Finite Group Databasebeta

← #374391

Subgroups — #374391

Order 512.

2108 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3C2×#1890C214yesmaximal
4C2×C4×#99C211yesmaximal, characteristic
5#13904C212yesmaximal, complemented
6#11070C214yesmaximal, complemented
7#10854C212yesmaximal, complemented
8C22×#123C2212yes—
9C2×#518C2212yes—
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×#99C2212yes—
13#2486—24no—
14#2486C418yescomplemented
15C2×#398—21no—
16C2×#397—21no—
17#2349—24no—
18D4×C4⋊C4—24no—
19#2316—21no—
20#2315—21no—
21#2315C2212yes—
22C2×#389—21no—
23#2285—21no—
24C2×#381C2218yes—
25C2×#381C2212yes—
26C2×#380—21no—
27C2×#380—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×C4×C22⋊C4C2212yes—
31#1890—216no—
32C4×#99—24no—
33C2×#342C2212yes—
34C2×#342C2218yes—
35#1445—24no—
36C4⋊#95—24no—
37C23×D4C2311yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4C2312yes—
40C24×C4C2311yescharacteristic
41C2×#123—24no—
42C2×#123—28no—
43C2×#123—28no—
44C2×#123C2×C418yes—
45#518—28no—
46#518—44no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—24no—
50C2×C4×D4—28no—
51C2×C4×D4C2×C414yes—
52C2×C4×D4D414yes—
53C2×#117—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2×C414yescomplemented
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D414yes—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4C2314yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2311yescharacteristic
73C22×C42—22no—
74C22×C42D411yescharacteristic
75C2×#99—24no—
76C2×#99—24no—
77C2×#99D418yes—
78#398—28no—
79#397—28no—
80#389—28no—
81#389—28no—
82#385—28no—
83#385—28no—
84#384—22no—
85#382—22no—
86#381—24no—
87#381—216no—
88#381—48no—
89#381C2×C418yescomplemented
90#380—22no—
91#380—22no—
92#380—28no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380C2×C414yescomplemented
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—216no—
110C2×#95—22no—
111C2×#95D414yes—
112#342—216no—
113#342—24no—
114C25C2×D411yescharacteristic
115C25C2411yescharacteristic
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4C22×C412yes—
121C22×D4C2×D412yes—
122C22×D4C2×D414yes—
123C22×D4#4114yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4C22⋊C414yes—
141C23×C4C22×C414yes—
142C23×C4C22×C412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D418yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4#4114yes—
153#123—232no—
154#123—416no—
155C4×D4—28no—
156C4×D4—28no—
157C4×D4—28no—
158C4×D4—28no—
159C4×D4—432no—
160#117—44no—
161#117—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—84no—
187C2×C4⋊C4C22⋊C414yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4C22⋊C414yes—
211C2×C22⋊C4C22×C414yes—
212C2×C22⋊C4C22×C412yes—
213C2×C22⋊C4C2×D418yes—
214C2×C22⋊C4C2×D412yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—48no—
220C2×C42—44no—
221C2×C42C22⋊C414yes—
222#99—216no—
223#95—24no—
224#95—24no—
225#95—28no—
226#95—24no—
227#95—28no—
228#95—24no—
229#95—24no—
230#95—432no—
231#95—44no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24C2×C22⋊C412yes—
241C24#12412yes—
242C24#12712yes—
243C24C23×C411yescharacteristic
244C24C22×D411yescharacteristic
245C24C22×D412yes—
246C24C22×D411yescharacteristic
247C24C22×D412yes—
248C24C2×#4112yes—
249C24C2×#4112yes—
250C24H(5,𝔽2)12yes—
251C242−1+412yes—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—28no—
257C2×D4—28no—
258C2×D4—28no—
259C2×D4—28no—
260C2×D4C4×D418yes—
261C2×D4#12118yes—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—28no—
281C22×C4—216no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—48no—
292C22×C4—44no—
293C22×C4—48no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—44no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—84no—
316C22×C4—82no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C4×D414yes—
325C22×C4C4×D414yes—
326C22×C4#12011yescharacteristic
327C22×C4#12012yes—
328C22×C4#12011yescharacteristic
329C22×C4#12114yes—
330C22×C4#12114yes—
331C22×C4#12312yes—
332C22×C4#12312yes—
333C22×C4#12412yes—
334C22×C4#12712yes—
335C4⋊C4—28no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—44no—
341C4⋊C4—416no—
342C4⋊C4—48no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—28no—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—28no—
352C22⋊C4—48no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—48no—
358C22⋊C4—44no—
359C22⋊C4—416no—
360C42—24no—
361C42—24no—
362C42—48no—
363C42—88no—
364C23—24no—
365C23—22no—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—44no—
379C23—41no—
380C23—44no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23—42no—
385C23—42no—
386C23—44no—
387C23—44no—
388C23—42no—
389C23—42no—
390C23—81no—
391C23—82no—
392C23—82no—
393C23—81no—
394C23#39014yes—
395C23C22×C22⋊C411yescharacteristic
396C23C2×C4×D412yes—
397C23C2×C4×D412yes—
398C23#52012yes—
399C23C2×#12011yescharacteristic
400C23C2×#12112yes—
401C23C2×#12112yes—
402C23C2×#12311yescharacteristic
403C23C2×#12411yescharacteristic
404C23C2×#12711yescharacteristic
405C23#53511yescharacteristic
406C23#53611yescharacteristic
407C23#53712yes—
408C23D4214yes—
409C23#54714yes—
410C23#54814yes—
411C23#55314yes—
412D4—216no—
413D4—216no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—28no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—48no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—416no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—416no—
442C2×C4—48no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—84no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—88no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—88no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—816no—
463C2×C4—168no—
464C2×C4#38612yes—
465C2×C4#38614yes—
466C2×C4#38612yes—
467C2×C4#39014yes—
468C2×C4#45814yes—
469C22—22no—
470C22—28no—
471C22—22no—
472C22—28no—
473C22—22no—
474C22—22no—
475C22—21no—
476C22—22no—
477C22—21no—
478C22—22no—
479C22—21no—
480C22—22no—
481C22—21no—
482C22—22no—
483C22—24no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—44no—
493C22—42no—
494C22—42no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22C2×#38611yescharacteristic
499C22C2×#39011yescharacteristic
500C22#231111yescharacteristic
501C22#236114yes—
502C22#241614yes—
503C22#243311yescharacteristic
504C22#243611yescharacteristic
505C22#250111yescharacteristic
506C22#250411yescharacteristic
507C22C2×#45812yes—
508C22#302812yes—
509C4—24no—
510C4—22no—
511C4—24no—
512C4—42no—
513C4—42no—
514C4—84no—
515C4—88no—
516C4—84no—
517C4—88no—
518C4#189014yescomplemented
519C2—21no—
520C2—24no—
521C2—21no—
522C2—21no—
523C2—21no—
524C2—24no—
525C2—44no—
526C2#1413711yescharacteristic
527C2C2×#189011yescharacteristic
528C2#2096211yescharacteristic
529C2#2577212yes—
530C2#2594812yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13904C2semidirect
6. #11070C2semidirect
7. #10854C2semidirect
14. #2486C4semidirect
63. C22×C4⋊C4C2×C4semidirect
89. #381C2×C4semidirect
100. #380C2×C4semidirect
518. C4#1890semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
#341812
C2×#18902
C2×C4×#992
#139042
#110702
#108542