Finite Group Databasebeta

← #497979

Subgroups — #497979

Order 512.

521 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#33428C211yesmaximal, characteristic, complemented
2#33401C211yesmaximal, characteristic, complemented
3#20510C211yesmaximal, characteristic
4#13127C212yesmaximal, complemented
5#13108C212yesmaximal, complemented
6#12986C212yesmaximal
7#12881C211yesmaximal, characteristic, complemented
8#12792C211yesmaximal, characteristic, complemented
9#12792C211yesmaximal, characteristic, complemented
10#12660C211yesmaximal, characteristic, complemented
11#12618C212yesmaximal, complemented
12#3407C2212yes—
13C2×#444C2211yescharacteristic
14C2×#443C411yescharacteristic
15#2943C411yescharacteristic
16C4×#131C2211yescharacteristic
17C4×#131C2211yescharacteristic
18#2166—22no—
19#2165—22no—
20#2153—22no—
21#2153C2212yes—
22C2×#364—22no—
23C2×#364C2211yescharacteristic
24C2×#364C2211yescharacteristic
25C2×#364C2211yescharacteristic
26C2×#364C2211yescharacteristic
27#2117C2212yes—
28#2117C2212yes—
29#2115C2212yes—
30#2110C2212yes—
31C2×#347C2212yes—
32#1778C411yescharacteristic
33#1773C2211yescharacteristic
34#1772C411yescharacteristic
35#1765C2211yescharacteristic
36#1765C2211yescharacteristic
37#1764C411yescharacteristic
38#1764C411yescharacteristic
39#1759C411yescharacteristic
40#1759C411yescharacteristic
41#1726C2211yescharacteristic
42#1378C2212yes—
43#1378C2212yes—
44#1377C2212yes—
45#1377C2212yes—
46#1375C2212yes—
47C2×#135C2×C411yescharacteristic
48C2×#131C2×C411yescharacteristic
49C2×#131C2×C411yescharacteristic
50C2×#131C2311yescharacteristic
51C2×#131C2311yescharacteristic
52C4×#41C2×C411yescharacteristic
53C4×#41C2×C411yescharacteristic
54#504—24no—
55#504—42no—
56C2×M32—24no—
57C2×M32C2312yes—
58C2×M32C2312yes—
59C2×M32C2312yes—
60C2×M32C2312yes—
61#444—22no—
62#444—22no—
63#443—24no—
64#442—21no—
65#441—21no—
66#440—21no—
67#440—21no—
68C2×#108C2×C411yescharacteristic
69C2×#108C2×C411yescharacteristic
70#427C2×C411yescharacteristic
71C2×#104C2×C411yescharacteristic
72C2×#104C2×C411yescharacteristic
73C2×#104C2×C411yescharacteristic
74C2×#104C2×C411yescharacteristic
75#416C2×C411yescharacteristic
76#416C2×C411yescharacteristic
77#405—21no—
78#405—21no—
79#405—21no—
80#405—21no—
81#405C2×C411yescharacteristic
82#405C2×C411yescharacteristic
83#405C2311yescharacteristic
84#405C2311yescharacteristic
85C4×M16—21no—
86C4×M16—21no—
87C4×M16C2×C411yescharacteristic
88C4×M16C2×C411yescharacteristic
89C2×#97C2×C411yescharacteristic
90C2×#97C2311yescharacteristic
91C2×#97C2311yescharacteristic
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2×C411yescharacteristic
95C2×C4×C8C2×C411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#376C2×C411yescharacteristic
98#364—24no—
99#364—21no—
100#364—21no—
101#364—21no—
102#364—21no—
103#364—21no—
104#364—21no—
105#364D412yes—
106#364D412yes—
107#350—24no—
108#350—42no—
109#349—24no—
110#347—24no—
111#347—42no—
112#341C2×C411yescharacteristic
113#341C2×C411yescharacteristic
114#339C2×C411yescharacteristic
115#339C2×C411yescharacteristic
116#339C2×C411yescharacteristic
117#339C2×C411yescharacteristic
118C2×#41C22×C411yescharacteristic
119C2×#41C22×C411yescharacteristic
120#135—22no—
121#135—22no—
122C2×Q16M1611yescharacteristic
123C2×SD16M1611yescharacteristic
124C2×SD16M1611yescharacteristic
125C2×D8M1611yescharacteristic
126#131—22no—
127#131—21no—
128#131—21no—
129#131—22no—
130#131—22no—
131#131—21no—
132#131—21no—
133#131—21no—
134#131—21no—
135#131M1614yes—
136#131#4112yes—
137#131#4112yes—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16C22×C411yescharacteristic
143C2×M16C22×C411yescharacteristic
144C2×M16C22×C411yescharacteristic
145C2×M16C22×C411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C22×C411yescharacteristic
151C22×C8C22×C411yescharacteristic
152C22×C8C22×C411yescharacteristic
153C22×C8C2411yescharacteristic
154C4×Q8—21no—
155C4×Q8—21no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—41no—
159C4×D4—41no—
160#117—21no—
161#117—21no—
162#117C22×C411yescharacteristic
163#117C22×C411yescharacteristic
164C2×C42—21no—
165C2×C42—21no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169M32—22no—
170M32—22no—
171M32—22no—
172M32—22no—
173M32—22no—
174M32—22no—
175M32—22no—
176M32—44no—
177M32C2×D414yes—
178C2×C16—22no—
179C2×C16—22no—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16—44no—
183#108—22no—
184#108—22no—
185#107—21no—
186#106—21no—
187#104—22no—
188#104—22no—
189#104—22no—
190#104—22no—
191#103—21no—
192#103—21no—
193#102—21no—
194#102—21no—
195#97—21no—
196#97—21no—
197#97—21no—
198#97—21no—
199#97—21no—
200#97—21no—
201#97—21no—
202#97—21no—
203#97—21no—
204#97—21no—
205#97—21no—
206#97—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8C2×D411yescharacteristic
216C4×C8C2×D411yescharacteristic
217C4×C8#4111yescharacteristic
218C4×C8#4111yescharacteristic
219#95—41no—
220#41—21no—
221#41—21no—
222#41—21no—
223#41—21no—
224#41—21no—
225#41—21no—
226#41#11712yes—
227#41C2×M1612yes—
228C2×Q8C2×M1611yescharacteristic
229C2×Q8C2×M1611yescharacteristic
230C2×D4—21no—
231C2×D4—21no—
232C2×D4C2×M1611yescharacteristic
233C2×D4C2×M1611yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4C23×C411yescharacteristic
242Q16—22no—
243SD16—22no—
244SD16—22no—
245D8—22no—
246M16—21no—
247M16—21no—
248M16—22no—
249M16—22no—
250M16—24no—
251M16—21no—
252M16—21no—
253M16—21no—
254M16—21no—
255M16—41no—
256M16—41no—
257M16—42no—
258M16#11712yes—
259M16C2×M1612yes—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—24no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8C4×D411yescharacteristic
276C2×C8C4×D411yescharacteristic
277C2×C8C2×M1611yescharacteristic
278C2×C8C2×M1611yescharacteristic
279C2×C8C2×M1612yes—
280C2×C8#13112yes—
281C2×C8C22×D411yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8H(5,𝔽2)11yescharacteristic
286C2×C82−1+411yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—41no—
303C42—41no—
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C16—24no—
307C16—42no—
308C16—42no—
309C16—42no—
310C23—21no—
311C23—21no—
312C23#56811yescharacteristic
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8#43212yes—
317D4—22no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—41no—
323D4—41no—
324D4#43212yes—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4C2×#11711yescharacteristic
343C2×C4C2×C4×D411yescharacteristic
344C2×C4#52011yescharacteristic
345C2×C4C22×M1611yescharacteristic
346C2×C4C22×M1611yescharacteristic
347C2×C4C2×#13111yescharacteristic
348C8—22no—
349C8—21no—
350C8—21no—
351C8—22no—
352C8—41no—
353C8—41no—
354C8—41no—
355C8#43512yes—
356C8#43512yes—
357C8#54611yescharacteristic
358C8#54811yescharacteristic
359C8#55211yescharacteristic
360C8#55311yescharacteristic
361C22—22no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—81no—
366C22#292511yescharacteristic
367C22D4×M1611yescharacteristic
368C22#298911yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4#235211yescharacteristic
378C4C2×#43211yescharacteristic
379C4C2×#43511yescharacteristic
380C4#299011yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—81no—
386C2#2275011yescharacteristic
subgroupG/H
centerC4#2352
commutatorC22×C4C23×C4
Frattini, cosocleC22×C8C24
socleC2#22750
prime radicalC2×#135C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33428C2semidirect
2. #33401C2semidirect
4. #13127C2semidirect
5. #13108C2semidirect
7. #12881C2semidirect
8. #12792C2semidirect
9. #12792C2semidirect
10. #12660C2semidirect
11. #12618C2semidirect

Maximal subgroups

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

subgroupindex
#334282
#334012
#205102
#131272
#131082
#129862
#128812
#127922
#126602
#126182