Finite Group Databasebeta

← #481942

Subgroups — #481942

Order 512.

859 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36030C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#11833C214yesmaximal, complemented
5#11831C214yesmaximal, complemented
6#11724C212yesmaximal, complemented
7#11697C212yesmaximal
8C2×#556C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2983—21no—
11#2905—21no—
12C22×#101—21no—
13#2632C2212yes—
14#2628C2211yescharacteristic
15#2610C412yes—
16#2599C412yes—
17#2567C411yescharacteristic
18#2536C411yescharacteristic
19#2479—22no—
20#2469—22no—
21#2423—21no—
22#2409—21no—
23#2399C2212yes—
24#2392C2211yescharacteristic
25#2362—21no—
26#2356C411yescharacteristic
27#2354—21no—
28#1801—22no—
29#1801C2214yes—
30#1742C2212yes—
31#1467C2214yes—
32#1464—22no—
33#1461—21no—
34#1458C2212yes—
35#1354C2218yes—
36#1346C2214yes—
37#1345C2214yes—
38C23×Q8D411yescharacteristic
39C22×M16—22no—
40#556—44no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#126—22no—
43C2×#124C2×C412yes—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46#519—44no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D412yes—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8C2312yes—
54C2×C4×D4—21no—
55C2×C4×D4C2×C411yescharacteristic
56C2×#117—21no—
57C2×#117C2×C412yes—
58C2×#117D412yes—
59C2×#117D411yescharacteristic
60C2×#117C2311yescharacteristic
61#446—44no—
62#432—24no—
63#431—22no—
64#424—44no—
65#423—24no—
66C2×#101—24no—
67C2×#101—44no—
68#400—21no—
69#399—22no—
70#399—21no—
71#398—22no—
72#398C2×C412yes—
73#398C2311yescharacteristic
74#397C2×C411yescharacteristic
75#396—21no—
76#395—22no—
77#395C2×C412yes—
78#395C2×C412yes—
79#395C2312yes—
80#394—21no—
81#394C2×C412yes—
82#394C2311yescharacteristic
83#393—22no—
84#393C2×C412yes—
85#392C2×C411yescharacteristic
86#391—21no—
87#391—21no—
88#390—22no—
89#389—22no—
90#389—21no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#386—22no—
101#385—22no—
102#384—21no—
103#383C2×C412yes—
104#382C2×C411yescharacteristic
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C22⋊C4—21no—
112#376C2×C412yes—
113#343—22no—
114#343C2314yes—
115#324—24no—
116#324C2314yes—
117#323—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C412yes—
125C22×Q8C2×D412yes—
126C22×D4—21no—
127C22×D4C22×C411yescharacteristic
128C23×C4—21no—
129C23×C4C22⋊C412yes—
130C23×C4C2×D411yescharacteristic
131C2×M16—24no—
132C2×M16—42no—
133C2×M16—44no—
134C2×M16—44no—
135C22×C8—42no—
136C4⋊Q8—42no—
137#126—48no—
138#124—44no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—44no—
144C4×Q8—44no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—44no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151#117—24no—
152#117—42no—
153#117—44no—
154#117—44no—
155#117—44no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22×C411yescharacteristic
187C2×C22⋊C4C2×D411yescharacteristic
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42C22⋊C412yes—
192C2×C42C22⋊C412yes—
193C2×C42C22×C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C4⋊C8—48no—
199C4⋊C8—42no—
200C4⋊C8—44no—
201#101—48no—
202C22⋊C8—24no—
203#97—44no—
204C4×C8—42no—
205#95—22no—
206#95—21no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95—22no—
216#95C22×C411yescharacteristic
217#95C22×C412yes—
218#95C22×C411yescharacteristic
219C24—21no—
220C24C2×C22⋊C411yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—48no—
232C2×Q8—48no—
233C2×Q8—42no—
234C2×Q8—82no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—81no—
239C2×D4—81no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4AGL(1,ℤ/8)12yes—
261C22×C4#13712yes—
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265M16—44no—
266M16—44no—
267M16—84no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—84no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4#9914yes—
291C22⋊C4—24no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—44no—
302C22⋊C4—82no—
303C22⋊C4—81no—
304C22⋊C4—82no—
305C42—22no—
306C42—24no—
307C42—24no—
308C42—22no—
309C42—24no—
310C42—42no—
311C42—42no—
312C42—44no—
313C42—44no—
314C42—42no—
315C23—21no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23#41712yes—
327C23C22×C22⋊C411yescharacteristic
328Q8—44no—
329Q8—44no—
330Q8—44no—
331Q8—44no—
332Q8—84no—
333D4—84no—
334D4—82no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—48no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—84no—
373C2×C4C2×#9914yes—
374C2×C4C2×#9912yes—
375C2×C4#41712yes—
376C2×C4#42112yes—
377C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
378C2×C4C2×#13711yescharacteristic
379C8—84no—
380C8—84no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22C22×#9911yescharacteristic
396C22C2×#41711yescharacteristic
397C22C2×#42111yescharacteristic
398C4—24no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—41no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—42no—
408C4—42no—
409C4—81no—
410C4—82no—
411C4—82no—
412C4—81no—
413C4#151314yes—
414C2—21no—
415C2—21no—
416C2—42no—
417C2—161no—
418C2C2×#151311yescharacteristic
419C2#3335111yescharacteristic
420C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#392C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20639C2semidirect
3. #17371C2semidirect
4. #11833C2semidirect
5. #11831C2semidirect
6. #11724C2semidirect

Maximal subgroups

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

subgroupindex
#360302
#206392
#173712
#118332
#118312
#117242
#116972