Finite Group Databasebeta

← #482959

Subgroups — #482959

Order 512.

780 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62249C211yesmaximal, characteristic
2#33486C211yesmaximal, characteristic, complemented
3#22708C211yesmaximal, characteristic, complemented
4#11885C214yesmaximal, complemented
5#11857C211yesmaximal, characteristic, complemented
6#11856C211yesmaximal, characteristic, complemented
7#11743C211yesmaximal, characteristic, complemented
8#11742C211yesmaximal, characteristic, complemented
9#11703C214yesmaximal, complemented
10#3506—21no—
11#3342C412yescomplemented
12#3341C412yescomplemented
13#3304—22no—
14#3303—22no—
15#3287C412yescomplemented
16#3283C412yescomplemented
17#3246—21no—
18#3191—22no—
19#3170C2211yescharacteristic
20#3169C2211yescharacteristic
21#3050C2214yes—
22C2×C8×D4C2211yescharacteristic
23#2354—21no—
24D4×C16—22no—
25D4×C16—22no—
26#2162C2211yescharacteristic
27#2162C2211yescharacteristic
28C2×C22⋊C16C2214yes—
29#1948—21no—
30#1854—22no—
31#1834C2211yescharacteristic
32#1807C2212yes—
33C4×#107C2211yescharacteristic
34#1735C2212yes—
35#1348C2214yes—
36#1348C2212yes—
37#1348C2212yes—
38#1347C2214yes—
39#1347C2212yes—
40#1347C2212yes—
41C2×#135C2×C414yes—
42C23×C8C2312yes—
43#559—42no—
44#556—22no—
45#555—22no—
46#553—42no—
47#550—22no—
48#547—22no—
49#538—42no—
50#535—41no—
51#533—21no—
52#532—21no—
53#525—24no—
54#520—41no—
55C2×C4×D4D411yescharacteristic
56C2×#117—22no—
57C22×C16—24no—
58#500C2×C411yescharacteristic
59#500C2×C411yescharacteristic
60#494C2×C411yescharacteristic
61#493C2×C411yescharacteristic
62#487—21no—
63#486—21no—
64#485—24no—
65#482—24no—
66#476—22no—
67#475—22no—
68#467—21no—
69#467C2×C414yes—
70#467C2312yes—
71#466—21no—
72#466C2×C414yes—
73#466C2312yes—
74#465—24no—
75#465—24no—
76#461—22no—
77#460—22no—
78#452—24no—
79#449—24no—
80C4×Q16—21no—
81C4×Q16C2×C412yes—
82C4×D8—21no—
83C4×D8C2×C412yes—
84C8×D4C2×C412yes—
85C8×D4C2×C412yes—
86C8×D4D412yes—
87C8×D4D412yes—
88#427C2×C414yes—
89C2×#107—21no—
90C2×#106—22no—
91C2×C4⋊C8D411yescharacteristic
92#416—24no—
93C2×C22⋊C8D412yes—
94C2×C4×C8C2311yescharacteristic
95#390—41no—
96#388—42no—
97#384—21no—
98#380—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—41no—
101C4⋊C16—21no—
102C4⋊C16—21no—
103C22⋊C16—24no—
104C22⋊C16C2312yes—
105C22⋊C16C2312yes—
106C4×C16—21no—
107C4×C16—21no—
108#340—21no—
109#340—22no—
110#340C2311yescharacteristic
111#340C2312yes—
112#340C2311yescharacteristic
113C2×#41—24no—
114C2×#41—41no—
115C22×D4D811yescharacteristic
116C23×C4C2×D412yes—
117#135—84no—
118C2×Q16—21no—
119C2×Q16C22×C412yes—
120C2×SD16—24no—
121C2×D8—21no—
122C2×D8C22×C412yes—
123C22×C8—82no—
124C22×C8C22×C414yes—
125C22×C8C2×D412yes—
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D414yes—
128C22×C8C2411yescharacteristic
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132#127—21no—
133#127—42no—
134#126—44no—
135#125—41no—
136#124—42no—
137#124—44no—
138#124—41no—
139#123—44no—
140#123—44no—
141C22⋊Q8—24no—
142C22⋊Q8—22no—
143C22⋊Q8—41no—
144#121—24no—
145#121—22no—
146#121—42no—
147#121—42no—
148#121—41no—
149#121—44no—
150#120—44no—
151C4×Q8—22no—
152C4×Q8—42no—
153C4×Q8—41no—
154C4×D4—22no—
155C4×D4—44no—
156C4×D4—44no—
157C4×D4—42no—
158C4×D4—41no—
159C4×D4C22⋊C412yes—
160C4×D4C22⋊C412yes—
161C4×D4SD1612yes—
162C4×D4SD1612yes—
163#117—24no—
164#117—41no—
165#117—42no—
166#117—44no—
167#117—82no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4D811yescharacteristic
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4D812yes—
178C2×C42—42no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—84no—
184#107—21no—
185#107—21no—
186#107—42no—
187#107C22×C412yes—
188#107C22×C412yes—
189#106—24no—
190#106—82no—
191C4⋊C8C22⋊C412yes—
192C4⋊C8C22⋊C412yes—
193#103—24no—
194#103—22no—
195#103—22no—
196#102—24no—
197#102—22no—
198#102—22no—
199C22⋊C8C22⋊C414yes—
200C22⋊C8C22⋊C414yes—
201C4×C8C22×C411yescharacteristic
202C4×C8C22×C411yescharacteristic
203C4×C8C2×D411yescharacteristic
204C4×C8C2×D411yescharacteristic
205#95—41no—
206#95—42no—
207C24C2×D812yes—
208#41—84no—
209#41—84no—
210#41—82no—
211#41—82no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—42no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4#10212yes—
224C2×D4#10212yes—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4C2×C22⋊C414yes—
233C22×C4C2×D812yes—
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×SD1614yes—
237C22×C4C22×D411yescharacteristic
238Q16—42no—
239Q16—81no—
240SD16—84no—
241D8—42no—
242D8—81no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—44no—
246C2×C8—41no—
247C2×C8—44no—
248C2×C8C2×C22⋊C411yescharacteristic
249C2×C8C2×C22⋊C411yescharacteristic
250C2×C8C2×C22⋊C414yes—
251C2×C8C23×C411yescharacteristic
252C2×C8C22×D411yescharacteristic
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—44no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—82no—
269C4⋊C4—161no—
270C4⋊C4#10212yes—
271C4⋊C4#10212yes—
272C22⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C22⋊C4—44no—
277C22⋊C4—42no—
278C22⋊C4—84no—
279C22⋊C4—162no—
280C22⋊C4#10214yes—
281C22⋊C4#10214yes—
282C42—44no—
283C42—42no—
284C42—84no—
285C42—82no—
286C42C2×C22⋊C411yescharacteristic
287C42C2×C22⋊C411yescharacteristic
288C42C2×SD1611yescharacteristic
289C42C2×SD1611yescharacteristic
290C16—42no—
291C16—42no—
292C23—22no—
293C23—22no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23C2×#10214yes—
298C23C22×D811yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—82no—
302Q8—161no—
303D4—24no—
304D4—24no—
305D4—42no—
306D4—42no—
307D4—82no—
308D4—82no—
309D4—82no—
310D4—161no—
311D4—164no—
312D4—164no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—24no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—164no—
334C2×C4C2×#10214yes—
335C2×C4C2×#10211yescharacteristic
336C2×C4C2×#10211yescharacteristic
337C2×C4C2×#10211yescharacteristic
338C2×C4C2×#10211yescharacteristic
339C2×C4#50812yes—
340C2×C4#50812yes—
341C2×C4C22×C22⋊C411yescharacteristic
342C2×C4C22×SD1611yescharacteristic
343C8—21no—
344C8—21no—
345C8—24no—
346C8—21no—
347C8—21no—
348C22—21no—
349C22—21no—
350C22—24no—
351C22—24no—
352C22—24no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22C22×#10211yescharacteristic
359C22C2×#50811yescharacteristic
360C22C2×#50811yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—24no—
365C4—82no—
366C4—82no—
367C4—82no—
368C4—82no—
369C4—161no—
370C4—162no—
371C4#213812yes—
372C4#213812yes—
373C4#288311yescharacteristic
374C4#290111yescharacteristic
375C2—21no—
376C2—21no—
377C2—44no—
378C2—162no—
379C2#1761311yescharacteristic
380C2C2×#213811yescharacteristic
381C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62249C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33486C2semidirect
3. #22708C2semidirect
4. #11885C2semidirect
5. #11857C2semidirect
6. #11856C2semidirect
7. #11743C2semidirect
8. #11742C2semidirect
9. #11703C2semidirect
11. #3342C4semidirect
12. #3341C4semidirect
15. #3287C4semidirect
16. #3283C4semidirect

Maximal subgroups

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

subgroupindex
#622492
#334862
#227082
#118852
#118572
#118562
#117432
#117422
#117032