Finite Group Databasebeta

← #215932

Subgroups — #215932

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2#21916C211yesmaximal, characteristic
3#13950C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9910C212yesmaximal
6#9889C211yesmaximal, characteristic
7#9889C211yesmaximal, characteristic
8#9888C211yesmaximal, characteristic
9#9888C211yesmaximal, characteristic
10#8714C212yesmaximal
11#8714C212yesmaximal
12#2857C2211yescharacteristic
13#2857C2211yescharacteristic
14#2851C2212yes—
15#2627—21no—
16#2627—21no—
17#2606—22no—
18#2605—22no—
19#2570C412yescomplemented
20#2570C412yescomplemented
21#2550C412yescomplemented
22#2550C412yescomplemented
23#2329—21no—
24#2329—21no—
25#2283C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1917—22no—
28#1917—22no—
29#1817C2212yes—
30#1797C2212yes—
31#1749C2211yescharacteristic
32#1749C2211yescharacteristic
33#1445—21no—
34#1443—21no—
35#1436—21no—
36#1434—21no—
37#1434C2211yescharacteristic
38#1434C2211yescharacteristic
39#1434C2211yescharacteristic
40#1434C2211yescharacteristic
41#1434C2211yescharacteristic
42C4×#95C2211yescharacteristic
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125C2×C412yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—22no—
63C2×C22⋊C8C2312yes—
64C2×C22⋊C8C2312yes—
65C2×#97—21no—
66C2×#97—21no—
67#400—22no—
68#400—22no—
69#398—22no—
70#398—22no—
71#398—22no—
72#398—22no—
73#395—22no—
74#395—22no—
75#391—21no—
76#391—21no—
77#391—22no—
78#391—22no—
79#389—21no—
80#389—21no—
81#389—22no—
82#389—22no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#385—22no—
95#384C2×C411yescharacteristic
96#384C2×C411yescharacteristic
97#383—22no—
98#382C2×C412yes—
99#382C2×C412yes—
100#382C2×C412yes—
101#381—22no—
102#380—21no—
103#380—21no—
104#380C2312yes—
105#380C2312yes—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C22⋊C4—22no—
113#376—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127C2×#95C2311yescharacteristic
128C2×#95C2311yescharacteristic
129C2×#95C2311yescharacteristic
130C2×#95C2311yescharacteristic
131#336—22no—
132#336—24no—
133#336—21no—
134#336—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—81no—
151C4⋊Q8—81no—
152#125—82no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—81no—
190C2×C42—81no—
191C2×C42—81no—
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C4⋊C8—82no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205#97—81no—
206#97—81no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—42no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—42no—
233#95—81no—
234#95—81no—
235#95—81no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24C22×D411yescharacteristic
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—81no—
244C2×Q8—81no—
245C2×Q8—81no—
246C2×Q8—81no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
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—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
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—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C42−1+412yes—
318C22×C42−1+412yes—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C42—24no—
350C42—24no—
351C42—41no—
352C42—41no—
353C42—42no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—82no—
358C42—82no—
359C42—84no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23C22×C22⋊C411yescharacteristic
365C23#53611yescharacteristic
366C23#53611yescharacteristic
367C23#57511yescharacteristic
368C23#57511yescharacteristic
369C23#57511yescharacteristic
370C23#57511yescharacteristic
371Q8—162no—
372Q8—162no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
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—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4#51912yes—
458C2×C4#53612yes—
459C8—82no—
460C8—82no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22#231011yescharacteristic
474C22#288311yescharacteristic
475C22#288411yescharacteristic
476C22#289611yescharacteristic
477C22#289611yescharacteristic
478C22#290111yescharacteristic
479C22#290111yescharacteristic
480C22#324011yescharacteristic
481C22#324011yescharacteristic
482C22#324011yescharacteristic
483C22#324011yescharacteristic
484C4—22no—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—42no—
493C4—41no—
494C4—82no—
495C4—82no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4—82no—
500C4—161no—
501C4—161no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—21no—
506C2#1762311yescharacteristic
507C2#1762411yescharacteristic
508C2#1762411yescharacteristic
509C2#1762511yescharacteristic
510C2#2182811yescharacteristic
511C2#2184211yescharacteristic
512C2#2184611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #2570C4semidirect
20. #2570C4semidirect
21. #2550C4semidirect
22. #2550C4semidirect

Maximal subgroups

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

subgroupindex
#351042
#219162
#139502
#99132
#99102
#98892
#98882
#87142