Finite Group Databasebeta

← #362431

Subgroups — #362431

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1938C212yesmaximal
2#20754C211yesmaximal, characteristic, complemented
3#20630C211yesmaximal, characteristic
4C4×#416C211yesmaximal, characteristic
5#11329C212yesmaximal, complemented
6#11125C212yesmaximal
7#10883C212yesmaximal, complemented
8#10864C211yesmaximal, characteristic, complemented
9#10808C211yesmaximal, characteristic, complemented
10#10790C211yesmaximal, characteristic
11#10696C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C22×#108C2212yes—
14C2×#424C2212yes—
15C2×#424C2212yes—
16C22×#104C2212yes—
17C2×#416C2211yescharacteristic
18C2×#416C2211yescharacteristic
19C2×#408C2212yes—
20C22×#97C2212yes—
21C22×C4×C8C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1938—28no—
24#1832—22no—
25#1831—22no—
26#1798—22no—
27#1797—22no—
28C4×#103C412yescomplemented
29C4×#103C412yescomplemented
30C4×#102C412yescomplemented
31C4×#102C412yescomplemented
32C2×#341C2212yes—
33#1735C2211yescharacteristic
34#1735C2211yescharacteristic
35C2×#339C2212yes—
36C2×#339C2212yes—
37C2×#339C2212yes—
38#1729C2212yes—
39#1728C2211yescharacteristic
40#1728C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C4×#95C2211yescharacteristic
44C22×#41C2311yescharacteristic
45C22×M16C2312yes—
46C22×M16C2312yes—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C4×#41—24no—
51C4×#41—42no—
52C2×C4×Q8C2×C412yes—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yes—
55C2×#117—21no—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C42—21no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×#108—24no—
63C2×#108—42no—
64C2×#108—42no—
65#424—24no—
66#424—24no—
67#424—42no—
68#424—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—24no—
71C2×#104—24no—
72C2×#104—24no—
73C2×#104—42no—
74#416—24no—
75#416D414yes—
76#416Q814yes—
77C2×#103C2×C411yescharacteristic
78C2×#103C2×C411yescharacteristic
79C2×#103C2×C412yes—
80C2×#102C2×C411yescharacteristic
81C2×#102C2×C411yescharacteristic
82C2×#102C2×C412yes—
83#408—24no—
84#408—42no—
85C2×C22⋊C8—24no—
86C2×#97—24no—
87C2×#97—24no—
88C2×#97—42no—
89C2×C4×C8—42no—
90C2×C4×C8C2×C412yes—
91C2×C4×C8C2×C412yes—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4C2×C412yes—
94C4×C22⋊C4—22no—
95#376—21no—
96C2×#95—22no—
97C43—21no—
98#341—24no—
99#340C2×C412yes—
100#340C2×C412yes—
101#340C2×C412yes—
102#340C2×C412yes—
103#339—24no—
104#339—24no—
105#339—24no—
106#336—24no—
107#336—24no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336D412yes—
113#336D412yes—
114C2×#41—21no—
115C2×#41—21no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41C2×D412yes—
121C2×#41C2×Q812yes—
122C22×Q8C22×C411yescharacteristic
123C22×D4—21no—
124C22×D4C22×C411yescharacteristic
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4#4112yes—
129C23×C4C2411yescharacteristic
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8C22×C412yes—
156C22×C8C22×C411yescharacteristic
157C22×C8C22×C411yescharacteristic
158C22×C8C2×D414yes—
159C22×C8C2×D412yes—
160C22×C8C2×Q812yes—
161C22×C8#4114yes—
162C4×Q8—24no—
163C4×Q8—42no—
164C4×D4—24no—
165C4×D4—42no—
166C4×D4—48no—
167#117—22no—
168#117—21no—
169#117—21no—
170#117—22no—
171#117—22no—
172#117—44no—
173#117C2×D412yes—
174#117C2×Q812yes—
175#117#4114yes—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—44no—
199C2×C42C22×C412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42#4114yes—
203#108—48no—
204C4⋊C8—48no—
205C4⋊C8—48no—
206#104—48no—
207#103—22no—
208#103—22no—
209#103C4⋊C418yes—
210#102—22no—
211#102—22no—
212#102C4⋊C418yes—
213C22⋊C8—48no—
214#97—48no—
215C4×C8—42no—
216C4×C8—42no—
217#95—22no—
218#95—22no—
219#95—44no—
220C24—21no—
221C24C2×#4111yescharacteristic
222#41—24no—
223#41—24no—
224#41—24no—
225#41—24no—
226#41—42no—
227#41—42no—
228#41—42no—
229#41—42no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8C2×C4⋊C412yes—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4C2×C4⋊C412yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—82no—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×Q811yescharacteristic
272C22×C4C2×#4112yes—
273C22×C4C2×#4112yes—
274C22×C4C2×#4112yes—
275C22×C4H(5,𝔽2)12yes—
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C42−1+411yescharacteristic
278M16—42no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16—42no—
285M16—42no—
286C2×C8—22no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8C2×C4⋊C414yes—
310C2×C8#11712yes—
311C2×C8#11712yes—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—44no—
319C4⋊C4C2×C4⋊C414yes—
320C4⋊C4#11714yes—
321C22⋊C4—22no—
322C22⋊C4—24no—
323C22⋊C4—22no—
324C22⋊C4—44no—
325C22⋊C4—42no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—24no—
334C42—24no—
335C42—44no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—44no—
340C42—82no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23C2×C4×D411yescharacteristic
349C23#52912yes—
350C23#54711yescharacteristic
351C23#55112yes—
352C23#55411yescharacteristic
353Q8—22no—
354Q8—22no—
355Q8—41no—
356Q8—41no—
357D4—22no—
358D4—22no—
359D4—44no—
360D4—44no—
361D4—41no—
362D4—41no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—48no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—88no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4#44414yes—
391C2×C4C22×C4⋊C411yescharacteristic
392C2×C4C2×#11711yescharacteristic
393C2×C4C2×#11711yescharacteristic
394C2×C4C4×#4111yescharacteristic
395C2×C4#51811yescharacteristic
396C2×C4#52011yescharacteristic
397C2×C4#53212yes—
398C2×C4#53212yes—
399C2×C4#53312yes—
400C2×C4#53812yes—
401C2×C4#53912yes—
402C2×C4D4211yescharacteristic
403C2×C4#54612yes—
404C2×C4#54612yes—
405C2×C4D4×Q811yescharacteristic
406C2×C4#55212yes—
407C2×C4#55412yes—
408C2×C4#55512yes—
409C8—22no—
410C8—22no—
411C8—24no—
412C8—44no—
413C8—82no—
414C8—82no—
415C22—22no—
416C22—22no—
417C22—24no—
418C22—22no—
419C22—22no—
420C22—42no—
421C22—44no—
422C22—81no—
423C22—81no—
424C22—82no—
425C22#229311yescharacteristic
426C22#234211yescharacteristic
427C22D4×C4⋊C411yescharacteristic
428C22#255312yes—
429C22#257412yes—
430C22C2×#44412yes—
431C22#295712yes—
432C4—24no—
433C4—42no—
434C4—44no—
435C4—42no—
436C4—44no—
437C4—84no—
438C4—81no—
439C4—81no—
440C4#228511yescharacteristic
441C4#231811yescharacteristic
442C4#234611yescharacteristic
443C4#234911yescharacteristic
444C4#255512yes—
445C4#255612yes—
446C2—22no—
447C2—22no—
448C2—82no—
449C2#1391511yescharacteristic
450C2#2129311yescharacteristic
451C2#2129411yescharacteristic
452C2#2258612yes—
453C2#2268612yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20754C2semidirect
5. #11329C2semidirect
7. #10883C2semidirect
8. #10864C2semidirect
9. #10808C2semidirect
11. #10696C2semidirect
28. C4×#103C4semidirect
29. C4×#103C4semidirect
30. C4×#102C4semidirect
31. C4×#102C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19382
#207542
#206302
C4×#4162
#113292
#111252
#108832
#108642
#108082
#107902
#106962