Finite Group Databasebeta

← #208183

Subgroups — #208183

Order 512.

1035 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic, complemented
3#20470C211yesmaximal, characteristic, complemented
4#9902C211yesmaximal, characteristic, complemented
5#9897C211yesmaximal, characteristic, complemented
6#9706C211yesmaximal, characteristic, complemented
7#9695C211yesmaximal, characteristic, complemented
8#9142C212yesmaximal, complemented
9#9134C212yesmaximal, complemented
10#8234C212yesmaximal, complemented
11C4×#331C212yesmaximal, complemented
12C2×#540—21no—
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2867C2211yescharacteristic
16#2859C2211yescharacteristic
17#2856C2212yes—
18#2641—21no—
19#2619—21no—
20#2445—22no—
21#2445C414yes—
22#2428—22no—
23#2428C414yes—
24#2328—21no—
25C4×#127C2212yes—
26C4×#117C2211yescharacteristic
27#1923—22no—
28#1916—22no—
29#1842C2211yescharacteristic
30#1831C2211yescharacteristic
31#1797C2212yes—
32#1767C2211yescharacteristic
33#1767C2211yescharacteristic
34C4×C4⋊C8C2211yescharacteristic
35C4×C4⋊C8C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37#1750—21no—
38#1750—21no—
39#1746—21no—
40#1726—21no—
41C2×#331C2212yes—
42C2×#331C2212yes—
43C2×#331C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#540—44no—
50#540—44no—
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#432—22no—
69#432—22no—
70#431—22no—
71#431—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—22no—
82C2×#97—21no—
83C2×#97—21no—
84C2×#97—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—22no—
89#394—24no—
90#394—24no—
91#394—24no—
92#392—24no—
93#390—22no—
94#390—22no—
95#390—22no—
96#390—22no—
97#390C2×C414yes—
98#390C2×C414yes—
99#389—22no—
100#389C2×C414yes—
101#386—24no—
102#386—24no—
103#384C2312yes—
104#382—21no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4D412yes—
107C4×C22⋊C4C2×C412yes—
108C4×C22⋊C4D412yes—
109#376D411yescharacteristic
110#376D411yescharacteristic
111C43C2311yescharacteristic
112C43C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#331—24no—
121#331—24no—
122#331—24no—
123#331—24no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4C22×C414yes—
130C23×C4—24no—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140#127—22no—
141#127—22no—
142#127—22no—
143#127—22no—
144#127—42no—
145#127—42no—
146#127—42no—
147#127—42no—
148#127—82no—
149#127—82no—
150#127—84no—
151#125—42no—
152#125—42no—
153#123—416no—
154#121—416no—
155#121—44no—
156#121—44no—
157#121—82no—
158#121—82no—
159#121—84no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—82no—
163C4×D4—82no—
164C4×D4—84no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22×C414yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C412yes—
189C2×C42C22×C412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2411yescharacteristic
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—82no—
208C4⋊C8#4112yes—
209C4⋊C8#4112yes—
210C4⋊C8#4112yes—
211C4⋊C8#4112yes—
212C22⋊C8—24no—
213C22⋊C8—24no—
214C22⋊C8—82no—
215#97—24no—
216#97—42no—
217#97—42no—
218C4×C8—24no—
219C4×C8—42no—
220C4×C8—42no—
221C4×C8—82no—
222#95—24no—
223#95C22⋊C412yes—
224#95C22⋊C412yes—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227C24—24no—
228C24C2×C22⋊C411yescharacteristic
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4—48no—
234C2×D4—48no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—48no—
248C2×D4—84no—
249C2×D4—84no—
250C2×D4—84no—
251C2×D4—84no—
252C2×D4—84no—
253C2×D4—84no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—42no—
268C22×C4—48no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—44no—
302C4⋊C4—48no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C22⋊C4—84no—
312C22⋊C4—84no—
313C22⋊C4—84no—
314C22⋊C4—84no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—42no—
331C42#13512yes—
332C42#13512yes—
333C42AGL(1,ℤ/8)12yes—
334C42AGL(1,ℤ/8)12yes—
335C42C2×#4111yescharacteristic
336C42C2×#4111yescharacteristic
337C42C2×#4111yescharacteristic
338C42C2×#4111yescharacteristic
339C23—22no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—21no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23C22×C22⋊C411yescharacteristic
352D4—84no—
353D4—84no—
354D4—82no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4—82no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—168no—
363D4—168no—
364D4—168no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—168no—
396C2×C4#41614yes—
397C2×C4#41814yes—
398C2×C4C4×#4111yescharacteristic
399C2×C4C4×#4111yescharacteristic
400C2×C4#52511yescharacteristic
401C2×C4#52511yescharacteristic
402C2×C4#53211yescharacteristic
403C2×C4#53211yescharacteristic
404C2×C4C2×#13511yescharacteristic
405C2×C4C2×#13511yescharacteristic
406C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
407C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
408C2×C4#57612yes—
409C2×C4#57712yes—
410C8—42no—
411C8—42no—
412C8—42no—
413C8—42no—
414C8—82no—
415C8—82no—
416C22—42no—
417C22—42no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—84no—
424C22—84no—
425C22—84no—
426C22—84no—
427C22—84no—
428C22#229711yescharacteristic
429C22#288812yes—
430C22C2×#41611yescharacteristic
431C22C2×#41811yescharacteristic
432C22#289712yes—
433C4—24no—
434C4—24no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—42no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—84no—
443C4#310611yescharacteristic
444C4#310611yescharacteristic
445C4#310611yescharacteristic
446C4#310611yescharacteristic
447C4#317411yescharacteristic
448C4#317611yescharacteristic
449C4#317912yes—
450C2—81no—
451C2—164no—
452C2#1772412yes—
453C2#1775812yes—
454C2#2139011yescharacteristic
455C2#2141411yescharacteristic
456C2#2142811yescharacteristic
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. #21922C2semidirect
3. #20470C2semidirect
4. #9902C2semidirect
5. #9897C2semidirect
6. #9706C2semidirect
7. #9695C2semidirect
8. #9142C2semidirect
9. #9134C2semidirect
10. #8234C2semidirect
11. C4×#331C2semidirect

Maximal subgroups

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

subgroupindex
#352782
#219222
#204702
#99022
#98972
#97062
#96952
#91422
#91342
#82342
C4×#3312