Finite Group Databasebeta

← #192111

Subgroups — #192111

Order 512.

966 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#34378C211yesmaximal, characteristic
2#21363C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic
4C2×#1470C211yesmaximal, characteristic
5#14289C211yesmaximal, characteristic
6#8857C214yesmaximal
7#8855C214yesmaximal
8#8025C211yesmaximal, characteristic
9#8018C211yesmaximal, characteristic
10C22×#125C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2382—22no—
14#2370—22no—
15#2345—21no—
16#2344—22no—
17#2344—21no—
18#2336—22no—
19#2330—22no—
20#2316C2212yes—
21#2313C2212yes—
22#2285C2211yescharacteristic
23#2283C2211yescharacteristic
24C2×#383C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1949—22no—
27#1948—22no—
28#1853C2214yes—
29#1853C2212yes—
30#1852C2212yes—
31#1776—22no—
32C4×#106—22no—
33#1470C418yescomplemented
34C2×#324C2214yes—
35#1445—21no—
36#1445—21no—
37#1441C2214yes—
38#1436—22no—
39#1294C2214yes—
40#1294C2212yes—
41#1294C2212yes—
42C2×#125—21no—
43C2×#125—21no—
44C2×#125—21no—
45C2×#125—21no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125C2×C414yes—
49C2×#125Q814yes—
50C2×#117—22no—
51C2×#117—21no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#432—22no—
60#432C2×C414yes—
61#431—22no—
62#431C2×C414yes—
63C2×#107—22no—
64C2×#106—24no—
65C2×#106—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C22⋊C8C2312yes—
69C2×C22⋊C8C2312yes—
70C2×#97—21no—
71C2×C4×C8—21no—
72#389—22no—
73#389—24no—
74#389—22no—
75#389—24no—
76#389—24no—
77#387—24no—
78#387—24no—
79#387—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#384D412yes—
84#383—21no—
85#383—22no—
86#383—22no—
87#382—21no—
88#382D412yes—
89#381—22no—
90#381—21no—
91#380—22no—
92#380—21no—
93#380C2311yescharacteristic
94#380C2312yes—
95#380C2311yescharacteristic
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—21no—
108#376—21no—
109C2×#95—21no—
110C2×#95—24no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—24no—
114C2×#95C2312yes—
115#340—22no—
116#340—24no—
117#340—22no—
118#324C2×C4116yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128#125—24no—
129#125—24no—
130#125—24no—
131#125—24no—
132#125—24no—
133#125—24no—
134#125—24no—
135#125—24no—
136#117—82no—
137#117—84no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4C4⋊C414yes—
159C2×C4⋊C4C4⋊C418yes—
160C2×C4⋊C4C4⋊C414yes—
161C2×C4⋊C4C22×C414yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2×Q814yes—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C42—22no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—82no—
176C2×C42C22×C412yes—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×Q812yes—
182C2×C42#4112yes—
183C2×C42#4112yes—
184#107—44no—
185#106—48no—
186#106—44no—
187C4⋊C8—24no—
188C4⋊C8—24no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8C22×C414yes—
192C22⋊C8C22×C414yes—
193#97—24no—
194C4×C8—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—21no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—48no—
209#95—41no—
210#95—41no—
211#95—42no—
212#95—42no—
213#95—84no—
214C24C22×D411yescharacteristic
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C2×C4⋊C414yes—
240C22×C4C2×C4⋊C412yes—
241C22×C4C2×C4⋊C414yes—
242C22×C4C2×C4⋊C412yes—
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12312yes—
248C22×C4#12312yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—42no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—28no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—28no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—82no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—42no—
284C42—44no—
285C42—82no—
286C42—82no—
287C42—82no—
288C42—84no—
289C42—84no—
290C42C4×D412yes—
291C42C4×D412yes—
292C42C4×D412yes—
293C42C4×D412yes—
294C42C22⋊Q812yes—
295C42C22⋊Q812yes—
296C42C22⋊Q812yes—
297C42C22⋊Q812yes—
298C23—22no—
299C23—22no—
300C23C22×C4⋊C411yescharacteristic
301C23C2×#12011yescharacteristic
302C23C2×#12311yescharacteristic
303C23#57712yes—
304C23#57712yes—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—28no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—48no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—84no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#38512yes—
350C2×C4#38512yes—
351C2×C4#38512yes—
352C2×C4#38512yes—
353C2×C4C2×C4×D411yescharacteristic
354C2×C4C2×C4×D411yescharacteristic
355C2×C4C2×C22⋊Q811yescharacteristic
356C2×C4C2×C22⋊Q811yescharacteristic
357C8—44no—
358C8—44no—
359C22—24no—
360C22—24no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22C2×#38511yescharacteristic
366C22#291212yes—
367C22#300911yescharacteristic
368C22#301011yescharacteristic
369C22#302012yes—
370C22#302512yes—
371C22#309912yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—44no—
381C4—44no—
382C4—82no—
383C4—84no—
384C4—82no—
385C4—84no—
386C4—162no—
387C2—22no—
388C2—22no—
389C2#1824512yes—
390C2#1824612yes—
391C2#2084811yescharacteristic
392C2#2085711yescharacteristic
393C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
33. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#343782
#213632
#213592
C2×#14702
#142892
#88572
#88552
#80252
#80182