Finite Group Databasebeta

← #215381

Subgroups — #215381

Order 512.

1172 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3C4×#382C211yesmaximal, characteristic, complemented
4#9713C214yesmaximal, complemented
5#9682C214yesmaximal, complemented
6#8333C212yesmaximal, complemented
7#8332C212yesmaximal
8C2×#533C2211yescharacteristic
9C2×#532C2211yescharacteristic
10#2862C411yescharacteristic, complemented
11#2862C411yescharacteristic, complemented
12#2861—21no—
13#2861—21no—
14#2858C411yescharacteristic, complemented
15#2858C411yescharacteristic, complemented
16#2850—21no—
17#2850—21no—
18#2850C2212yes—
19#2850C2212yes—
20C4×#124—21no—
21C4×#124—21no—
22C4×#124C412yescomplemented
23C4×#124C412yescomplemented
24#2297—21no—
25#2297—21no—
26C2×#382—21no—
27C2×C4×C4⋊C4—21no—
28C2×C43C2211yescharacteristic
29#1839—21no—
30#1831—21no—
31#1817C2212yes—
32#1797C2212yes—
33#1767—21no—
34#1749C2212yes—
35#1745—21no—
36C2×#328C2218yes—
37C2×#327C2218yes—
38C4⋊#95C2212yes—
39#1434—21no—
40#1434C2212yes—
41C4×#95—21no—
42C4×#95C2212yes—
43C22×#41—21no—
44#533—44no—
45#532—44no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#127C2×C411yescharacteristic
50C2×#125—21no—
51C2×#124—21no—
52C2×#124—21no—
53C2×#124—21no—
54C2×#124C2×C412yes—
55C2×#124C2×C412yes—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2312yes—
58C2×#121—21no—
59C2×#121C2312yes—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C412yes—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×#117—21no—
65C22×C4⋊C4—22no—
66C22×C42—22no—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—22no—
75C2×C22⋊C8C2314yes—
76C2×#97—22no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391C2×C412yes—
84#391C2×C412yes—
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#390C2×C412yes—
92#390C2×C412yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#386—22no—
105#382—28no—
106#382—21no—
107#382—21no—
108#382C2×C412yes—
109#382C2×C412yes—
110C4×C4⋊C4—44no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95C2314yes—
120C43C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#328—48no—
133#327—48no—
134C2×#41—44no—
135C2×#41—44no—
136C2×#41—42no—
137C2×#41—42no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8C22×C412yes—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C22×C412yes—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—24no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158#127—42no—
159#125—44no—
160#124—44no—
161#124—44no—
162#124—44no—
163#124—44no—
164#124—44no—
165C22⋊Q8—28no—
166C22⋊Q8—44no—
167#121—28no—
168#121—44no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×D4—44no—
172C4×D4—44no—
173#117—44no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—44no—
206C2×C42—44no—
207C2×C42—44no—
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×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C4⋊C8—44no—
237C22⋊C8—28no—
238#97—44no—
239#95—28no—
240#95—28no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—28no—
250#95—44no—
251#95—44no—
252C24—22no—
253C24C22×D411yescharacteristic
254#41—88no—
255#41—88no—
256C2×Q8—24no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×D4—24no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—42no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—28no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4AGL(1,ℤ/8)12yes—
325C22×C4#13712yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4H(5,𝔽2)14yes—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—416no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—44no—
380C42—88no—
381C42#10414yes—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23C22×C22⋊C411yescharacteristic
394C23#53512yes—
395C23C2×AGL(1,ℤ/8)11yescharacteristic
396C23C2×#13711yescharacteristic
397C23#57512yes—
398Q8—84no—
399Q8—84no—
400Q8—84no—
401D4—84no—
402D4—84no—
403D4—84no—
404D4—84no—
405D4—84no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
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—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—88no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4#41012yes—
466C2×C4#41012yes—
467C2×C4#41712yes—
468C2×C4#41712yes—
469C2×C4C2×#10412yes—
470C2×C4C2×#10412yes—
471C2×C4C2×#10412yes—
472C2×C4#42112yes—
473C2×C4#51812yes—
474C2×C4#53512yes—
475C8—84no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—21no—
483C22—21no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22#230911yescharacteristic
492C22C2×#41011yescharacteristic
493C22C2×#41011yescharacteristic
494C22C2×#41711yescharacteristic
495C22#289611yescharacteristic
496C22C22×#10411yescharacteristic
497C22C2×#42111yescharacteristic
498C22#321312yes—
499C22#321512yes—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—88no—
515C4—82no—
516C4—82no—
517C4—82no—
518C2—22no—
519C2—22no—
520C2—162no—
521C2#1754311yescharacteristic
522C2#1754611yescharacteristic
523C2#1755811yescharacteristic
524C2#1758811yescharacteristic
525C2#2179411yescharacteristic
526C2#2180811yescharacteristic
527C2#2181511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. C4×#382C2semidirect
4. #9713C2semidirect
5. #9682C2semidirect
6. #8333C2semidirect
10. #2862C4semidirect
11. #2862C4semidirect
14. #2858C4semidirect
15. #2858C4semidirect
22. C4×#124C4semidirect
23. C4×#124C4semidirect

Maximal subgroups

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

subgroupindex
#366392
#207782
C4×#3822
#97132
#96822
#83332
#83322