Finite Group Databasebeta

← #192441

Subgroups — #192441

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4C2×#1502C211yesmaximal, characteristic
5#13911C211yesmaximal, characteristic, complemented
6#8869C218yesmaximal, complemented
7#8014C211yesmaximal, characteristic, complemented
8#7908C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C2×#520—21no—
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2367—22no—
14#2366—21no—
15#2365—21no—
16#2354—21no—
17#2352—22no—
18D4×C4⋊C4—21no—
19#2325C2214yes—
20#2311—21no—
21C4×#121—22no—
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990—21no—
28#1957—21no—
29#1947—22no—
30#1874C2214yes—
31#1852C2214yes—
32#1845—21no—
33#1834—21no—
34#1802—22no—
35#1502C418yescomplemented
36C2×#327C2214yes—
37#1442C2214yes—
38C4⋊#95—22no—
39C4⋊#95—21no—
40#1294C2214yes—
41#1280C2214yes—
42C22×#41—21no—
43#532—22no—
44#532—22no—
45#532C2×C414yes—
46#532Q814yes—
47C2×C4⋊Q8D411yescharacteristic
48C2×#127D411yescharacteristic
49C2×#124—21no—
50C2×#121—21no—
51C2×#121C2312yes—
52#520—44no—
53#520—44no—
54C2×C4×Q8—22no—
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4D412yes—
60C2×#117—22no—
61C2×#117—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C42—21no—
70C22×C42D412yes—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431—22no—
74#431—22no—
75#431C2×C414yescomplemented
76C2×#107—24no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×#102—24no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#391—22no—
85#390—22no—
86#390—22no—
87#389—22no—
88#388—24no—
89#388—24no—
90#387—24no—
91#386—24no—
92#386—24no—
93#385—24no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#382—21no—
101#382—44no—
102#380—22no—
103#380—22no—
104#380—21no—
105#380C2312yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C414yescomplemented
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—24no—
116C2×#95—24no—
117C2×#95—24no—
118C2×#95C2312yes—
119C2×#95C2312yes—
120#340—24no—
121#340—24no—
122#327C2×C4116yescomplemented
123C2×#41—24no—
124C2×#41—24no—
125C2×#41—42no—
126C2×#41—42no—
127C2×#41—42no—
128C2×#41—42no—
129C22×Q8—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C2×D412yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—22no—
146C4⋊Q8C4⋊C414yes—
147#127—22no—
148#127C4⋊C414yes—
149#124—24no—
150#124—24no—
151#121—24no—
152#121—24no—
153#121—22no—
154#121—22no—
155#121C22×C414yes—
156#121C2×Q814yes—
157C4×Q8—48no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—48no—
162C4×D4—48no—
163C4×D4—48no—
164C4×D4—84no—
165C4×D4C4⋊C418yes—
166#117—48no—
167#117—42no—
168#117—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
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—22no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4D814yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—44no—
214C2×C42C22×C412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222#107—84no—
223C4⋊C8—24no—
224C4⋊C8—42no—
225#102—84no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8C22×C414yes—
230C4×C8—24no—
231C4×C8—42no—
232#95—24no—
233#95—24no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—48no—
240#95—84no—
241#95C22×C414yes—
242C24—22no—
243C24C22×D411yescharacteristic
244#41—48no—
245#41—48no—
246#41—48no—
247#41—48no—
248C2×Q8—24no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4C2×C4⋊C414yes—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—28no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—48no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4C4×D412yes—
302C22×C4C4×D412yes—
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4C22⋊Q812yes—
308C22×C4C22⋊Q812yes—
309C22×C4C2×D814yes—
310C22×C4C2×D812yes—
311C22×C4#13512yes—
312C22×C4AGL(1,ℤ/8)12yes—
313C22×C4#13712yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q811yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4C2×C4⋊C414yes—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—28no—
350C22⋊C4—48no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C42—48no—
356C42—48no—
357C42—44no—
358C42—44no—
359C42—82no—
360C42C2×C4⋊C412yes—
361C42C2×C4⋊C412yes—
362C42C4×D412yes—
363C42C4×D412yes—
364C42C22⋊Q812yes—
365C42C22⋊Q812yes—
366C42#12312yes—
367C42#12312yes—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—24no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23C2×C4×D411yescharacteristic
380C23C2×#12011yescharacteristic
381C23C2×C22⋊Q811yescharacteristic
382C23C22×D811yescharacteristic
383C23C2×#13511yescharacteristic
384C23C2×AGL(1,ℤ/8)11yescharacteristic
385C23C2×#13711yescharacteristic
386Q8—48no—
387D4—44no—
388D4—44no—
389D4—44no—
390D4—44no—
391D4—44no—
392D4—48no—
393D4—48no—
394D4—82no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—416no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—168no—
440C2×C4#38512yes—
441C2×C4#38512yes—
442C2×C4#38512yes—
443C2×C4#38512yes—
444C2×C4C4×D814yes—
445C2×C4#44014yes—
446C2×C4#44712yes—
447C2×C4#44712yes—
448C2×C4#45212yes—
449C2×C4#45212yes—
450C2×C4#45412yes—
451C2×C4#45512yes—
452C2×C4#47414yes—
453C2×C4#47814yes—
454C2×C4C22×C4⋊C411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C22⋊Q811yescharacteristic
457C2×C4C2×#12311yescharacteristic
458C8—44no—
459C8—82no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22C2×#38511yescharacteristic
480C22C2×C4×D811yescharacteristic
481C22C2×#44011yescharacteristic
482C22C2×#44711yescharacteristic
483C22C2×#45211yescharacteristic
484C22C2×#45511yescharacteristic
485C22C2×#45411yescharacteristic
486C22#302711yescharacteristic
487C22#302811yescharacteristic
488C22C2×#47411yescharacteristic
489C22C2×#47811yescharacteristic
490C4—22no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—44no—
496C4—44no—
497C4—84no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—162no—
502C4#186014yes—
503C4#186214yes—
504C4#186214yes—
505C2—21no—
506C2—21no—
507C2—21no—
508C2—21no—
509C2—84no—
510C2#1839411yescharacteristic
511C2#1839911yescharacteristic
512C2#1840111yescharacteristic
513C2#1840611yescharacteristic
514C2C2×#186011yescharacteristic
515C2C2×#186211yescharacteristic
516C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21982C2semidirect
5. #13911C2semidirect
6. #8869C2semidirect
7. #8014C2semidirect
8. #7908C2semidirect
35. #1502C4semidirect
75. #431C2×C4semidirect
113. C4×C4⋊C4C2×C4semidirect
122. #327C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#219822
#218592
C2×#15022
#139112
#88692
#80142
#79082