Finite Group Databasebeta

← #503520

Subgroups — #503520

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic, complemented
2C2×#2244C211yesmaximal, characteristic, complemented
3C2×#2240C211yesmaximal, characteristic, complemented
4#33179C211yesmaximal, characteristic
5#13365C212yesmaximal, complemented
6#13363C212yesmaximal, complemented
7#13283C211yesmaximal, characteristic, complemented
8#13281C211yesmaximal, characteristic, complemented
9#12992C212yesmaximal, complemented
10#12991C212yesmaximal, complemented
11#12600C211yesmaximal, characteristic, complemented
12C2×#544—21no—
13#3164—24no—
14#3164—24no—
15C2×#501—21no—
16C2×#501—21no—
17C2×#500C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#499C2211yescharacteristic
20C2×#499C2211yescharacteristic
21C2×#493C2211yescharacteristic
22C2×#469—21no—
23C2×#466C2212yes—
24C2×#428—21no—
25C2×#427C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C2×#405C2211yescharacteristic
28#2619—21no—
29#2244—24no—
30#2240—24no—
31C2×#357C2212yes—
32C2×#357C2212yes—
33C2×#357C2212yes—
34C2×#357C2212yes—
35C2×C22⋊C16C2212yes—
36C2×#346C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38#1972—21no—
39#1970C2212yes—
40#1961—21no—
41#1846C2211yescharacteristic
42#1380C2212yes—
43#1380C2212yes—
44#1380C2212yes—
45#1380C2212yes—
46#1374C2212yes—
47C22×D8—21no—
48C22×D8C2312yes—
49C22×M16D411yescharacteristic
50C23×C8C2311yescharacteristic
51#544—28no—
52#544—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—41no—
56C2×#127—21no—
57C2×#125—21no—
58C2×#125—21no—
59C2×#125—41no—
60C2×C22⋊Q8—41no—
61C2×C22⋊Q8—41no—
62C2×#121—22no—
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C16C2312yes—
67C22×C16C2312yes—
68#501—28no—
69#501—28no—
70#500—22no—
71#500—22no—
72#500—22no—
73#500D414yescomplemented
74#499—22no—
75#499—22no—
76#499—22no—
77#499D414yescomplemented
78#493—22no—
79#493—22no—
80#469—44no—
81#466—28no—
82#428—28no—
83#427—22no—
84#427—22no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107C2312yes—
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#106—21no—
96C2×#106—21no—
97C2×#106—21no—
98C2×#106—21no—
99C2×#102—22no—
100C2×#102—22no—
101#405—22no—
102#405—22no—
103C2×#97D411yescharacteristic
104C2×#97D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#394—41no—
108#394—41no—
109#392—41no—
110#392—41no—
111#390—22no—
112#382—21no—
113#357—24no—
114#357—24no—
115#357—24no—
116#357—24no—
117C22⋊C16—24no—
118#346—22no—
119C4×C16—22no—
120C22×Q8—41no—
121C22×Q8—41no—
122C22×D4—22no—
123C22×D4—42no—
124C22×D4—41no—
125C22×D4—41no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C2×D8—22no—
129C2×D8—22no—
130C2×D8—24no—
131C2×D8—24no—
132C2×D8—42no—
133C2×D8—42no—
134C2×D8—42no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16SD1614yes—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8#4114yes—
145C22×C8C2411yescharacteristic
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—42no—
149C4⋊Q8—48no—
150C4⋊Q8—42no—
151#127—42no—
152#127—42no—
153#125—24no—
154#125—24no—
155#125—42no—
156#125—48no—
157#125—42no—
158C22⋊Q8—48no—
159C22⋊Q8—48no—
160#121—44no—
161#121—44no—
162#117—22no—
163#117—22no—
164#117—24no—
165#117—42no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4D811yescharacteristic
181C2×C4⋊C4D811yescharacteristic
182C2×C22⋊C4—41no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4D811yescharacteristic
187C2×C22⋊C4D811yescharacteristic
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C16—22no—
192C2×C16—22no—
193C2×C16—22no—
194C2×C16—22no—
195C2×C16—22no—
196C2×C16—22no—
197#107—24no—
198#107—24no—
199#107—24no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—24no—
205#107C2×D412yes—
206#107C2×D412yes—
207#107C2×D412yes—
208#107C2×D412yes—
209#106—24no—
210#106—24no—
211#106—24no—
212#106—42no—
213#102—44no—
214#102—44no—
215#97—22no—
216#97—22no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8C2×D412yes—
220C4×C8C2×D412yes—
221#95—41no—
222#95—41no—
223C24—42no—
224C24C2×D811yescharacteristic
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—162no—
228C2×Q8—162no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—84no—
236C2×D4—82no—
237C2×D4—82no—
238C2×D4—162no—
239C2×D4—162no—
240C2×D4—162no—
241C2×D4—162no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—162no—
253C22×C4#12412yes—
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×SD1612yes—
260C22×C4C22×D411yescharacteristic
261D8—44no—
262D8—44no—
263D8—84no—
264M16—28no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8#12012yes—
272C2×C8#12412yes—
273C2×C8C2×SD1612yes—
274C2×C8C2×SD1612yes—
275C2×C8#13712yes—
276C2×C8C22×D411yescharacteristic
277C2×C8C22×D411yescharacteristic
278C2×C8C2×#4112yes—
279C2×C82−1+412yes—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—44no—
302C22⋊C4—84no—
303C22⋊C4—162no—
304C22⋊C4—162no—
305C42—21no—
306C42—21no—
307C42—44no—
308C42#12012yes—
309C42AGL(1,ℤ/8)12yes—
310C16—42no—
311C16—42no—
312C23—22no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23—82no—
317C23—82no—
318C23—82no—
319C23—82no—
320C23—82no—
321C23—82no—
322C23—82no—
323C23#48612yes—
324C23C22×D811yescharacteristic
325Q8—164no—
326Q8—164no—
327D4—82no—
328D4—82no—
329D4—82no—
330D4—82no—
331D4—164no—
332D4—164no—
333D4—168no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—164no—
358C2×C4#44712yes—
359C2×C4#44712yes—
360C2×C4#48612yes—
361C2×C4C2×#12011yescharacteristic
362C2×C4C2×#12411yescharacteristic
363C2×C4#53611yescharacteristic
364C2×C4C22×SD1611yescharacteristic
365C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
366C2×C4C2×#13711yescharacteristic
367C8—22no—
368C8—22no—
369C8#44814yes—
370C8#54812yes—
371C8#54812yes—
372C22—22no—
373C22—24no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—162no—
378C22—162no—
379C22—162no—
380C22—162no—
381C22—162no—
382C22—162no—
383C22—162no—
384C22C2×#44711yescharacteristic
385C22C2×#48611yescharacteristic
386C22#322611yescharacteristic
387C22#341512yes—
388C22#341612yes—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—82no—
395C4—82no—
396C4—82no—
397C4—82no—
398C4—162no—
399C4#243611yescharacteristic
400C4C2×#44811yescharacteristic
401C4#314111yescharacteristic
402C4#322711yescharacteristic
403C2—42no—
404C2—322no—
405C2#2590311yescharacteristic
406C2#3352211yescharacteristic
407C2#3352411yescharacteristic
408C2#3370212yes—
409C2#3380712yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#33179C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3164C2semidirect
2. C2×#2244C2semidirect
3. C2×#2240C2semidirect
5. #13365C2semidirect
6. #13363C2semidirect
7. #13283C2semidirect
8. #13281C2semidirect
9. #12992C2semidirect
10. #12991C2semidirect
11. #12600C2semidirect
73. #500D4semidirect
77. #499D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C2×#22442
C2×#22402
#331792
#133652
#133632
#132832
#132812
#129922
#129912
#126002