Finite Group Databasebeta

← #194137

Subgroups — #194137

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3037C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4#21386C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic, complemented
6#8987C214yesmaximal, complemented
7#8979C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C22×#137—21no—
11C2×#532C2211yescharacteristic
12C22×C4×Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14#3037—24no—
15#3037C418yescomplemented
16C2×#464—22no—
17C2×#462C2212yes—
18C2×#460C2212yes—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#417—21no—
23Q8×C4⋊C4—22no—
24Q8×C4⋊C4—22no—
25Q8×C4⋊C4—22no—
26Q8×C4⋊C4C2214yes—
27C2×#389—22no—
28C2×#389—21no—
29C2×#384—21no—
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990C2212yes—
34#1988C2212yes—
35#1982C2212yes—
36#1957C2212yes—
37#1874—22no—
38#1874—21no—
39#1874—21no—
40#1802—22no—
41C4×#102—22no—
42#1479—24no—
43C2×Q8⋊C8C2214yes—
44#1296C2214yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8D411yescharacteristic
48C2×#137—28no—
49C2×#137—44no—
50C2×#137—44no—
51C22×Q16—22no—
52C22×SD16—22no—
53C22×M16—21no—
54#532—22no—
55#532—22no—
56#532—22no—
57#532C2×C414yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×#127C2311yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—44no—
70C2×C4×Q8—44no—
71C2×C4×Q8C2×C414yes—
72C2×C4×Q8D414yes—
73C2×C4×Q8D412yes—
74C2×C4×Q8C2312yes—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4D412yes—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90#464—216no—
91#462—24no—
92#462C2×C418yescomplemented
93#460—24no—
94#460C2×C418yescomplemented
95#431—22no—
96#431—22no—
97#431—22no—
98#431—22no—
99#423—22no—
100#423C2×C414yes—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105#417—28no—
106C2×#103—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—24no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2312yes—
115#389—24no—
116#389—28no—
117#389—24no—
118#389—28no—
119#384—24no—
120#384—24no—
121#384—21no—
122#384—21no—
123#384D412yes—
124#384C2312yes—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4C2312yes—
131C2×#95—22no—
132C2×#95—21no—
133#340—24no—
134#340—22no—
135#340—22no—
136Q8⋊C8—28no—
137C2×#41—24no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8C22⋊C414yes—
149C22×Q8D814yes—
150C22×Q8SD1614yes—
151C22×Q8C2×D412yes—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—41no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—41no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165#137—432no—
166C2×Q16—28no—
167C2×Q16—44no—
168C2×Q16—44no—
169C2×SD16—28no—
170C2×SD16—44no—
171C2×SD16—44no—
172C2×M16—24no—
173C2×M16—42no—
174C2×M16—42no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C22×C412yes—
182#127—21no—
183#127—21no—
184#127—21no—
185#127C22×C412yes—
186#124—24no—
187#124—42no—
188#121—24no—
189#121—24no—
190#121—42no—
191#121—42no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—48no—
200C4×Q8—416no—
201C4×Q8—48no—
202C4×Q8C22⋊C418yes—
203C4×Q8C22×C414yes—
204C4×Q8C2×D414yes—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—42no—
208C4×D4—42no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—84no—
234C2×C4⋊C4C22⋊C414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—28no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—48no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42C22×C412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42#4112yes—
257C2×C42#4112yes—
258C2×C42C2411yescharacteristic
259C4⋊C8—24no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—42no—
264C4⋊C8C22×C414yes—
265#103—28no—
266#102—28no—
267#102—44no—
268#102—44no—
269#102—84no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—42no—
275#95—28no—
276#95—22no—
277#95—22no—
278C24—41no—
279C24#12011yescharacteristic
280#41—48no—
281#41—48no—
282#41—84no—
283#41—84no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—28no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—28no—
292C2×Q8—28no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8#102116yescomplemented
300C2×Q8C2×C22⋊C414yes—
301C2×Q8C2×D814yes—
302C2×Q8C2×SD1614yes—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—44no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—42no—
318C2×D4—82no—
319C2×D4—82no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4C2×C22⋊C412yes—
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#12312yes—
357C22×C4C2×D812yes—
358C22×C4C2×SD1612yes—
359C22×C4AGL(1,ℤ/8)12yes—
360C22×C4#13712yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364Q16—416no—
365SD16—416no—
366M16—48no—
367C2×C8—24no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—416no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—82no—
398C4⋊C4—88no—
399C4⋊C4C2×C22⋊C414yes—
400C4⋊C4#13714yescomplemented
401C22⋊C4—44no—
402C22⋊C4—44no—
403C42—24no—
404C42—24no—
405C42—28no—
406C42—416no—
407C42—42no—
408C42—42no—
409C42—44no—
410C42C2×C22⋊C412yes—
411C42C4×D412yes—
412C42C4×D412yes—
413C42#12112yes—
414C42#12112yes—
415C42#12312yes—
416C42C23×C411yescharacteristic
417C42C22×D411yescharacteristic
418C42C2×#4111yescharacteristic
419C42C2×#4111yescharacteristic
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—42no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23#38612yes—
432C23#44712yes—
433C23#44812yes—
434C23C2×#12011yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—24no—
438Q8—216no—
439Q8—42no—
440Q8—42no—
441Q8—81no—
442Q8—81no—
443Q8C2×#10218yes—
444D4—44no—
445D4—42no—
446D4—42no—
447D4—44no—
448D4—82no—
449D4—88no—
450D4—81no—
451D4—81no—
452D4—82no—
453C2×C4—216no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—84no—
483C2×C4—88no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—88no—
487C2×C4#38612yes—
488C2×C4#38612yes—
489C2×C4#38612yes—
490C2×C4C2×#10214yes—
491C2×C4#41714yes—
492C2×C4#44712yes—
493C2×C4#44812yes—
494C2×C4#48014yes—
495C2×C4#48114yes—
496C2×C4C22×C22⋊C411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4C2×#12311yescharacteristic
502C2×C4C22×D811yescharacteristic
503C2×C4C22×SD1611yescharacteristic
504C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
505C2×C4C2×#13711yescharacteristic
506C2×C4C2×#13712yes—
507C2×C4#57512yes—
508C8—44no—
509C8—82no—
510C8—82no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22#187414yes—
525C22C2×#38611yescharacteristic
526C22C2×#44711yescharacteristic
527C22C2×#44811yescharacteristic
528C22#300411yescharacteristic
529C22#300511yescharacteristic
530C22#301911yescharacteristic
531C22#302011yescharacteristic
532C4—28no—
533C4—22no—
534C4—21no—
535C4—21no—
536C4—22no—
537C4—44no—
538C4—44no—
539C4—82no—
540C4—84no—
541C4—82no—
542C4#187414yes—
543C4C22×#10211yescharacteristic
544C4C2×#41711yescharacteristic
545C4C4×#13712yes—
546C4#303712yes—
547C4C2×#48011yescharacteristic
548C4C2×#48111yescharacteristic
549C2—24no—
550C2—162no—
551C2#1876611yescharacteristic
552C2#1876811yescharacteristic
553C2#1877611yescharacteristic
554C2#1877811yescharacteristic
555C2C2×#187411yescharacteristic
556C2#2092611yescharacteristic
557C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ 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
2. C2×Q8×C4⋊C4C2semidirect
5. C2×#1479C2semidirect
6. #8987C2semidirect
7. #8979C2semidirect
8. #8043C2semidirect
9. #7901C2semidirect
15. #3037C4semidirect
92. #462C2×C4semidirect
94. #460C2×C4semidirect
299. C2×Q8#102semidirect
400. C4⋊C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#30372
C2×Q8×C4⋊C42
#218592
#213862
C2×#14792
#89872
#89792
#80432
#79012