Finite Group Databasebeta

← #302378

Subgroups — #302378

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#33310C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic
4#28689C211yesmaximal, characteristic, complemented
5#10577C214yesmaximal, complemented
6#9896C211yesmaximal, characteristic, complemented
7#9870C211yesmaximal, characteristic, complemented
8#8889C214yesmaximal
9C4×D4⋊C8C211yesmaximal, characteristic
10C2×#500—21no—
11C2×#498—22no—
12C2×#474—21no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2857—21no—
16#2571C2212yes—
17#2563—21no—
18#2538—21no—
19#2497—22no—
20#2492—22no—
21#2439—22no—
22#2369—21no—
23#2355—21no—
24#2354C2212yes—
25C4×C4⋊Q8C2211yescharacteristic
26#2285—21no—
27C42×D4C2211yescharacteristic
28#2083—22no—
29#1990—21no—
30#1988—21no—
31#1947—21no—
32#1947C2214yes—
33#1947C2212yes—
34#1844—22no—
35#1844C2214yes—
36#1842C2211yescharacteristic
37#1802—21no—
38#1802C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42#1296C2214yes—
43#1279C2212yes—
44#1279C2214yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D412yes—
52C2×C22⋊Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C2×#117—22no—
58C22×C42—21no—
59#500—44no—
60#498—48no—
61#474—44no—
62C2×#107—24no—
63C2×#107—22no—
64C2×#106—28no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×#102—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8C2312yes—
74#400—24no—
75#396—24no—
76#393—24no—
77#391—22no—
78#391—24no—
79#391—21no—
80#390—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—21no—
85#389D412yes—
86#388—24no—
87#387—24no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384D411yescharacteristic
94#384D412yes—
95#384C2312yes—
96#384C2312yes—
97#382—22no—
98#382—22no—
99#380—22no—
100#380—22no—
101#380—22no—
102#380—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D411yescharacteristic
109C4×C4⋊C4C2312yes—
110C4×C4⋊C4C2311yescharacteristic
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C43C2311yescharacteristic
115#340—22no—
116#340—22no—
117#340—24no—
118#336—21no—
119D4⋊C8Q818yescomplemented
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—41no—
123C22×D4C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—82no—
141C22⋊Q8—84no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4C2×Q814yes—
150#117—84no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C4⋊C4C2×D414yes—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—24no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2411yescharacteristic
192#107—44no—
193#107—48no—
194#106—416no—
195C4⋊C8—22no—
196C4⋊C8—24no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8C2×Q814yes—
200#102—44no—
201C4×C8—24no—
202C4×C8—24no—
203C4×C8—22no—
204C4×C8—42no—
205C4×C8C2×Q814yes—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—21no—
211#95—21no—
212#95—22no—
213#95—21no—
214#95—22no—
215C24—21no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—44no—
219C2×Q8—42no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4C4⋊Q812yes—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—24no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—84no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4C4⋊Q812yes—
293C22⋊C4—24no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—84no—
297C22⋊C4—84no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—41no—
302C42—42no—
303C42—41no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—84no—
308C42#13512yes—
309C42#13512yes—
310C42#13512yes—
311C42AGL(1,ℤ/8)12yes—
312C42AGL(1,ℤ/8)12yes—
313C42#13712yes—
314C42C22×Q812yes—
315C422−1+412yes—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23C2×#12011yescharacteristic
324Q8—84no—
325Q8—164no—
326Q8—164no—
327D4—22no—
328D4—22no—
329D4—41no—
330D4—41no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4#44912yes—
363C2×C4#44912yes—
364C2×C4#44912yes—
365C2×C4#44912yes—
366C2×C4#45212yes—
367C2×C4#45212yes—
368C2×C4#45312yes—
369C2×C4#45612yes—
370C2×C4C2×C4⋊Q811yescharacteristic
371C2×C4#53611yescharacteristic
372C2×C4D4×Q812yes—
373C2×C4D4×Q812yes—
374C2×C4C2×#13511yescharacteristic
375C2×C4C2×#13511yescharacteristic
376C2×C4C2×#13511yescharacteristic
377C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
378C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
379C2×C4C2×#13711yescharacteristic
380C2×C4#57712yes—
381C8—28no—
382C8—44no—
383C8—82no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#243411yescharacteristic
392C22C2×#44911yescharacteristic
393C22C2×#44911yescharacteristic
394C22C2×#45211yescharacteristic
395C22#300911yescharacteristic
396C22C2×#45311yescharacteristic
397C22C2×#45611yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—82no—
405C4—84no—
406C4—82no—
407C4—162no—
408C4#162414yes—
409C4#165214yes—
410C4#316011yescharacteristic
411C4#316511yescharacteristic
412C4#322811yescharacteristic
413C4#323111yescharacteristic
414C4#338112yes—
415C4#338412yes—
416C2—82no—
417C2C2×#165211yescharacteristic
418C2C2×#162411yescharacteristic
419C2#1862511yescharacteristic
420C2#1865611yescharacteristic
421C2#2581211yescharacteristic
422C2#2583011yescharacteristic
423C2#2583511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33310C2semidirect
4. #28689C2semidirect
5. #10577C2semidirect
6. #9896C2semidirect
7. #9870C2semidirect
119. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#361942
#333102
#331952
#286892
#105772
#98962
#98702
#88892
C4×D4⋊C82