Finite Group Databasebeta

← #482143

Subgroups — #482143

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35711C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic
3#17352C211yesmaximal, characteristic
4#11907C214yesmaximal, complemented
5#11905C214yesmaximal, complemented
6#11723C212yesmaximal
7#11721C212yesmaximal
8C2×Q82C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2982—21no—
11#2905—21no—
12C22×#101—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—
28#1801—22no—
29#1801C2214yes—
30#1742C2212yes—
31#1461—21no—
32#1461C2214yes—
33#1458—22no—
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
60C8×Q8—44no—
61#431—24no—
62#431—22no—
63#424—44no—
64#423—24no—
65C2×#101—24no—
66C2×#101—44no—
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—21no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8SD1614yes—
125C22×Q8Q1614yes—
126C22×Q8C22×C411yescharacteristic
127C22×Q8C2×D412yes—
128C23×C4C22⋊C412yes—
129C23×C4C2×D411yescharacteristic
130C2×M16—24no—
131C2×M16—42no—
132C2×M16—44no—
133C2×M16—44no—
134C22×C8—42no—
135C4⋊Q8—44no—
136C4⋊Q8—42no—
137C4⋊Q8—48no—
138C4⋊Q8—44no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—44no—
142C4×Q8—44no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—44no—
146C4×Q8—42no—
147C4×Q8—44no—
148C4×Q8—44no—
149#117—24no—
150#117—42no—
151#117—44no—
152#117—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22×C411yescharacteristic
180C2×C22⋊C4C2×D411yescharacteristic
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C4⋊C8—48no—
193C4⋊C8—44no—
194C4⋊C8—42no—
195#101—48no—
196C22⋊C8—24no—
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—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—48no—
225C2×Q8—48no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—41no—
229C2×Q8—44no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8#10314yes—
234C2×Q8#10314yes—
235C2×Q8C2×SD1614yes—
236C2×Q8C2×Q1614yes—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4#10318yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×SD1612yes—
256C22×C4C2×Q1612yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260M16—44no—
261M16—44no—
262M16—84no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—84no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C4⋊C4#9914yes—
288C22⋊C4—24no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—82no—
293C22⋊C4—81no—
294C42—24no—
295C42—22no—
296C42—24no—
297C42—24no—
298C42—22no—
299C42—42no—
300C42—42no—
301C42—44no—
302C42—44no—
303C42—42no—
304C42—42no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23C2×#10312yes—
310C23C22×C22⋊C411yescharacteristic
311Q8—28no—
312Q8—28no—
313Q8—44no—
314Q8—44no—
315Q8—42no—
316Q8—42no—
317Q8—164no—
318Q8—162no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—48no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4C2×#9914yes—
358C2×C4C2×#9912yes—
359C2×C4C2×#10314yes—
360C2×C4C2×#10314yes—
361C2×C4C2×#10312yes—
362C2×C4#42112yes—
363C2×C4C22×SD1611yescharacteristic
364C2×C4C22×Q1611yescharacteristic
365C8—84no—
366C8—84no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22C22×#9911yescharacteristic
375C22C22×#10311yescharacteristic
376C22C2×#42111yescharacteristic
377C4—24no—
378C4—21no—
379C4—21no—
380C4—24no—
381C4—24no—
382C4—41no—
383C4—42no—
384C4—42no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—82no—
390C4—81no—
391C4—81no—
392C4—82no—
393C4#152014yes—
394C2—21no—
395C2—21no—
396C2—42no—
397C2C2×#152011yescharacteristic
398C2#3334911yescharacteristic
399C2#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. #11907C2semidirect
5. #11905C2semidirect

Maximal subgroups

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

subgroupindex
#357112
#206392
#173522
#119072
#119052
#117232
#117212