Finite Group Databasebeta

← #365513

Subgroups — #365513

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21323C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4#20801C211yesmaximal, characteristic
5#13909C211yesmaximal, characteristic
6#11340C212yesmaximal
7#11339C212yesmaximal
8#10899C212yesmaximal
9#10899C212yesmaximal
10#10851C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×C4×M16C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2370—22no—
18#2369—21no—
19#2365—21no—
20#2355—21no—
21#2355—22no—
22Q8×C4⋊C4—21no—
23#2329—22no—
24#2317C2212yes—
25C4×C22⋊Q8—22no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30#2274C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1993—21no—
34#1993—21no—
35#1991—22no—
36#1986C418yescomplemented
37#1847—21no—
38#1847—21no—
39#1846—22no—
40C2×#343C2212yes—
41C2×#343C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48#1441—22no—
49#1436—22no—
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2211yescharacteristic
52C22×M16C2312yes—
53C22×M16C2312yes—
54#533—22no—
55#533—22no—
56#533C2×C414yes—
57#533Q814yes—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D411yescharacteristic
60C2×#125D412yes—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117C2312yes—
67C2×#117C2312yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#428—24no—
79#428—24no—
80#428—42no—
81#428—42no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#106—22no—
85C2×#106—22no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16C2×C414yes—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#391—24no—
95#391—22no—
96#389—24no—
97#389—22no—
98#389—22no—
99#389—22no—
100#387—24no—
101#387—24no—
102#387—24no—
103#385—24no—
104#384—22no—
105#384—42no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#382—22no—
109#382—21no—
110#382—21no—
111#382—21no—
112#382—21no—
113#382—42no—
114#382D412yes—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2×C414yes—
131C4×C22⋊C4—24no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95D412yes—
139#343—24no—
140#343Q818yescomplemented
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#339—42no—
146#339—42no—
147#339C2×C418yes—
148#339C2×C418yes—
149C22×Q8—21no—
150C22×Q8—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16C22×C414yes—
169C2×M16C2×Q814yes—
170C2×M16C2×Q814yes—
171C2×M16#4114yes—
172C22×C8—22no—
173C22×C8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8C4⋊C414yescomplemented
177C4⋊Q8C4⋊C414yescomplemented
178#125—24no—
179#125C4⋊C418yescomplemented
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—84no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—22no—
193#117—22no—
194#117—22no—
195#117C22×C414yes—
196#117C2×Q814yes—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×Q814yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—82no—
248C2×C42C22×C412yes—
249C2×C42C22×C412yes—
250C2×C42C22×C412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×Q812yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258#107—44no—
259#107—44no—
260#106—44no—
261#106—44no—
262#97—24no—
263#97—42no—
264C4×C8—24no—
265C4×C8—42no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—44no—
276#95—44no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—82no—
282C24C22×D411yescharacteristic
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—81no—
288C2×Q8—81no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
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—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4C22⋊Q812yes—
337C22×C4C22⋊Q812yes—
338C22×C4C4⋊Q814yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×Q811yescharacteristic
343C22×C4C22×Q812yes—
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4112yes—
346C22×C4C2×#4111yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349M16—24no—
350M16—24no—
351M16—24no—
352M16—24no—
353M16—24no—
354M16—42no—
355M16—42no—
356M16C4×Q818yescomplemented
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—28no—
368C4⋊C4—28no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—48no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—81no—
385C4⋊C4—81no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4—88no—
389C4⋊C4C2×C4⋊C414yes—
390C4⋊C4C2×C4⋊C414yes—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C42—24no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—44no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—82no—
405C42—84no—
406C42—84no—
407C42C2×C4⋊C412yes—
408C42C2×C4⋊C412yes—
409C42C4×D412yes—
410C42C4×D412yes—
411C42C22⋊Q812yes—
412C42C22⋊Q812yes—
413C42#12312yes—
414C42#12312yes—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23C2×C4×D411yescharacteristic
420C23C2×#12011yescharacteristic
421C23C2×C22⋊Q811yescharacteristic
422C23C2×C4⋊Q811yescharacteristic
423C23#53211yescharacteristic
424C23#53511yescharacteristic
425C23#53611yescharacteristic
426Q8—48no—
427Q8—84no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4#38512yes—
487C2×C4#38512yes—
488C2×C4#38512yes—
489C2×C4#38512yes—
490C2×C4C22×C4⋊C411yescharacteristic
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×C4×Q812yes—
493C2×C4#52012yes—
494C2×C4C2×C22⋊Q811yescharacteristic
495C2×C4C2×#12311yescharacteristic
496C2×C4#53312yes—
497C2×C4#54212yes—
498C2×C4#54612yes—
499C2×C4D4×Q812yes—
500C2×C4D4×Q812yes—
501C2×C4#55012yes—
502C2×C4#55212yes—
503C2×C4#55412yes—
504C2×C4#55712yes—
505C2×C4Q8212yes—
506C8—28no—
507C8—44no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22C2×#38511yescharacteristic
521C22C4×C4⋊Q811yescharacteristic
522C22#232611yescharacteristic
523C22#243111yescharacteristic
524C22#243411yescharacteristic
525C22#256511yescharacteristic
526C22#256711yescharacteristic
527C22#301811yescharacteristic
528C22#301811yescharacteristic
529C22#301911yescharacteristic
530C22#302011yescharacteristic
531C4—22no—
532C4—22no—
533C4—22no—
534C4—22no—
535C4—44no—
536C4—44no—
537C4—44no—
538C4—44no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—161no—
544C4—161no—
545C4Q8×C4⋊C412yes—
546C4#236012yes—
547C4#242312yes—
548C4#245112yes—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2#1397611yescharacteristic
554C2#2086311yescharacteristic
555C2#2086411yescharacteristic
556C2#2562511yescharacteristic
557C2#2562611yescharacteristic
558C2#2582711yescharacteristic
559C2#2582811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
36. #1986C4semidirect
140. #343Q8semidirect
176. C4⋊Q8C4⋊C4semidirect
177. C4⋊Q8C4⋊C4semidirect
179. #125C4⋊C4semidirect
356. M16C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#342132
#213232
C2×#19862
#208012
#139092
#113402
#113392
#108992
#108512
C4×#3432