Finite Group Databasebeta

← #161808

Subgroups — #161808

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3#20777C211yesmaximal, characteristic, complemented
4#7730C214yesmaximal, complemented
5#7702C212yesmaximal, complemented
6#7699C212yesmaximal, complemented
7#7584C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C2×#532C411yescharacteristic
10#2862C411yescharacteristic
11#2862C2211yescharacteristic
12#2861—21no—
13#2861—21no—
14#2858C411yescharacteristic
15#2858C2211yescharacteristic
16#2850—21no—
17#2850—21no—
18#2850C412yes—
19#2850C2212yes—
20C4×#124—21no—
21C4×#124—21no—
22C4×#124C412yes—
23C4×#124C2212yes—
24#2297—21no—
25#2297—21no—
26C2×C43C2211yescharacteristic
27#1839C2212yes—
28#1839C2212yes—
29#1831—21no—
30#1831—21no—
31#1817—21no—
32#1817C2212yes—
33#1797—21no—
34#1797C2212yes—
35#1767—21no—
36C4×C4⋊C8C2212yes—
37#1749—21no—
38C4×C22⋊C8—21no—
39#1745C2212yes—
40#1744—21no—
41#1272C2218yes—
42#1270C2218yes—
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⋊Q8C2×C412yes—
58C2×#121—21no—
59C2×#121C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C412yes—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×#117—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×#97—22no—
77C2×C4×C8—22no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391C2×C412yes—
85#391C2312yes—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#390—21no—
92#390C2×C412yes—
93#390C2312yes—
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#386—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#382—21no—
107#382—21no—
108#382C2×C412yes—
109#382C2×C412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
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#336C2314yes—
129#336C2314yes—
130C2×#41—44no—
131C2×#41—44no—
132C2×#41—44no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8C22×C412yes—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C22×C412yes—
140C23×C4—22no—
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C22×C8—24no—
149C22×C8—24no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153#127—42no—
154#125—44no—
155#124—28no—
156#124—44no—
157#124—44no—
158#124—44no—
159#124—44no—
160C22⋊Q8—44no—
161C22⋊Q8—44no—
162#121—44no—
163#121—44no—
164C4×Q8—44no—
165C4×Q8—44no—
166C4×D4—44no—
167C4×D4—44no—
168#117—44no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
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—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—44no—
225C4⋊C8—44no—
226C22⋊C8—44no—
227C22⋊C8—44no—
228#97—44no—
229C4×C8—44no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238C24—22no—
239C24C2×C22⋊C411yescharacteristic
240#41—816no—
241C2×Q8—24no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×D4—24no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—42no—
261C2×D4—42no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
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—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4#10014yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—44no—
356C42#13114yes—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23C2×#10012yes—
371C23C22×C22⋊C411yescharacteristic
372Q8—84no—
373Q8—84no—
374Q8—84no—
375D4—84no—
376D4—84no—
377D4—84no—
378D4—88no—
379C2×C4—28no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
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—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—88no—
428C2×C4#40812yes—
429C2×C4#40812yes—
430C2×C4#41012yes—
431C2×C4C2×#10012yes—
432C2×C4C2×#10012yes—
433C2×C4#42112yes—
434C2×C4#42112yes—
435C2×C4C2×#13112yes—
436C2×C4#57512yes—
437C2×C4#57512yes—
438C8—84no—
439C8—84no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22C2×#40811yescharacteristic
455C22C2×#41011yescharacteristic
456C22C22×#10011yescharacteristic
457C22#289611yescharacteristic
458C22#289611yescharacteristic
459C22C2×#42111yescharacteristic
460C22C2×#42111yescharacteristic
461C4—22no—
462C4—22no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—22no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—82no—
476C4—82no—
477C4—82no—
478C2—42no—
479C2—162no—
480C2#1738411yescharacteristic
481C2#1746711yescharacteristic
482C2#1747711yescharacteristic
483C2#1758611yescharacteristic
484C2#1759411yescharacteristic
485C2#1766611yescharacteristic
486C2#1775411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. #20777C2semidirect
4. #7730C2semidirect
5. #7702C2semidirect
6. #7699C2semidirect
7. #7584C2semidirect

Maximal subgroups

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

subgroupindex
#366392
#207782
#207772
#77302
#77022
#76992
#75842