Finite Group Databasebeta

← #391428

Subgroups — #391428

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2083C211yesmaximal, characteristic
2#14712C211yesmaximal, characteristic
3#14699C211yesmaximal, characteristic
4#11377C212yesmaximal
5#11357C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11293C212yesmaximal
8#11267C211yesmaximal, characteristic
9#11219C212yesmaximal
10#11207C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic
12C22×#107C2212yes—
13C2×#389—21no—
14C2×#389—21no—
15C2×#389C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#389C2212yes—
18C2×#387—21no—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×C43C2211yescharacteristic
22#2083C418yescomplemented
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34#1436—21no—
35#1436—21no—
36C4⋊#95—21no—
37C4⋊#95—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434—21no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#107—42no—
67C2×#107—42no—
68C2×#107C2×C418yes—
69#389—22no—
70#389—42no—
71#389—42no—
72#389—42no—
73#389—44no—
74#389—44no—
75#389C2×C414yes—
76#387—44no—
77#384—22no—
78#384—42no—
79#384C2×C414yes—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C43—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#340C2×C418yes—
101#336—22no—
102#336—42no—
103#336—42no—
104#336C2×C414yes—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8C22×C414yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C414yes—
182C2×C4⋊C4C22×C412yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42Q1614yes—
205C2×C42C22×C412yes—
206C2×C42#4112yes—
207#107—48no—
208#95—24no—
209#95—42no—
210#95—42no—
211#95—42no—
212#95—42no—
213#95—42no—
214#95—42no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—42no—
219#95—42no—
220#95—42no—
221C24C22×D411yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4C4×D412yes—
297C22×C4C4×D412yes—
298C22×C4C2×Q1612yes—
299C22×C4C2×Q1612yes—
300C22×C4C2×Q1612yes—
301C22×C4#13512yes—
302C22×C4AGL(1,ℤ/8)12yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4#13712yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C4H(5,𝔽2)12yes—
311C22×C4H(5,𝔽2)11yescharacteristic
312C22×C42−1+412yes—
313C22×C42−1+412yes—
314C22×C42−1+412yes—
315C22×C42−1+411yescharacteristic
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—48no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C4⋊C4—88no—
355C42—24no—
356C42—24no—
357C42—44no—
358C42—44no—
359C42—44no—
360C42—44no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C23C2×C4×D411yescharacteristic
366C23#53411yescharacteristic
367C23#53511yescharacteristic
368C23#53611yescharacteristic
369C23#53611yescharacteristic
370C23#53611yescharacteristic
371C23#53711yescharacteristic
372C23C22×Q1611yescharacteristic
373C23C2×#13511yescharacteristic
374C23C2×AGL(1,ℤ/8)11yescharacteristic
375C23C2×AGL(1,ℤ/8)11yescharacteristic
376C23C2×AGL(1,ℤ/8)11yescharacteristic
377C23C2×#13711yescharacteristic
378C23#57511yescharacteristic
379C23#57511yescharacteristic
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
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—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4C4×Q1614yes—
463C2×C4#44214yes—
464C2×C4#51811yescharacteristic
465C2×C4#51912yes—
466C2×C4#52012yes—
467C2×C4#52011yescharacteristic
468C2×C4#53912yes—
469C2×C4#54011yescharacteristic
470C2×C4#54112yes—
471C2×C4#54211yescharacteristic
472C2×C4#56212yes—
473C2×C4#56212yes—
474C2×C4#56212yes—
475C2×C4#56312yes—
476C2×C4#56312yes—
477C2×C4#56312yes—
478C2×C4#56312yes—
479C2×C4#56412yes—
480C8—44no—
481C8—82no—
482C22#232211yescharacteristic
483C22#232411yescharacteristic
484C22#232711yescharacteristic
485C22#281411yescharacteristic
486C22#281611yescharacteristic
487C22#282111yescharacteristic
488C22#282211yescharacteristic
489C22C2×C4×Q1611yescharacteristic
490C22C2×#44211yescharacteristic
491C22C4×AGL(1,ℤ/8)11yescharacteristic
492C22C4×#13711yescharacteristic
493C22#318811yescharacteristic
494C22#318911yescharacteristic
495C22#319311yescharacteristic
496C22#319411yescharacteristic
497C22#320911yescharacteristic
498C22#321211yescharacteristic
499C22#321411yescharacteristic
500C22#321711yescharacteristic
501C22#322811yescharacteristic
502C22#322811yescharacteristic
503C22#323011yescharacteristic
504C22#323011yescharacteristic
505C22#323011yescharacteristic
506C22#323211yescharacteristic
507C22#323411yescharacteristic
508C22#323411yescharacteristic
509C22#323711yescharacteristic
510C22#323811yescharacteristic
511C22#323911yescharacteristic
512C22#324011yescharacteristic
513C22#324811yescharacteristic
514C22#324911yescharacteristic
515C22#325111yescharacteristic
516C22#325311yescharacteristic
517C4—24no—
518C4—24no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—84no—
525C4—82no—
526C4—162no—
527C4—161no—
528C4—161no—
529C4#237812yes—
530C4#238212yes—
531C4#283612yes—
532C4#283912yes—
533C2#1463111yescharacteristic
534C2#2207911yescharacteristic
535C2#2208111yescharacteristic
536C2#2214811yescharacteristic
537C2#2215111yescharacteristic
538C2#2224811yescharacteristic
539C2#2225011yescharacteristic
540C2#3273911yescharacteristic
541C2#3274411yescharacteristic
542C2#3277411yescharacteristic
543C2#3277611yescharacteristic
544C2#3283611yescharacteristic
545C2#3283811yescharacteristic
546C2#3285011yescharacteristic
547C2#3285211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #2083C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20832
#147122
#146992
#113772
#113572
#112932
#112672
#112192
#112072
#107952