Finite Group Databasebeta

← #6344339

Subgroups — #6344339

Order 512.

1667 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2C2×#2988C211yesmaximal, characteristic
3#60123C211yesmaximal, characteristic
4#17812C218yesmaximal, complemented
5#17811C218yesmaximal, complemented
6#17808C214yesmaximal, complemented
7#17807C214yesmaximal, complemented
8#17787C212yesmaximal, complemented
9C2×#1565C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11#2989C2214yes—
12#2989C2214yes—
13#2988—24no—
14#2988C2218yes—
15#2983—21no—
16#2975—21no—
17#2973—22no—
18#2965—24no—
19C2×C8×Q8—21no—
20C2×C8×D4—22no—
21C2×C8×D4—21no—
22#2923C418yes—
23#2922C2218yes—
24#2921C414yes—
25#2920C2214yes—
26#2920C2214yes—
27C2×#431C2211yescharacteristic
28C2×#431C2212yes—
29C2×#431C2212yes—
30#2905C412yes—
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2212yes—
33#2879C412yes—
34C2×C4×M16C2212yes—
35#1574C2218yescomplemented
36#1574C2218yescomplemented
37#1573C22116yescomplemented
38#1572C22116yescomplemented
39#1571C2218yescomplemented
40#1570C2218yescomplemented
41#1565C2218yes—
42#1565C2214yes—
43#1565C2214yes—
44#1565C2218yescomplemented
45#1565C2214yescomplemented
46#1565C2214yescomplemented
47#1564C2218yescomplemented
48C2×#335C2214yescomplemented
49C2×#334C2214yescomplemented
50C22×#41C2×C411yescharacteristic
51C22×M16C2×C414yes—
52C23×C8—21no—
53C23×C8—22no—
54C4×#41C2×C414yes—
55C4×#41D414yes—
56C4×#41Q814yes—
57C4×#41C2314yes—
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8Q811yescharacteristic
60C2×C4×D4D412yes—
61C2×C4×D4D411yescharacteristic
62C2×C4×D4Q812yes—
63C2×C4×D4Q811yescharacteristic
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C412yes—
66C22×C42C2311yescharacteristic
67C22×C42C2312yes—
68#446—24no—
69#446—24no—
70C8×Q8—28no—
71#436—24no—
72#436—24no—
73#435—28no—
74#435—28no—
75C8×D4—216no—
76C8×D4—28no—
77#433C2×C4132yes—
78#432—24no—
79#432C2×C418yes—
80#432C2×C418yes—
81#432C2318yes—
82#431—22no—
83#431—22no—
84#431—22no—
85#431—24no—
86#431—24no—
87#431C2314yes—
88#431C2314yes—
89#431C2314yes—
90#431C2318yes—
91#431C2318yes—
92#431C2318yes—
93#424C2×C4116yes—
94#423C2×C418yes—
95#423C2314yes—
96#423C2314yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2318yes—
108#408C2×C4116yes—
109#407—24no—
110#407C2×C418yes—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C4×M16—24no—
115C4×M16C2×C414yes—
116C4×M16C2×C414yes—
117C4×M16C2314yes—
118C4×M16C2314yes—
119C2×#97C2×C414yes—
120C2×C4×C8—21no—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8C2314yes—
124#335C23116yes—
125#335C2318yes—
126#335C2318yes—
127#334C23116yes—
128#334C2318yes—
129#334C2318yes—
130C2×#41—28no—
131C2×#41C4⋊C414yes—
132C2×#41C22×C414yes—
133C22×Q8C4⋊C411yescharacteristic
134C22×D4C4⋊C412yes—
135C22×D4C4⋊C411yescharacteristic
136C23×C4C22×C412yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×Q812yes—
141C23×C4C2×Q811yescharacteristic
142C2×M16—216no—
143C2×M16—24no—
144C2×M16—24no—
145C2×M16C22×C418yes—
146C2×M16C22×C418yes—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—24no—
151C22×C8—28no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—82no—
155C22×C8—84no—
156C22×C8C22×C418yes—
157C4×Q8C4⋊C414yes—
158C4×Q8C4⋊C414yes—
159C4×Q8C2×D414yes—
160C4×Q8C2×Q814yes—
161C4×D4C4⋊C418yes—
162C4×D4C4⋊C414yes—
163C4×D4C4⋊C418yes—
164C4×D4C4⋊C414yes—
165C4×D4C2×D418yes—
166C4×D4C2×D414yes—
167C4×D4C2×Q818yes—
168C4×D4C2×Q814yes—
169#117C4⋊C418yes—
170#117C4⋊C414yes—
171#117C22×C414yes—
172#117C22×C418yes—
173C2×C4⋊C4C4⋊C412yes—
174C2×C4⋊C4C4⋊C411yescharacteristic
175C2×C4⋊C4C4⋊C412yes—
176C2×C4⋊C4C4⋊C411yescharacteristic
177C2×C22⋊C4C4⋊C412yes—
178C2×C22⋊C4C4⋊C412yes—
179C2×C22⋊C4C4⋊C412yes—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C412yes—
182C2×C42C22×C412yes—
183C2×C42C22×C414yes—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D414yes—
188C2×C42C2×Q812yes—
189C2×C42C2×Q811yescharacteristic
190C2×C42C2×Q814yes—
191C2×C42C2×Q812yes—
192C2×C42C2412yes—
193C2×C42C2414yes—
194C2×C42C2411yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—24no—
198C4⋊C8—28no—
199C4⋊C8—21no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—24no—
203C4⋊C8—24no—
204C4⋊C8C22×C4116yes—
205C4⋊C8C22×C4116yes—
206C4⋊C8C2414yes—
207C4⋊C8C2414yes—
208C4⋊C8C2412yes—
209C4⋊C8C2412yes—
210C4⋊C8C2414yes—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—28no—
214C22⋊C8—24no—
215C22⋊C8—28no—
216C22⋊C8C22×C4116yes—
217C22⋊C8C22×C4116yes—
218#97—28no—
219#97C22×C418yes—
220#97C22×C418yes—
221C4×C8—24no—
222C4×C8—24no—
223C4×C8—28no—
224C4×C8—24no—
225C4×C8C22×C418yes—
226C4×C8C2418yes—
227C24C2×C4⋊C412yes—
228C24C2×C4⋊C411yescharacteristic
229#41—216no—
230#41—216no—
231C2×Q8—24no—
232C2×Q8C2×C4⋊C414yes—
233C2×D4—28no—
234C2×D4—24no—
235C2×D4C2×C4⋊C418yes—
236C2×D4C2×C4⋊C414yes—
237C22×C4—28no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4—82no—
244C22×C4C2×C4⋊C412yes—
245C22×C4C2×C4⋊C414yes—
246C22×C4C2×C4⋊C411yescharacteristic
247C22×C4C2×C4⋊C412yes—
248C22×C4C2×C4⋊C414yes—
249C22×C4C2×C4⋊C412yes—
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4C2×C4⋊C412yes—
252C22×C4C2×C4⋊C414yes—
253C22×C4C2×C4⋊C412yes—
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C2×C4⋊C412yes—
256C22×C4#13118yes—
257C22×C4C23×C414yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C23×C412yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D412yes—
262C22×C4C22×D414yes—
263C22×C4C22×Q811yescharacteristic
264C22×C4C22×Q814yes—
265C22×C4C22×Q812yes—
266M16—216no—
267M16—216no—
268M16—84no—
269M16—84no—
270C2×C8—216no—
271C2×C8—22no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—42no—
277C2×C8—44no—
278C2×C8—42no—
279C2×C8—82no—
280C2×C8—82no—
281C2×C8—84no—
282C2×C8—88no—
283C2×C8C23×C4116yes—
284C4⋊C4C2×C4⋊C418yes—
285C4⋊C4C2×C4⋊C414yes—
286C4⋊C4C2×C4⋊C418yes—
287C4⋊C4C2×C4⋊C414yes—
288C22⋊C4C2×C4⋊C418yes—
289C22⋊C4C2×C4⋊C418yes—
290C22⋊C4C2×C4⋊C418yes—
291C42C2×C4⋊C412yes—
292C42C2×C4⋊C414yes—
293C42C2×C4⋊C414yes—
294C42C2×C4⋊C412yes—
295C42C2×C4⋊C414yes—
296C42C2×C4⋊C412yes—
297C42C23×C412yes—
298C42C23×C411yescharacteristic
299C42C23×C414yes—
300C42C22×D411yescharacteristic
301C42C22×D412yes—
302C42C22×D414yes—
303C42C22×Q811yescharacteristic
304C42C22×Q814yes—
305C42C22×Q812yes—
306C42C2511yescharacteristic
307C23—24no—
308C23—41no—
309C23—42no—
310C23—41no—
311C23—42no—
312C23#42414yes—
313C23C22×C4⋊C414yes—
314C23C22×C4⋊C412yes—
315C23C22×C4⋊C411yescharacteristic
316Q8—28no—
317D4—216no—
318D4—28no—
319C2×C4—24no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4#42414yes—
340C2×C4#42418yes—
341C2×C4C22×C4⋊C411yescharacteristic
342C2×C4C22×C4⋊C411yescharacteristic
343C2×C4C22×C4⋊C412yes—
344C2×C4C22×C4⋊C414yes—
345C2×C4C22×C4⋊C412yes—
346C2×C4C22×C4⋊C414yes—
347C2×C4C22×C4⋊C414yes—
348C2×C4C22×C4⋊C411yescharacteristic
349C2×C4C22×C4⋊C412yes—
350C2×C4C2×#13118yes—
351C2×C4C2×#13114yes—
352C2×C4C24×C411yescharacteristic
353C2×C4C23×D411yescharacteristic
354C2×C4C23×Q811yescharacteristic
355C8—216no—
356C8—44no—
357C8—42no—
358C8—44no—
359C8—42no—
360C8—44no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—22no—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—44no—
369C22—48no—
370C22C2×#42412yes—
371C22C2×#42414yes—
372C22C23×C4⋊C411yescharacteristic
373C4—24no—
374C4—21no—
375C4—22no—
376C4—24no—
377C4—21no—
378C4—21no—
379C4—22no—
380C4—24no—
381C4C2×#42412yes—
382C4C2×#42414yes—
383C4C22×#13112yes—
384C4#358511yescharacteristic
385C4#358611yescharacteristic
386C2—22no—
387C2—24no—
388C2—44no—
389C2C22×#42411yescharacteristic
390C2#6010911yescharacteristic
391C2#6011311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C4⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17812C2semidirect
5. #17811C2semidirect
6. #17808C2semidirect
7. #17807C2semidirect
8. #17787C2semidirect
9. C2×#1565C2semidirect
35. #1574C22semidirect
36. #1574C22semidirect
37. #1573C22semidirect
38. #1572C22semidirect
39. #1571C22semidirect
40. #1570C22semidirect
44. #1565C22semidirect
45. #1565C22semidirect
46. #1565C22semidirect
47. #1564C22semidirect
48. C2×#335C22semidirect
49. C2×#334C22semidirect

Maximal subgroups

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

subgroupindex
#602772
C2×#29882
#601232
#178122
#178112
#178082
#178072
#177872
C2×#15652