Finite Group Databasebeta

← #114134

Subgroups — #114134

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic
6#10837C211yesmaximal, characteristic
7#7918C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3165—22no—
11#3165—22no—
12#3165—22no—
13#3165—22no—
14C2×#501—22no—
15C2×#500C412yescomplemented
16C2×#498C412yescomplemented
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#428—22no—
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#1990—22no—
24#1987—22no—
25#1981—22no—
26#1956—22no—
27#1824—22no—
28#1787—22no—
29C2×H(3,ℤ/4)—22no—
30#1520—22no—
31#1520—22no—
32#1518—22no—
33#1518—22no—
34#1503—22no—
35#1503—22no—
36#1281—22no—
37C23×Q8—21no—
38C22×M16D412yes—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533D412yes—
46#533D412yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
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⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8D412yes—
60C2×C4⋊Q8D412yes—
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42C2311yescharacteristic
72#501—28no—
73#501—28no—
74#500—24no—
75#500—24no—
76#498—24no—
77#498—24no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#428—28no—
87#428—28no—
88C2×#107—24no—
89C2×#106—24no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×#103—28no—
93C2×C22⋊C8D412yes—
94C2×C22⋊C8D412yes—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16—21no—
99C2×#97D412yes—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2×C412yes—
103#384—21no—
104#384—21no—
105#382—21no—
106#382—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115#340—28no—
116H(3,ℤ/4)—44no—
117#328—28no—
118#328—28no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—84no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16SD1614yes—
137C2×M16Q1614yes—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C22⋊C414yes—
141C4⋊Q8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—22no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—22no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8C22⋊C414yes—
163C4⋊Q8C22⋊C412yes—
164C4⋊Q8C22⋊C412yes—
165#125—22no—
166#125—22no—
167#125—41no—
168#125—41no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—41no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185#117—22no—
186#117—22no—
187#117—41no—
188#117—41no—
189#117—42no—
190#117—42no—
191#117—42no—
192#117—42no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—41no—
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42#4111yescharacteristic
235#107—216no—
236#106—216no—
237C4⋊C8—41no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241#103—88no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246#97—22no—
247#97—22no—
248C4×C8—22no—
249C4×C8—22no—
250C4×C8—41no—
251C4×C8—41no—
252C4×C8—41no—
253C4×C8—41no—
254#95—22no—
255#95—22no—
256#95—41no—
257#95—41no—
258C24#12711yescharacteristic
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—48no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—84no—
273C2×Q8—84no—
274C2×Q8—84no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—84no—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12411yescharacteristic
303C22×C4C2×SD1612yes—
304C22×C4C2×Q1612yes—
305C22×C4#13712yes—
306C22×C4#13712yes—
307M16—24no—
308M16—24no—
309M16—24no—
310M16—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8#10318yescomplemented
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—81no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4—81no—
350C42—42no—
351C42—42no—
352C42—44no—
353C42—44no—
354C42C2×C22⋊C411yescharacteristic
355C42C2×C22⋊C411yescharacteristic
356C42C4×D411yescharacteristic
357C42C4×D411yescharacteristic
358C42#12011yescharacteristic
359C42#12011yescharacteristic
360C42#12111yescharacteristic
361C42#12111yescharacteristic
362C42#12111yescharacteristic
363C42#12111yescharacteristic
364C42#12311yescharacteristic
365C42#12311yescharacteristic
366C42#12311yescharacteristic
367C42#12311yescharacteristic
368C42#12411yescharacteristic
369C42#12711yescharacteristic
370C23—41no—
371C23—41no—
372C23#39011yescharacteristic
373C23#39211yescharacteristic
374C23#39211yescharacteristic
375C23#49211yescharacteristic
376C23#49411yescharacteristic
377C23#49711yescharacteristic
378C23#49711yescharacteristic
379Q8—48no—
380Q8—82no—
381Q8—82no—
382Q8—84no—
383Q8—84no—
384Q8—84no—
385Q8—82no—
386Q8—82no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4#37911yescharacteristic
414C2×C4#38611yescharacteristic
415C2×C4#38611yescharacteristic
416C2×C4#38611yescharacteristic
417C2×C4#38611yescharacteristic
418C2×C4#39011yescharacteristic
419C2×C4#39211yescharacteristic
420C2×C4#39211yescharacteristic
421C2×C4#39411yescharacteristic
422C2×C4#39411yescharacteristic
423C2×C4#39411yescharacteristic
424C2×C4#39411yescharacteristic
425C2×C4C2×#10312yes—
426C2×C4#41912yes—
427C2×C4#44812yes—
428C2×C4#44812yes—
429C2×C4#45112yes—
430C2×C4#45112yes—
431C2×C4#46112yes—
432C2×C4#46112yes—
433C2×C4#46212yes—
434C2×C4#46212yes—
435C2×C4#48712yes—
436C2×C4#49112yes—
437C8—28no—
438C8—82no—
439C8—82no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22#143811yescharacteristic
451C22#195611yescharacteristic
452C22#196011yescharacteristic
453C22#201311yescharacteristic
454C22#201311yescharacteristic
455C22#201611yescharacteristic
456C22#201611yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—84no—
464C4—82no—
465C4—82no—
466C4—162no—
467C4—162no—
468C4#178712yes—
469C4#190412yes—
470C4#190512yes—
471C4#198112yes—
472C4#204312yes—
473C4#204312yes—
474C2—21no—
475C2—21no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2#1101011yescharacteristic
481C2#1101211yescharacteristic
482C2#1101311yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
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
15. C2×#500C4semidirect
16. C2×#498C4semidirect
317. C2×C8#103semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#218342
#218042
#175752
#108372
#79182