Finite Group Databasebeta

← #215349

Subgroups — #215349

Order 512.

1052 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34946C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic, complemented
3#20786C211yesmaximal, characteristic, complemented
4#9676C212yesmaximal, complemented
5#9675C212yesmaximal, complemented
6#9645C212yesmaximal, complemented
7#9639C212yesmaximal, complemented
8#8314C212yesmaximal, complemented
9#8310C212yesmaximal
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#2563C412yes—
13#2562—22no—
14#2561—22no—
15#2547C412yes—
16#2503C412yes—
17#2435C412yes—
18#2398—21no—
19#2398—21no—
20C4×#124C2212yes—
21C4×#124C2212yes—
22#2280—21no—
23#2280—21no—
24C4×#117C2211yescharacteristic
25#1923—22no—
26#1916—22no—
27#1840—21no—
28#1831C2212yescomplemented
29#1818—21no—
30#1797C2212yes—
31#1767C2211yescharacteristic, complemented
32C4×C4⋊C8C2211yescharacteristic, complemented
33C4×C4⋊C8C2212yescomplemented
34#1750—21no—
35C4×C22⋊C8C2212yes—
36#1746—21no—
37#1744C2211yescharacteristic, complemented
38C42×C8C2211yescharacteristic, complemented
39C2×Q8⋊C8C2212yescomplemented
40C2×Q8⋊C8C2212yescomplemented
41C2×Q8⋊C8C2212yescomplemented
42C2×Q8⋊C8C2212yescomplemented
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C23×D4C2×C411yescharacteristic
48C24×C4—21no—
49#533—42no—
50#533—42no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×#124C2×C412yes—
55C2×#124C2×C412yes—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8C2312yes—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2312yes—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—21no—
78C2×C4×C8—22no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81C2×C4×C8C2312yes—
82#400—22no—
83#400—22no—
84#397—22no—
85#397—22no—
86#394—22no—
87#394—22no—
88#393—22no—
89#393—22no—
90#391C2×C412yes—
91#391C2×C412yes—
92#390C2×C412yes—
93#390C2×C412yes—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#382C2×C412yes—
99#382C2×C412yes—
100#381—22no—
101#381—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106#379—22no—
107#379—22no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4D412yes—
116#376D411yescharacteristic
117#376D411yescharacteristic
118C43C2311yescharacteristic
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—21no—
123#336—21no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128Q8⋊C8—42no—
129Q8⋊C8—42no—
130Q8⋊C8—42no—
131Q8⋊C8—42no—
132D4⋊C8—42no—
133D4⋊C8—42no—
134D4⋊C8—42no—
135D4⋊C8—42no—
136C25—21no—
137C22×Q8—21no—
138C22×Q8C22×C412yes—
139C22×D4—24no—
140C22×D4—21no—
141C22×D4—41no—
142C22×D4—41no—
143C22×D4—42no—
144C22×D4C22×C412yes—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—44no—
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163#125—44no—
164#124—24no—
165#124—24no—
166#124—42no—
167#124—42no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C22⋊Q8—44no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—44no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—44no—
183C4×D4—44no—
184#117—24no—
185#117—41no—
186#117—41no—
187#117—44no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C412yes—
219C2×C42C22×C412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2411yescharacteristic
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—44no—
234C22⋊C8—44no—
235C22⋊C8—44no—
236#97—42no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—42no—
242C4×C8—44no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255C24—22no—
256C24—44no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—82no—
261C24C2×C22⋊C411yescharacteristic
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—42no—
269C2×Q8—82no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—48no—
279C2×D4—44no—
280C2×D4#13114yes—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—88no—
306C22×C4—88no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4#13514yes—
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C2×C8—24no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C42—21no—
358C42—21no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42H(5,𝔽2)12yes—
376C42H(5,𝔽2)12yes—
377C23—21no—
378C23—22no—
379C23—21no—
380C23—22no—
381C23—24no—
382C23—48no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—88no—
390C23—84no—
391C23—88no—
392C23—84no—
393C23#41612yes—
394C23C22×C22⋊C411yescharacteristic
395Q8—42no—
396Q8—42no—
397Q8—42no—
398Q8—42no—
399Q8—84no—
400D4—24no—
401D4—24no—
402D4—48no—
403D4—42no—
404D4—42no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—88no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#41612yes—
446C2×C4#51811yescharacteristic
447C2×C4#51811yescharacteristic
448C2×C4#53512yes—
449C2×C4#53512yes—
450C2×C4C2×#13112yes—
451C2×C4#56812yes—
452C2×C4C2×#13512yes—
453C2×C4#57511yescharacteristic
454C2×C4#57511yescharacteristic
455C8—44no—
456C8—82no—
457C8—82no—
458C8—82no—
459C22—22no—
460C22—24no—
461C22—41no—
462C22—41no—
463C22—42no—
464C22—44no—
465C22—44no—
466C22—44no—
467C22—816no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22#230911yescharacteristic
473C22#288012yes—
474C22C2×#41611yescharacteristic
475C22#289611yescharacteristic
476C22#290112yes—
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—82no—
486C4—82no—
487C4—82no—
488C4#297612yes—
489C4#297711yescharacteristic
490C4#297811yescharacteristic
491C4#321112yes—
492C4#321112yes—
493C2—42no—
494C2—84no—
495C2#1750812yes—
496C2#1751312yes—
497C2#2179211yescharacteristic
498C2#2180611yescharacteristic
499C2#2181711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×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
2. #21430C2semidirect
3. #20786C2semidirect
4. #9676C2semidirect
5. #9675C2semidirect
6. #9645C2semidirect
7. #9639C2semidirect
8. #8314C2semidirect
28. #1831C22semidirect
31. #1767C22semidirect
32. C4×C4⋊C8C22semidirect
33. C4×C4⋊C8C22semidirect
37. #1744C22semidirect
38. C42×C8C22semidirect
39. C2×Q8⋊C8C22semidirect
40. C2×Q8⋊C8C22semidirect
41. C2×Q8⋊C8C22semidirect
42. C2×Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#349462
#214302
#207862
#96762
#96752
#96452
#96392
#83142
#83102