Finite Group Databasebeta

← #223095

Subgroups — #223095

Order 512.

936 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36191C211yesmaximal, characteristic
2#20491C212yesmaximal
3#9890C212yesmaximal, complemented
4#9122C212yesmaximal, complemented
5#9121C212yesmaximal, complemented
6#8722C211yesmaximal, characteristic, complemented
7#8719C211yesmaximal, characteristic, complemented
8#8584C211yesmaximal, characteristic, complemented
9C4×#330C211yesmaximal, characteristic, complemented
10#8028C211yesmaximal, characteristic, complemented
11C2×#1295C211yesmaximal, characteristic, complemented
12C2×#435—22no—
13C2×C4×M16C2212yes—
14C22×#97C2212yes—
15#2862C2211yescharacteristic
16#2851C2211yescharacteristic
17#2851C2211yescharacteristic
18#2574—22no—
19#2570C412yes—
20#2520—22no—
21#2450—22no—
22#2440C412yes—
23#2439C412yes—
24#2368—22no—
25#2352C412yes—
26C4×#125C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1771—22no—
31#1767C2212yes—
32#1753—22no—
33C8×C22⋊C4—22no—
34#1749C2212yes—
35#1749C2212yes—
36C4×#97C2212yes—
37C2×#330C2212yes—
38C2×#330C2212yes—
39#1295C2211yescharacteristic
40#1295C2211yescharacteristic
41#1295C2212yes—
42#1295C2211yescharacteristic
43#1295C2212yes—
44#1295C2211yescharacteristic
45#1295C2212yes—
46#1295C2212yes—
47C22×M16—22no—
48C23×C8—22no—
49C2×C4⋊Q8C2×C412yes—
50C2×#125C2311yescharacteristic
51C2×C22⋊Q8—22no—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×#117C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#435—216no—
59C2×C4⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C4×M16—24no—
63C4×M16—42no—
64C2×#97—22no—
65C2×#97—24no—
66C2×#97—22no—
67C2×#97—42no—
68C2×#97C2312yes—
69C2×#97C2312yes—
70C2×#97C2312yes—
71C2×#97C2312yes—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74#400—22no—
75#398—22no—
76#398—22no—
77#396—22no—
78#395—22no—
79#393—22no—
80#391—22no—
81#391—22no—
82#391—22no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#388—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#386—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#384C2×C412yes—
99#382—22no—
100#382C2×C412yes—
101#382C2×C412yes—
102#382C2311yescharacteristic
103#382C2311yescharacteristic
104#382C2311yescharacteristic
105#380—22no—
106#380C2×C412yes—
107#380C2×C412yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C411yescharacteristic
112C4×C4⋊C4D412yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4C2×C411yescharacteristic
115C4×C22⋊C4C2×C411yescharacteristic
116C4×C22⋊C4D412yes—
117C4×C22⋊C4D412yes—
118C2×#95—22no—
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—22no—
123#330—24no—
124#330—42no—
125C22×Q8—22no—
126C22×D4C22⋊C411yescharacteristic
127C23×C4—22no—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C2×M16—28no—
133C2×M16—44no—
134C2×M16—44no—
135C22×C8—22no—
136C22×C8—28no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—44no—
141C22×C8—44no—
142C4⋊Q8—82no—
143#125—22no—
144#125—21no—
145#125—21no—
146C22⋊Q8—48no—
147C4×D4—24no—
148C4×D4—28no—
149#117—82no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4C22⋊C411yescharacteristic
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C22⋊C412yes—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2411yescharacteristic
189C4⋊C8—28no—
190C22⋊C8—28no—
191C22⋊C8—28no—
192#97—28no—
193#97—22no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—44no—
201#97#4114yes—
202C4×C8—44no—
203C4×C8—44no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95C22⋊C412yes—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220C24C2×C22⋊C411yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C2×Q8—44no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×D4—22no—
226C2×D4—24no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D412yes—
259M16—416no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—416no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C22⋊C4—22no—
290C22⋊C4—24no—
291C22⋊C4—44no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4#10114yes—
295C22⋊C4#10418yes—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—24no—
306C42—42no—
307C42—41no—
308C42—82no—
309C42—82no—
310C42C2×#4112yes—
311C42H(5,𝔽2)11yescharacteristic
312C422−1+411yescharacteristic
313C23—21no—
314C23—21no—
315C23—22no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—22no—
321C23C2×#10112yes—
322C23C2×#10414yes—
323C23C22×C22⋊C411yescharacteristic
324Q8—164no—
325D4—44no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—164no—
366C2×C4C2×#10112yes—
367C2×C4C2×#10112yes—
368C2×C4C2×#10414yes—
369C2×C4C2×#10414yes—
370C2×C4C4×#4111yescharacteristic
371C2×C4#52011yescharacteristic
372C2×C4#54612yes—
373C2×C4#54712yes—
374C8—28no—
375C8—44no—
376C8—44no—
377C8—44no—
378C22—22no—
379C22—22no—
380C22—24no—
381C22—24no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22#152614yes—
388C22#233911yescharacteristic
389C22C22×#10111yescharacteristic
390C22#288811yescharacteristic
391C22C22×#10412yes—
392C22#290112yes—
393C4—22no—
394C4—21no—
395C4—21no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—24no—
400C4—24no—
401C4—41no—
402C4—41no—
403C4—42no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—82no—
408C2—24no—
409C2—81no—
410C2C2×#152611yescharacteristic
411C2#1765212yes—
412C2#1768911yescharacteristic
413C2#2248111yescharacteristic
414C2#2250512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9890C2semidirect
4. #9122C2semidirect
5. #9121C2semidirect
6. #8722C2semidirect
7. #8719C2semidirect
8. #8584C2semidirect
9. C4×#330C2semidirect
10. #8028C2semidirect
11. C2×#1295C2semidirect

Maximal subgroups

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

subgroupindex
#361912
#204912
#98902
#91222
#91212
#87222
#87192
#85842
C4×#3302
#80282
C2×#12952