Finite Group Databasebeta

← #363425

Subgroups — #363425

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic, complemented
3#20393C211yesmaximal, characteristic, complemented
4#11154C212yesmaximal, complemented
5#10979C212yesmaximal
6#10949C212yesmaximal, complemented
7#10891C212yesmaximal, complemented
8#10816C212yesmaximal, complemented
9#10775C212yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#520C2211yescharacteristic
13C2×#421C2214yes—
14C2×#410C2212yes—
15#2879C2212yes—
16#2361—24no—
17#2354—22no—
18D4×C4⋊C4—22no—
19#2326C412yescomplemented
20#2326C412yescomplemented
21#2311C2211yescharacteristic
22#2311C2211yescharacteristic
23C4×C4⋊Q8C412yescomplemented
24C4×#127C412yescomplemented
25C2×#384C2211yescharacteristic
26#2274C2212yes—
27C4×#117C2211yescharacteristic
28#1822—22no—
29#1821—22no—
30C4×#99—24no—
31#1742C2212yes—
32C2×#343C2212yes—
33#1740C2212yes—
34#1736C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C22×#41C2311yescharacteristic
39C22×M16C2314yes—
40#532—24no—
41#532—24no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#127C2×C411yescharacteristic
44C2×#124C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#121—22no—
47#520—44no—
48#520—44no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C412yes—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C412yes—
56C2×C4×D4D414yes—
57C2×#117—21no—
58C2×#117D412yes—
59C2×#117C2314yes—
60C2×#117C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#421—28no—
69#421—28no—
70C2×#104—44no—
71#410—28no—
72C2×#99—24no—
73#408—44no—
74#407—24no—
75#391C2×C412yes—
76#391C2×C412yes—
77#390—24no—
78#390C2×C412yes—
79#390C2×C412yes—
80#389—24no—
81#388—24no—
82#388—24no—
83#386—22no—
84#386—22no—
85#385—24no—
86#385—24no—
87#384—22no—
88#384C2×C412yes—
89#384C2×C412yes—
90#380—22no—
91#380—22no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—44no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—24no—
98#376C2×C412yes—
99C43C2×C412yes—
100#343—22no—
101#343—24no—
102#343—22no—
103#342—24no—
104#340—44no—
105#339—28no—
106H(3,ℤ/4)—28no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—24no—
111C2×#41—24no—
112C2×#41—24no—
113C2×#41—24no—
114C2×#41#4114yes—
115C22×Q8C22×C411yescharacteristic
116C22×Q8C22×C411yescharacteristic
117C22×D4—22no—
118C22×D4—21no—
119C22×D4C22×C411yescharacteristic
120C22×D4C22×C411yescharacteristic
121C22×D4C2×D412yes—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4C22⋊C414yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C22×C8—24no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139#127—22no—
140#127—22no—
141#124—22no—
142#124—22no—
143#124—22no—
144#124—22no—
145#121—216no—
146C4×Q8—44no—
147C4×Q8—44no—
148C4×D4—216no—
149C4×D4—44no—
150C4×D4—44no—
151C4×D4—48no—
152C4×D4—48no—
153C4×D4—88no—
154#117—24no—
155#117—24no—
156#117—24no—
157#117—24no—
158#117—24no—
159#117—44no—
160#117—44no—
161#117#4118yes—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—84no—
174C2×C4⋊C4C22⋊C414yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4C22⋊C414yes—
188C2×C22⋊C4C2×D412yes—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—44no—
196C2×C42C22⋊C414yes—
197C2×C42C22×C412yes—
198C2×C42C22×C412yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42#4114yes—
202#104—416no—
203#99—216no—
204C22⋊C8—48no—
205#95—22no—
206#95—22no—
207#95—24no—
208#95—24no—
209#95—44no—
210C24—21no—
211C24C2×C22⋊C412yes—
212C24C22×D411yescharacteristic
213#41—28no—
214#41—28no—
215#41—44no—
216#41—44no—
217#41—44no—
218#41—44no—
219#41—44no—
220#41—44no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8#11712yes—
229C2×D4—28no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4#11712yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C414yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4114yes—
278C22×C4H(5,𝔽2)14yes—
279M16—44no—
280M16—44no—
281M16—44no—
282M16—44no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—44no—
286C4⋊C4—28no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—28no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—48no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—84no—
300C22⋊C4—28no—
301C22⋊C4—28no—
302C22⋊C4—24no—
303C22⋊C4—28no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—42no—
308C22⋊C4—84no—
309C22⋊C4—84no—
310C42—21no—
311C42—21no—
312C42—24no—
313C42—24no—
314C42—24no—
315C42—28no—
316C42—44no—
317C42—44no—
318C42—44no—
319C42—44no—
320C42—44no—
321C42—164no—
322C42#11712yes—
323C42#11712yes—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—41no—
336C23C22×C22⋊C411yescharacteristic
337C23#52512yes—
338C23#52512yes—
339C23#53412yes—
340Q8—24no—
341Q8—42no—
342Q8—42no—
343Q8—42no—
344D4—24no—
345D4—416no—
346D4—48no—
347D4—48no—
348D4—42no—
349D4—42no—
350D4—42no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—48no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—164no—
382C2×C4—164no—
383C2×C4—162no—
384C2×C4C2×#11711yescharacteristic
385C2×C4C2×#11711yescharacteristic
386C2×C4C4×#4112yes—
387C2×C4#51812yes—
388C2×C4#52512yes—
389C2×C4#53212yes—
390C2×C4#53412yes—
391C2×C4#55114yes—
392C2×C4#55314yes—
393C2×C4#55314yes—
394C2×C4#55914yes—
395C8—84no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—22no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—44no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—82no—
409C22—82no—
410C22—81no—
411C22—81no—
412C22#228011yescharacteristic
413C22#229711yescharacteristic
414C22#230811yescharacteristic
415C22#260014yes—
416C4—24no—
417C4—42no—
418C4—44no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—82no—
424C4—88no—
425C4—82no—
426C4#235312yes—
427C4#235712yes—
428C4#260214yes—
429C2—21no—
430C2—21no—
431C2—24no—
432C2—82no—
433C2—82no—
434C2#1394011yescharacteristic
435C2#2178211yescharacteristic
436C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20695C2semidirect
3. #20393C2semidirect
4. #11154C2semidirect
6. #10949C2semidirect
7. #10891C2semidirect
8. #10816C2semidirect
9. #10775C2semidirect
19. #2326C4semidirect
20. #2326C4semidirect
23. C4×C4⋊Q8C4semidirect
24. C4×#127C4semidirect

Maximal subgroups

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

subgroupindex
#342072
#206952
#203932
#111542
#109792
#109492
#108912
#108162
#107752