Finite Group Databasebeta

← #215181

Subgroups — #215181

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic
3#20463C211yesmaximal, characteristic
4#9686C214yesmaximal, complemented
5#9686C214yesmaximal, complemented
6#8338C214yesmaximal
7C2×#533C2212yes—
8C2×#432—21no—
9C2×#431—21no—
10#2868C411yescharacteristic
11#2868C411yescharacteristic
12#2862C412yes—
13#2859C2212yes—
14#2859C2212yes—
15#2857—21no—
16#2857—21no—
17#2857—21no—
18#2857—21no—
19C4×C4⋊Q8—21no—
20C4×C4⋊Q8—21no—
21C4×C4⋊Q8—21no—
22C4×C4⋊Q8—21no—
23#2285C412yes—
24#2285C412yes—
25C2×C43C2211yescharacteristic
26#1767—21no—
27#1767C2212yes—
28#1767C2212yes—
29C4×C4⋊C8—21no—
30#1749—21no—
31#1749C2212yes—
32#1749C2212yes—
33C4×C22⋊C8—21no—
34#1745—21no—
35#1744—21no—
36C4×#97C2212yes—
37C4×#97C2212yes—
38C2×#330C22116yes—
39#533—48no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#125C2314yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×#117C2×C414yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59#432—28no—
60#431—28no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×#97—22no—
66C2×#97—22no—
67C2×#97C2314yes—
68C2×#97C2314yes—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71#391—22no—
72#391—22no—
73#391—22no—
74#391—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384C2×C412yes—
88#384C2×C412yes—
89#384C2×C412yes—
90#384C2×C412yes—
91#382C2×C414yes—
92#382C2×C414yes—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C43C2×C411yescharacteristic
102C43C2×C411yescharacteristic
103C43D411yescharacteristic
104C43D411yescharacteristic
105C43D411yescharacteristic
106C43D411yescharacteristic
107C43C2311yescharacteristic
108C43C2311yescharacteristic
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#330—416no—
114C22×Q8—22no—
115C22×Q8—22no—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—24no—
124C22×C8—24no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131#125—216no—
132C22⋊Q8—48no—
133C22⋊Q8—48no—
134C4×Q8—48no—
135C4×Q8—48no—
136#117—48no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4C22×C414yes—
150C2×C4⋊C4C22×C414yes—
151C2×C22⋊C4—24no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2411yescharacteristic
188C4⋊C8—28no—
189C4⋊C8—28no—
190C22⋊C8—28no—
191C22⋊C8—28no—
192#97—28no—
193#97—28no—
194#97—28no—
195#97—44no—
196C4×C8—28no—
197C4×C8—44no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202C24C2×C22⋊C411yescharacteristic
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—44no—
208C2×Q8—44no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4#10114yes—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C2×C8—28no—
247C2×C8—28no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—44no—
252C4⋊C4—28no—
253C4⋊C4—28no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C22⋊C4—48no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—41no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—48no—
292C42#10114yes—
293C42#10414yes—
294C42#10414yes—
295C42H(5,𝔽2)14yes—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23C2×#10112yes—
303C23C22×C22⋊C411yescharacteristic
304Q8—88no—
305Q8—88no—
306C2×C4—28no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—28no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4C2×#10112yes—
346C2×C4C2×#10112yes—
347C2×C4C2×#10112yes—
348C2×C4C2×#10112yes—
349C2×C4C2×#10112yes—
350C2×C4C2×#10412yes—
351C2×C4C2×#10412yes—
352C2×C4C2×#10412yes—
353C2×C4C2×#10412yes—
354C2×C4C2×#10412yes—
355C2×C4C2×#10412yes—
356C2×C4#42112yes—
357C2×C4#42112yes—
358C2×C4#51812yes—
359C2×C4#53512yes—
360C2×C4#53512yes—
361C8—48no—
362C8—48no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22#230911yescharacteristic
371C22C22×#10111yescharacteristic
372C22C22×#10111yescharacteristic
373C22C22×#10411yescharacteristic
374C22C22×#10411yescharacteristic
375C22C2×#42111yescharacteristic
376C22C2×#42111yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—41no—
388C4—41no—
389C4—84no—
390C4—84no—
391C4#152614yes—
392C4#153114yes—
393C2—42no—
394C2C2×#152611yescharacteristic
395C2C2×#153111yescharacteristic
396C2#1764811yescharacteristic
397C2#1767211yescharacteristic
398C2#2179911yescharacteristic
399C2#2181511yescharacteristic
400C2#2181511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9686C2semidirect
5. #9686C2semidirect

Maximal subgroups

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

subgroupindex
#366452
C4×#4322
#204632
#96862
#83382