Finite Group Databasebeta

← #216196

Subgroups — #216196

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
1C2×#2335C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3#21912C211yesmaximal, characteristic, complemented
4#9880C212yesmaximal, complemented
5#9649C212yesmaximal
6#9080C212yesmaximal, complemented
7#9080C212yesmaximal, complemented
8#8559C212yesmaximal, complemented
9#8557C212yesmaximal, complemented
10C22×#123C411yescharacteristic
11C2×#431C2212yes—
12#2335—24no—
13#2335—24no—
14C2×#390C2211yescharacteristic
15C2×#389C411yescharacteristic
16C2×#388C411yescharacteristic
17C2×#388C411yescharacteristic
18C2×#388C2211yescharacteristic
19C2×#388C2211yescharacteristic
20C2×#387C411yescharacteristic
21C2×#387C2211yescharacteristic
22C2×#386C411yescharacteristic
23C2×#385C411yescharacteristic
24C2×#380C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1939—24no—
29#1916—24no—
30#1817C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#323C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C2×#123—24no—
43C2×#123—24no—
44C2×#123C818yescomplemented
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67#431—44no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8C2314yes—
73C2×C22⋊C8C2314yes—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76#390—22no—
77#390—22no—
78#389—24no—
79#388—24no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—24no—
85#387—24no—
86#387—22no—
87#387—22no—
88#386—24no—
89#385—24no—
90#380—24no—
91C4×C4⋊C4D414yes—
92C4×C4⋊C4D414yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107#336—22no—
108#336—22no—
109#336—22no—
110#336—24no—
111#336—22no—
112#323—44no—
113#323—44no—
114C25C22×C411yescharacteristic
115C25C22×C411yescharacteristic
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—81no—
119C22×D4—81no—
120C22×D4C2×C814yes—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—42no—
125C23×C4—81no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4—81no—
129C23×C4C2×C814yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—24no—
144C22×C8—24no—
145#123—232no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—81no—
154C2×C4⋊C4—81no—
155C2×C4⋊C4—81no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C2×C814yes—
160C2×C4⋊C4C2×C814yes—
161C2×C4⋊C4M1614yes—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—81no—
183C2×C22⋊C4—81no—
184C2×C22⋊C4—81no—
185C2×C22⋊C4—81no—
186C2×C22⋊C4C2×C814yes—
187C2×C22⋊C4C2×C814yes—
188C2×C22⋊C4C2×C814yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—81no—
194C2×C42—81no—
195C2×C42—81no—
196C2×C42—81no—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C4⋊C8—44no—
204C4⋊C8—82no—
205C4⋊C8—82no—
206C4⋊C8—82no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C4×C8—44no—
212C4×C8—82no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95C22⋊C414yes—
224#95C22⋊C414yes—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—41no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24C22×C812yes—
242C24C22×C812yes—
243C24C23×C411yescharacteristic
244C2×D4—28no—
245C2×D4—82no—
246C2×D4—82no—
247C2×D4—84no—
248C2×D4—84no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—28no—
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—41no—
268C22×C4—41no—
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—44no—
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—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—88no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C22×C812yes—
306C22×C4C22×C812yes—
307C22×C4C22×C812yes—
308C22×C4C22×C812yes—
309C22×C4C22×C812yes—
310C22×C4C2×M1612yes—
311C22×C4C2×M1612yes—
312C22×C4C2×M1612yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4H(5,𝔽2)12yes—
316C22×C42−1+412yes—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—44no—
323C4⋊C4—28no—
324C4⋊C4—44no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4C22⋊C8116yes—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—22no—
345C42—22no—
346C42—24no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—82no—
352C42—82no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—22no—
359C23—24no—
360C23—24no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23C22×C22⋊C411yescharacteristic
392C23#52012yes—
393C23C23×C811yescharacteristic
394C23C22×M1611yescharacteristic
395C23#56812yes—
396D4—88no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
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—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—88no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—88no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4C2×C22⋊C814yes—
456C2×C4C2×C22⋊C814yes—
457C2×C4C2×C22⋊C814yes—
458C2×C4#53511yescharacteristic
459C2×C4#53611yescharacteristic
460C2×C4#53712yes—
461C8—84no—
462C8—84no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—84no—
478C22—84no—
479C22—84no—
480C22—84no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22#231111yescharacteristic
490C22C22×C22⋊C811yescharacteristic
491C22#287911yescharacteristic
492C22#288311yescharacteristic
493C22#288411yescharacteristic
494C22#288811yescharacteristic
495C22#288811yescharacteristic
496C22#298812yes—
497C22#298912yes—
498C4—24no—
499C4—21no—
500C4—22no—
501C4—21no—
502C4—24no—
503C4—24no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4—81no—
513C2—22no—
514C2—22no—
515C2—82no—
516C2—82no—
517C2#1735011yescharacteristic
518C2#1735311yescharacteristic
519C2#1740511yescharacteristic
520C2#1740911yescharacteristic
521C2#2184711yescharacteristic
522C2#2187711yescharacteristic
523C2#2188211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ 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
2. #21951C2semidirect
3. #21912C2semidirect
4. #9880C2semidirect
6. #9080C2semidirect
7. #9080C2semidirect
8. #8559C2semidirect
9. #8557C2semidirect
44. C2×#123C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23352
#219512
#219122
#98802
#96492
#90802
#85592
#85572