Finite Group Databasebeta

← #355511

Subgroups — #355511

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#532C211yesmaximal, characteristic
2C2×#1980C212yesmaximal
3#20364C211yesmaximal, characteristic, complemented
4C42×C4⋊C4C211yesmaximal, characteristic, complemented
5#10923C214yesmaximal, complemented
6#10851C212yesmaximal, complemented
7#10790C214yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×C4×#41C2212yes—
10C22×#104C2218yes—
11C2×C4×M16C2212yes—
12C4×C4⋊Q8C412yescomplemented
13C4×#127C412yescomplemented
14C4×#124C414yescomplemented
15C4×#121—22no—
16C4×#121—22no—
17#2297C2212yes—
18#2293—21no—
19#2293—21no—
20#2285C2211yescharacteristic
21C42×D4—21no—
22C42×D4—21no—
23C2×C4×C4⋊C4—22no—
24C2×C4×C4⋊C4C2212yes—
25C2×C43C2212yes—
26C2×C43C2211yescharacteristic
27#1980C4116yescomplemented
28#1745—21no—
29#1745—21no—
30#1744—21no—
31#1744—21no—
32C2×#339C2218yes—
33#1728C2212yes—
34#1728C2212yes—
35C4×#95—22no—
36C4×#95C2212yes—
37C22×#41C2312yes—
38C22×M16C2314yes—
39#532—24no—
40#532C2×C418yes—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#127C2×C411yescharacteristic
43C2×#124C2×C412yes—
44C2×#121—21no—
45C2×#121—21no—
46C4×#41—48no—
47C2×C4×Q8C2×C414yes—
48C2×C4×D4—24no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4C2×C414yes—
54C2×#117—22no—
55C2×#117C2312yes—
56C22×C4⋊C4—22no—
57C22×C42—24no—
58C22×C42—22no—
59C22×C42D412yes—
60C22×C42D412yes—
61C22×C42C2314yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#104—48no—
66C2×#104—48no—
67C2×#104C2×C4132yescomplemented
68C4×M16—24no—
69C4×M16C2×C418yescomplemented
70C2×#97—22no—
71C2×C4×C8—22no—
72#391C2×C414yes—
73#390—22no—
74#390—22no—
75#390C2×C414yes—
76#389—22no—
77#389—22no—
78#388—24no—
79#388—24no—
80#386—24no—
81#384C2×C412yes—
82#382C2×C412yes—
83#380—21no—
84#380—21no—
85C4×C4⋊C4—216no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4C2×C418yescomplemented
92C4×C22⋊C4—24no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C2×#95—22no—
97C2×#95—24no—
98C43—44no—
99C43C2×C412yes—
100C43C2×C412yes—
101C43D411yescharacteristic
102C43D411yescharacteristic
103C43D411yescharacteristic
104C43D411yescharacteristic
105#339—48no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110C2×#41—24no—
111C2×#41—28no—
112C2×#41—24no—
113C2×#41—24no—
114C2×#41C22×C418yes—
115C22×Q8C22×C412yes—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4C22×C412yes—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16C22×C418yes—
134C22×C8—24no—
135C4⋊Q8—22no—
136C4⋊Q8C4214yescomplemented
137#127—22no—
138#127C4214yescomplemented
139#124—24no—
140#124C4218yescomplemented
141#121—28no—
142#121—28no—
143C4×Q8—48no—
144C4×D4—28no—
145C4×D4—28no—
146C4×D4—48no—
147C4×D4—48no—
148C4×D4—416no—
149C4×D4—48no—
150#117—28no—
151#117—48no—
152C2×C4⋊C4—216no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C22×C414yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—28no—
170C2×C42—28no—
171C2×C42—28no—
172C2×C42—216no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42—44no—
177C2×C42—48no—
178C2×C42—48no—
179C2×C42—48no—
180C2×C42C22⋊C412yes—
181C2×C42C22⋊C412yes—
182C2×C42C22⋊C412yes—
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C22×C418yes—
189C2×C42C22×C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C414yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4114yes—
201#104—432no—
202#97—28no—
203C4×C8—28no—
204#95—216no—
205#95—22no—
206#95—22no—
207#95—24no—
208#95—24no—
209#95—28no—
210C24—22no—
211C24C22×D411yescharacteristic
212#41—216no—
213#41—216no—
214#41—48no—
215#41—48no—
216C2×Q8—22no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8C2×C4214yes—
220C2×D4—28no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—44no—
231C2×D4C2×C4214yes—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—216no—
238C22×C4—22no—
239C22×C4—28no—
240C22×C4—24no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—48no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C4×D414yes—
263C22×C4C4×D414yes—
264C22×C4C23×C411yescharacteristic
265C22×C4C23×C412yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4H(5,𝔽2)14yes—
269C22×C4H(5,𝔽2)12yes—
270C22×C42−1+412yes—
271M16—28no—
272M16—28no—
273M16—44no—
274M16—44no—
275C2×C8—28no—
276C2×C8—44no—
277C2×C8—44no—
278C4⋊C4—232no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—28no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C22⋊C4—28no—
286C22⋊C4—28no—
287C22⋊C4—48no—
288C22⋊C4—44no—
289C22⋊C4—48no—
290C22⋊C4—48no—
291C42—24no—
292C42—24no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—216no—
303C42—42no—
304C42—42no—
305C42—432no—
306C42—48no—
307C42—44no—
308C42—48no—
309C42#10418yescomplemented
310C42C2×C4212yes—
311C42C2×C4212yes—
312C42#11712yes—
313C42#11712yes—
314C42C4×D414yes—
315C42C4×D414yes—
316C23—22no—
317C23—22no—
318C23—24no—
319C23—42no—
320C23—42no—
321C23—44no—
322C23—44no—
323C23C22×C22⋊C411yescharacteristic
324C23C2×C4×D412yes—
325C23#53512yes—
326C23#53511yescharacteristic
327C23#53611yescharacteristic
328Q8—28no—
329Q8—44no—
330D4—28no—
331D4—48no—
332D4—48no—
333D4—48no—
334D4—44no—
335D4—48no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—28no—
348C2×C4—216no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—416no—
356C2×C4—416no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—48no—
361C2×C4—416no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—816no—
366C2×C4C4×C22⋊C412yes—
367C2×C4C4×C22⋊C412yes—
368C2×C4C4×C22⋊C412yes—
369C2×C4C4×C22⋊C412yes—
370C2×C4C2×#10414yes—
371C2×C4C2×#10414yes—
372C2×C4C2×#10414yes—
373C2×C4#42114yes—
374C2×C4C22×C4211yescharacteristic
375C2×C4C2×#11711yescharacteristic
376C2×C4C2×C4×D412yes—
377C2×C4#51812yes—
378C2×C4#51814yes—
379C2×C4#52012yes—
380C2×C4#52014yes—
381C2×C4#53512yes—
382C2×C4#53712yes—
383C2×C4#54014yes—
384C2×C4#54214yes—
385C8—48no—
386C22—24no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—84no—
395C22—84no—
396C22C2×C4×C22⋊C411yescharacteristic
397C22#230911yescharacteristic
398C22#231111yescharacteristic
399C22#232612yes—
400C22#232812yes—
401C22C22×#10412yes—
402C22C2×#42112yes—
403C4—24no—
404C4—24no—
405C4—24no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—416no—
410C4—42no—
411C4—44no—
412C4—84no—
413C4C4×#10414yes—
414C4#227512yes—
415C4#231912yes—
416C4#232414yes—
417C2—22no—
418C2—22no—
419C2—84no—
420C2#1378511yescharacteristic
421C2C2×C4×#10411yescharacteristic
422C2C4×#42111yescharacteristic
423C2#2181512yes—
424C2#2189912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20364C2semidirect
4. C42×C4⋊C4C2semidirect
5. #10923C2semidirect
6. #10851C2semidirect
7. #10790C2semidirect
12. C4×C4⋊Q8C4semidirect
13. C4×#127C4semidirect
14. C4×#124C4semidirect
27. #1980C4semidirect
67. C2×#104C2×C4semidirect
69. C4×M16C2×C4semidirect
91. C4×C4⋊C4C2×C4semidirect
136. C4⋊Q8C42semidirect
138. #127C42semidirect
140. #124C42semidirect
309. C42#104semidirect

Maximal subgroups

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

subgroupindex
C4×#5322
C2×#19802
#203642
C42×C4⋊C42
#109232
#108512
#107902