Finite Group Databasebeta

← #216444

Subgroups — #216444

Order 512.

861 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#541C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#21350C211yesmaximal, characteristic
4#9867C212yesmaximal, complemented
5#9677C212yesmaximal, complemented
6#9077C212yesmaximal
7#9075C212yesmaximal
8#8566C212yesmaximal, complemented
9#8564C212yesmaximal, complemented
10C2×#541C411yescharacteristic
11C2×#431C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2336—22no—
14#2334—22no—
15#2330C411yescharacteristic
16#2327C411yescharacteristic
17#2323C412yes—
18#2320C412yes—
19#2314C411yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C4×#125—22no—
22C4×#125C2211yescharacteristic
23C4×#126—22no—
24C4×C22⋊Q8C2212yes—
25C42×Q8C2212yes—
26C4×#117C2211yescharacteristic
27#1940—22no—
28#1924—22no—
29#1917—22no—
30#1851—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34#1767C2212yes—
35C4×C4⋊C8C2212yes—
36C4×C4⋊C8C2212yes—
37C2×Q8⋊C8C2212yes—
38C2×Q8⋊C8C2212yes—
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C22×M16—21no—
46#541—28no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#126—22no—
49C2×#125—22no—
50C2×#125C2×C411yescharacteristic
51C2×C22⋊Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2312yes—
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C22×C42C2311yescharacteristic
58#431—41no—
59#431—41no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—22no—
68C4×M16—41no—
69C4×M16—41no—
70C2×#97—21no—
71C2×#97—22no—
72C2×C4×C8C2312yes—
73C2×C4×C8C2312yes—
74#391C2×C412yes—
75#391C2×C412yes—
76#389—22no—
77#389—22no—
78#389—22no—
79#389C2×C412yes—
80#389C2×C412yes—
81#388—22no—
82#388—22no—
83#388—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#384C2×C411yescharacteristic
95#383—22no—
96#382—22no—
97#382C2×C411yescharacteristic
98#380C2×C412yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4C2312yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4D412yes—
112C4×C22⋊C4D412yes—
113#376C2×C411yescharacteristic
114#376C2×C411yescharacteristic
115C43—22no—
116C43C2311yescharacteristic
117C43C2311yescharacteristic
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123Q8⋊C8—42no—
124Q8⋊C8—42no—
125Q8⋊C8—42no—
126Q8⋊C8—42no—
127C22×Q8C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C2×M16—44no—
132C2×M16—42no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C4⋊Q8—24no—
138#126—216no—
139#125—216no—
140#125—24no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C4×Q8—24no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8M1618yescomplemented
151#117—24no—
152#117—42no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C412yes—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2411yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—42no—
198C4⋊C8—44no—
199C4⋊C8—44no—
200C4⋊C8—82no—
201C22⋊C8—42no—
202C22⋊C8—82no—
203#97—42no—
204#97—82no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223#95C22⋊C412yes—
224C24C2×C22⋊C411yescharacteristic
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8C2×M1614yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256M16—48no—
257C2×C8—24no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—22no—
273C4⋊C4—28no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4C2×M1614yes—
287C22⋊C4—28no—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—82no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—28no—
305C42—42no—
306C42—42no—
307C42—44no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—82no—
313C42—82no—
314C42C2×M1614yes—
315C42H(5,𝔽2)11yescharacteristic
316C42H(5,𝔽2)11yescharacteristic
317C422−1+411yescharacteristic
318C422−1+411yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—22no—
323C23—22no—
324C23C22×C22⋊C411yescharacteristic
325Q8—24no—
326Q8—24no—
327Q8—42no—
328Q8—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4#40714yes—
368C2×C4#52011yescharacteristic
369C2×C4#52011yescharacteristic
370C2×C4#53511yescharacteristic
371C2×C4#53611yescharacteristic
372C2×C4#53711yescharacteristic
373C2×C4#53711yescharacteristic
374C2×C4C22×M1612yes—
375C2×C4#56812yes—
376C2×C4#57712yes—
377C2×C4#57812yes—
378C8—44no—
379C8—82no—
380C8—82no—
381C8—82no—
382C22—42no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22#231111yescharacteristic
388C22C2×#40711yescharacteristic
389C22#287911yescharacteristic
390C22#289812yes—
391C22#290112yes—
392C4—24no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—44no—
397C4—41no—
398C4—44no—
399C4—42no—
400C4—42no—
401C4—41no—
402C4—41no—
403C4—82no—
404C4—82no—
405C4#298912yes—
406C4#298912yes—
407C4#325812yes—
408C4#326011yescharacteristic
409C4#326111yescharacteristic
410C2—81no—
411C2#1745412yes—
412C2#1750612yes—
413C2#2185311yescharacteristic
414C2#2189811yescharacteristic
415C2#2190111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9867C2semidirect
5. #9677C2semidirect
8. #8566C2semidirect
9. #8564C2semidirect
150. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5412
#219262
#213502
#98672
#96772
#90772
#90752
#85662
#85642