Finite Group Databasebeta

← #495429

Subgroups — #495429

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35333C211yesmaximal, characteristic
2#20637C211yesmaximal, characteristic
3C4×#413C211yesmaximal, characteristic
4#13303C218yesmaximal, complemented
5#12587C214yesmaximal
6C22×#125C411yescharacteristic
7C2×C4×#41C2211yescharacteristic
8C2×#423—21no—
9C2×#413C2212yes—
10C2×#413C2212yes—
11C2×C4×M16—21no—
12#2851—21no—
13#2851—21no—
14#2629—21no—
15#2629—21no—
16#2579—22no—
17#2579—22no—
18#2566C414yes—
19#2565C2214yes—
20#2317C412yes—
21#2311C2212yes—
22#2285C411yescharacteristic
23#1801C2214yes—
24#1800—22no—
25C4×#101C2214yes—
26C4×#100—22no—
27#1742—21no—
28#1742—21no—
29#1394C22116yes—
30C22×#41C2311yescharacteristic
31C22×M16—22no—
32C22×M16—22no—
33C2×#125—21no—
34C2×#125—21no—
35C2×#125—41no—
36C2×#125—41no—
37C2×#125—42no—
38C2×#125C2×C414yes—
39C4×#41—41no—
40C4×#41—41no—
41C4×#41—42no—
42C2×C4×Q8C2312yes—
43C2×C4×D4C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C412yes—
48C2×#117D411yescharacteristic
49C2×#117D411yescharacteristic
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2×C412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55#423—24no—
56#423—41no—
57#423—41no—
58C2×C4⋊C8—42no—
59#413—24no—
60#413—28no—
61#413Q818yes—
62C2×#101C2314yes—
63C2×#101C2314yes—
64C2×#100—22no—
65C2×#100—22no—
66C4×M16—24no—
67C4×M16—42no—
68C2×#97—41no—
69C2×C4×C8—41no—
70#398—24no—
71#398—24no—
72#396—24no—
73#396—24no—
74#391C2314yes—
75#390C2×C414yes—
76#389—22no—
77#389—22no—
78#387—24no—
79#387—24no—
80#386—22no—
81#386—22no—
82#385—24no—
83#385—24no—
84#385—22no—
85#385—22no—
86#384—21no—
87#384C2×C414yes—
88#382—21no—
89#382C2×C414yes—
90#380—22no—
91#380—22no—
92#380—22no—
93C4×C4⋊C4C2×C414yes—
94#343—24no—
95#343—24no—
96C2×#41—41no—
97C2×#41—41no—
98C2×#41—42no—
99C2×#41C2×Q814yes—
100C2×#41#4114yes—
101C22×Q8C2411yescharacteristic
102C22×D4C22×C411yescharacteristic
103C22×D4C2×D411yescharacteristic
104C22×D4C2×D411yescharacteristic
105C23×C4—22no—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C2×M16—28no—
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—44no—
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—44no—
119C2×M16—44no—
120C22×C8—42no—
121C22×C8—42no—
122#125—44no—
123#125—44no—
124#125—42no—
125#125—42no—
126#125—41no—
127#125—41no—
128#125—42no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×D4—42no—
132C4×D4—42no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136C4×D4—42no—
137#117—41no—
138#117—41no—
139#117—41no—
140#117—41no—
141#117—41no—
142#117—41no—
143#117—44no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C22×C414yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22⋊C412yes—
164C2×C22⋊C4C22⋊C412yes—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22×C412yes—
179C2×C42C22×C414yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C4⋊C8—44no—
183C4⋊C8—44no—
184#101—24no—
185#101—28no—
186#101C2×Q818yes—
187#100—28no—
188#100—44no—
189#97—44no—
190C4×C8—44no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196C24C2×C22⋊C411yescharacteristic
197C24C2×C22⋊C411yescharacteristic
198C24C2×C22⋊C411yescharacteristic
199#41—44no—
200#41—44no—
201#41—44no—
202#41—44no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C22×Q812yes—
206C2×Q8C2×#4112yes—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4C4×Q814yes—
214C2×D4C22⋊Q814yes—
215C2×D4#12412yes—
216C2×D4C4⋊Q812yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—44no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—84no—
238C22×C4—82no—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C4×Q814yes—
245C22×C4C22⋊Q814yes—
246C22×C4#12412yes—
247C22×C4C4⋊Q812yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251M16—28no—
252M16—44no—
253M16—48no—
254M16—44no—
255M16—44no—
256M16—84no—
257C2×C8—44no—
258C2×C8—48no—
259C2×C8—44no—
260C2×C8—84no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C4⋊C4—84no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—84no—
279C22⋊C4—84no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—22no—
291C42—22no—
292C42—44no—
293C42#9911yescharacteristic
294C42#9911yescharacteristic
295C42#9912yes—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23#39112yes—
306C23#39112yes—
307C23#39112yes—
308C23C22×C22⋊C411yescharacteristic
309Q8—42no—
310Q8—42no—
311D4—44no—
312D4—44no—
313D4—42no—
314D4—42no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—88no—
349C2×C4#39112yes—
350C2×C4C2×#9912yes—
351C2×C4C2×#9912yes—
352C2×C4C2×#9911yescharacteristic
353C2×C4C2×#9911yescharacteristic
354C2×C4#41012yes—
355C2×C4C2×C4×Q812yes—
356C2×C4C2×C22⋊Q812yes—
357C2×C4C2×#12411yescharacteristic
358C2×C4C2×C4⋊Q811yescharacteristic
359C8—48no—
360C8—84no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—81no—
373C22—81no—
374C22C2×#39111yescharacteristic
375C22C22×#9911yescharacteristic
376C22C2×#41011yescharacteristic
377C4—22no—
378C4—21no—
379C4—22no—
380C4—21no—
381C4—22no—
382C4—22no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—84no—
387C4—84no—
388C4#199612yes—
389C4#199612yes—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—42no—
394C2—42no—
395C2C2×#199611yescharacteristic
396C2#3336912yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13303C2semidirect

Maximal subgroups

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

subgroupindex
#353332
#206372
C4×#4132
#133032
#125872