Finite Group Databasebeta

← #206859

Subgroups — #206859

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic
2#20815C211yesmaximal, characteristic
3#20472C211yesmaximal, characteristic
4#9876C212yesmaximal, complemented
5#9875C212yesmaximal, complemented
6#9872C212yesmaximal, complemented
7#9871C212yesmaximal, complemented
8#7904C211yesmaximal, characteristic
9#7900C211yesmaximal, characteristic
10#7896C211yesmaximal, characteristic
11#7892C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2571C412yes—
17#2570—22no—
18#2569—22no—
19#2552C412yes—
20#2442C412yes—
21#2442C412yes—
22#2402—21no—
23#2402—21no—
24C4×#125C2212yes—
25C4×#125C2212yes—
26#2296—21no—
27#2285—21no—
28#2285C2211yescharacteristic
29C4×#117C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1842C2212yes—
32#1831C2212yes—
33#1798—21no—
34#1798—21no—
35#1771—22no—
36#1768—21no—
37#1768—21no—
38#1753—22no—
39#1744C2211yescharacteristic
40#1744C2211yescharacteristic
41C4×#97C2211yescharacteristic
42C42×C8C2211yescharacteristic
43#1279C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×M16—21no—
52C2×#125C2×C412yes—
53C2×#125C2×C412yes—
54C2×#117—21no—
55C2×#117D411yescharacteristic
56C2×#117D411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#432—22no—
61#432—22no—
62#431—22no—
63#431—22no—
64#431—22no—
65#431Q814yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C4×M16—22no—
75C4×M16Q814yes—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—22no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#400—22no—
86#400—22no—
87#400—22no—
88#400—22no—
89#398—22no—
90#398—22no—
91#398—22no—
92#398—22no—
93#389—22no—
94#389—22no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#384—22no—
106#384C2×C412yes—
107#382—22no—
108#382—21no—
109#382C2×C412yes—
110#382C2×C412yes—
111#380—21no—
112#380—22no—
113#380—22no—
114#380C2×C411yescharacteristic
115#380C2×C411yescharacteristic
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4C2×C412yes—
130C4×C22⋊C4D412yes—
131#376D411yescharacteristic
132#376D411yescharacteristic
133C2×#95—21no—
134C43C2311yescharacteristic
135C43C2311yescharacteristic
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140C23×C4—22no—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C2×M16—44no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152#125—24no—
153#125—24no—
154#125—42no—
155#125—42no—
156#117—42no—
157#117—42no—
158#117—44no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22⋊C412yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—44no—
187C2×C42C22×C412yes—
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×Q812yes—
199C2×C42C2×Q812yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—24no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C22⋊C8—24no—
211C22⋊C8—42no—
212C22⋊C8—42no—
213#97—24no—
214#97—44no—
215#97—42no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—44no—
221C4×C8—42no—
222C4×C8C2×Q812yes—
223C4×C8C2×Q812yes—
224C4×C8C2×Q812yes—
225C4×C8C2×Q812yes—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244C24C2×C22⋊C411yescharacteristic
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—82no—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C4×Q812yes—
265C22×C4C4×Q812yes—
266C22×C4C22⋊Q812yes—
267C22×C4C22⋊Q812yes—
268C22×C4#12412yes—
269C22×C4C4⋊Q812yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273M16—48no—
274C2×C8—24no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4#13114yes—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—44no—
326C42—44no—
327C42C4×Q812yes—
328C42C4×Q812yes—
329C42C22⋊Q812yes—
330C42C22⋊Q812yes—
331C42#12412yes—
332C42C4⋊Q812yes—
333C42C22×Q811yescharacteristic
334C42C22×Q811yescharacteristic
335C42C2×#4111yescharacteristic
336C42C2×#4111yescharacteristic
337C23—22no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23#39112yes—
342C23C22×C22⋊C411yescharacteristic
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—88no—
380C2×C4#39112yes—
381C2×C4#39112yes—
382C2×C4#39112yes—
383C2×C4C2×C4×Q811yescharacteristic
384C2×C4C2×C4×Q811yescharacteristic
385C2×C4C2×C22⋊Q811yescharacteristic
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4C2×#12411yescharacteristic
388C2×C4C2×C4⋊Q811yescharacteristic
389C2×C4C2×#13112yes—
390C2×C4#56812yes—
391C2×C4#57512yes—
392C2×C4#57512yes—
393C8—24no—
394C8—42no—
395C8—42no—
396C8—42no—
397C8—82no—
398C8—82no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—42no—
403C22—42no—
404C22C2×#39111yescharacteristic
405C22#288012yes—
406C22#289611yescharacteristic
407C22#289611yescharacteristic
408C22#290112yes—
409C4—22no—
410C4—22no—
411C4—21no—
412C4—21no—
413C4—22no—
414C4—42no—
415C4—42no—
416C4—44no—
417C4—42no—
418C4—42no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4#296011yescharacteristic
423C4#296111yescharacteristic
424C4#296212yes—
425C4#314212yes—
426C4#316612yes—
427C2—42no—
428C2#1751312yes—
429C2#1751612yes—
430C2#2130411yescharacteristic
431C2#2132511yescharacteristic
432C2#2133011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9876C2semidirect
5. #9875C2semidirect
6. #9872C2semidirect
7. #9871C2semidirect

Maximal subgroups

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

subgroupindex
#349482
#208152
#204722
#98762
#98752
#98722
#98712
#79042
#79002
#78962
#78922