Finite Group Databasebeta

← #116837

Subgroups — #116837

Order 512.

698 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21860C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4C2×#1481C211yesmaximal, characteristic
5C2×#1301C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7#7908C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×#107—21no—
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1990—21no—
17#1989—22no—
18#1988—21no—
19#1987—21no—
20#1956—21no—
21#1947—22no—
22#1947—22no—
23#1872—22no—
24#1803—22no—
25#1736—21no—
26#1736—21no—
27C2×#340—21no—
28#1481—22no—
29#1481C414yescomplemented
30C2×Q8⋊C8—22no—
31#1301—22no—
32#1296—22no—
33#1280—22no—
34C22×M16—21no—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533Q812yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67C2×#107—24no—
68C2×#107—42no—
69C2×#107—42no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2×C412yes—
81C2×C4⋊C8D412yes—
82C2×C4⋊C8D412yes—
83C2×C4⋊C8D412yes—
84C2×#103—24no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8D412yes—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384D412yes—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C2×#95—21no—
98#340—24no—
99#340—24no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104Q8⋊C8—28no—
105C22×Q8—21no—
106C22×Q8—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—41no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8C4⋊C412yes—
136C4⋊Q8C4⋊C412yes—
137#125—22no—
138#125—22no—
139#125—41no—
140#125—41no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157#117—22no—
158#117—22no—
159#117—41no—
160#117—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4SD1612yes—
190C2×C4⋊C4Q1612yes—
191C2×C22⋊C4—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42—82no—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×Q811yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208#107—48no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8—41no—
220C4⋊C8C22⋊C414yes—
221#103—84no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—41no—
226C4×C8—22no—
227C4×C8—41no—
228C4×C8—41no—
229C4×C8—41no—
230#95—41no—
231#95—41no—
232C24#12111yescharacteristic
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—42no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4#11711yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4#12311yescharacteristic
276C22×C4#12411yescharacteristic
277C22×C4#12611yescharacteristic
278C22×C4#12611yescharacteristic
279C22×C4#12711yescharacteristic
280C22×C4C2×SD1611yescharacteristic
281C22×C4C2×Q1611yescharacteristic
282C22×C4#13511yescharacteristic
283C22×C4#13511yescharacteristic
284C22×C4AGL(1,ℤ/8)11yescharacteristic
285C22×C4#13711yescharacteristic
286C22×C4#13711yescharacteristic
287C22×C4#13711yescharacteristic
288M16—81no—
289M16—81no—
290M16—81no—
291M16—81no—
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—81no—
307C2×C8—81no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—82no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C42—42no—
339C42—42no—
340C42—82no—
341C42—82no—
342C42C2×C22⋊C411yescharacteristic
343C42C2×C4⋊C411yescharacteristic
344C42C4×D411yescharacteristic
345C42C4×D411yescharacteristic
346C42#12111yescharacteristic
347C42#12111yescharacteristic
348C42#12111yescharacteristic
349C42C22⋊Q811yescharacteristic
350C42C22⋊Q811yescharacteristic
351C42C22⋊Q811yescharacteristic
352C42#12311yescharacteristic
353C42#12311yescharacteristic
354C42#12311yescharacteristic
355C42#12311yescharacteristic
356C42#12411yescharacteristic
357C42#12411yescharacteristic
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23#38811yescharacteristic
363C23#39211yescharacteristic
364C23#39611yescharacteristic
365C23#46511yescharacteristic
366C23#46711yescharacteristic
367C23#46811yescharacteristic
368C23#47011yescharacteristic
369Q8—44no—
370Q8—44no—
371Q8—82no—
372Q8—82no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—162no—
407C2×C4—162no—
408C2×C4#38011yescharacteristic
409C2×C4#38511yescharacteristic
410C2×C4#38511yescharacteristic
411C2×C4#38611yescharacteristic
412C2×C4#38811yescharacteristic
413C2×C4#39011yescharacteristic
414C2×C4#39311yescharacteristic
415C2×C4#39411yescharacteristic
416C2×C4#39411yescharacteristic
417C2×C4#39611yescharacteristic
418C2×C4#39711yescharacteristic
419C2×C4#39911yescharacteristic
420C2×C4C4×SD1611yescharacteristic
421C2×C4C4×Q1611yescharacteristic
422C2×C4#44011yescharacteristic
423C2×C4#44111yescharacteristic
424C2×C4#44811yescharacteristic
425C2×C4#44911yescharacteristic
426C2×C4#45011yescharacteristic
427C2×C4#45111yescharacteristic
428C2×C4#45111yescharacteristic
429C2×C4#45211yescharacteristic
430C2×C4#45211yescharacteristic
431C2×C4#45211yescharacteristic
432C2×C4#46111yescharacteristic
433C2×C4#46211yescharacteristic
434C2×C4#46311yescharacteristic
435C2×C4#46411yescharacteristic
436C2×C4#47212yes—
437C2×C4#47312yes—
438C2×C4#47511yescharacteristic
439C2×C4#47711yescharacteristic
440C2×C4#47911yescharacteristic
441C2×C4#47911yescharacteristic
442C2×C4#48711yescharacteristic
443C2×C4#48811yescharacteristic
444C2×C4#48911yescharacteristic
445C2×C4#49111yescharacteristic
446C8—44no—
447C8—82no—
448C8—82no—
449C8—82no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22#143911yescharacteristic
463C22#193311yescharacteristic
464C22#193511yescharacteristic
465C22#200011yescharacteristic
466C22#200111yescharacteristic
467C22#200811yescharacteristic
468C22#200911yescharacteristic
469C22#201811yescharacteristic
470C22#202011yescharacteristic
471C22#206611yescharacteristic
472C22#206811yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—82no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—162no—
484C4#181612yes—
485C4#186311yescharacteristic
486C4#186511yescharacteristic
487C4#187711yescharacteristic
488C4#187811yescharacteristic
489C4#202411yescharacteristic
490C4#202611yescharacteristic
491C4#204211yescharacteristic
492C4#204311yescharacteristic
493C4#207812yes—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—21no—
498C2#1017811yescharacteristic
499C2#1018311yescharacteristic
500C2#1018711yescharacteristic
501C2#1019011yescharacteristic
502C2#1106711yescharacteristic
503C2#1110711yescharacteristic
504C2#1111711yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #1481C4semidirect

Maximal subgroups

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

subgroupindex
#218602
#213122
C2×#19472
C2×#14812
C2×#13012
#80432
#79082