Finite Group Databasebeta

← #4414345

Subgroups — #4414345

Order 512.

2382 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2845C211yesmaximal, characteristic, complemented
2C22×#382C211yesmaximal, characteristic
3#16687C214yesmaximal, complemented
4#16571C214yesmaximal, complemented
5#16525C214yesmaximal, complemented
6#14766C214yesmaximal
7#14736C214yesmaximal
8#14728C214yesmaximal
9#14682C212yesmaximal
10#14677C211yesmaximal, characteristic, complemented
11#14677C212yesmaximal, complemented
12C23×C4⋊C4C2212yes—
13C23×C42C2211yescharacteristic
14#2845—28no—
15C2×#398C2214yes—
16C2×#398C2214yes—
17C2×#398C2214yes—
18C2×#398C2214yes—
19C2×#385C2214yes—
20C2×#385C2214yes—
21C2×#382—28no—
22C2×#382C2212yes—
23C2×#382C2212yes—
24C2×#382C2212yes—
25C2×#382C2211yescharacteristic
26C2×#382C2214yes—
27C2×#382C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2214yes—
31C22×#95C2214yes—
32#1445C2214yes—
33#1445C2214yes—
34#1444C2214yes—
35#1444C2214yes—
36#1444C2214yes—
37#1444C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52C4⋊#95C2212yes—
53C4⋊#95C2212yes—
54#1434C2214yes—
55#1434C2214yes—
56C22⋊#95C2214yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C24×C4C2311yescharacteristic
60C24×C4C2314yes—
61C24×C4C2312yes—
62C22×C4⋊C4—28no—
63C22×C4⋊C4—44no—
64C22×C4⋊C4—44no—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C42—24no—
81C22×C42—24no—
82C22×C42Q818yes—
83C22×C42C2311yescharacteristic
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86#398—216no—
87#398—216no—
88#398—216no—
89#398—48no—
90#385—48no—
91#385—48no—
92#382—28no—
93#382—28no—
94#382—24no—
95#382—28no—
96#382—216no—
97#382D418yes—
98#380—44no—
99#380—48no—
100#380—44no—
101C2×#95—216no—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C25C2411yescharacteristic
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—28no—
138C23×C4—28no—
139C23×C4—28no—
140C23×C4—28no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4C2×Q814yes—
147C23×C4C2×Q818yes—
148C23×C4C2414yes—
149C23×C4C2414yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2412yes—
155C23×C4C2411yescharacteristic
156C23×C4C2412yes—
157C23×C4C2411yescharacteristic
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—432no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4C2×D418yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—232no—
209C2×C42C2×D414yes—
210C2×C42C2×Q814yes—
211C2×C42C2×Q818yes—
212C2×C42C2×Q814yes—
213C2×C42#4114yes—
214#95—28no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95C2×D4116yes—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24C22×Q812yes—
245C24H(5,𝔽2)14yes—
246C24H(5,𝔽2)12yes—
247C242−1+412yes—
248C242−1+414yes—
249C24C2511yescharacteristic
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—216no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—216no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—416no—
324C22×C4—416no—
325C22×C4—416no—
326C22×C4—416no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4—88no—
330C22×C4C22×D418yes—
331C22×C4C22×D414yes—
332C22×C4C22×D412yes—
333C22×C4C22×Q814yes—
334C22×C4C22×Q812yes—
335C22×C4C22×Q814yes—
336C22×C4C22×Q812yes—
337C22×C4C2×#4112yes—
338C22×C4C2×#4114yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C42−1+412yes—
341C4⋊C4—48no—
342C4⋊C4—416no—
343C4⋊C4—48no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—88no—
349C4⋊C4—88no—
350C4⋊C4—88no—
351C4⋊C4—88no—
352C4⋊C4—88no—
353C4⋊C4—88no—
354C22⋊C4—416no—
355C22⋊C4—416no—
356C42—216no—
357C42—28no—
358C42C22⋊Q8116yes—
359C23—22no—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23#54414yes—
378C23#54418yes—
379C23#56214yes—
380C23#56218yes—
381C23#56414yes—
382C23C23×D411yescharacteristic
383C23C23×Q811yescharacteristic
384C23C22×#4111yescharacteristic
385C23C2×H(5,𝔽2)11yescharacteristic
386C23C2×H(5,𝔽2)11yescharacteristic
387C23C2×H(5,𝔽2)12yes—
388C23C2×2−1+411yescharacteristic
389C23C2×2−1+411yescharacteristic
390C23C2×2−1+412yes—
391C23#58512yes—
392C23#58512yes—
393C2×C4—232no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—432no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—832no—
463C2×C4C2×C22⋊Q814yes—
464C2×C4C2×C22⋊Q818yes—
465C2×C4#54414yes—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—28no—
470C22—24no—
471C22—24no—
472C22—28no—
473C22—28no—
474C22—28no—
475C22—24no—
476C22#284518yescomplemented
477C22C22×C22⋊Q811yescharacteristic
478C22C2×#54411yescharacteristic
479C22C2×#54412yes—
480C22C2×#54411yescharacteristic
481C22#345812yes—
482C22#348411yescharacteristic
483C22#348611yescharacteristic
484C22#348812yes—
485C22#348912yes—
486C22#349211yescharacteristic
487C22#349211yescharacteristic
488C22#349412yes—
489C22#349512yes—
490C22C2×#56212yes—
491C22C2×#56211yescharacteristic
492C22C2×#56411yescharacteristic
493C22#354512yes—
494C22#355311yescharacteristic
495C22#355311yescharacteristic
496C22#355812yes—
497C22#355912yes—
498C22#356112yes—
499C22#356312yes—
500C4—24no—
501C4—24no—
502C4—28no—
503C4—48no—
504C4—44no—
505C4—84no—
506C4—84no—
507C4—84no—
508C4—84no—
509C4—84no—
510C2—28no—
511C2#3522411yescharacteristic
512C2#3525312yes—
513C2C2×#284511yescharacteristic
514C2#3756112yes—
515C2#3760911yescharacteristic
516C2#4597612yes—
517C2#4597811yescharacteristic
518C2#4598811yescharacteristic
519C2#4599012yes—
520C2C22⋊#56411yescharacteristic
521C2#4614411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2845C2semidirect
3. #16687C2semidirect
4. #16571C2semidirect
5. #16525C2semidirect
10. #14677C2semidirect
11. #14677C2semidirect
476. C22#2845semidirect

Maximal subgroups

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

subgroupindex
C2×#28452
C22×#3822
#166872
#165712
#165252
#147662
#147362
#147282
#146822
#146772