharacteristic
132C2×SD16—22no—
133C2×SD16—21no—
134C2×SD16—22no—
135C2×D8—21no—
136C2×D8C22×C412yes—
137C2×D8C22×C411yescharacteristic
138#131—44no—
139#131—82no—
140C2×M16—21no—
141C2×M16—22no—
142C2×M16—21no—
143C2×M16—42no—
144C22×C8—42no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D412yes—
147C22×C8C2411yescharacteristic
148C4⋊Q8—21no—
149#127—21no—
150#124—22no—
151#121—41no—
152#121—41no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—41no—
158C4×D4—41no—
159C4×D4—41no—
160C4×D4—41no—
161#117—21no—
162#117—41no—
163C2×C4⋊C4—21no—
164C2×C42—41no—
165C2×C42C2×D412yes—
166C2×C42C2×D411yescharacteristic
167Q32—42no—
168SD32—42no—
169SD32—42no—
170D16—42no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—22no—
174C2×C16—22no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16C22×C412yes—
179C2×C16C22×C412yes—
180C2×C16#4112yes—
181#108—22no—
182#108—21no—
183#108—21no—
184#108—22no—
185#108—21no—
186#108—81no—
187#108#4112yes—
188#107C22×C411yescharacteristic
189#107C22×C411yescharacteristic
190#106—21no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8D811yescharacteristic
194C4⋊C8D811yescharacteristic
195C4⋊C8D812yes—
196#104—44no—
197#104—82no—
198#104—82no—
199#103—21no—
200#103—21no—
201#102—21no—
202#102—21no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207#97—42no—
208#97—41no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—42no—
212C4×C8—41no—
213C4×C8C22×C411yescharacteristic
214C4×C8C22×C411yescharacteristic
215C4×C8#4111yescharacteristic
216C4×C8#4111yescharacteristic
217#95—21no—
218#41—42no—
219#41—41no—
220#41—42no—
221#41—41no—
222#41—41no—
223#41—42no—
224#41—42no—
225#41—41no—
226C2×Q8—21no—
227C2×Q8—22no—
228C2×D4—21no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—42no—
232C2×D4—41no—
233C2×D4—41no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4#13511yescharacteristic
240C22×C4#13511yescharacteristic
241C22×C4C22×D411yescharacteristic
242Q16—22no—
243Q16—21no—
244Q16—21no—
245Q16—22no—
246Q16—81no—
247SD16—41no—
248SD16—42no—
249SD16—41no—
250SD16—42no—
251SD16—82no—
252D8—22no—
253D8—21no—
254D8—21no—
255D8—22no—
256D8—81no—
257M16—22no—
258M16—41no—
259M16—42no—
260M16—41no—
261M16—41no—
262M16—42no—
263M16—84no—
264C2×C8—22no—
265C2×C8—21no—
266C2×C8—22no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—84no—
272C2×C8C4×D411yescharacteristic
273C2×C8C4×D411yescharacteristic
274C2×C8C2×D812yes—
275C2×C8C2×D812yes—
276C2×C8C23×C411yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C2×C8C2×#4111yescharacteristic
279C2×C8C2×#4111yescharacteristic
280C2×C8H(5,𝔽2)11yescharacteristic
281C2×C8H(5,𝔽2)11yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C42—22no—
295C42—22no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42C2×D811yescharacteristic
302C42C2×D811yescharacteristic
303C16—22no—
304C16—22no—
305C16—22no—
306C16—41no—
307C23—41no—
308C23—42no—
309C23C2×#13511yescharacteristic
310Q8—41no—
311Q8—42no—
312Q8—42no—
313Q8—41no—
314D4—41no—
315D4—42no—
316D4—42no—
317D4—41no—
318D4—81no—
319D4—82no—
320D4—81no—
321D4—82no—
322D4—82no—
323D4—161no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—161no—
339C2×C4C4×D811yescharacteristic
340C2×C4C4×D811yescharacteristic
341C2×C4C4×D811yescharacteristic
342C2×C4C4×D811yescharacteristic
343C2×C4C2×C4×D411yescharacteristic
344C2×C4#54611yescharacteristic
345C2×C4#55011yescharacteristic
346C2×C4C22×D811yescharacteristic
347C2×C4#57611yescharacteristic
348C2×C4#57711yescharacteristic
349C8—21no—
350C8—21no—
351C8—22no—
352C8—22no—
353C8—42no—
354C8—81no—
355C8—82no—
356C8C4×#4111yescharacteristic
357C8#51811yescharacteristic
358C8#55311yescharacteristic
359C8#55511yescharacteristic
360C22—21no—
361C22—21no—
362C22—82no—
363C22—81no—
364C22—81no—
365C22—82no—
366C22—81no—
367C22—82no—
368C22C2×C4×D811yescharacteristic
369C22#330511yescharacteristic
370C22#339411yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—81no—
376C4—82no—
377C4—81no—
378C4—81no—
379C4#217912yes—
380C4#236311yescharacteristic
381C4#294611yescharacteristic
382C4#329411yescharacteristic
383C4#339111yescharacteristic
384C2—21no—
385C2—21no—
386C2—161no—
387C2—162no—
388C2#2309311yescharacteristic
389C2C2×#217911yescharacteristic
390C2#3346411yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radical#33579C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23179C2semidirect
3. #22802C2semidirect
4. #13242C2semidirect
5. #13208C2semidirect
6. #13179C2semidirect
7. #13134C2semidirect
8. #13011C2semidirect
9. #12662C2semidirect
10. #12642C2semidirect
11. C4×#359C2semidirect
18. #2211C4semidirect
19. #2210C4semidirect
20. #2209C4semidirect
21. #2208C4semidirect
22. #2207C4semidirect
23. #2206C4semidirect

Maximal subgroups

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

subgroupindex
#335792
#231792
#228022
#132422
#132082
#131792
#131342
#130112
#126622
#126422
C4×#3592