Finite Group Databasebeta

← #190203

Subgroups — #190203

Order 512.

1171 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#8761C218yesmaximal, complemented
5#8014C212yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×C4×#41—21no—
8C2×C4×#41C2211yescharacteristic
9C2×#431C2212yes—
10C2×#410—21no—
11#2351C412yescomplemented
12Q8×C4⋊C4—21no—
13#2349C412yescomplemented
14#2349C2214yes—
15D4×C4⋊C4—21no—
16D4×C4⋊C4—22no—
17#2296—22no—
18#2296—21no—
19#2285C412yescomplemented
20#2285C2212yes—
21#2285C2211yescharacteristic
22C2×C4×C4⋊C4C412yescomplemented
23C2×C4×C4⋊C4C2211yescharacteristic
24#1799—22no—
25#1799C2214yes—
26#1740—21no—
27#1461—21no—
28#1460—21no—
29#1459C2214yes—
30#1458—22no—
31#1294C2218yes—
32#1289C2218yes—
33C22×#41D411yescharacteristic
34C4×#41—22no—
35C4×#41—22no—
36C4×#41—22no—
37C4×#41—22no—
38C4×#41—48no—
39C2×C4×Q8—22no—
40C2×C4×Q8C2×C411yescharacteristic
41C2×C4×Q8D411yescharacteristic
42C2×C4×D4—22no—
43C2×C4×D4—24no—
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4D412yes—
46C2×C4×D4D411yescharacteristic
47C2×C4×D4C2312yes—
48C2×#117—22no—
49C2×#117—21no—
50C2×#117—22no—
51C2×#117D411yescharacteristic
52C2×#117D412yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C42—22no—
56C22×C42—21no—
57C22×C42C2×C412yes—
58C22×C42C2311yescharacteristic
59#431—22no—
60#431—22no—
61#431—22no—
62#431—22no—
63C2×C4⋊C8—22no—
64#410—44no—
65C2×#99—24no—
66C2×C22⋊C8C2314yes—
67C2×C4×C8—22no—
68#389—22no—
69#389—24no—
70#389—22no—
71#389—24no—
72#385—24no—
73#385—24no—
74#385—24no—
75#384—22no—
76#384—21no—
77#384—22no—
78#382—22no—
79#382C2×C412yes—
80#382C2×C414yes—
81#380—22no—
82#380—21no—
83#380—22no—
84#380C2×C414yes—
85#380C2314yes—
86#380C2312yes—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C414yes—
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2×C411yescharacteristic
100C2×#95C2×C414yes—
101C2×#95C2312yes—
102#342—24no—
103#324—24no—
104#323—24no—
105C2×#41—22no—
106C2×#41—22no—
107C2×#41—24no—
108C2×#41SD1614yes—
109C2×#41Q1614yes—
110C22×Q8C22⋊C411yescharacteristic
111C22×D4C22⋊C411yescharacteristic
112C22×D4C2×D412yes—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4C22⋊C412yes—
116C23×C4C22⋊C412yes—
117C23×C4C22×C412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8—24no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—48no—
127C4×D4—24no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—22no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—416no—
136C4×D4—48no—
137#117—22no—
138#117—22no—
139#117—24no—
140#117—24no—
141#117—48no—
142#117—44no—
143#117—44no—
144#117—44no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C2×D412yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C22⋊C4C2×D412yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—48no—
179C2×C42—44no—
180C2×C42—42no—
181C2×C42C22⋊C412yes—
182C2×C42C22⋊C412yes—
183C2×C42C22×C412yes—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4112yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4112yes—
192C4⋊C8—42no—
193C4⋊C8—42no—
194#99—48no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C4×C8—42no—
198C4×C8—42no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—22no—
203#95—24no—
204#95—42no—
205#95C22×C412yes—
206#95C22×C414yes—
207#95C22×C412yes—
208C24C2×C22⋊C412yes—
209C24C22×D411yescharacteristic
210#41—28no—
211#41—28no—
212#41—28no—
213#41—28no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8#10314yes—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4#10314yes—
222C2×D4C2×SD1614yes—
223C2×D4C2×Q1614yes—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—48no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—81no—
242C22×C4—82no—
243C22×C4—84no—
244C22×C4—82no—
245C22×C4#10318yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#12312yes—
254C22×C4#12312yes—
255C22×C4#12312yes—
256C22×C4#12312yes—
257C22×C4C2×SD1612yes—
258C22×C4C2×Q1612yes—
259C22×C4AGL(1,ℤ/8)14yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4112yes—
264C2×C8—44no—
265C2×C8—84no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—84no—
278C4⋊C4—82no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—48no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—48no—
296C42—42no—
297C42—44no—
298C42—42no—
299C42—44no—
300C42—84no—
301C42#11711yescharacteristic
302C42#11711yescharacteristic
303C42#11712yes—
304C42#11712yes—
305C42#11711yescharacteristic
306C42#11711yescharacteristic
307C42#12312yes—
308C42#12312yes—
309C42#12312yes—
310C42#12312yes—
311C23—22no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23C2×#10314yes—
318C23C22×C22⋊C411yescharacteristic
319C23C2×#12312yes—
320C23C22×SD1611yescharacteristic
321C23C22×Q1611yescharacteristic
322C23C2×AGL(1,ℤ/8)12yes—
323Q8—28no—
324D4—28no—
325D4—28no—
326D4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—28no—
330C2×C4—28no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—88no—
357C2×C4—816no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4#38111yescharacteristic
362C2×C4#38112yes—
363C2×C4#38111yescharacteristic
364C2×C4#38112yes—
365C2×C4#38111yescharacteristic
366C2×C4#38111yescharacteristic
367C2×C4C2×#10314yes—
368C2×C4C2×#10312yes—
369C2×C4C2×#10312yes—
370C2×C4#41812yes—
371C2×C4#41812yes—
372C2×C4#48114yes—
373C2×C4#48114yes—
374C2×C4#48414yes—
375C2×C4#48414yes—
376C2×C4C2×#11711yescharacteristic
377C2×C4C2×#11711yescharacteristic
378C2×C4C2×#12312yes—
379C8—84no—
380C22—21no—
381C22—22no—
382C22—21no—
383C22—22no—
384C22—24no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—44no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22C2×#38111yescharacteristic
393C22C22×#10311yescharacteristic
394C22C2×#41811yescharacteristic
395C22C2×#48112yes—
396C22C2×#48412yes—
397C4—24no—
398C4—21no—
399C4—22no—
400C4—22no—
401C4—21no—
402C4—21no—
403C4—44no—
404C4—42no—
405C4—42no—
406C4—84no—
407C4—82no—
408C4—164no—
409C4#182412yes—
410C4#182412yes—
411C4#182414yes—
412C4#182712yes—
413C2—21no—
414C2—22no—
415C2—21no—
416C2—22no—
417C2—44no—
418C2C2×#182411yescharacteristic
419C2C2×#182711yescharacteristic
420C2#2071011yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8761C2semidirect
5. #8014C2semidirect
6. #7981C2semidirect
11. #2351C4semidirect
13. #2349C4semidirect
19. #2285C4semidirect
22. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
#206322
#173492
#87612
#80142
#79812