Finite Group Databasebeta

← #215923

Subgroups — #215923

Order 512.

774 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic
3#13950C211yesmaximal, characteristic
4#9912C211yesmaximal, characteristic
5#9911C211yesmaximal, characteristic
6#9909C211yesmaximal, characteristic
7#9908C211yesmaximal, characteristic
8#9886C212yesmaximal
9#9886C212yesmaximal
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#1916—22no—
28#1916—22no—
29#1817C2212yes—
30#1797C2211yescharacteristic
31#1797C2211yescharacteristic
32C4×C22⋊C8C2212yes—
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—21no—
63C2×C4⋊C8—21no—
64C2×C22⋊C8C2312yes—
65C2×C22⋊C8C2312yes—
66C2×C4×C8—22no—
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—21no—
132#336—21no—
133#336—24no—
134#336—22no—
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—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
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—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—81no—
192C2×C42—81no—
193C2×C42—81no—
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C4⋊C8—81no—
203C4⋊C8—81no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C4×C8—82no—
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—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—81no—
240#95—81no—
241#95—81no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246C24C22×D411yescharacteristic
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—81no—
250C2×Q8—81no—
251C2×Q8—81no—
252C2×Q8—81no—
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—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
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×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4#13512yes—
320C22×C4#13512yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C42−1+411yescharacteristic
324C22×C42−1+411yescharacteristic
325C22×C42−1+411yescharacteristic
326C22×C42−1+411yescharacteristic
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—42no—
362C42—41no—
363C42—41no—
364C42—42no—
365C42—41no—
366C42—41no—
367C42—82no—
368C42—82no—
369C42—84no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23C22×C22⋊C411yescharacteristic
375C23#53611yescharacteristic
376C23#53611yescharacteristic
377C23C2×#13511yescharacteristic
378C23C2×#13511yescharacteristic
379C23#57511yescharacteristic
380C23#57511yescharacteristic
381Q8—162no—
382Q8—162no—
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—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
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—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
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—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4#41612yes—
468C2×C4#41612yes—
469C2×C4#51911yescharacteristic
470C2×C4#51911yescharacteristic
471C2×C4#53611yescharacteristic
472C2×C4#53611yescharacteristic
473C8—82no—
474C8—82no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22#231011yescharacteristic
488C22#288311yescharacteristic
489C22#288411yescharacteristic
490C22C2×#41611yescharacteristic
491C22#289611yescharacteristic
492C22#290111yescharacteristic
493C22#290111yescharacteristic
494C22#323311yescharacteristic
495C22#323311yescharacteristic
496C22#323311yescharacteristic
497C22#323311yescharacteristic
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—41no—
508C4—42no—
509C4—41no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—161no—
517C4—161no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—21no—
522C2#1761511yescharacteristic
523C2#1761611yescharacteristic
524C2#1762411yescharacteristic
525C2#1762511yescharacteristic
526C2#2182811yescharacteristic
527C2#2183711yescharacteristic
528C2#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
#219122
#139502
#99122
#99112
#99092
#99082
#98862
#87142