Finite Group Databasebeta

← #495994

Subgroups — #495994

Order 512.

552 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33626C211yesmaximal, characteristic
2#21732C211yesmaximal, characteristic, complemented
3C4×#443C211yesmaximal, characteristic
4#13274C211yesmaximal, characteristic, complemented
5#13179C211yesmaximal, characteristic, complemented
6#13113C212yesmaximal, complemented
7#12939C212yesmaximal
8#12815C211yesmaximal, characteristic
9#12670C212yesmaximal, complemented
10C4×C4⋊C16C211yesmaximal, characteristic, complemented
11#12641C212yesmaximal
12#3160C2211yescharacteristic
13#2960—21no—
14C2×#443C2211yescharacteristic
15C2×#443C2211yescharacteristic
16C4×#135C2211yescharacteristic
17C4×#131—21no—
18#2229C412yescomplemented
19#2228C412yescomplemented
20#2227C411yescharacteristic, complemented
21#2226C411yescharacteristic, complemented
22#2225C411yescharacteristic, complemented
23#2224C411yescharacteristic, complemented
24C2×#368C2212yes—
25#2158C2212yes—
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#359C2212yes—
29#2138C2212yes—
30C2×C4×C16C2212yes—
31#1994—21no—
32#1945—21no—
33#1945C2212yes—
34#1848C2211yescharacteristic
35C4×#108C2211yescharacteristic
36#1772—21no—
37#1772C2211yescharacteristic
38#1772C2211yescharacteristic
39#1772C2212yes—
40#1764—21no—
41#1764—21no—
42C4×#104—21no—
43C42×C8C2211yescharacteristic
44#1382C2212yes—
45#1382C2212yes—
46#1382C2212yes—
47#1381C2212yes—
48#1374C2212yes—
49C2×#135C2311yescharacteristic
50C2×#131—21no—
51C2×#131—21no—
52#533—21no—
53C4×#41—21no—
54C22×C16C2312yes—
55C22×C16C2312yes—
56#500C2×C411yescharacteristic
57#500C2×C411yescharacteristic
58#499—21no—
59#499C2×C412yes—
60#498—21no—
61#446—41no—
62#446—41no—
63C8×Q8—41no—
64C8×Q8—41no—
65#443—22no—
66#443—22no—
67#443—22no—
68#443Q814yescomplemented
69C4×Q16C2×C411yescharacteristic
70C4×Q16C2×C411yescharacteristic
71C4×SD16—21no—
72C4×SD16—21no—
73C4×D8C2×C411yescharacteristic
74C4×D8C2×C411yescharacteristic
75C2×#108—21no—
76C2×#108C2312yes—
77C2×#108C2311yescharacteristic
78C2×#108C2311yescharacteristic
79#427—21no—
80#427C2312yes—
81#427C2311yescharacteristic
82#424—22no—
83#423—21no—
84C2×#104—21no—
85C2×#104—21no—
86#416—21no—
87#405—21no—
88#405—21no—
89#405—22no—
90#405—41no—
91#405—41no—
92C4×M16—21no—
93C4×M16—41no—
94C2×C4×C8—41no—
95C2×C4×C8D411yescharacteristic
96C2×C4×C8D411yescharacteristic
97C2×C4×C8D411yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C43D411yescharacteristic
102#368—24no—
103#367C2×C412yes—
104#367C2×C412yes—
105#366C2×C412yes—
106#366C2×C412yes—
107C4⋊C16—21no—
108C4⋊C16—21no—
109C4⋊C16C2×C411yescharacteristic
110C4⋊C16C2×C411yescharacteristic
111C4⋊C16C2×C412yes—
112#359—24no—
113#358C2×C412yes—
114#358C2×C412yes—
115#357C2×C412yes—
116#357C2×C412yes—
117C4×C16—24no—
118#341—21no—
119#339—22no—
120#339—22no—
121#339—21no—
122#339—21no—
123C2×#41—21no—
124#135—21no—
125#135—21no—
126#135C2×Q812yes—
127#135#4112yes—
128C2×Q16C22×C411yescharacteristic
129C2×SD16—21no—
130C2×D8C22×C411yescharacteristic
131#131—24no—
132#131—24no—
133#131—42no—
134#131—42no—
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—22no—
138C2×M16—41no—
139C2×M16—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2411yescharacteristic
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148#125—22no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—41no—
158C4×D4—41no—
159#117—21no—
160#117—22no—
161#117—41no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—41no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C16—22no—
169C2×C16—22no—
170C2×C16—22no—
171C2×C16—22no—
172C2×C16—22no—
173C2×C16C22×C412yes—
174C2×C16C22×C412yes—
175#108—21no—
176#108—21no—
177#108—22no—
178#108—22no—
179#108—42no—
180#108C2×Q812yes—
181#108#4112yes—
182#107—21no—
183#107—21no—
184#107C22×C412yes—
185#107C22×C412yes—
186#107C22×C411yescharacteristic
187#107C22×C411yescharacteristic
188#106—22no—
189#106—21no—
190#106—22no—
191C4⋊C8—42no—
192C4⋊C8—42no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195#104—24no—
196#104—24no—
197#103—21no—
198#103—21no—
199#102—21no—
200#102—21no—
201#97—41no—
202#97—41no—
203#97—42no—
204#97—41no—
205C4×C8—22no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—41no—
209C4×C8—41no—
210C4×C8—41no—
211C4×C8—42no—
212C4×C8—41no—
213C4×C8D811yescharacteristic
214C4×C8D811yescharacteristic
215C4×C8D811yescharacteristic
216C4×C8D811yescharacteristic
217C4×C8C22×C411yescharacteristic
218C4×C8C22×C411yescharacteristic
219C4×C8C2×Q812yes—
220C4×C8#4111yescharacteristic
221C4×C8#4111yescharacteristic
222C4×C8#4112yes—
223#41—22no—
224#41—22no—
225#41—41no—
226#41—41no—
227C2×Q8—21no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×D4—21no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4#13511yescharacteristic
238C22×C4#13511yescharacteristic
239C22×C4C22×D411yescharacteristic
240Q16—21no—
241Q16C4×Q812yes—
242SD16—22no—
243SD16—22no—
244D8—21no—
245D8C4×Q812yes—
246M16—22no—
247M16—22no—
248M16—44no—
249M16—41no—
250M16—41no—
251M16—44no—
252M16—42no—
253M16—42no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—44no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—44no—
262C2×C8—42no—
263C2×C8C4×D411yescharacteristic
264C2×C8C4×D411yescharacteristic
265C2×C8C22⋊Q812yes—
266C2×C8C2×D811yescharacteristic
267C2×C8C2×D811yescharacteristic
268C2×C8C2×D811yescharacteristic
269C2×C8C2×D811yescharacteristic
270C2×C8#13712yes—
271C2×C8C23×C411yescharacteristic
272C2×C8C22×Q811yescharacteristic
273C2×C8C2×#4111yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42—24no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—42no—
296C42—41no—
297C42—42no—
298C42—41no—
299C42—41no—
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42C22⋊Q812yes—
303C42C2×D811yescharacteristic
304C42C2×D811yescharacteristic
305C16—22no—
306C16—22no—
307C16—42no—
308C23—41no—
309C23C2×#13511yescharacteristic
310Q8—22no—
311Q8—41no—
312Q8—82no—
313Q8—82no—
314D4—22no—
315D4—42no—
316D4—42no—
317D4—41no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C4×D811yescharacteristic
336C2×C4C4×D811yescharacteristic
337C2×C4C4×D811yescharacteristic
338C2×C4C4×D811yescharacteristic
339C2×C4C2×C4×D411yescharacteristic
340C2×C4C2×C22⋊Q811yescharacteristic
341C2×C4#52511yescharacteristic
342C2×C4C22×D811yescharacteristic
343C2×C4C2×#13711yescharacteristic
344C2×C4#57511yescharacteristic
345C8—21no—
346C8—21no—
347C8—21no—
348C8—21no—
349C8—42no—
350C8—42no—
351C8—82no—
352C8#47412yes—
353C8#47412yes—
354C8C2×C4×Q811yescharacteristic
355C8C4×#4111yescharacteristic
356C8#52911yescharacteristic
357C8#53311yescharacteristic
358C22—21no—
359C22—21no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22C2×C4×D811yescharacteristic
364C22#307611yescharacteristic
365C22#310711yescharacteristic
366C4—24no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—42no—
371C4—42no—
372C4—81no—
373C4—82no—
374C4—81no—
375C4—82no—
376C4#217812yes—
377C4C4×C22⋊Q811yescharacteristic
378C4C4×#13711yescharacteristic
379C4C2×#47411yescharacteristic
380C4#310011yescharacteristic
381C2—21no—
382C2—21no—
383C2—82no—
384C2C4×#47411yescharacteristic
385C2C2×#217811yescharacteristic
386C2#3346311yescharacteristic
subgroupG/H
centerC2×C8C2×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radicalC2×#135C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21732C2semidirect
4. #13274C2semidirect
5. #13179C2semidirect
6. #13113C2semidirect
9. #12670C2semidirect
10. C4×C4⋊C16C2semidirect
18. #2229C4semidirect
19. #2228C4semidirect
20. #2227C4semidirect
21. #2226C4semidirect
22. #2225C4semidirect
23. #2224C4semidirect
68. #443Q8semidirect

Maximal subgroups

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

subgroupindex
#336262
#217322
C4×#4432
#132742
#131792
#131132
#129392
#128152
#126702
C4×C4⋊C162
#126412