Finite Group Databasebeta

← #217099

Subgroups — #217099

Order 512.

1019 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#9902C211yesmaximal, characteristic, complemented
5#9897C211yesmaximal, characteristic, complemented
6#9706C211yesmaximal, characteristic, complemented
7#9695C211yesmaximal, characteristic, complemented
8#9142C212yesmaximal, complemented
9#9141C212yesmaximal, complemented
10#8588C212yesmaximal, complemented
11#8585C212yesmaximal, complemented
12C2×#540—21no—
13C2×#433—21no—
14C2×#432C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2867C2211yescharacteristic
17#2859C2211yescharacteristic
18#2856C2212yes—
19#2641—21no—
20#2619—21no—
21#2445—22no—
22#2445C414yes—
23#2428—22no—
24#2428C414yes—
25#2328—21no—
26C4×#127C2212yes—
27C4×#117C2211yescharacteristic
28#1940—22no—
29#1851—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1832—21no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1798—21no—
37#1797C2212yes—
38#1768—21no—
39#1767C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C2×#331C2212yes—
42C2×#331C2212yes—
43C2×#331C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×M16—21no—
50#540—48no—
51C2×#127—21no—
52C2×#127—21no—
53C2×#127—22no—
54C2×#127C2×C414yes—
55C2×#127C2312yes—
56C2×#125—21no—
57C2×#123—24no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—24no—
61C2×#121C2×C414yes—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C414yes—
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C22×C42C2311yescharacteristic
68#433—44no—
69#432—42no—
70#423—42no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C22⋊C8—22no—
83C2×C22⋊C8—21no—
84C2×#97—21no—
85C2×#97—21no—
86C2×C4×C8—21no—
87#394—24no—
88#394—24no—
89#394—24no—
90#392—24no—
91#390—22no—
92#390—22no—
93#390—22no—
94#390—22no—
95#390C2×C414yes—
96#390C2×C414yes—
97#389—22no—
98#389C2×C414yes—
99#386—24no—
100#386—24no—
101#384C2312yes—
102#382—21no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4D412yes—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4D412yes—
107#376D411yescharacteristic
108#376D411yescharacteristic
109C43C2311yescharacteristic
110C43C2311yescharacteristic
111#336—22no—
112#336—21no—
113#336—21no—
114#336—22no—
115#336—22no—
116#331—42no—
117#331—42no—
118#331—42no—
119#331—42no—
120C22×D4—24no—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4C22×C414yes—
126C23×C4—24no—
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—42no—
131C2×M16—82no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138#127—22no—
139#127—22no—
140#127—22no—
141#127—22no—
142#127—44no—
143#127—42no—
144#127—42no—
145#127—82no—
146#127—82no—
147#127—84no—
148#125—44no—
149#123—416no—
150#121—416no—
151#121—44no—
152#121—44no—
153#121—82no—
154#121—82no—
155#121—84no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4—82no—
159C4×D4—82no—
160C4×D4—84no—
161#117—24no—
162#117—22no—
163#117—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22×C414yes—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C412yes—
182C2×C42C22×C412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2411yescharacteristic
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—42no—
201C4⋊C8—44no—
202C4⋊C8—44no—
203C4⋊C8—82no—
204C4⋊C8—82no—
205C22⋊C8—42no—
206C22⋊C8—44no—
207#97—42no—
208#97—42no—
209C4×C8—42no—
210#95—24no—
211#95C22⋊C412yes—
212#95C22⋊C412yes—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215C24—24no—
216C24C2×C22⋊C411yescharacteristic
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—48no—
222C2×D4—48no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—48no—
236C2×D4—84no—
237C2×D4—84no—
238C2×D4—84no—
239C2×D4—84no—
240C2×D4—84no—
241C2×D4—84no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266M16—84no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—44no—
284C4⋊C4—48no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—48no—
292C22⋊C4—48no—
293C22⋊C4—84no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C22⋊C4—84no—
297C42—22no—
298C42—21no—
299C42—21no—
300C42—22no—
301C42—21no—
302C42—22no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—42no—
309C42—42no—
310C42AGL(1,ℤ/8)12yes—
311C42AGL(1,ℤ/8)14yes—
312C42#13712yes—
313C42H(5,𝔽2)12yes—
314C422−1+412yes—
315C23—21no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23C22×C22⋊C411yescharacteristic
328D4—84no—
329D4—84no—
330D4—82no—
331D4—82no—
332D4—82no—
333D4—82no—
334D4—82no—
335D4—82no—
336D4—82no—
337D4—82no—
338D4—168no—
339D4—168no—
340D4—168no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—82no—
368C2×C4—168no—
369C2×C4#41714yes—
370C2×C4#41814yes—
371C2×C4#52012yes—
372C2×C4#53511yescharacteristic
373C2×C4#53611yescharacteristic
374C2×C4#53712yes—
375C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
376C2×C4C2×AGL(1,ℤ/8)12yes—
377C2×C4C2×#13711yescharacteristic
378C2×C4#57612yes—
379C2×C4#57712yes—
380C8—81no—
381C8—81no—
382C8—81no—
383C8—81no—
384C8—82no—
385C8—82no—
386C22—42no—
387C22—42no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—84no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22—84no—
396C22—84no—
397C22—84no—
398C22#231111yescharacteristic
399C22#288812yes—
400C22C2×#41811yescharacteristic
401C22C2×#41711yescharacteristic
402C22#289712yes—
403C4—24no—
404C4—24no—
405C4—22no—
406C4—42no—
407C4—42no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—84no—
412C4#322011yescharacteristic
413C4#322111yescharacteristic
414C4#324112yes—
415C4#325312yes—
416C4#325412yes—
417C2—81no—
418C2—164no—
419C2#1775812yes—
420C2#1776212yes—
421C2#2187211yescharacteristic
422C2#2188211yescharacteristic
423C2#2189311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35278C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21978C2semidirect
3. #20812C2semidirect
4. #9902C2semidirect
5. #9897C2semidirect
6. #9706C2semidirect
7. #9695C2semidirect
8. #9142C2semidirect
9. #9141C2semidirect
10. #8588C2semidirect
11. #8585C2semidirect

Maximal subgroups

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

subgroupindex
#352782
#219782
#208122
#99022
#98972
#97062
#96952
#91422
#91412
#85882
#85852