Finite Group Databasebeta

← #239792

Subgroups — #239792

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic
7#10422C214yesmaximal, complemented
8#10414C214yesmaximal, complemented
9#7893C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103—21no—
16#2855—22no—
17#2627—22no—
18#2582—21no—
19#2582—21no—
20#2581—22no—
21#2568C2212yes—
22#2567C2212yes—
23#2565—22no—
24#2317—21no—
25#2310—21no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#1990C2212yes—
29#1956C2212yes—
30#1947C2214yes—
31#1872—22no—
32#1852—22no—
33#1845—21no—
34#1834—21no—
35#1803—22no—
36#1803—22no—
37#1803C2214yes—
38C2×#340—21no—
39#1542—24no—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1479—24no—
43C2×Q8⋊C8C2214yes—
44#1279C2214yes—
45C23×Q8D411yescharacteristic
46C23×C8—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8D412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8D412yes—
64C2×C4×Q8D412yes—
65C2×C4×Q8C2312yes—
66C2×#117—21no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#431—22no—
76#431—22no—
77#431—22no—
78#431D414yes—
79C2×#107—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—44no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2312yes—
93#400—24no—
94#398—24no—
95#398—24no—
96#395—24no—
97#391—22no—
98#391—22no—
99#391—22no—
100#391—22no—
101#391—22no—
102#391—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#387—24no—
108#387—24no—
109#385—24no—
110#384—22no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384D411yescharacteristic
115#384D411yescharacteristic
116#384C2312yes—
117#382—22no—
118#382—21no—
119#381—24no—
120#380—21no—
121#380D412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2312yes—
124#340—24no—
125#340—24no—
126#340—24no—
127#333—24no—
128#332—24no—
129Q8⋊C8—28no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8Q1614yes—
138C22×Q8C2×D412yes—
139C22×Q8C2×D412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—42no—
150C22×C8—42no—
151C22×C8—42no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160#125—42no—
161#125—42no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180#117—42no—
181#117—42no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42C2411yescharacteristic
232#107—84no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8C2×D414yes—
239#103—48no—
240#103—84no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243C4×C8—24no—
244C4×C8—22no—
245C4×C8—22no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254C24#12011yescharacteristic
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—28no—
264C2×Q8—28no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8C2×Q1614yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4#12412yes—
305C22×C4C2×Q1612yes—
306C22×C4#13512yes—
307C22×C4AGL(1,ℤ/8)12yes—
308C22×C4#13712yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
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—84no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4—82no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—44no—
360C42—44no—
361C42#12112yes—
362C42#12112yes—
363C42#12112yes—
364C42#12312yes—
365C42#12312yes—
366C42#12412yes—
367C42C22×D411yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23#39412yes—
376C23#44912yes—
377C23#45112yes—
378C23C2×#12011yescharacteristic
379Q8—24no—
380Q8—24no—
381Q8—44no—
382Q8—42no—
383Q8—42no—
384Q8—48no—
385Q8—82no—
386Q8—82no—
387Q8—82no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—48no—
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—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—84no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—164no—
422C2×C4#39412yes—
423C2×C4#39412yes—
424C2×C4#39412yes—
425C2×C4#44912yes—
426C2×C4#45112yes—
427C2×C4#46714yes—
428C2×C4#46814yes—
429C2×C4#48414yes—
430C2×C4#48514yes—
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12111yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4C2×#12311yescharacteristic
436C2×C4C2×#12411yescharacteristic
437C2×C4C22×Q1611yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4C2×#13711yescharacteristic
441C2×C4#57712yes—
442C2×C4#57812yes—
443C8—44no—
444C8—82no—
445C8—82no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22#203614yes—
453C22C2×#39411yescharacteristic
454C22C2×#45111yescharacteristic
455C22C2×#44911yescharacteristic
456C22#301011yescharacteristic
457C22#301111yescharacteristic
458C22#302111yescharacteristic
459C22#302211yescharacteristic
460C4—21no—
461C4—21no—
462C4—22no—
463C4—22no—
464C4—44no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4—82no—
470C4—162no—
471C4#203614yes—
472C4#304612yes—
473C4C2×#46711yescharacteristic
474C4C2×#46811yescharacteristic
475C4C2×#48511yescharacteristic
476C4C2×#48411yescharacteristic
477C4#313712yes—
478C2—24no—
479C2#1889811yescharacteristic
480C2#1890811yescharacteristic
481C2#1891511yescharacteristic
482C2#1892511yescharacteristic
483C2C2×#203611yescharacteristic
484C2#2349511yescharacteristic
485C2#2351011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10422C2semidirect
8. #10414C2semidirect

Maximal subgroups

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

subgroupindex
#351792
#219822
#218602
C2×#18032
C2×#15422
C2×#14792
#104222
#104142
#78932