Finite Group Databasebeta

← #508619

Subgroups — #508619

Order 512.

453 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33318C211yesmaximal, characteristic, complemented
2#21294C211yesmaximal, characteristic
3#13543C211yesmaximal, characteristic, complemented
4#13542C211yesmaximal, characteristic, complemented
5#13534C211yesmaximal, characteristic
6#13533C211yesmaximal, characteristic
7#13458C211yesmaximal, characteristic, complemented
8#13457C211yesmaximal, characteristic, complemented
9#13372C211yesmaximal, characteristic
10#13371C211yesmaximal, characteristic
11#13184C211yesmaximal, characteristic, complemented
12#13183C211yesmaximal, characteristic, complemented
13#12884C211yesmaximal, characteristic, complemented
14#12817C211yesmaximal, characteristic
15#12816C211yesmaximal, characteristic
16C2×#501—21no—
17C2×#499C2211yescharacteristic
18C2×#498C2211yescharacteristic
19C4×#135C2211yescharacteristic
20#2923—21no—
21#2660—21no—
22C2×#365C2211yescharacteristic
23C2×#365C2211yescharacteristic
24C2×#365C2211yescharacteristic
25C2×#365C2211yescharacteristic
26#2150C2211yescharacteristic
27#2150C2211yescharacteristic
28C2×#360C2211yescharacteristic
29C2×#360C2211yescharacteristic
30C2×#360C2211yescharacteristic
31C2×#360C2211yescharacteristic
32#1974C2211yescharacteristic
33#1974C2211yescharacteristic
34#1945C2211yescharacteristic
35#1945C2211yescharacteristic
36#1938—21no—
37#1938—21no—
38#1847—21no—
39#1847—21no—
40#1846C2211yescharacteristic
41#1846C2211yescharacteristic
42#1844C2211yescharacteristic
43#1844C2211yescharacteristic
44C4×#108C2211yescharacteristic
45#1759—21no—
46#1726C2211yescharacteristic
47#1386C2211yescharacteristic
48#1386C2211yescharacteristic
49#1386C2211yescharacteristic
50#1386C2211yescharacteristic
51#1383C2211yescharacteristic
52#1383C2211yescharacteristic
53#1383C2211yescharacteristic
54#1383C2211yescharacteristic
55#1376C2211yescharacteristic
56#1376C2211yescharacteristic
57#1376C2211yescharacteristic
58#1376C2211yescharacteristic
59C2×#135C2311yescharacteristic
60C2×C4⋊Q8—21no—
61C2×#125—21no—
62C4×#41—21no—
63C2×M32C2311yescharacteristic
64C2×M32C2311yescharacteristic
65C2×M32C2311yescharacteristic
66C2×M32C2311yescharacteristic
67#501—44no—
68#499—41no—
69#499—41no—
70#498—41no—
71#498—41no—
72C4×Q16Q812yescomplemented
73C4×SD16—22no—
74C4×D8Q812yescomplemented
75#433—42no—
76#432—42no—
77C2×#108C2311yescharacteristic
78C2×#108C2311yescharacteristic
79#427C2311yescharacteristic
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106C2311yescharacteristic
87C2×#106C2311yescharacteristic
88C2×#106C2311yescharacteristic
89C2×#106C2311yescharacteristic
90#424—21no—
91#424—21no—
92C2×#104—21no—
93C2×#104—21no—
94#416—21no—
95#408—21no—
96#408—21no—
97C4×M16—21no—
98C2×#97D411yescharacteristic
99C2×#97D411yescharacteristic
100C2×#97D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#398—42no—
105#395—42no—
106#384—21no—
107#384—21no—
108#382—21no—
109#382—21no—
110#376D411yescharacteristic
111#365—41no—
112#365—41no—
113#365—41no—
114#365—41no—
115C4⋊C16—22no—
116C4⋊C16—22no—
117#360—41no—
118#360—41no—
119#360—41no—
120#360—41no—
121#341—21no—
122#339—21no—
123#339—21no—
124C2×#41—21no—
125C22×Q8—41no—
126#135—21no—
127#135—21no—
128#135—21no—
129#135—21no—
130C2×Q16C2×Q811yescharacteristic
131C2×SD16—21no—
132C2×D8C2×Q811yescharacteristic
133C2×M16—21no—
134C2×M16—21no—
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2411yescharacteristic
139C4⋊Q8—42no—
140C4⋊Q8—42no—
141#125—42no—
142#125—42no—
143C4×Q8—22no—
144C4×D4—22no—
145C4×D4—42no—
146#117—21no—
147#117—41no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C42—41no—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161M32—21no—
162M32—21no—
163M32—21no—
164M32—21no—
165M32—21no—
166M32—21no—
167M32—21no—
168M32—21no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173#108—21no—
174#108—21no—
175#108—21no—
176#108—21no—
177#107—42no—
178#107—42no—
179#107—42no—
180#107C2×Q812yes—
181#106—21no—
182#106—21no—
183#106—21no—
184#106—21no—
185#106—21no—
186#106—21no—
187#106—21no—
188#106—21no—
189#106—21no—
190#106—42no—
191#106—42no—
192#106—42no—
193C4⋊C8—42no—
194C4⋊C8—42no—
195#104—42no—
196#104—42no—
197#103—22no—
198#102—22no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201#97—22no—
202#97—22no—
203#97—41no—
204#97SD1612yes—
205#97SD1612yes—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—41no—
211C4×C8C2×Q812yes—
212C4×C8#4112yes—
213#95—22no—
214#95—42no—
215#95—42no—
216#41—41no—
217#41—41no—
218#41—41no—
219#41—41no—
220C2×Q8—21no—
221C2×Q8—82no—
222C2×Q8—162no—
223C2×D4—21no—
224C2×D4—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4AGL(1,ℤ/8)12yes—
234C22×C4C22×D411yescharacteristic
235Q16—21no—
236Q16—21no—
237SD16—41no—
238SD16—41no—
239D8—21no—
240D8—21no—
241M16—41no—
242M16—41no—
243M16—41no—
244M16—41no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8C22⋊Q812yes—
254C2×C8C2×SD1612yes—
255C2×C8C2×SD1612yes—
256C2×C8C22×Q811yescharacteristic
257C2×C8C2×#4111yescharacteristic
258C2×C8H(5,𝔽2)11yescharacteristic
259C2×C8H(5,𝔽2)11yescharacteristic
260C2×C82−1+411yescharacteristic
261C2×C82−1+411yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—162no—
276C4⋊C4—162no—
277C22⋊C4—41no—
278C22⋊C4—42no—
279C42—21no—
280C42—21no—
281C42—41no—
282C42—42no—
283C42C22⋊Q812yes—
284C42C2×SD1611yescharacteristic
285C42C2×SD1611yescharacteristic
286C42AGL(1,ℤ/8)12yes—
287C42#13712yes—
288C16—41no—
289C16—41no—
290C16—41no—
291C16—41no—
292C23—41no—
293C23C2×AGL(1,ℤ/8)11yescharacteristic
294Q8—41no—
295Q8—41no—
296Q8—164no—
297D4—41no—
298D4—41no—
299D4—81no—
300D4—81no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—162no—
319C2×C4#47612yes—
320C2×C4#47612yes—
321C2×C4C2×C22⋊Q811yescharacteristic
322C2×C4#53511yescharacteristic
323C2×C4#53611yescharacteristic
324C2×C4C22×SD1611yescharacteristic
325C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
326C2×C4C2×#13711yescharacteristic
327C8—22no—
328C8—22no—
329C8—22no—
330C8—81no—
331C8—81no—
332C8#54211yescharacteristic
333C8#54211yescharacteristic
334C8#54411yescharacteristic
335C8#54411yescharacteristic
336C22—22no—
337C22—81no—
338C22—82no—
339C22C2×#47611yescharacteristic
340C22#321711yescharacteristic
341C22#323911yescharacteristic
342C4—22no—
343C4—22no—
344C4—42no—
345C4—42no—
346C4—81no—
347C4—82no—
348C4—161no—
349C4—161no—
350C4—161no—
351C4—161no—
352C4#262811yescharacteristic
353C4#307911yescharacteristic
354C4#320611yescharacteristic
355C4#322711yescharacteristic
356C2—21no—
357C2—21no—
358C2—161no—
359C2#2998111yescharacteristic
360C2#3363812yes—
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#135C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33318C2semidirect
3. #13543C2semidirect
4. #13542C2semidirect
7. #13458C2semidirect
8. #13457C2semidirect
11. #13184C2semidirect
12. #13183C2semidirect
13. #12884C2semidirect
72. C4×Q16Q8semidirect
74. C4×D8Q8semidirect

Maximal subgroups

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

subgroupindex
#333182
#212942
#135432
#135422
#135342
#135332
#134582
#134572
#133722
#133712
#131842
#131832
#128842
#128172
#128162