Finite Group Databasebeta

← #116819

Subgroups — #116819

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21322C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic
3C2×#1550C211yesmaximal, characteristic
4C2×#1481C211yesmaximal, characteristic
5#8043C211yesmaximal, characteristic
6C2×#1296C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C22×#108—21no—
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C2×#384C2211yescharacteristic
16#1992—22no—
17#1990—21no—
18#1990—21no—
19#1950—22no—
20#1950—22no—
21C2×#341—21no—
22#1736—21no—
23#1736—21no—
24#1550—22no—
25#1550—22no—
26C2×#333—21no—
27C2×#332—21no—
28#1481—22no—
29#1481C414yescomplemented
30C2×Q8⋊C8—22no—
31#1296—22no—
32#1296—22no—
33#1283—22no—
34C22×M16—21no—
35C22×M16D411yescharacteristic
36C22×M16D411yescharacteristic
37C23×C8—21no—
38C23×C8—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533C2×C412yes—
46#533Q812yes—
47C2×C4⋊Q8D411yescharacteristic
48C2×C4⋊Q8D411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×#117—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C42C2311yescharacteristic
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63C2×#108—24no—
64C2×#108—42no—
65C2×#108—42no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423D412yes—
70C2×C4⋊C8—21no—
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⋊C8C2×C412yes—
82C2×C4⋊C8D412yes—
83C2×C4⋊C8D412yes—
84C2×C4⋊C8D412yes—
85C2×C22⋊C8—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C2×#97—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384D412yes—
97#341—24no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#333—24no—
103#332—24no—
104Q8⋊C8—28no—
105C22×Q8—21no—
106C22×Q8—21no—
107C23×C4—41no—
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—21no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16—21no—
121C2×M16—22no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16SD1612yes—
128C2×M16Q1612yes—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
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—81no—
144C22×C8—81no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8C4⋊C412yes—
152C4⋊Q8C4⋊C412yes—
153#125—22no—
154#125—22no—
155#125—41no—
156#125—41no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—41no—
162C22⋊Q8—41no—
163C22⋊Q8—41no—
164C22⋊Q8—41no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—41no—
173#117—22no—
174#117—22no—
175#117—41no—
176#117—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C22⋊C4—41no—
197C2×C42—41no—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q811yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210#108—48no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8C22⋊C414yes—
225C22⋊C8—22no—
226C22⋊C8—41no—
227#97—41no—
228#97—41no—
229C4×C8—22no—
230C4×C8—41no—
231C4×C8—41no—
232C4×C8—41no—
233C24#12411yescharacteristic
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—81no—
243C2×Q8—81no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4#11711yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4C22⋊Q811yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4#12311yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12411yescharacteristic
273C22×C4#12411yescharacteristic
274C22×C4#12611yescharacteristic
275C22×C4C2×SD1611yescharacteristic
276C22×C4C2×Q1611yescharacteristic
277C22×C4#13511yescharacteristic
278C22×C4#13511yescharacteristic
279C22×C4AGL(1,ℤ/8)11yescharacteristic
280C22×C4#13711yescharacteristic
281C22×C4#13711yescharacteristic
282C22×C4#13711yescharacteristic
283M16—22no—
284M16—22no—
285M16—22no—
286M16—22no—
287M16—41no—
288M16—41no—
289M16—41no—
290M16—41no—
291M16—42no—
292M16—42no—
293M16—81no—
294M16—81no—
295C2×C8—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—81no—
312C2×C8—81no—
313C2×C8—82no—
314C2×C8—81no—
315C2×C8—81no—
316C2×C8—81no—
317C2×C8—81no—
318C2×C8—82no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C42—42no—
343C42—42no—
344C42C2×C22⋊C411yescharacteristic
345C42C2×C4⋊C411yescharacteristic
346C42C4×D411yescharacteristic
347C42C4×D411yescharacteristic
348C42#12011yescharacteristic
349C42#12111yescharacteristic
350C42#12111yescharacteristic
351C42#12111yescharacteristic
352C42#12111yescharacteristic
353C42C22⋊Q811yescharacteristic
354C42C22⋊Q811yescharacteristic
355C42C22⋊Q811yescharacteristic
356C42#12311yescharacteristic
357C42#12311yescharacteristic
358C42#12611yescharacteristic
359C42#12711yescharacteristic
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23#38811yescharacteristic
365C23#39311yescharacteristic
366C23#39411yescharacteristic
367C23#48711yescharacteristic
368C23#48811yescharacteristic
369C23#48911yescharacteristic
370C23#49111yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—82no—
374Q8—82no—
375Q8—81no—
376Q8—81no—
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—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—162no—
405C2×C4—162no—
406C2×C4#38011yescharacteristic
407C2×C4#38511yescharacteristic
408C2×C4#38511yescharacteristic
409C2×C4#38611yescharacteristic
410C2×C4#38811yescharacteristic
411C2×C4#39011yescharacteristic
412C2×C4#39211yescharacteristic
413C2×C4#39411yescharacteristic
414C2×C4#39611yescharacteristic
415C2×C4#39611yescharacteristic
416C2×C4#39711yescharacteristic
417C2×C4#39911yescharacteristic
418C2×C4C4×SD1611yescharacteristic
419C2×C4C4×Q1611yescharacteristic
420C2×C4#44011yescharacteristic
421C2×C4#44111yescharacteristic
422C2×C4#44811yescharacteristic
423C2×C4#44911yescharacteristic
424C2×C4#45011yescharacteristic
425C2×C4#45111yescharacteristic
426C2×C4#45111yescharacteristic
427C2×C4#45211yescharacteristic
428C2×C4#45211yescharacteristic
429C2×C4#45211yescharacteristic
430C2×C4#46111yescharacteristic
431C2×C4#46211yescharacteristic
432C2×C4#46311yescharacteristic
433C2×C4#46411yescharacteristic
434C2×C4#46511yescharacteristic
435C2×C4#46711yescharacteristic
436C2×C4#46811yescharacteristic
437C2×C4#47011yescharacteristic
438C2×C4#47111yescharacteristic
439C2×C4#47111yescharacteristic
440C2×C4#47211yescharacteristic
441C2×C4#47311yescharacteristic
442C2×C4#47511yescharacteristic
443C2×C4#47711yescharacteristic
444C2×C4#47911yescharacteristic
445C2×C4#47911yescharacteristic
446C8—44no—
447C8—42no—
448C8—42no—
449C8—82no—
450C8—82no—
451C8—82no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22#143911yescharacteristic
465C22#193311yescharacteristic
466C22#193511yescharacteristic
467C22#202411yescharacteristic
468C22#202611yescharacteristic
469C22#204211yescharacteristic
470C22#204311yescharacteristic
471C22#204811yescharacteristic
472C22#204911yescharacteristic
473C22#205111yescharacteristic
474C22#205211yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—82no—
482C4—82no—
483C4—162no—
484C4#181511yescharacteristic
485C4#181611yescharacteristic
486C4#186311yescharacteristic
487C4#186511yescharacteristic
488C4#187711yescharacteristic
489C4#187811yescharacteristic
490C4#200011yescharacteristic
491C4#200111yescharacteristic
492C4#205111yescharacteristic
493C4#205211yescharacteristic
494C4#206611yescharacteristic
495C4#206811yescharacteristic
496C2—21no—
497C2—21no—
498C2—21no—
499C2—21no—
500C2#1055311yescharacteristic
501C2#1055411yescharacteristic
502C2#1055511yescharacteristic
503C2#1055611yescharacteristic
504C2#1106711yescharacteristic
505C2#1110511yescharacteristic
506C2#1112011yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
#213222
C2×#19502
C2×#15502
C2×#14812
#80432
C2×#12962
C2×#12832