Finite Group Databasebeta

← #6344446

Subgroups — #6344446

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#60276C211yesmaximal, characteristic
3C2×#2990C211yesmaximal, characteristic
4#17836C218yesmaximal, complemented
5#17834C218yesmaximal, complemented
6#17818C214yesmaximal, complemented
7#17817C212yesmaximal, complemented
8#17806C212yesmaximal, complemented
9C2×#1573C214yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11#2990—24no—
12#2990C2218yes—
13#2990C2214yes—
14#2990C2214yes—
15#2988C418yes—
16#2983—21no—
17#2982C412yes—
18#2973—22no—
19#2972—21no—
20#2972C414yes—
21#2972C412yes—
22#2970—24no—
23C8×#41C2218yes—
24C2×#446—21no—
25C2×#435—22no—
26C2×C8×D4—21no—
27#2920C2212yes—
28C2×#432C2212yes—
29C2×#432C2214yes—
30C2×#431C2211yescharacteristic
31C2×#431C2214yes—
32C2×#431C2212yes—
33C2×#423C2211yescharacteristic
34C22×C4⋊C8C2212yes—
35#1574C2214yescomplemented
36#1574C2214yescomplemented
37#1574C2218yescomplemented
38#1573C2214yes—
39#1573C2218yes—
40#1573C2214yes—
41#1573C22116yescomplemented
42#1573C2218yescomplemented
43#1573C2218yescomplemented
44#1568C22116yescomplemented
45#1567C2214yescomplemented
46#1566C2214yescomplemented
47#1564C2218yescomplemented
48C2×#335C2218yescomplemented
49C2×#334C2218yescomplemented
50C22×#41Q811yescharacteristic
51C22×M16—21no—
52C23×C8—22no—
53C4×#41C2×C414yes—
54C4×#41D414yes—
55C4×#41Q814yes—
56C4×#41C2314yes—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D412yes—
63C2×#117Q811yescharacteristic
64C2×#117Q812yes—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67#446—28no—
68#446—24no—
69#446—24no—
70C8×Q8C2×C4116yes—
71#435—216no—
72#435—28no—
73#435—28no—
74C8×D4—28no—
75C8×D4—24no—
76C8×D4—24no—
77C8×D4C2×C4132yes—
78C8×D4C2×C4116yes—
79#432—24no—
80#432—24no—
81#432—24no—
82#432—22no—
83#432C2318yes—
84#432C2318yes—
85#432C2314yes—
86#431—22no—
87#431—22no—
88#431C2×C418yes—
89#431C2×C414yes—
90#431C2×C418yes—
91#431C2×C414yes—
92#431C2314yes—
93#431C2314yes—
94#431C2314yes—
95#431C2318yes—
96#431C2314yes—
97#423—22no—
98#423C2312yes—
99#423C2314yes—
100#423C2312yes—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—24no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2×C418yes—
105C2×C4⋊C8C2×C414yes—
106C2×C4⋊C8C2314yes—
107C2×C4⋊C8C2318yes—
108C2×C4⋊C8C2314yes—
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2311yescharacteristic
112#407—22no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—24no—
116C2×C22⋊C8C2×C418yes—
117C2×C22⋊C8C2×C414yes—
118C4×M16—22no—
119C2×#97—22no—
120C2×#97—24no—
121C2×C4×C8—21no—
122C2×C4×C8C2318yes—
123C2×C4×C8C2314yes—
124#335C2318yes—
125#335C23116yes—
126#335C2318yes—
127#334C2318yes—
128#334C23116yes—
129#334C2318yes—
130C2×#41—28no—
131C2×#41C4⋊C414yes—
132C2×#41C2×Q814yes—
133C22×Q8C4⋊C411yescharacteristic
134C22×D4C4⋊C412yes—
135C22×D4C4⋊C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×Q812yes—
141C23×C4C2×Q811yescharacteristic
142C2×M16—24no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—82no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—28no—
150C22×C8—24no—
151C22×C8—24no—
152C22×C8—84no—
153C22×C8C22×C4116yes—
154C22×C8C22×C418yes—
155C4×Q8C4⋊C414yes—
156C4×Q8C4⋊C414yes—
157C4×Q8C22×C414yes—
158C4×Q8C2×D414yes—
159C4×D4C4⋊C418yes—
160C4×D4C4⋊C414yes—
161C4×D4C4⋊C418yes—
162C4×D4C4⋊C414yes—
163C4×D4C22×C418yes—
164C4×D4C22×C414yes—
165C4×D4C2×D414yes—
166C4×D4C2×D418yes—
167#117C4⋊C414yes—
168#117C4⋊C418yes—
169#117C2×Q814yes—
170#117C2×Q818yes—
171C2×C4⋊C4C4⋊C412yes—
172C2×C4⋊C4C4⋊C411yescharacteristic
173C2×C4⋊C4C4⋊C412yes—
174C2×C4⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C4⋊C412yes—
176C2×C22⋊C4C4⋊C412yes—
177C2×C22⋊C4C4⋊C412yes—
178C2×C42C22×C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C414yes—
181C2×C42C22×C412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D412yes—
184C2×C42C2×D414yes—
185C2×C42C2×D412yes—
186C2×C42C2×Q812yes—
187C2×C42C2×Q811yescharacteristic
188C2×C42C2×Q814yes—
189C2×C42C2×Q812yes—
190C2×C42C2414yes—
191C2×C42C2412yes—
192C2×C42C2411yescharacteristic
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—28no—
196C4⋊C8—24no—
197C4⋊C8—21no—
198C4⋊C8—24no—
199C4⋊C8—24no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8C22×C418yes—
203C4⋊C8C22×C4116yes—
204C4⋊C8C22×C418yes—
205C4⋊C8C2414yes—
206C4⋊C8C2412yes—
207C4⋊C8C2414yes—
208C4⋊C8C2412yes—
209C4⋊C8C2414yes—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—28no—
213C22⋊C8—28no—
214C22⋊C8—24no—
215C22⋊C8C22×C418yes—
216C22⋊C8C22×C4116yes—
217C22⋊C8C22×C418yes—
218#97—24no—
219#97—28no—
220#97—28no—
221C4×C8—24no—
222C4×C8—24no—
223C4×C8C22×C4116yes—
224C4×C8C22×C418yes—
225C4×C8C2418yes—
226C24C2×C4⋊C411yescharacteristic
227C24C2×C4⋊C412yes—
228#41—216no—
229#41—216no—
230C2×Q8—24no—
231C2×Q8C2×C4⋊C414yes—
232C2×D4—28no—
233C2×D4#10818yes—
234C2×D4C2×C4⋊C418yes—
235C2×D4C2×C4⋊C414yes—
236C22×C4—28no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4C2×C4⋊C412yes—
245C22×C4C2×C4⋊C414yes—
246C22×C4C2×C4⋊C412yes—
247C22×C4C2×C4⋊C412yes—
248C22×C4C2×C4⋊C411yescharacteristic
249C22×C4C2×C4⋊C412yes—
250C22×C4C2×C4⋊C414yes—
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C412yes—
253C22×C4C2×C4⋊C412yes—
254C22×C4C2×C4⋊C414yes—
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C23×C411yescharacteristic
257C22×C4C23×C412yes—
258C22×C4C23×C414yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D414yes—
261C22×C4C22×D412yes—
262C22×C4C22×Q811yescharacteristic
263C22×C4C22×Q814yes—
264C22×C4C22×Q812yes—
265M16—82no—
266M16—82no—
267M16—84no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—416no—
274C2×C8—48no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—44no—
278C2×C8—84no—
279C2×C8—84no—
280C2×C8—88no—
281C2×C8C23×C4116yes—
282C4⋊C4C2×C4⋊C418yes—
283C4⋊C4C2×C4⋊C414yes—
284C4⋊C4C2×C4⋊C418yes—
285C4⋊C4C2×C4⋊C414yes—
286C22⋊C4C2×C4⋊C418yes—
287C22⋊C4C2×C4⋊C418yes—
288C22⋊C4C2×C4⋊C418yes—
289C42C2×C4⋊C414yes—
290C42C2×C4⋊C412yes—
291C42C2×C4⋊C414yes—
292C42C2×C4⋊C412yes—
293C42C2×C4⋊C414yes—
294C42C2×C4⋊C412yes—
295C42C23×C414yes—
296C42C23×C412yes—
297C42C23×C411yescharacteristic
298C42C22×D412yes—
299C42C22×D411yescharacteristic
300C42C22×D414yes—
301C42C22×Q811yescharacteristic
302C42C22×Q814yes—
303C42C22×Q812yes—
304C42C2511yescharacteristic
305C23—22no—
306C23—42no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23C2×#10818yes—
311C23C22×C4⋊C412yes—
312C23C22×C4⋊C414yes—
313C23C22×C4⋊C411yescharacteristic
314Q8—28no—
315D4—216no—
316D4C2×#108116yes—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—21no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—28no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4C2×#10814yes—
341C2×C4C22×C4⋊C411yescharacteristic
342C2×C4C22×C4⋊C411yescharacteristic
343C2×C4C22×C4⋊C411yescharacteristic
344C2×C4C22×C4⋊C414yes—
345C2×C4C22×C4⋊C414yes—
346C2×C4C22×C4⋊C412yes—
347C2×C4C22×C4⋊C412yes—
348C2×C4C22×C4⋊C414yes—
349C2×C4C22×C4⋊C412yes—
350C2×C4C24×C411yescharacteristic
351C2×C4C23×D411yescharacteristic
352C2×C4C23×Q811yescharacteristic
353C8—216no—
354C8—42no—
355C8—44no—
356C8—44no—
357C8—42no—
358C8—44no—
359C22—21no—
360C22—24no—
361C22—21no—
362C22—24no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—48no—
367C22—82no—
368C22C22×#10812yes—
369C22C22×#10818yes—
370C22C23×C4⋊C411yescharacteristic
371C4—21no—
372C4—24no—
373C4—22no—
374C4—21no—
375C4—24no—
376C4—22no—
377C4—21no—
378C4—21no—
379C4—24no—
380C4C22×#10814yes—
381C4#357312yes—
382C4#358511yescharacteristic
383C4#358611yescharacteristic
384C2—22no—
385C2—28no—
386C2—42no—
387C2#6010011yescharacteristic
388C2#6010911yescharacteristic
389C2C23×#10811yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C4⋊C4
prime radicalC22×#41Q8
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. #17836C2semidirect
5. #17834C2semidirect
6. #17818C2semidirect
7. #17817C2semidirect
8. #17806C2semidirect
9. C2×#1573C2semidirect
35. #1574C22semidirect
36. #1574C22semidirect
37. #1574C22semidirect
41. #1573C22semidirect
42. #1573C22semidirect
43. #1573C22semidirect
44. #1568C22semidirect
45. #1567C22semidirect
46. #1566C22semidirect
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
#602822
#602762
C2×#29902
#178362
#178342
#178182
#178172
#178062
C2×#15732