Finite Group Databasebeta

← #114187

Subgroups — #114187

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3139C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4C2×#1546C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic
6#10855C211yesmaximal, characteristic
7#7882C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3139—22no—
11#3139—22no—
12#3139—22no—
13#3139C414yescomplemented
14C2×#491—22no—
15C2×#487—21no—
16C2×#487—21no—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#419—22no—
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23#1990—22no—
24#1988—22no—
25#1823—22no—
26C2×#344—22no—
27#1550—22no—
28#1549—21no—
29#1548—21no—
30#1546—22no—
31#1546—22no—
32#1542—22no—
33C2×#332—22no—
34#1520—22no—
35#1518—22no—
36#1503—22no—
37#1278—22no—
38C23×Q8—21no—
39C22×M16—21no—
40C22×M16D412yes—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533C2×C412yes—
48#533D412yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8D412yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D412yes—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C42C2311yescharacteristic
73#491—28no—
74#491—28no—
75#487—24no—
76#487—24no—
77#487—24no—
78#487—24no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431D412yes—
87#423—21no—
88#423—21no—
89#423—21no—
90#423—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8D412yes—
94C2×C4⋊C8D412yes—
95#419—28no—
96#419—28no—
97C2×#103—24no—
98C2×#103—24no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C4×M16—21no—
102C4×M16—21no—
103C4×M16C2×C412yes—
104C4×M16D412yes—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109#384—21no—
110#382—21no—
111#380—21no—
112#380D412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115#344—44no—
116#340—28no—
117#333—24no—
118#332—28no—
119#332—24no—
120#328—28no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C23×C4—41no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4#4111yescharacteristic
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—41no—
143C2×M16—41no—
144C2×M16—42no—
145C2×M16—42no—
146C2×M16C22⋊C414yes—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—41no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—21no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8C22⋊C414yes—
173C4⋊Q8C22⋊C412yes—
174C4⋊Q8C22⋊C412yes—
175#125—22no—
176#125—22no—
177#125—41no—
178#125—41no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195#117—22no—
196#117—22no—
197#117—41no—
198#117—41no—
199#117—42no—
200#117—42no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—41no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—41no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246#103—216no—
247#103—216no—
248C22⋊C8—22no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251C22⋊C8—41no—
252#97—22no—
253#97—22no—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—41no—
258C4×C8—41no—
259C4×C8—41no—
260#95—22no—
261#95—41no—
262C24#12111yescharacteristic
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—82no—
278C2×Q8—82no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8—81no—
284C2×Q8—81no—
285C2×Q8—81no—
286C2×Q8—81no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12411yescharacteristic
312C22×C4#12411yescharacteristic
313C22×C4#12711yescharacteristic
314C22×C4#12711yescharacteristic
315C22×C4#13512yes—
316C22×C4#13512yes—
317C22×C4AGL(1,ℤ/8)12yes—
318C22×C4#13712yes—
319M16—24no—
320M16—24no—
321M16—42no—
322M16—42no—
323M16—82no—
324M16—81no—
325M16—81no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—44no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—44no—
334C2×C8—82no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—44no—
361C42—42no—
362C42—42no—
363C42—81no—
364C42—81no—
365C42C2×C22⋊C411yescharacteristic
366C42C2×C22⋊C411yescharacteristic
367C42C4×D411yescharacteristic
368C42C4×D411yescharacteristic
369C42#12011yescharacteristic
370C42#12011yescharacteristic
371C42#12011yescharacteristic
372C42#12011yescharacteristic
373C42#12111yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42#12111yescharacteristic
378C42#12111yescharacteristic
379C42#12311yescharacteristic
380C42#12311yescharacteristic
381C23—41no—
382C23—41no—
383C23#39011yescharacteristic
384C23#39211yescharacteristic
385C23#39411yescharacteristic
386C23#46312yes—
387C23#46412yes—
388Q8—48no—
389Q8—82no—
390Q8—82no—
391Q8—82no—
392Q8—82no—
393Q8—82no—
394Q8—82no—
395Q8—82no—
396Q8—82no—
397Q8—84no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—84no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4#37911yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#39011yescharacteristic
428C2×C4#39211yescharacteristic
429C2×C4#39211yescharacteristic
430C2×C4#39211yescharacteristic
431C2×C4#39411yescharacteristic
432C2×C4#39411yescharacteristic
433C2×C4#39411yescharacteristic
434C2×C4#41612yes—
435C2×C4#41712yes—
436C2×C4#44912yes—
437C2×C4#44912yes—
438C2×C4#45212yes—
439C2×C4#45212yes—
440C2×C4#46312yes—
441C2×C4#46412yes—
442C2×C4#48812yes—
443C2×C4#48912yes—
444C2×C4#49511yescharacteristic
445C2×C4#49511yescharacteristic
446C2×C4#49611yescharacteristic
447C2×C4#49711yescharacteristic
448C8—44no—
449C8—82no—
450C8—82no—
451C8—82no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—42no—
459C22—42no—
460C22#143811yescharacteristic
461C22#198312yes—
462C22#201211yescharacteristic
463C22#201511yescharacteristic
464C22#204212yes—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—84no—
472C4—82no—
473C4—82no—
474C4—162no—
475C4#178812yes—
476C4#195811yescharacteristic
477C4#195911yescharacteristic
478C4#200912yes—
479C4#201211yescharacteristic
480C4#201511yescharacteristic
481C4#204212yes—
482C4#204912yes—
483C2—21no—
484C2—22no—
485C2—21no—
486C2—22no—
487C2#1100911yescharacteristic
488C2#1103111yescharacteristic
489C2#1103511yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3139C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
#218332
#177082
C2×#15462
#175752
#108552
#78822