Finite Group Databasebeta

← #99657

Subgroups — #99657

Order 512.

663 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2072C211yesmaximal, characteristic
2#14467C211yesmaximal, characteristic
3#11373C211yesmaximal, characteristic
4#11357C211yesmaximal, characteristic
5C2×#1272C211yesmaximal, characteristic
6#7025C211yesmaximal, characteristic
7#7025C211yesmaximal, characteristic
8C22×#103—21no—
9C2×#391—21no—
10C2×#391C2211yescharacteristic
11C2×#384C2211yescharacteristic
12#2072—22no—
13#2072C414yescomplemented
14C2×#340—21no—
15C2×#340—21no—
16C2×#340—21no—
17C2×#340—21no—
18C2×#340—21no—
19C2×#336C2211yescharacteristic
20C2×#336C2211yescharacteristic
21C2×#336C2211yescharacteristic
22#1444—21no—
23#1444—21no—
24#1441—21no—
25#1436—21no—
26#1436—21no—
27C4⋊#95C2211yescharacteristic
28C4⋊#95C2211yescharacteristic
29#1272—22no—
30C23×Q8D411yescharacteristic
31C23×C8—21no—
32C23×C8—21no—
33C23×C8—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46C2×#103—24no—
47C2×#103—42no—
48C2×#103—42no—
49#391—21no—
50#391—21no—
51#391—42no—
52#391—42no—
53#391C2×C412yes—
54#391D412yes—
55#384—21no—
56#384—21no—
57#384—21no—
58#384C2×C412yes—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70#340—24no—
71#340—42no—
72#340—42no—
73#340—42no—
74#340—42no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336C2×C412yes—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8C22⋊C412yes—
89C22×Q8D812yes—
90C22×Q8SD1612yes—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—81no—
159C2×C4⋊C4—81no—
160C2×C4⋊C4—81no—
161C2×C4⋊C4—81no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4SD1612yes—
165C2×C4⋊C4Q1612yes—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—42no—
169C2×C42—81no—
170C2×C42—81no—
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175#103—48no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—42no—
181#95—42no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—81no—
191#95—81no—
192#95—81no—
193C24#12011yescharacteristic
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—81no—
204C2×Q8—81no—
205C2×Q8—81no—
206C2×Q8—81no—
207C2×Q8—81no—
208C2×Q8—81no—
209C2×Q8#10214yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C4×D411yescharacteristic
273C22×C4C4×D411yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4#12311yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×SD1611yescharacteristic
279C22×C4C2×SD1611yescharacteristic
280C22×C4C2×Q1611yescharacteristic
281C22×C4#13511yescharacteristic
282C22×C4#13511yescharacteristic
283C22×C4AGL(1,ℤ/8)11yescharacteristic
284C22×C4AGL(1,ℤ/8)11yescharacteristic
285C22×C4#13711yescharacteristic
286C22×C4#13711yescharacteristic
287C22×C4#13711yescharacteristic
288C22×C4#13711yescharacteristic
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—81no—
299C2×C8—81no—
300C2×C8—81no—
301C2×C8—81no—
302C2×C8—81no—
303C2×C8—81no—
304C2×C8—81no—
305C2×C8—81no—
306C2×C8—81no—
307C2×C8—81no—
308C2×C8—81no—
309C2×C8—81no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—81no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4—81no—
326C4⋊C4—81no—
327C4⋊C4—81no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C4⋊C4—81no—
331C4⋊C4—81no—
332C4⋊C4—81no—
333C4⋊C4—81no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—84no—
339C23#38611yescharacteristic
340C23#44711yescharacteristic
341C23#44811yescharacteristic
342C23#44811yescharacteristic
343C23#44911yescharacteristic
344C23#45111yescharacteristic
345C23#45211yescharacteristic
346C23#45311yescharacteristic
347C23#45411yescharacteristic
348C23#45511yescharacteristic
349C23#45511yescharacteristic
350C23#45511yescharacteristic
351C23#45611yescharacteristic
352C23#45811yescharacteristic
353C23#45811yescharacteristic
354Q8—44no—
355Q8—82no—
356Q8—82no—
357Q8—82no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—161no—
440C2×C4—161no—
441C2×C4—161no—
442C2×C4—161no—
443C2×C4C2×#10211yescharacteristic
444C2×C4#41711yescharacteristic
445C2×C4C4×SD1611yescharacteristic
446C2×C4C4×Q1611yescharacteristic
447C2×C4#44011yescharacteristic
448C2×C4#44111yescharacteristic
449C2×C4#46011yescharacteristic
450C2×C4#46211yescharacteristic
451C2×C4#46411yescharacteristic
452C2×C4#46411yescharacteristic
453C2×C4#48011yescharacteristic
454C2×C4#48111yescharacteristic
455C8—82no—
456C8—82no—
457C8—82no—
458C22#160711yescharacteristic
459C22#160711yescharacteristic
460C22#160811yescharacteristic
461C22#160811yescharacteristic
462C22#160811yescharacteristic
463C22#160911yescharacteristic
464C22#161111yescharacteristic
465C22#161211yescharacteristic
466C22#161411yescharacteristic
467C22#161511yescharacteristic
468C22#161711yescharacteristic
469C22#161811yescharacteristic
470C22#162111yescharacteristic
471C22#162311yescharacteristic
472C22#162611yescharacteristic
473C22#162711yescharacteristic
474C22#164111yescharacteristic
475C22#164311yescharacteristic
476C22#164411yescharacteristic
477C22#164511yescharacteristic
478C22#164511yescharacteristic
479C22#164611yescharacteristic
480C22#164911yescharacteristic
481C22#165311yescharacteristic
482C22#165911yescharacteristic
483C22#166111yescharacteristic
484C22#166211yescharacteristic
485C22#166411yescharacteristic
486C22#187411yescharacteristic
487C22#187811yescharacteristic
488C22#188011yescharacteristic
489C22#188211yescharacteristic
490C22#188311yescharacteristic
491C22#188511yescharacteristic
492C22#189011yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—44no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4—162no—
510C4—161no—
511C4—161no—
512C2#778711yescharacteristic
513C2#778911yescharacteristic
514C2#779411yescharacteristic
515C2#779611yescharacteristic
516C2#781211yescharacteristic
517C2#781311yescharacteristic
518C2#781411yescharacteristic
519C2#781611yescharacteristic
520C2#897911yescharacteristic
521C2#898711yescharacteristic
522C2#899911yescharacteristic
523C2#902511yescharacteristic
524C2#904411yescharacteristic
525C2#904611yescharacteristic
526C2#906011yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2072C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20722
#144672
#113732
#113572
C2×#12722
#70252