Finite Group Databasebeta

← #2745697

Subgroups — #2745697

Order 512.

1950 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2843C212yesmaximal
2#17221C212yesmaximal
3#17114C212yesmaximal
4#17099C214yesmaximal
5#16670C214yesmaximal
6#16522C212yesmaximal
7#15151C212yesmaximal
8#15145C212yesmaximal
9#15143C212yesmaximal
10#15139C212yesmaximal
11#15137C212yesmaximal
12#15128C211yesmaximal, characteristic
13#13921C212yesmaximal
14#13910C211yesmaximal, characteristic
15C4×#400C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17#2843—216no—
18C2×#400C2214yes—
19C2×#400C2211yescharacteristic
20C2×#398C2214yes—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#398C2214yes—
25C2×#395C2212yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#384C2212yes—
33C2×#384C2212yes—
34C2×#384C2212yes—
35C2×#384C2211yescharacteristic
36C2×#383C2214yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#1446C2212yes—
41#1445C2214yes—
42#1445C2214yes—
43#1445C2211yescharacteristic
44#1444C2212yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2212yes—
51#1444C2212yes—
52#1443C2214yes—
53#1443C2212yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2211yescharacteristic
61#1441C2212yes—
62#1441C2214yes—
63#1441C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C4×#95C2211yescharacteristic
74C4×#95C2212yes—
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104#400—216no—
105#400D418yes—
106#398—216no—
107#398—216no—
108#398—216no—
109#398—48no—
110#398—48no—
111#395—28no—
112#395—216no—
113#395—48no—
114#389—44no—
115#389—44no—
116#389—44no—
117#389—44no—
118#384—28no—
119#384—42no—
120#384—44no—
121#384—44no—
122#383—216no—
123C4×C4⋊C4—42no—
124C4×C4⋊C4—42no—
125C4×C4⋊C4Q818yescomplemented
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2311yescharacteristic
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8—42no—
163C22×Q8—44no—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2411yescharacteristic
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C23×C4C2412yes—
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C4⋊Q8—416no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×Q818yes—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42C2×Q814yes—
249C2×C42C2×Q818yes—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—24no—
256#95—28no—
257#95—24no—
258#95—28no—
259#95—24no—
260#95—28no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—28no—
267#95—24no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—42no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95C2×D418yes—
280#95C2×D418yes—
281#95C2×D414yes—
282#95C2×Q818yes—
283C24C2511yescharacteristic
284C2×Q8—48no—
285C2×Q8—84no—
286C2×Q8—88no—
287C2×Q8—84no—
288C2×Q8—88no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4C22×D414yes—
366C22×C4C22×D414yes—
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C22×Q812yes—
371C22×C4C22×Q818yes—
372C22×C4C22×Q812yes—
373C22×C4C22×Q812yes—
374C22×C4H(5,𝔽2)14yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)14yes—
377C22×C42−1+414yes—
378C22×C42−1+412yes—
379C22×C42−1+412yes—
380C22×C42−1+412yes—
381C22×C42−1+414yes—
382C4⋊C4—216no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—84no—
392C4⋊C4—88no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4—88no—
399C4⋊C4—84no—
400C4⋊C4—88no—
401C42—28no—
402C42—216no—
403C42—44no—
404C42—44no—
405C42—48no—
406C42—48no—
407C23C23×D411yescharacteristic
408C23C23×Q811yescharacteristic
409C23C2×H(5,𝔽2)11yescharacteristic
410C23C2×H(5,𝔽2)12yes—
411C23C2×H(5,𝔽2)12yes—
412C23C2×2−1+412yes—
413C23C2×2−1+411yescharacteristic
414C23C2×2−1+411yescharacteristic
415C23C2×2−1+411yescharacteristic
416C23C2×2−1+412yes—
417C23#58511yescharacteristic
418Q8—816no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4#54414yes—
507C2×C4#54418yes—
508C2×C4D4×Q814yes—
509C2×C4#56014yes—
510C2×C4#56314yes—
511C2×C4#56318yes—
512C22C2×#54412yes—
513C22C2×#54411yescharacteristic
514C22C2×D4×Q811yescharacteristic
515C22#349611yescharacteristic
516C22#349812yes—
517C22#349912yes—
518C22#349911yescharacteristic
519C22#350211yescharacteristic
520C22#350312yes—
521C22C2×#56011yescharacteristic
522C22C2×#56311yescharacteristic
523C22C2×#56312yes—
524C22#353711yescharacteristic
525C22#353811yescharacteristic
526C22#354412yes—
527C22#354412yes—
528C22#354612yes—
529C22#355011yescharacteristic
530C22#355212yes—
531C22#355212yes—
532C22#355212yes—
533C22#356311yescharacteristic
534C22#356312yes—
535C4—24no—
536C4—42no—
537C4—44no—
538C4—44no—
539C4—44no—
540C4—44no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—48no—
545C4—81no—
546C4—84no—
547C4—84no—
548C4—84no—
549C4#284318yescomplemented
550C2C2×#284311yescharacteristic
551C2#3936811yescharacteristic
552C2#3936912yes—
553C2#4282312yes—
554C2#4293311yescharacteristic
555C2#4548711yescharacteristic
556C2#4599312yes—
557C2#4614711yescharacteristic
558C2#5300811yescharacteristic
559C2#5303111yescharacteristic
560C2#5304012yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
125. C4×C4⋊C4Q8semidirect
549. C4#2843semidirect

Maximal subgroups

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

subgroupindex
C2×#28432
#172212
#171142
#170992
#166702
#165222
#151512
#151452
#151432
#151392
#151372
#151282
#139212
#139102
C4×#4002