Finite Group Databasebeta

← #221947

Subgroups — #221947

Order 512.

957 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36132C211yesmaximal, characteristic
2#21108C212yesmaximal
3C2×#1537C211yesmaximal, characteristic, complemented
4C2×#1526C211yesmaximal, characteristic, complemented
5#9690C212yesmaximal, complemented
6#9125C212yesmaximal, complemented
7#9122C212yesmaximal, complemented
8#8584C211yesmaximal, characteristic, complemented
9#8338C211yesmaximal, characteristic, complemented
10#8338C211yesmaximal, characteristic, complemented
11#8225C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×#533C2211yescharacteristic
14C2×#432C2212yes—
15C2×C4×M16C2212yes—
16#2859C2211yescharacteristic
17#2851C2211yescharacteristic
18#2851C2211yescharacteristic
19#2551C412yes—
20#2465—22no—
21#2465—22no—
22#2452—22no—
23#2439C412yes—
24#2439C412yes—
25#2365—22no—
26#2354C412yes—
27C4×#125C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1923—22no—
30#1916—22no—
31#1870—22no—
32#1851—22no—
33#1767C2212yes—
34#1749C2212yes—
35#1749C2212yes—
36#1745C2212yes—
37#1537—24no—
38#1526—24no—
39C2×#330C2211yescharacteristic
40C2×#330C2211yescharacteristic
41C2×#330C2211yescharacteristic
42C2×#330C2212yes—
43C2×#330C2212yes—
44C2×#330C2212yes—
45C2×#330C2212yes—
46C2×#330C2211yescharacteristic
47#1295C2212yes—
48#1295C2212yes—
49C22×M16—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—42no—
55#533—42no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2×C412yes—
59C2×#125C2311yescharacteristic
60C2×#125C2311yescharacteristic
61C2×#125C2311yescharacteristic
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D412yes—
68C2×#117C2×C412yes—
69C2×#117C2×C412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#432—24no—
73#432—42no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C4×M16—24no—
80C4×M16—42no—
81C2×#97C2312yes—
82C2×#97C2312yes—
83C2×#97C2312yes—
84C2×#97C2312yes—
85C2×C4×C8—22no—
86#396—22no—
87#396—22no—
88#395—22no—
89#395—22no—
90#393—22no—
91#393—22no—
92#391—22no—
93#391—22no—
94#390—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#388—22no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#385—22no—
109#385—22no—
110#384C2×C412yes—
111#382C2×C412yes—
112#382C2311yescharacteristic
113#380C2×C412yes—
114#380C2×C412yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4D412yes—
120C4×C22⋊C4C2×C411yescharacteristic
121C4×C22⋊C4C2×C411yescharacteristic
122C4×C22⋊C4D412yes—
123C4×C22⋊C4D412yes—
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#330—22no—
131#330—22no—
132#330—22no—
133#330—24no—
134#330—24no—
135#330—42no—
136#330—42no—
137#330D414yescomplemented
138C22×Q8—22no—
139C22×Q8—22no—
140C22×D4C22⋊C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C2×M16—44no—
146C2×M16—44no—
147C2×M16—44no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—48no—
157#125—21no—
158#125—21no—
159#125—21no—
160#125—21no—
161#125—22no—
162#125—22no—
163#125—22no—
164#125C2×D412yes—
165#125#4112yes—
166C22⋊Q8—44no—
167C22⋊Q8—44no—
168C22⋊Q8—44no—
169C22⋊Q8—44no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×Q8—44no—
173C4×Q8—44no—
174C4×D4—24no—
175C4×D4—44no—
176#117—24no—
177#117—24no—
178#117—42no—
179#117—42no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊C411yescharacteristic
201C2×C22⋊C4C22⋊C411yescharacteristic
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2411yescharacteristic
222C4⋊C8—44no—
223C4⋊C8—44no—
224C22⋊C8—44no—
225C22⋊C8—44no—
226C22⋊C8—44no—
227#97—22no—
228#97—22no—
229#97—22no—
230#97—22no—
231#97—22no—
232#97—22no—
233#97—22no—
234#97C2×D414yes—
235C4×C8—44no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248C24C2×C22⋊C411yescharacteristic
249C24C2×C22⋊C411yescharacteristic
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—82no—
255C2×Q8—82no—
256C2×D4—22no—
257C2×D4—42no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D412yes—
287M16—416no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4#10114yes—
321C4⋊C4C4×D414yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—24no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—42no—
340C42—41no—
341C42—42no—
342C42—44no—
343C42—44no—
344C42—44no—
345C42C22×D411yescharacteristic
346C42C2×#4111yescharacteristic
347C42H(5,𝔽2)12yes—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23C22×C22⋊C411yescharacteristic
359Q8—88no—
360Q8—88no—
361D4—44no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4C2×#10112yes—
402C2×C4C2×#10112yes—
403C2×C4C2×#10112yes—
404C2×C4#42114yes—
405C2×C4C2×C4×D411yescharacteristic
406C2×C4#51811yescharacteristic
407C2×C4D4212yes—
408C2×C4#54612yes—
409C8—44no—
410C8—44no—
411C8—82no—
412C8—82no—
413C22—41no—
414C22—41no—
415C22—42no—
416C22—42no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—42no—
422C22—42no—
423C22D4×C22⋊C411yescharacteristic
424C22C22×#10111yescharacteristic
425C22#288911yescharacteristic
426C22C2×#42112yes—
427C22#290112yes—
428C4—22no—
429C4—21no—
430C4—21no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—24no—
436C4—42no—
437C4—41no—
438C4—42no—
439C4—44no—
440C4—82no—
441C4—82no—
442C4—82no—
443C2—81no—
444C2—81no—
445C2#1766411yescharacteristic
446C2#1768312yes—
447C2#1769211yescharacteristic
448C2D4×#10111yescharacteristic
449C2#2243412yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1537C2semidirect
4. C2×#1526C2semidirect
5. #9690C2semidirect
6. #9125C2semidirect
7. #9122C2semidirect
8. #8584C2semidirect
9. #8338C2semidirect
10. #8338C2semidirect
11. #8225C2semidirect
137. #330D4semidirect

Maximal subgroups

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

subgroupindex
#361322
#211082
C2×#15372
C2×#15262
#96902
#91252
#91222
#85842
#83382
#82252