Finite Group Databasebeta

← #496916

Subgroups — #496916

Order 512.

469 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33626C211yesmaximal, characteristic
2#21732C211yesmaximal, characteristic, complemented
3#21292C211yesmaximal, characteristic
4#13274C211yesmaximal, characteristic, complemented
5#13179C211yesmaximal, characteristic, complemented
6#13170C212yesmaximal, complemented
7#13152C212yesmaximal
8#13134C212yesmaximal
9#13113C212yesmaximal, complemented
10#12815C211yesmaximal, characteristic
11#12653C211yesmaximal, characteristic, complemented
12#3160C2211yescharacteristic
13#2960—21no—
14C4×#135C2211yescharacteristic
15#2922—21no—
16#2229C412yescomplemented
17#2228C412yescomplemented
18#2227C411yescharacteristic, complemented
19#2226C411yescharacteristic, complemented
20#2225C411yescharacteristic, complemented
21#2224C411yescharacteristic, complemented
22C2×#368C2212yes—
23#2158C2212yes—
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#359C2212yes—
27#2138C2212yes—
28C2×#346C2212yes—
29#1994—21no—
30#1974C2211yescharacteristic
31#1974C2211yescharacteristic
32#1945—21no—
33#1945C2211yescharacteristic
34#1945C2211yescharacteristic
35#1945C2212yes—
36#1937—21no—
37#1937—21no—
38#1848C2211yescharacteristic
39C4×#108C2211yescharacteristic
40#1772—21no—
41#1772C2212yes—
42C4×#104—21no—
43C42×C8C2211yescharacteristic
44#1382C2212yes—
45#1382C2212yes—
46#1382C2212yes—
47#1381C2212yes—
48#1374C2212yes—
49C2×#135C2311yescharacteristic
50#533—21no—
51C4×#41—21no—
52C22×C16C2312yes—
53C22×C16C2312yes—
54#500C2×C411yescharacteristic
55#500C2×C411yescharacteristic
56#499—21no—
57#499C2×C412yes—
58#498—21no—
59#446—41no—
60#446—41no—
61C8×Q8—41no—
62C8×Q8—41no—
63C4×Q16C2×C411yescharacteristic
64C4×Q16C2×C411yescharacteristic
65C4×SD16—21no—
66C4×SD16—21no—
67C4×D8C2×C411yescharacteristic
68C4×D8C2×C411yescharacteristic
69#433—41no—
70#433—41no—
71#431—41no—
72#431—41no—
73C2×#108—21no—
74C2×#108C2312yes—
75C2×#108C2311yescharacteristic
76C2×#108C2311yescharacteristic
77#427—21no—
78#427C2312yes—
79#427C2311yescharacteristic
80#424—21no—
81#424—21no—
82#424—22no—
83#423—21no—
84C2×#104—21no—
85C2×#104—21no—
86#416—21no—
87#408—21no—
88#408—21no—
89#405—22no—
90C4×M16—21no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C43D411yescharacteristic
98#368—42no—
99#367C2×C412yes—
100#367C2×C412yes—
101#366C2×C412yes—
102#366C2×C412yes—
103C4⋊C16—21no—
104C4⋊C16—21no—
105C4⋊C16C2×C411yescharacteristic
106C4⋊C16C2×C411yescharacteristic
107C4⋊C16C2×C412yes—
108#359—42no—
109#358C2×C412yes—
110#358C2×C412yes—
111#357C2×C412yes—
112#357C2×C412yes—
113#346—42no—
114#341—21no—
115#339—21no—
116#339—21no—
117#339—22no—
118#339—22no—
119C2×#41—21no—
120#135—21no—
121#135—21no—
122#135—21no—
123#135—21no—
124C2×Q16C22×C411yescharacteristic
125C2×SD16—21no—
126C2×D8C22×C411yescharacteristic
127C2×M16—22no—
128C2×M16—21no—
129C2×M16—21no—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2411yescharacteristic
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136#125—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—41no—
146C4×D4—41no—
147#117—21no—
148#117—22no—
149#117—41no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—41no—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C16—22no—
157C2×C16—22no—
158C2×C16—22no—
159C2×C16—22no—
160C2×C16—22no—
161C2×C16C22×C412yes—
162C2×C16C22×C412yes—
163#108—22no—
164#108—21no—
165#108—22no—
166#108—21no—
167#108—21no—
168#108—21no—
169#108—42no—
170#107—21no—
171#107—21no—
172#107C22×C412yes—
173#107C22×C412yes—
174#107C22×C411yescharacteristic
175#107C22×C411yescharacteristic
176#106—22no—
177#106—21no—
178#106—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—42no—
184C4⋊C8—42no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187#104—42no—
188#104—42no—
189#103—21no—
190#103—21no—
191#102—21no—
192#102—21no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#97—42no—
198#97—41no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—22no—
204C4×C8—41no—
205C4×C8—42no—
206C4×C8—41no—
207C4×C8D811yescharacteristic
208C4×C8D811yescharacteristic
209C4×C8D811yescharacteristic
210C4×C8D811yescharacteristic
211C4×C8C22×C411yescharacteristic
212C4×C8C22×C411yescharacteristic
213C4×C8#4111yescharacteristic
214C4×C8#4111yescharacteristic
215#41—41no—
216#41—41no—
217#41—41no—
218#41—41no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—21no—
223C2×D4—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4#13511yescharacteristic
230C22×C4#13511yescharacteristic
231C22×C4C22×D411yescharacteristic
232Q16—21no—
233Q16—21no—
234SD16—41no—
235SD16—41no—
236D8—21no—
237D8—21no—
238M16—41no—
239M16—42no—
240M16—41no—
241M16—41no—
242M16—41no—
243M16—42no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8C4×D411yescharacteristic
254C2×C8C4×D411yescharacteristic
255C2×C8C2×D811yescharacteristic
256C2×C8C2×D811yescharacteristic
257C2×C8C2×D811yescharacteristic
258C2×C8C2×D811yescharacteristic
259C2×C8C23×C411yescharacteristic
260C2×C8C2×#4111yescharacteristic
261C2×C8H(5,𝔽2)11yescharacteristic
262C2×C82−1+411yescharacteristic
263C2×C82−1+411yescharacteristic
264C2×C82−1+411yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C2×D811yescharacteristic
291C42C2×D811yescharacteristic
292C42#13712yes—
293C16—22no—
294C16—22no—
295C16—42no—
296C23—41no—
297C23C2×#13511yescharacteristic
298Q8—41no—
299Q8—41no—
300Q8—82no—
301Q8—82no—
302D4—41no—
303D4—41no—
304D4—81no—
305D4—81no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C4×D811yescharacteristic
324C2×C4C4×D811yescharacteristic
325C2×C4C4×D811yescharacteristic
326C2×C4C4×D811yescharacteristic
327C2×C4C2×C4×D411yescharacteristic
328C2×C4#53611yescharacteristic
329C2×C4#53711yescharacteristic
330C2×C4C22×D811yescharacteristic
331C2×C4C2×#13711yescharacteristic
332C2×C4#57511yescharacteristic
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C8—21no—
339C8—81no—
340C8—81no—
341C8—82no—
342C8#51911yescharacteristic
343C8#52011yescharacteristic
344C8#54111yescharacteristic
345C8#54211yescharacteristic
346C22—21no—
347C22—21no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22C2×C4×D811yescharacteristic
352C22#323311yescharacteristic
353C22#324811yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—42no—
359C4—81no—
360C4—81no—
361C4—82no—
362C4—81no—
363C4—82no—
364C4#217912yes—
365C4#232311yescharacteristic
366C4C4×#13711yescharacteristic
367C4#322611yescharacteristic
368C4#324511yescharacteristic
369C2—21no—
370C2—21no—
371C2—161no—
372C2#2211011yescharacteristic
373C2C2×#217911yescharacteristic
374C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radicalC2×#135C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #13170C2semidirect
9. #13113C2semidirect
11. #12653C2semidirect
16. #2229C4semidirect
17. #2228C4semidirect
18. #2227C4semidirect
19. #2226C4semidirect
20. #2225C4semidirect
21. #2224C4semidirect

Maximal subgroups

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

subgroupindex
#336262
#217322
#212922
#132742
#131792
#131702
#131522
#131342
#131132
#128152
#126532