Finite Group Databasebeta

← #215697

Subgroups — #215697

Order 512.

926 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3#20778C211yesmaximal, characteristic
4#9907C214yesmaximal, complemented
5#9904C214yesmaximal, complemented
6#8725C212yesmaximal
7#8724C212yesmaximal
8C22×#125—21no—
9C2×#432—21no—
10#2859—21no—
11#2859—21no—
12#2859—21no—
13#2859—21no—
14#2859C412yes—
15#2859C412yes—
16#2859C2212yes—
17#2859C2212yes—
18C2×#389—21no—
19C2×#389—21no—
20C2×#389C412yes—
21C2×#389C412yes—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×C43C2211yescharacteristic
26#1842C2212yes—
27#1839—21no—
28#1831—21no—
29#1831C2212yes—
30#1817—21no—
31#1797—21no—
32#1797C2212yes—
33#1797C2212yes—
34#1767—21no—
35#1767C2212yes—
36#1767C2212yes—
37#1749—21no—
38#1745—21no—
39#1296C2218yes—
40#1296C2218yes—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—22no—
44C2×#125—22no—
45C2×#125—48no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61#432—44no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2314yes—
65C2×C4⋊C8C2314yes—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×#97—22no—
69C2×#97—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389C2×C414yes—
83#389C2×C414yes—
84#389C2×C414yes—
85#389C2×C414yes—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384C2×C412yes—
91#384C2×C412yes—
92#384C2312yes—
93#384C2312yes—
94#382—22no—
95#382—22no—
96#382—22no—
97#382—22no—
98C2×#95—22no—
99C2×#95—22no—
100C43C2×C411yescharacteristic
101C43C2×C411yescharacteristic
102C43D411yescharacteristic
103C43D411yescharacteristic
104C43D411yescharacteristic
105C43D411yescharacteristic
106C43C2311yescharacteristic
107C43C2311yescharacteristic
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114C23×C4—22no—
115C23×C4—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—
125#125—48no—
126#125—48no—
127#125—48no—
128#125—48no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
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—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C414yes—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—42no—
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×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2411yescharacteristic
190C4⋊C8—28no—
191C4⋊C8—28no—
192C4⋊C8—44no—
193C4⋊C8—44no—
194C22⋊C8—44no—
195C22⋊C8—44no—
196#97—44no—
197#97—44no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202C24C2×C22⋊C411yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
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—24no—
219C22×C4—24no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
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×C4AGL(1,ℤ/8)12yes—
242C22×C4#13712yes—
243C22×C4#13714yes—
244C22×C4C23×C411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—48no—
260C4⋊C4—48no—
261C4⋊C4—42no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—48no—
269C4⋊C4—48no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—21no—
274C42—21no—
275C42—22no—
276C42—21no—
277C42—22no—
278C42—22no—
279C42—21no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—41no—
291C42—41no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42AGL(1,ℤ/8)14yes—
299C422−1+414yes—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23#41312yes—
305C23#41712yes—
306C23#41912yes—
307C23C22×C22⋊C411yescharacteristic
308C2×C4—28no—
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—24no—
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—88no—
353C2×C4—88no—
354C2×C4#41312yes—
355C2×C4#41712yes—
356C2×C4#41812yes—
357C2×C4#41812yes—
358C2×C4#41912yes—
359C2×C4#51912yes—
360C2×C4#53612yes—
361C2×C4#53612yes—
362C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
363C2×C4C2×AGL(1,ℤ/8)12yes—
364C2×C4C2×#13711yescharacteristic
365C2×C4C2×#13712yes—
366C2×C4#57512yes—
367C8—84no—
368C8—84no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22#155914yes—
376C22#231011yescharacteristic
377C22C2×#41311yescharacteristic
378C22C2×#41311yescharacteristic
379C22C2×#41811yescharacteristic
380C22C2×#41911yescharacteristic
381C22C2×#41711yescharacteristic
382C22#289611yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—84no—
396C4—84no—
397C4#323412yes—
398C4#323612yes—
399C4#323914yes—
400C2—24no—
401C2C2×#155911yescharacteristic
402C2#1774411yescharacteristic
403C2#1774511yescharacteristic
404C2#1774711yescharacteristic
405C2#2183011yescharacteristic
406C2#2183811yescharacteristic
407C2#2184111yescharacteristic
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. #9907C2semidirect
5. #9904C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
#208102
#207782
#99072
#99042
#87252
#87242