Finite Group Databasebeta

← #512422

Subgroups — #512422

Order 512.

560 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
1C2×#2231C211yesmaximal, characteristic
2C2×#2230C211yesmaximal, characteristic
3#33318C211yesmaximal, characteristic, complemented
4#20514C211yesmaximal, characteristic
5#13273C211yesmaximal, characteristic, complemented
6#13272C211yesmaximal, characteristic, complemented
7#13187C211yesmaximal, characteristic, complemented
8#13186C211yesmaximal, characteristic, complemented
9#13184C211yesmaximal, characteristic, complemented
10#13183C211yesmaximal, characteristic, complemented
11#12901C211yesmaximal, characteristic, complemented
12#12824C211yesmaximal, characteristic
13#12823C211yesmaximal, characteristic
14#12817C211yesmaximal, characteristic
15#12816C211yesmaximal, characteristic
16C2×#501—21no—
17C2×#499C2211yescharacteristic
18C2×#498C2211yescharacteristic
19C2×#443C2211yescharacteristic
20C2×#443C2211yescharacteristic
21C4×#135C2211yescharacteristic
22#2875—21no—
23#2660—21no—
24#2231—24no—
25#2230—24no—
26C2×#365C2211yescharacteristic
27C2×#365C2211yescharacteristic
28C2×#365C2211yescharacteristic
29C2×#365C2211yescharacteristic
30C2×#364C2211yescharacteristic
31C2×#364C2211yescharacteristic
32C2×#364C2211yescharacteristic
33C2×#364C2211yescharacteristic
34#2150C2211yescharacteristic
35#2150C2211yescharacteristic
36C2×#360C2211yescharacteristic
37C2×#360C2211yescharacteristic
38C2×#360C2211yescharacteristic
39C2×#360C2211yescharacteristic
40#1847—21no—
41#1847—21no—
42#1846C2211yescharacteristic
43#1846C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46C4×#108C2211yescharacteristic
47#1772C2211yescharacteristic
48#1772C2211yescharacteristic
49#1765—21no—
50#1765—21no—
51#1759—21no—
52#1726C2211yescharacteristic
53#1386C2211yescharacteristic
54#1386C2211yescharacteristic
55#1386C2211yescharacteristic
56#1386C2211yescharacteristic
57#1383C2211yescharacteristic
58#1383C2211yescharacteristic
59#1383C2211yescharacteristic
60#1383C2211yescharacteristic
61C2×#135C2311yescharacteristic
62C2×#131—21no—
63C2×#131—21no—
64C2×C4⋊Q8—21no—
65C2×#125—21no—
66C4×#41—21no—
67C2×M32C2311yescharacteristic
68C2×M32C2311yescharacteristic
69C2×M32C2311yescharacteristic
70C2×M32C2311yescharacteristic
71#501—44no—
72#499—22no—
73#499—22no—
74#498—22no—
75#498—22no—
76#443—22no—
77#443—22no—
78#443D414yes—
79#443Q814yescomplemented
80C4×Q16Q812yescomplemented
81C4×SD16—22no—
82C4×D8Q812yescomplemented
83C2×#108C2311yescharacteristic
84C2×#108C2311yescharacteristic
85#427C2311yescharacteristic
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106C2311yescharacteristic
93C2×#106C2311yescharacteristic
94C2×#106C2311yescharacteristic
95C2×#106C2311yescharacteristic
96C2×#104—21no—
97C2×#104—21no—
98#416—21no—
99#405—21no—
100#405—21no—
101#405—41no—
102#405—41no—
103C4×M16—21no—
104C4×M16—41no—
105C2×#97—41no—
106C2×#97D411yescharacteristic
107C2×#97D411yescharacteristic
108C2×#97D411yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#398—42no—
113#395—42no—
114#384—21no—
115#384—21no—
116#382—21no—
117#382—21no—
118#376D411yescharacteristic
119#365—22no—
120#365—22no—
121#365—22no—
122#365—22no—
123#364—22no—
124#364—22no—
125#364—22no—
126#364—22no—
127C4⋊C16—22no—
128C4⋊C16—22no—
129#360—22no—
130#360—22no—
131#360—22no—
132#360—22no—
133#341—21no—
134#339—21no—
135#339—21no—
136C2×#41—21no—
137C22×Q8—41no—
138#135—21no—
139#135—21no—
140#135C2×D412yes—
141#135C2×Q812yes—
142C2×Q16C2×Q811yescharacteristic
143C2×SD16—21no—
144C2×D8C2×Q811yescharacteristic
145#131—24no—
146#131—24no—
147#131—42no—
148#131—42no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—41no—
152C2×M16—41no—
153C22×C8—41no—
154C22×C8—41no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2411yescharacteristic
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161#125—42no—
162#125—42no—
163C4×Q8—22no—
164C4×D4—22no—
165C4×D4—42no—
166#117—21no—
167#117—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C42—41no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181M32—21no—
182M32—21no—
183M32—21no—
184M32—21no—
185M32—21no—
186M32—21no—
187M32—21no—
188M32—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193#108—21no—
194#108—21no—
195#108C2×D412yes—
196#108C2×Q812yes—
197#107—42no—
198#107—42no—
199#107—42no—
200#107C2×Q812yes—
201#106—21no—
202#106—21no—
203#106—21no—
204#106—21no—
205#106—21no—
206#106—21no—
207#106—21no—
208#106—21no—
209#106—21no—
210#106—42no—
211#106—42no—
212#106—42no—
213#104—24no—
214#104—24no—
215#103—22no—
216#102—22no—
217#97—22no—
218#97—22no—
219#97—41no—
220#97—41no—
221#97—41no—
222#97SD1611yescharacteristic
223#97SD1611yescharacteristic
224#97SD1611yescharacteristic
225#97SD1611yescharacteristic
226C4×C8—41no—
227C4×C8—41no—
228C4×C8—41no—
229C4×C8C2×D412yes—
230C4×C8C2×Q812yes—
231C4×C8C2×Q812yes—
232C4×C8#4112yes—
233C4×C8#4112yes—
234C4×C8#4112yes—
235#95—22no—
236#95—42no—
237#95—42no—
238#41—22no—
239#41—22no—
240#41—41no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—82no—
244C2×Q8—162no—
245C2×D4—21no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4AGL(1,ℤ/8)12yes—
256C22×C4C22×D411yescharacteristic
257Q16—21no—
258Q16C4⋊Q812yes—
259SD16—22no—
260SD16—22no—
261D8—21no—
262D8C4⋊Q812yes—
263M16—22no—
264M16—22no—
265M16—41no—
266M16—41no—
267M16—44no—
268M16—44no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8#12112yes—
276C2×C8C22⋊Q812yes—
277C2×C8C22⋊Q812yes—
278C2×C8C2×SD1611yescharacteristic
279C2×C8C2×SD1611yescharacteristic
280C2×C8C2×SD1611yescharacteristic
281C2×C8C2×SD1611yescharacteristic
282C2×C8AGL(1,ℤ/8)12yes—
283C2×C8#13712yes—
284C2×C8C22×D411yescharacteristic
285C2×C8C22×Q811yescharacteristic
286C2×C8C22×Q811yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—162no—
304C4⋊C4—162no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C42—21no—
308C42—21no—
309C42—41no—
310C42—42no—
311C42#12112yes—
312C42C22⋊Q812yes—
313C42C22⋊Q812yes—
314C42C2×SD1611yescharacteristic
315C42C2×SD1611yescharacteristic
316C16—41no—
317C16—41no—
318C16—41no—
319C16—41no—
320C23—41no—
321C23C2×AGL(1,ℤ/8)11yescharacteristic
322Q8—22no—
323Q8—41no—
324Q8—164no—
325D4—22no—
326D4—42no—
327D4—42no—
328D4—41no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—162no—
347C2×C4#47612yes—
348C2×C4#47612yes—
349C2×C4C2×#12111yescharacteristic
350C2×C4C2×C22⋊Q811yescharacteristic
351C2×C4C2×C22⋊Q811yescharacteristic
352C2×C4C22×SD1611yescharacteristic
353C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C8—22no—
356C8—42no—
357C8—42no—
358C8#46112yes—
359C8#46112yes—
360C8#47612yes—
361C8#47612yes—
362C8C2×C4⋊Q811yescharacteristic
363C8#53211yescharacteristic
364C8#53311yescharacteristic
365C8#53311yescharacteristic
366C22—22no—
367C22—42no—
368C22—82no—
369C22#303811yescharacteristic
370C22C2×#47611yescharacteristic
371C22#307911yescharacteristic
372C4—22no—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—82no—
378C4—161no—
379C4—161no—
380C4—161no—
381C4—161no—
382C4#285711yescharacteristic
383C4C2×#46111yescharacteristic
384C4C2×#47611yescharacteristic
385C4#307911yescharacteristic
386C2—21no—
387C2—21no—
388C2—82no—
389C2#3319711yescharacteristic
390C2#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
3. #33318C2semidirect
5. #13273C2semidirect
6. #13272C2semidirect
7. #13187C2semidirect
8. #13186C2semidirect
9. #13184C2semidirect
10. #13183C2semidirect
11. #12901C2semidirect
79. #443Q8semidirect
80. C4×Q16Q8semidirect
82. C4×D8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22312
C2×#22302
#333182
#205142
#132732
#132722
#131872
#131862
#131842
#131832
#129012
#128242
#128232
#128172
#128162