Finite Group Databasebeta

← #481839

Subgroups — #481839

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35700C211yesmaximal, characteristic
2#20691C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#11801C214yesmaximal, complemented
5#11798C214yesmaximal
6#11694C212yesmaximal, complemented
7#11691C212yesmaximal
8C2×D4×Q8—21no—
9C2×D4×Q8C411yescharacteristic, complemented
10C2×#519C2211yescharacteristic
11#2983—21no—
12#2621C2212yes—
13#2621C2211yescharacteristic
14#2546C411yescharacteristic, complemented
15#2546C412yescomplemented
16#2536C412yescomplemented
17#2536C411yescharacteristic, complemented
18#2429—21no—
19#2429—22no—
20#2415—22no—
21#2415—21no—
22#2392C2212yes—
23#2392C2211yescharacteristic
24#2351C411yescharacteristic, complemented
25Q8×C22⋊C4—21no—
26#2310—21no—
27#1820—22no—
28#1820C2214yes—
29#1740—21no—
30#1740C2212yes—
31#1467—22no—
32#1464C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1344C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C23×Q8—21no—
39C23×Q8D411yescharacteristic
40D4×Q8—44no—
41D4×Q8—48no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—21no—
52#519—44no—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C411yescharacteristic
64C2×#117—22no—
65C2×#117D412yes—
66C2×#117D411yescharacteristic
67C2×#117C2312yes—
68C2×#117C2311yescharacteristic
69#446—44no—
70#432—24no—
71#431—22no—
72#398—22no—
73#398—21no—
74#398C2×C412yes—
75#398C2×C412yes—
76#398C2×C412yes—
77#398C2×C412yes—
78#398C2×C411yescharacteristic
79#397—22no—
80#397—21no—
81#397C2311yescharacteristic
82#397C2312yes—
83#396—22no—
84#396—21no—
85#394—22no—
86#394—21no—
87#394C2×C412yes—
88#394C2×C411yescharacteristic
89#392C2311yescharacteristic
90#391—22no—
91#391—21no—
92#391—21no—
93#391—22no—
94#391—44no—
95#389—22no—
96#389—21no—
97#389—22no—
98#389—22no—
99#389—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#382C2×C412yes—
104#382C2×C411yescharacteristic
105#381—24no—
106#380—22no—
107#380—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—21no—
111C4×C22⋊C4—22no—
112#342—22no—
113#342—24no—
114#342C2314yes—
115#324—22no—
116#324—24no—
117#323C2314yes—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—24no—
122C22×Q8—42no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22⋊C412yes—
128C22×Q8C22⋊C411yescharacteristic
129C22×D4—21no—
130C22×D4C22×C411yescharacteristic
131C23×C4—22no—
132C23×C4—21no—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C4⋊Q8—44no—
136C4⋊Q8—42no—
137C4⋊Q8—48no—
138C4⋊Q8—44no—
139C22⋊Q8—48no—
140C22⋊Q8—44no—
141C22⋊Q8—48no—
142C22⋊Q8—44no—
143C22⋊Q8—48no—
144C22⋊Q8—44no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—44no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8—44no—
151C4×Q8—44no—
152C4×D4—44no—
153C4×D4—42no—
154C4×D4—48no—
155C4×D4—44no—
156#117—42no—
157#117—44no—
158#117—42no—
159#117—44no—
160#117—44no—
161#117—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4C2×D411yescharacteristic
195C2×C22⋊C4C2×D412yes—
196C2×C22⋊C4C2411yescharacteristic
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—42no—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C4⋊C8—42no—
208C4⋊C8—44no—
209C22⋊C8—24no—
210#97—44no—
211C4×C8—42no—
212#95—21no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—48no—
222#95C22×C411yescharacteristic
223#95C22×C412yes—
224#95C22×C411yescharacteristic
225C24—21no—
226C24C22×D411yescharacteristic
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—42no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—816no—
244C2×Q8#9914yes—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4AGL(1,ℤ/8)14yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C2×C8—44no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—84no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—42no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C42—22no—
316C42—24no—
317C42—24no—
318C42—22no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—44no—
323C42—42no—
324C42—84no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23C22×C22⋊C411yescharacteristic
337C23C2×AGL(1,ℤ/8)12yes—
338Q8—28no—
339Q8—48no—
340Q8—44no—
341Q8—88no—
342Q8—84no—
343D4—42no—
344D4—48no—
345D4—42no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—48no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—88no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4C2×#9914yes—
389C2×C4C2×#9912yes—
390C2×C4#41812yes—
391C2×C4#41812yes—
392C2×C4#42112yes—
393C8—84no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—42no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22C22×#9911yescharacteristic
409C22C2×#41811yescharacteristic
410C22C2×#42111yescharacteristic
411C4—22no—
412C4—24no—
413C4—42no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—41no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—82no—
422C4—81no—
423C4—82no—
424C4—81no—
425C4—164no—
426C4#151014yes—
427C2—21no—
428C2—21no—
429C2—42no—
430C2—82no—
431C2C2×#151011yescharacteristic
432C2#3334911yescharacteristic
433C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#392C23
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20691C2semidirect
3. #17371C2semidirect
4. #11801C2semidirect
6. #11694C2semidirect
9. C2×D4×Q8C4semidirect
14. #2546C4semidirect
15. #2546C4semidirect
16. #2536C4semidirect
17. #2536C4semidirect
24. #2351C4semidirect

Maximal subgroups

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

subgroupindex
#357002
#206912
#173712
#118012
#117982
#116942
#116912