Finite Group Databasebeta

← #116697

Subgroups — #116697

Order 512.

732 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1803C211yesmaximal, characteristic
2C2×#1801C211yesmaximal, characteristic
3C2×#1570C211yesmaximal, characteristic
4C2×#1479C211yesmaximal, characteristic
5C2×#1301C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7#8043C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103—21no—
15C22×#101—21no—
16C2×#384C2211yescharacteristic
17#1803—22no—
18#1803—22no—
19#1801—22no—
20#1801—22no—
21C2×#343—21no—
22C2×#340—21no—
23#1570—22no—
24#1570—22no—
25C2×#334—22no—
26#1479—22no—
27#1479C414yescomplemented
28C2×Q8⋊C8—22no—
29#1301—22no—
30#1296—22no—
31#1296—22no—
32C23×Q8D411yescharacteristic
33C22×M16—21no—
34C22×M16—21no—
35C23×C8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8Q812yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431C2×C412yes—
61#431C2×C412yes—
62#431D412yes—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423D412yes—
70#423D412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—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⋊C8—21no—
81C2×C4⋊C8D412yes—
82C2×C4⋊C8D412yes—
83C2×#103—24no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#101—24no—
87C2×#101—42no—
88C2×#101—42no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384D412yes—
97#343—24no—
98#340—24no—
99#334—28no—
100Q8⋊C8—28no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8C4⋊C412yes—
113C22×Q8SD1612yes—
114C22×Q8SD1612yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—41no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C4⋊C412yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×Q811yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8C22⋊C414yes—
218#103—48no—
219#101—48no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—41no—
228C24#12311yescharacteristic
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8#10614yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4C2×C4⋊C411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4C22⋊Q811yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4#12411yescharacteristic
293C22×C4#12411yescharacteristic
294C22×C4#12611yescharacteristic
295C22×C4#12611yescharacteristic
296C22×C4C2×SD1611yescharacteristic
297C22×C4C2×SD1611yescharacteristic
298C22×C4#13511yescharacteristic
299C22×C4#13511yescharacteristic
300C22×C4AGL(1,ℤ/8)11yescharacteristic
301C22×C4AGL(1,ℤ/8)11yescharacteristic
302C22×C4#13711yescharacteristic
303C22×C4#13711yescharacteristic
304M16—42no—
305M16—42no—
306M16—42no—
307M16—42no—
308M16—81no—
309M16—81no—
310M16—81no—
311M16—81no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—81no—
322C2×C8—81no—
323C2×C8—81no—
324C2×C8—81no—
325C2×C8—81no—
326C2×C8—81no—
327C2×C8—81no—
328C2×C8—81no—
329C2×C8—81no—
330C2×C8—81no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—82no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42C2×C22⋊C411yescharacteristic
352C42#11711yescharacteristic
353C42C4×D411yescharacteristic
354C42C4×D411yescharacteristic
355C42#12011yescharacteristic
356C42#12111yescharacteristic
357C42#12111yescharacteristic
358C42#12111yescharacteristic
359C42#12111yescharacteristic
360C42C22⋊Q811yescharacteristic
361C42C22⋊Q811yescharacteristic
362C42C22⋊Q811yescharacteristic
363C42C22⋊Q811yescharacteristic
364C42#12311yescharacteristic
365C42#12411yescharacteristic
366C42#12711yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23#38511yescharacteristic
372C23#39411yescharacteristic
373C23#39711yescharacteristic
374C23#48111yescharacteristic
375C23#48211yescharacteristic
376C23#48311yescharacteristic
377C23#48511yescharacteristic
378Q8—24no—
379Q8—24no—
380Q8—42no—
381Q8—42no—
382Q8—42no—
383Q8—42no—
384Q8—42no—
385Q8—42no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—162no—
416C2×C4—162no—
417C2×C4#38011yescharacteristic
418C2×C4#38511yescharacteristic
419C2×C4#38611yescharacteristic
420C2×C4#38811yescharacteristic
421C2×C4#38811yescharacteristic
422C2×C4#39011yescharacteristic
423C2×C4#39211yescharacteristic
424C2×C4#39311yescharacteristic
425C2×C4#39411yescharacteristic
426C2×C4#39611yescharacteristic
427C2×C4#39611yescharacteristic
428C2×C4#39911yescharacteristic
429C2×C4C2×#10611yescharacteristic
430C2×C4#42711yescharacteristic
431C2×C4#42811yescharacteristic
432C2×C4#42811yescharacteristic
433C2×C4#44811yescharacteristic
434C2×C4#44811yescharacteristic
435C2×C4#44911yescharacteristic
436C2×C4#44911yescharacteristic
437C2×C4#45011yescharacteristic
438C2×C4#45011yescharacteristic
439C2×C4#45211yescharacteristic
440C2×C4#45211yescharacteristic
441C2×C4#46511yescharacteristic
442C2×C4#46511yescharacteristic
443C2×C4#46511yescharacteristic
444C2×C4#46511yescharacteristic
445C2×C4#46811yescharacteristic
446C2×C4#46811yescharacteristic
447C2×C4#46911yescharacteristic
448C2×C4#47011yescharacteristic
449C2×C4#47112yes—
450C2×C4#47312yes—
451C2×C4#48111yescharacteristic
452C2×C4#48211yescharacteristic
453C2×C4#48311yescharacteristic
454C2×C4#48511yescharacteristic
455C8—44no—
456C8—82no—
457C8—82no—
458C8—82no—
459C8—82no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22#143911yescharacteristic
473C22#185311yescharacteristic
474C22#185411yescharacteristic
475C22#203411yescharacteristic
476C22#203511yescharacteristic
477C22#204611yescharacteristic
478C22#204711yescharacteristic
479C22#204811yescharacteristic
480C22#204911yescharacteristic
481C22#207711yescharacteristic
482C22#207711yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—44no—
490C4—82no—
491C4—162no—
492C4#180711yescharacteristic
493C4#180911yescharacteristic
494C4#185311yescharacteristic
495C4#185411yescharacteristic
496C4#190812yes—
497C4#200011yescharacteristic
498C4#200011yescharacteristic
499C4#203411yescharacteristic
500C4#203511yescharacteristic
501C4#206412yes—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—21no—
506C2#1044311yescharacteristic
507C2#1044711yescharacteristic
508C2#1045011yescharacteristic
509C2#1045411yescharacteristic
510C2#1106211yescharacteristic
511C2#1109511yescharacteristic
512C2#1112011yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
27. #1479C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18032
C2×#18012
C2×#15702
C2×#14792
C2×#13012
#80432