Finite Group Databasebeta

← #480493

Subgroups — #480493

Order 512.

871 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35672C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#11546C214yesmaximal
5#11544C214yesmaximal, complemented
6#11454C212yesmaximal, complemented
7#11425C212yesmaximal
8C22×C22⋊Q8C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2983—21no—
11#2982—21no—
12#2484C412yes—
13#2484C2212yes—
14#2482C412yes—
15#2482C2212yes—
16#2423—22no—
17#2423—21no—
18#2423—21no—
19#2421—21no—
20#2421—22no—
21#2421—21no—
22#2317C411yescharacteristic
23#2317C2211yescharacteristic
24#2310C411yescharacteristic
25#2280C411yescharacteristic
26#2280C2211yescharacteristic
27#1467—22no—
28#1464C2214yes—
29#1461—22no—
30#1461—21no—
31#1461—21no—
32#1458C2214yes—
33#1458C2212yes—
34#1458C2212yes—
35#1326C2218yes—
36#1317C2214yes—
37#1316C2214yes—
38C23×Q8C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—44no—
44#519—44no—
45C2×C4×Q8D411yescharacteristic
46C2×C4×Q8D412yes—
47C2×C4×Q8D412yes—
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C412yes—
55C2×#117C2311yescharacteristic
56C2×#117C2312yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60#446—44no—
61C8×Q8—44no—
62#432—24no—
63#431—22no—
64#431—24no—
65#431—22no—
66#400—22no—
67#400—22no—
68#399—22no—
69#399—22no—
70#398—22no—
71#398—22no—
72#397—22no—
73#397—22no—
74#391—21no—
75#391—22no—
76#391—21no—
77#389—22no—
78#389—21no—
79#389—21no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#385C2×C412yes—
89#385C2×C412yes—
90#385C2×C412yes—
91#385C2312yes—
92#384—21no—
93#384—21no—
94#382—21no—
95#382—21no—
96#381C2×C412yes—
97#381C2×C412yes—
98#381C2×C411yescharacteristic
99#381C2×C412yes—
100#381C2311yescharacteristic
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#379C2×C411yescharacteristic
106#379C2×C412yes—
107#379C2311yescharacteristic
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111#324—22no—
112#324—24no—
113#324—22no—
114#324—24no—
115#323C2314yes—
116#323C2314yes—
117C25C22×C411yescharacteristic
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—42no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22⋊C412yes—
123C22×Q8C22⋊C411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—24no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C412yes—
133C22⋊Q8—44no—
134C22⋊Q8—48no—
135C22⋊Q8—44no—
136C4×Q8—24no—
137C4×Q8—24no—
138C4×Q8—42no—
139C4×Q8—44no—
140C4×Q8—42no—
141C4×Q8—44no—
142#117—24no—
143#117—42no—
144#117—44no—
145#117—42no—
146#117—44no—
147#117—42no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4C22×C412yes—
178C2×C22⋊C4C22×C411yescharacteristic
179C2×C22⋊C4C22×C411yescharacteristic
180C2×C22⋊C4C22×C412yes—
181C2×C22⋊C4C22×C411yescharacteristic
182C2×C22⋊C4C2411yescharacteristic
183C2×C42—21no—
184C2×C42—21no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C4⋊C8—44no—
193C4⋊C8—42no—
194C4⋊C8—42no—
195C4⋊C8—44no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198#97—44no—
199C4×C8—44no—
200C4×C8—42no—
201C4×C8—42no—
202#95—22no—
203#95—21no—
204#95—22no—
205#95—21no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—22no—
216C24—21no—
217C24—22no—
218C24—41no—
219C24—42no—
220C24—41no—
221C24—81no—
222C24C23×C411yescharacteristic
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—24no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—416no—
232C2×Q8#9914yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—88no—
254C22×C4—82no—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4#13114yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C2×C8—44no—
267C2×C8—44no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C22⋊C4—24no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—41no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—44no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—82no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C42—24no—
301C42—22no—
302C42—22no—
303C42—24no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—42no—
309C23—21no—
310C23—21no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—82no—
323C23—84no—
324C23—84no—
325C23—82no—
326C23C22×C22⋊C411yescharacteristic
327C23C2×#13112yes—
328Q8—28no—
329Q8—48no—
330Q8—44no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—48no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4#40812yes—
372C2×C4#40812yes—
373C2×C4C2×#9914yes—
374C2×C4C2×#9912yes—
375C2×C4#41312yes—
376C8—84no—
377C8—84no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—42no—
388C22—81no—
389C22—81no—
390C22—84no—
391C22—84no—
392C22—81no—
393C22—81no—
394C22C2×#40811yescharacteristic
395C22C22×#9911yescharacteristic
396C22C2×#41311yescharacteristic
397C4—22no—
398C4—24no—
399C4—41no—
400C4—42no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—42no—
405C4—42no—
406C4—82no—
407C4—81no—
408C4—81no—
409C4—82no—
410C4—81no—
411C4#146814yes—
412C2—21no—
413C2—21no—
414C2—42no—
415C2—82no—
416C2C2×#146811yescharacteristic
417C2#3334911yescharacteristic
418C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ 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. #17371C2semidirect
3. #17352C2semidirect
5. #11544C2semidirect
6. #11454C2semidirect

Maximal subgroups

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

subgroupindex
#356722
#173712
#173522
#115462
#115442
#114542
#114252