Finite Group Databasebeta

← #266719

Subgroups — #266719

Order 512.

1102 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59817C211yesmaximal, characteristic
2#29286C211yesmaximal, characteristic, complemented
3#28283C211yesmaximal, characteristic, complemented
4C4×#486C211yesmaximal, characteristic
5#10575C212yesmaximal, complemented
6#10350C212yesmaximal, complemented
7#9870C211yesmaximal, characteristic, complemented
8#9870C211yesmaximal, characteristic, complemented
9#9057C212yesmaximal
10#8921C212yesmaximal
11#8585C211yesmaximal, characteristic
12C2×D4×Q8—21no—
13C2×D4×Q8—21no—
14C2×#486C2211yescharacteristic
15C2×#475—21no—
16#2683—22no—
17#2551C2212yes—
18#2538—21no—
19#2538—21no—
20#2452—22no—
21#2452—22no—
22#2409—22no—
23#2366C2212yes—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×#127C2211yescharacteristic
28#2086—22no—
29#2066—22no—
30#1990C2211yescharacteristic
31#1988—21no—
32#1988C2212yes—
33#1981C2212yes—
34#1957C2211yescharacteristic
35#1948C2212yes—
36#1947—21no—
37#1947C2212yes—
38#1934—22no—
39#1842C2211yescharacteristic
40#1842C2211yescharacteristic
41#1834C2211yescharacteristic
42#1803—21no—
43C4×#102—22no—
44C42×C8C2211yescharacteristic
45C2×#331C2212yes—
46#1296C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C23×Q8—22no—
52D4×Q8—44no—
53D4×Q8—44no—
54D4×Q8—44no—
55D4×Q8—44no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×#127C2311yescharacteristic
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—24no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4D412yes—
74#486—22no—
75#486Q814yescomplemented
76#475—44no—
77C2×#107—22no—
78C2×#106—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×#103—22no—
86C2×#103—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#396—24no—
94#396—24no—
95#395—24no—
96#393—24no—
97#391—21no—
98#391—21no—
99#391—21no—
100#390—22no—
101#390—22no—
102#390—21no—
103#390—21no—
104#390D412yes—
105#389—22no—
106#389—22no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389D412yes—
112#389D412yes—
113#386—24no—
114#385—24no—
115#384D411yescharacteristic
116#384D411yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#380—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2312yes—
125C4×C4⋊C4C2312yes—
126C43C2311yescharacteristic
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#331—24no—
134C22×Q8—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—48no—
139C22×Q8—48no—
140C22×Q8—84no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C2×D412yes—
144C23×C4—24no—
145C23×C4—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C4⋊Q8D814yes—
160C4⋊Q8SD1614yes—
161C4⋊Q8C2×Q812yescomplemented
162#127—21no—
163#127—21no—
164#127C2×Q812yes—
165#127#4112yes—
166C22⋊Q8—416no—
167C22⋊Q8—416no—
168C22⋊Q8—48no—
169C22⋊Q8—48no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—41no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224#107—82no—
225#106—44no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—81no—
233C4⋊C8—81no—
234#103—44no—
235#103—82no—
236#102—28no—
237#102—44no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—42no—
242C4×C8—42no—
243C4×C8C2×Q812yescomplemented
244#95—24no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—41no—
250#95—41no—
251C24—22no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—48no—
256C2×Q8—48no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—84no—
267C2×Q8—832no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4C22⋊Q814yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—41no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—84no—
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—42no—
327C4⋊C4—48no—
328C4⋊C4—48no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—82no—
347C4⋊C4C2×D814yes—
348C4⋊C4C2×SD1614yes—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—82no—
354C22⋊C4—84no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—22no—
360C42—41no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—82no—
366C42C2×D812yes—
367C42C2×SD1612yes—
368C42AGL(1,ℤ/8)12yes—
369C42AGL(1,ℤ/8)12yes—
370C42C22×Q811yescharacteristic
371C42C2×#4111yescharacteristic
372C422−1+411yescharacteristic
373C422−1+411yescharacteristic
374C23—24no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23C2×#12011yescharacteristic
382Q8—48no—
383Q8—48no—
384Q8—42no—
385Q8—42no—
386Q8—816no—
387Q8—81no—
388Q8—81no—
389D4—44no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—82no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—816no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—88no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4#44712yes—
432C2×C4#44712yes—
433C2×C4#45012yes—
434C2×C4#45012yes—
435C2×C4#45512yes—
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4#53611yescharacteristic
438C2×C4#54811yescharacteristic
439C2×C4#54811yescharacteristic
440C2×C4D4×Q811yescharacteristic
441C2×C4D4×Q811yescharacteristic
442C2×C4C22×D811yescharacteristic
443C2×C4C22×SD1611yescharacteristic
444C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C2×C4#57811yescharacteristic
450C8—42no—
451C8—42no—
452C8—82no—
453C8—82no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22#240911yescharacteristic
462C22C2×#44711yescharacteristic
463C22C2×#45011yescharacteristic
464C22#301311yescharacteristic
465C22#301311yescharacteristic
466C22C2×#45511yescharacteristic
467C22#301811yescharacteristic
468C4—22no—
469C4—22no—
470C4—21no—
471C4—21no—
472C4—41no—
473C4—44no—
474C4—44no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—161no—
480C4#163914yes—
481C4#335811yescharacteristic
482C4#335911yescharacteristic
483C4#336711yescharacteristic
484C4#337311yescharacteristic
485C4Q8×D811yescharacteristic
486C4Q8×SD1611yescharacteristic
487C4#338411yescharacteristic
488C4#338611yescharacteristic
489C2—84no—
490C2C2×#163911yescharacteristic
491C2#1867411yescharacteristic
492C2#1867511yescharacteristic
493C2#1869311yescharacteristic
494C2#2431411yescharacteristic
495C2#2432611yescharacteristic
496C2#2433011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29286C2semidirect
3. #28283C2semidirect
5. #10575C2semidirect
6. #10350C2semidirect
7. #9870C2semidirect
8. #9870C2semidirect
75. #486Q8semidirect
161. C4⋊Q8C2×Q8semidirect
243. C4×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598172
#292862
#282832
C4×#4862
#105752
#103502
#98702
#90572
#89212
#85852