Finite Group Databasebeta

← #481665

Subgroups — #481665

Order 512.

876 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35711C211yesmaximal, characteristic
2#20399C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#11794C214yesmaximal, complemented
5#11792C214yesmaximal, complemented
6#11723C212yesmaximal
7#11721C212yesmaximal
8C2×Q82C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2983—21no—
11C22×#101—21no—
12#2873—21no—
13#2628C412yes—
14#2628C2211yescharacteristic
15#2581C411yescharacteristic
16#2581C2212yes—
17#2568C411yescharacteristic
18#2568C2212yes—
19#2451—22no—
20#2451—21no—
21#2439—22no—
22#2439—21no—
23#2396C412yes—
24#2396C2211yescharacteristic
25#2351—21no—
26#2351C411yescharacteristic
27Q8×C22⋊C4—21no—
28C4×#101—22no—
29C4×#101C2214yes—
30#1742C2212yes—
31#1467C2214yes—
32#1464—22no—
33#1461—21no—
34#1458C2212yes—
35#1354C2214yes—
36#1353C2218yes—
37#1353C2214yes—
38C23×Q8D411yescharacteristic
39C22×M16—22no—
40Q82—44no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C412yes—
45#519—44no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D412yes—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2312yes—
56C2×#117C2×C412yes—
57C2×#117D412yes—
58C2×#117D411yescharacteristic
59C2×#117C2311yescharacteristic
60#446—44no—
61#432—24no—
62#431—22no—
63C2×#101—28no—
64C2×#101—24no—
65#405—44no—
66C4×M16—24no—
67#400C2×C412yes—
68#400C2×C412yes—
69#400C2×C412yes—
70#400C2311yescharacteristic
71#400C2312yes—
72#399C2×C411yescharacteristic
73#399C2×C412yes—
74#398—22no—
75#398—21no—
76#396—22no—
77#396—21no—
78#395—22no—
79#395—21no—
80#395C2×C412yes—
81#395C2311yescharacteristic
82#393—22no—
83#393—21no—
84#393C2×C411yescharacteristic
85#391—21no—
86#391—21no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391—22no—
91#391—22no—
92#389—21no—
93#389—22no—
94#389—21no—
95#389—22no—
96#389—22no—
97#389—21no—
98#387—22no—
99#387—22no—
100#387—22no—
101#382—21no—
102#382—22no—
103#382C2×C411yescharacteristic
104#382C2×C412yes—
105#380—22no—
106#380—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C4⋊C4C2×C412yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113#343—22no—
114#343C2314yes—
115#324—24no—
116#324C2314yes—
117#323—22no—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—24no—
122C22×Q8—42no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126C22×Q8C2×D412yes—
127C23×C4C22⋊C412yes—
128C23×C4C2×D411yescharacteristic
129C2×M16—28no—
130C2×M16—24no—
131C2×M16—42no—
132C2×M16—44no—
133C22×C8—42no—
134C4⋊Q8—44no—
135C4⋊Q8—42no—
136C4⋊Q8—48no—
137C4⋊Q8—44no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—44no—
141C4×Q8—44no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×Q8—44no—
145C4×Q8—42no—
146C4×Q8—44no—
147C4×Q8—44no—
148#117—24no—
149#117—42no—
150#117—44no—
151#117—44no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2411yescharacteristic
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22×C411yescharacteristic
179C2×C22⋊C4C2×D411yescharacteristic
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42C22⋊C412yes—
185C2×C42C22⋊C412yes—
186C2×C42C22×C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C4⋊C8—42no—
192C4⋊C8—44no—
193#101—216no—
194C22⋊C8—24no—
195#97—44no—
196#97—44no—
197C4×C8—44no—
198C4×C8—42no—
199#95—22no—
200#95—22no—
201#95—21no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—21no—
208#95—22no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—22no—
213#95—22no—
214#95C22×C411yescharacteristic
215#95C22×C412yes—
216#95C22×C411yescharacteristic
217C24C2×C22⋊C411yescharacteristic
218C2×Q8—24no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—24no—
224C2×Q8—22no—
225C2×Q8—42no—
226C2×Q8—48no—
227C2×Q8—48no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—44no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—82no—
235C2×Q8#10414yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4#13714yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258M16—28no—
259M16—48no—
260M16—44no—
261C2×C8—44no—
262C2×C8—48no—
263C2×C8—44no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C4⋊C4#9914yes—
286C22⋊C4—24no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—82no—
291C22⋊C4—81no—
292C42—24no—
293C42—22no—
294C42—24no—
295C42—24no—
296C42—22no—
297C42—42no—
298C42—42no—
299C42—44no—
300C42—42no—
301C42—44no—
302C42—42no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23#41912yes—
308C23C22×C22⋊C411yescharacteristic
309Q8—28no—
310Q8—48no—
311Q8—42no—
312Q8—42no—
313Q8—44no—
314Q8—164no—
315Q8—162no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—48no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4C2×#9914yes—
356C2×C4C2×#9912yes—
357C2×C4#41912yes—
358C2×C4C2×#10414yes—
359C2×C4C2×#10412yes—
360C2×C4C2×#13712yes—
361C8—48no—
362C8—84no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—42no—
367C22—41no—
368C22—41no—
369C22—81no—
370C22C22×#9911yescharacteristic
371C22C2×#41911yescharacteristic
372C22C22×#10411yescharacteristic
373C4—24no—
374C4—22no—
375C4—24no—
376C4—41no—
377C4—42no—
378C4—42no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—42no—
383C4—81no—
384C4—82no—
385C4—81no—
386C4—81no—
387C4—82no—
388C4#150814yes—
389C2—21no—
390C2—21no—
391C2—42no—
392C2C2×#150811yescharacteristic
393C2#3334911yescharacteristic
394C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11794C2semidirect
5. #11792C2semidirect

Maximal subgroups

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

subgroupindex
#357112
#203992
#173712
#117942
#117922
#117232
#117212