Finite Group Databasebeta

← #480333

Subgroups — #480333

Order 512.

995 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34266C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic
3#17370C211yesmaximal, characteristic
4#11528C214yesmaximal, complemented
5#11526C214yesmaximal, complemented
6#11452C212yesmaximal, complemented
7#11428C212yesmaximal, complemented
8C2×#536C411yescharacteristic
9C2×#520C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic
11#2983—21no—
12#2975—21no—
13#2367—21no—
14#2367—21no—
15#2365—21no—
16#2365—21no—
17#2362—22no—
18#2360—22no—
19#2322C412yes—
20#2322C2212yes—
21#2310C2211yescharacteristic
22C4×#123C412yes—
23C4×#123C2212yes—
24#2297C411yescharacteristic
25#2296C2211yescharacteristic
26#2274C411yescharacteristic
27#1466C2214yes—
28#1465C2214yes—
29#1464—22no—
30#1463—22no—
31#1461—21no—
32#1460—21no—
33#1458C2212yes—
34#1458C2212yes—
35#1326C2214yes—
36#1326C2214yes—
37#1317C2218yes—
38C22×#41C2×C411yescharacteristic
39C23×Q8C2×C411yescharacteristic
40#536—24no—
41#536—24no—
42C2×C4⋊Q8—21no—
43C2×#124—21no—
44C2×#123C2×C412yes—
45C2×#123C2×C412yes—
46C2×C22⋊Q8—22no—
47#520—44no—
48C2×C4×Q8—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—44no—
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×D4C2×C412yes—
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2312yes—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C2×#117D412yes—
61C2×#117C2311yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4D412yes—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C412yes—
67#446—44no—
68#436—44no—
69#432—24no—
70#431—22no—
71#431—22no—
72#407—24no—
73#391—21no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#390—21no—
79#389—22no—
80#389—22no—
81#389—21no—
82#389—21no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#386C2×C412yes—
92#386C2×C412yes—
93#385—22no—
94#385—22no—
95#385C2×C411yescharacteristic
96#385C2×C411yescharacteristic
97#385C2311yescharacteristic
98#385C2311yescharacteristic
99#381C2×C412yes—
100#381C2312yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4C2×C412yes—
113#324—22no—
114#324—22no—
115#324—24no—
116#323—24no—
117#323C2314yes—
118#323C2314yes—
119C2×#41—22no—
120C2×#41—22no—
121C2×#41—42no—
122C2×#41—42no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8C22⋊C411yescharacteristic
129C22×Q8M1614yescomplemented
130C22×D4C22⋊C411yescharacteristic
131C22×D4C22×C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C412yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D412yes—
140C2×M16—44no—
141C4⋊Q8—28no—
142#124—28no—
143#123—24no—
144#123—24no—
145#123—24no—
146#123—24no—
147C22⋊Q8—216no—
148C4×Q8—24no—
149C4×Q8—42no—
150C4×Q8—48no—
151C4×Q8—44no—
152C4×Q8—44no—
153C4×D4—24no—
154C4×D4—42no—
155C4×D4—44no—
156C4×D4—44no—
157#117—22no—
158#117—42no—
159#117—44no—
160#117—42no—
161#117M1614yescomplemented
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22×C411yescharacteristic
188C2×C22⋊C4C22×C411yescharacteristic
189C2×C22⋊C4C22×C412yes—
190C2×C22⋊C4C22×C411yescharacteristic
191C2×C22⋊C4C2×D412yes—
192C2×C22⋊C4C2411yescharacteristic
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42—44no—
203C2×C42C22⋊C412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C4⋊C8—42no—
208C4⋊C8—42no—
209C4⋊C8—44no—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—44no—
213#97—44no—
214C4×C8—42no—
215C4×C8—42no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—21no—
228C24C2×C22⋊C412yes—
229C24C23×C411yescharacteristic
230#41—48no—
231#41—48no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—48no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—48no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—42no—
249C2×D4—42no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—82no—
271C22×C4—81no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×M1612yes—
282C22×C4C2×M1612yes—
283C22×C4C2×M1614yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286M16—164no—
287C2×C8—44no—
288C2×C8—44no—
289C4⋊C4—28no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4#9914yes—
306C22⋊C4—28no—
307C22⋊C4—28no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—24no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—44no—
315C22⋊C4—42no—
316C22⋊C4—82no—
317C22⋊C4C2×M1612yes—
318C22⋊C4C2×M1612yes—
319C42—24no—
320C42—22no—
321C42—22no—
322C42—24no—
323C42—42no—
324C42—42no—
325C42—44no—
326C42—42no—
327C42—44no—
328C42—42no—
329C42—82no—
330C42—82no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23C22×C22⋊C411yescharacteristic
341C23C22×M1611yescharacteristic
342C23C22×M1611yescharacteristic
343Q8—48no—
344Q8—44no—
345Q8—48no—
346D4—44no—
347D4—44no—
348D4—44no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—48no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—162no—
394C2×C4#40712yes—
395C2×C4#40712yes—
396C2×C4C2×#9914yes—
397C2×C4C2×#9912yes—
398C2×C4#41312yes—
399C8—84no—
400C8—84no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—41no—
406C22—41no—
407C22—42no—
408C22—44no—
409C22—41no—
410C22—41no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—82no—
415C22C2×#40711yescharacteristic
416C22C22×#9911yescharacteristic
417C22C2×#41311yescharacteristic
418C4—24no—
419C4—22no—
420C4—42no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—42no—
429C4—44no—
430C4—82no—
431C4—81no—
432C4#146514yes—
433C2—21no—
434C2—21no—
435C2—41no—
436C2—41no—
437C2—82no—
438C2C2×#146511yescharacteristic
439C2#3334911yescharacteristic
440C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11528C2semidirect
5. #11526C2semidirect
6. #11452C2semidirect
7. #11428C2semidirect
129. C22×Q8M16semidirect
161. #117M16semidirect

Maximal subgroups

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

subgroupindex
#342662
#173722
#173702
#115282
#115262
#114522
#114282