Finite Group Databasebeta

← #215183

Subgroups — #215183

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3#20778C211yesmaximal, characteristic, complemented
4#9688C214yesmaximal, complemented
5#9688C214yesmaximal, complemented
6#8335C212yesmaximal, complemented
7#8334C212yesmaximal
8C22×#125C2211yescharacteristic
9C22×#124C2211yescharacteristic
10#2861—21no—
11#2861—21no—
12#2861—21no—
13#2861—21no—
14#2861C412yes—
15#2861C412yes—
16#2861C2212yes—
17#2861C2212yes—
18C2×#391C411yescharacteristic
19C2×#391C411yescharacteristic
20C2×#390C411yescharacteristic
21C2×#390C411yescharacteristic
22C2×#388—21no—
23C2×#388—21no—
24C2×#388—21no—
25C2×#388—21no—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1839—21no—
29#1831—21no—
30#1831—21no—
31#1817—21no—
32#1817—21no—
33#1797—21no—
34#1797—21no—
35#1767C2212yescomplemented
36#1767C2212yescomplemented
37#1749C2212yes—
38#1749C2212yes—
39#1745C2212yescomplemented
40#1745C2212yescomplemented
41C2×#330C2218yescomplemented
42C2×#329C2218yes—
43C23×Q8C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C2×#125—21no—
46C2×#125—21no—
47C2×#125—44no—
48C2×#125C2×C412yes—
49C2×#125C2312yes—
50C2×#124—21no—
51C2×#124—21no—
52C2×#124—44no—
53C2×#124C2×C412yes—
54C2×#124C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×#97C2314yes—
77C2×#97C2314yes—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C412yes—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#390C2×C412yes—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C43C2×C411yescharacteristic
115C43C2×C411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43C2311yescharacteristic
121C43C2311yescharacteristic
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#330—48no—
131#329—48no—
132C25—21no—
133C22×Q8—21no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8C22×C412yes—
138C22×D4—21no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4C22×C412yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—24no—
154C22×C8—24no—
155#125—28no—
156#125—44no—
157#125—44no—
158#125—44no—
159#124—28no—
160#124—44no—
161#124—44no—
162#124—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C42—24no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2411yescharacteristic
228C4⋊C8—44no—
229C4⋊C8—44no—
230C22⋊C8—44no—
231C22⋊C8—44no—
232#97—28no—
233#97—28no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242C24—22no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24C2×C22⋊C411yescharacteristic
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—88no—
258C2×D4—24no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—88no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42H(5,𝔽2)14yes—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—84no—
378C23—84no—
379C23—84no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23#41312yes—
385C23#42112yes—
386C23#42112yes—
387C23C22×C22⋊C411yescharacteristic
388Q8—84no—
389Q8—84no—
390D4—84no—
391D4—84no—
392C2×C4—28no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4#41312yes—
448C2×C4#42112yes—
449C2×C4#42112yes—
450C2×C4#51812yes—
451C2×C4#53512yes—
452C2×C4#53512yes—
453C8—84no—
454C8—84no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—168no—
469C22#153514yes—
470C22#230911yescharacteristic
471C22C2×#41311yescharacteristic
472C22C2×#41311yescharacteristic
473C22C2×#42111yescharacteristic
474C22C2×#42111yescharacteristic
475C22C2×#42111yescharacteristic
476C22C2×#42111yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—82no—
492C4—82no—
493C4—82no—
494C2—24no—
495C2—162no—
496C2C2×#153511yescharacteristic
497C2#1766811yescharacteristic
498C2#1766911yescharacteristic
499C2#1767011yescharacteristic
500C2#2179811yescharacteristic
501C2#2181611yescharacteristic
502C2#2181611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×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
2. #20778C2semidirect
3. #20778C2semidirect
4. #9688C2semidirect
5. #9688C2semidirect
6. #8335C2semidirect
35. #1767C22semidirect
36. #1767C22semidirect
39. #1745C22semidirect
40. #1745C22semidirect
41. C2×#330C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
#207782
#96882
#83352
#83342