Finite Group Databasebeta

← #215437

Subgroups — #215437

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3#14689C211yesmaximal, characteristic, complemented
4#9716C214yesmaximal, 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×C43C2211yescharacteristic
28#1839—21no—
29#1831—21no—
30#1817C2212yes—
31#1797C2212yes—
32#1767—21no—
33#1749C2212yes—
34#1745—21no—
35C2×#328C2218yes—
36C2×#327C2218yes—
37C4⋊#95—21no—
38C4⋊#95C2212yes—
39#1434—21no—
40#1434—21no—
41#1434C2212yes—
42#1434C2212yes—
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×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C22⋊C8C2314yes—
75C2×#97—22no—
76#391—21no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391C2×C412yes—
83#391C2×C412yes—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390C2×C412yes—
91#390C2×C412yes—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#382—21no—
105#382—21no—
106#382—44no—
107#382C2×C412yes—
108#382C2×C412yes—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95C2314yes—
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#328—48no—
132#327—48no—
133C2×#41—44no—
134C2×#41—44no—
135C2×#41—42no—
136C2×#41—42no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8C22×C412yes—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C22×C412yes—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—24no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157#127—42no—
158#125—44no—
159#124—44no—
160#124—44no—
161#124—44no—
162#124—44no—
163#124—44no—
164C22⋊Q8—28no—
165C22⋊Q8—44no—
166#121—28no—
167#121—44no—
168C4×Q8—44no—
169C4×Q8—44no—
170C4×D4—44no—
171C4×D4—44no—
172#117—44no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C22⋊C4—22no—
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×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
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×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×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C4⋊C8—44no—
233C22⋊C8—28no—
234#97—44no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—28no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249C24—22no—
250C24C22×D411yescharacteristic
251#41—88no—
252#41—88no—
253C2×Q8—24no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×D4—24no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—44no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—42no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
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×C4AGL(1,ℤ/8)12yes—
322C22×C4#13712yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4H(5,𝔽2)14yes—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—44no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—88no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊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—
349C42—22no—
350C42—22no—
351C42—22no—
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—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—44no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23C22×C22⋊C411yescharacteristic
390C23#53512yes—
391C23C2×AGL(1,ℤ/8)11yescharacteristic
392C23C2×#13711yescharacteristic
393C23#57512yes—
394Q8—84no—
395Q8—84no—
396Q8—84no—
397D4—84no—
398D4—84no—
399D4—84no—
400D4—84no—
401D4—84no—
402C2×C4—24no—
403C2×C4—24no—
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—24no—
421C2×C4—24no—
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—21no—
431C2×C4—21no—
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—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—88no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4#41012yes—
464C2×C4#41012yes—
465C2×C4#41712yes—
466C2×C4#41712yes—
467C2×C4#42112yes—
468C2×C4#42112yes—
469C2×C4#51812yes—
470C2×C4#53512yes—
471C8—84no—
472C22—22no—
473C22—22no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22#230911yescharacteristic
488C22C2×#41011yescharacteristic
489C22C2×#41011yescharacteristic
490C22C2×#41711yescharacteristic
491C22#289611yescharacteristic
492C22C2×#42111yescharacteristic
493C22C2×#42111yescharacteristic
494C22#321312yes—
495C22#321512yes—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—164no—
514C2—22no—
515C2—22no—
516C2—162no—
517C2#1756311yescharacteristic
518C2#1758611yescharacteristic
519C2#1758811yescharacteristic
520C2#1759411yescharacteristic
521C2#2179411yescharacteristic
522C2#2180811yescharacteristic
523C2#2181611yescharacteristic
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. #14689C2semidirect
4. #9716C2semidirect
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
#146892
#97162
#96822
#83332
#83322