Finite Group Databasebeta

← #181679

Subgroups — #181679

Order 512.

1343 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#60123C211yesmaximal, characteristic
3#34125C211yesmaximal, characteristic
4#8074C212yesmaximal, complemented
5#8071C214yesmaximal, complemented
6#8069C214yesmaximal, complemented
7#8068C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2990C418yescomplemented
10#2983—21no—
11#2973—22no—
12#2972—21no—
13#2970—24no—
14#2923C418yescomplemented
15#2922—24no—
16#2921—22no—
17#2920C2212yes—
18#2920C2214yes—
19C2×#432C2214yes—
20#2905—21no—
21#2879—21no—
22C2×C4×M16C2212yes—
23#2357C412yes—
24#2357C414yes—
25#2351C412yes—
26#2347—21no—
27#2347—22no—
28#2341—21no—
29#2293—21no—
30#2293—22no—
31#2283C2212yes—
32#2283C2211yescharacteristic
33C42×Q8—21no—
34C4×#117C2211yescharacteristic
35C4×#117C2212yes—
36#1300C2214yes—
37#1300C2214yes—
38#1300C2218yes—
39C22×#41D411yescharacteristic
40C22×M16—22no—
41C4×#41C2×C414yes—
42C4×#41C2×C414yes—
43C4×#41D414yes—
44C4×#41Q814yes—
45C2×C4×Q8—22no—
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8D411yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4C2×C412yes—
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D412yes—
52C2×#117—22no—
53C2×#117—21no—
54C2×#117D411yescharacteristic
55C2×#117D412yes—
56C22×C42C2311yescharacteristic
57C22×C42C2312yes—
58#446—28no—
59#435—216no—
60C8×D4—28no—
61#433—216no—
62#432—24no—
63#432—24no—
64#432—22no—
65#432C2×C418yes—
66#432C2×C4116yes—
67#432C2314yes—
68#431—28no—
69#431—22no—
70#431C2×C414yes—
71#424—28no—
72#423—24no—
73#423C2×C414yes—
74C2×C4⋊C8C2×C418yes—
75#408—28no—
76#407—24no—
77#407—22no—
78C2×C22⋊C8—24no—
79C4×M16—22no—
80C4×M16—22no—
81C4×M16—22no—
82C4×M16C2×C414yes—
83C4×M16C2314yes—
84C2×#97—24no—
85C2×#97C2×C414yes—
86C2×C4×C8—22no—
87#390—22no—
88#390—24no—
89#389—24no—
90#389—22no—
91#388—24no—
92#388—24no—
93#388—24no—
94#383—21no—
95#383—22no—
96#382C2×C414yes—
97#382C2×C412yes—
98#381C2×C414yes—
99#381C2×C412yes—
100#380—21no—
101#380—22no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C414yes—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4C2×C414yes—
116#376C2311yescharacteristic
117#376C2311yescharacteristic
118#376C2312yes—
119C43—21no—
120C43—22no—
121C2×#41—44no—
122C2×#41C22⋊C414yes—
123C2×#41C4⋊C414yes—
124C22×Q8C22⋊C411yescharacteristic
125C22×D4C22⋊C411yescharacteristic
126C22×D4C22⋊C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—48no—
137C22×C8—24no—
138C22×C8—28no—
139C4×Q8—48no—
140C4×Q8C4214yes—
141C4×Q8C22⋊C414yes—
142C4×Q8C4⋊C414yes—
143C4×Q8C4⋊C414yes—
144C4×D4C4214yes—
145C4×D4C4218yes—
146C4×D4C22⋊C414yes—
147C4×D4C22⋊C418yes—
148C4×D4C4⋊C414yes—
149C4×D4C4⋊C418yes—
150C4×D4C4⋊C414yes—
151C4×D4C4⋊C418yes—
152#117—48no—
153#117—44no—
154#117C22⋊C414yes—
155#117C22⋊C418yes—
156#117C4⋊C414yes—
157#117C4⋊C418yes—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C22⋊C412yes—
172C2×C22⋊C4C22⋊C412yes—
173C2×C42—21no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C22×C414yes—
184C2×C42C22×C412yes—
185C2×C42C22×C414yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D414yes—
192C2×C42C2×Q812yes—
193C2×C42C2×Q814yes—
194C2×C42C2411yescharacteristic
195C4⋊C8—28no—
196C4⋊C8—216no—
197C4⋊C8—24no—
198C4⋊C8C22×C418yes—
199C22⋊C8—28no—
200C22⋊C8—216no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203#97—28no—
204#97—24no—
205#97—24no—
206#97C22×C418yes—
207C4×C8—24no—
208C4×C8—24no—
209C4×C8—24no—
210#95—22no—
211#95—24no—
212#95—22no—
213#95—24no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95C22×C412yes—
218#95C22×C412yes—
219#95C22×C414yes—
220C24C2×C22⋊C411yescharacteristic
221C24C2×C22⋊C412yes—
222#41—48no—
223#41—48no—
224C2×Q8—24no—
225C2×Q8#9514yes—
226C2×D4—24no—
227C2×D4—44no—
228C2×D4#9514yes—
229C2×D4#9518yes—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—81no—
238C22×C4—82no—
239C22×C4C2×C4212yes—
240C22×C4C2×C4214yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C414yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C4⋊C412yes—
252C22×C4C2×C4⋊C414yes—
253C22×C4C2×C4⋊C412yes—
254C22×C4C2×C4⋊C414yes—
255C22×C4C2×C4⋊C412yes—
256C22×C4C2×C4⋊C414yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260M16—48no—
261M16—48no—
262M16—84no—
263C2×C8—28no—
264C2×C8—28no—
265C2×C8—48no—
266C2×C8—88no—
267C4⋊C4—48no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—44no—
271C4⋊C4#9518yes—
272C4⋊C4#9514yes—
273C4⋊C4#9518yes—
274C4⋊C4#9514yes—
275C22⋊C4—48no—
276C22⋊C4—44no—
277C22⋊C4—82no—
278C22⋊C4—84no—
279C22⋊C4#9518yes—
280C22⋊C4#9518yes—
281C22⋊C4#9518yes—
282C42—24no—
283C42—44no—
284C42—42no—
285C42—48no—
286C42—44no—
287C42—82no—
288C42—84no—
289C42C2×C4214yes—
290C42C2×C4212yes—
291C42C2×C22⋊C412yes—
292C42C2×C22⋊C414yes—
293C42C2×C22⋊C412yes—
294C42C2×C22⋊C414yes—
295C42C2×C4⋊C414yes—
296C42C2×C4⋊C412yes—
297C42C2×C4⋊C414yes—
298C42C2×C4⋊C412yes—
299C42C2×C4⋊C414yes—
300C42C2×C4⋊C412yes—
301C42C23×C411yescharacteristic
302C42C23×C411yescharacteristic
303C42C22×D411yescharacteristic
304C42C22×Q811yescharacteristic
305C23—22no—
306C23—24no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23C2×#9512yes—
312C23C2×#9514yes—
313C23C22×C22⋊C411yescharacteristic
314Q8—28no—
315D4—28no—
316D4—48no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—28no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4C2×#9512yes—
347C2×C4C2×#9514yes—
348C2×C4C2×#9514yes—
349C2×C4C2×#9512yes—
350C2×C4C2×#9514yes—
351C2×C4C2×#9512yes—
352C2×C4#42114yes—
353C2×C4C22×C4211yescharacteristic
354C2×C4C22×C22⋊C411yescharacteristic
355C2×C4C22×C22⋊C411yescharacteristic
356C2×C4C22×C4⋊C411yescharacteristic
357C2×C4C22×C4⋊C411yescharacteristic
358C2×C4C22×C4⋊C411yescharacteristic
359C8—48no—
360C8—48no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—22no—
368C22—24no—
369C22—82no—
370C22—84no—
371C22C22×#9511yescharacteristic
372C22C2×#42112yes—
373C4—21no—
374C4—21no—
375C4—24no—
376C4—21no—
377C4—24no—
378C4—22no—
379C4—21no—
380C4—21no—
381C4—22no—
382C4—24no—
383C4—44no—
384C4—84no—
385C4—82no—
386C4#173314yes—
387C4#290112yes—
388C2—22no—
389C2—42no—
390C2—44no—
391C2C2×#173311yescharacteristic
392C2#2034311yescharacteristic
393C2#2036011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC22C22×#95
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8074C2semidirect
5. #8071C2semidirect
6. #8069C2semidirect
7. #8068C2semidirect
9. #2990C4semidirect
14. #2923C4semidirect

Maximal subgroups

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

subgroupindex
#602822
#601232
#341252
#80742
#80712
#80692
#80682