Finite Group Databasebeta

← #234700

Subgroups — #234700

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic
2#35178C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic, complemented
4#21833C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic
6C2×#1461C211yesmaximal, characteristic, complemented
7#9919C218yesmaximal, complemented
8#8014C211yesmaximal, characteristic, complemented
9C2×#536—21no—
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8—21no—
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3226—24no—
15#3226—24no—
16C2×#500—21no—
17C2×#486—21no—
18C2×#480C2214yes—
19C2×#474—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2854—22no—
23#2618C2214yes—
24#2548—21no—
25#2546—24no—
26#2546—21no—
27#2546—24no—
28#2545—22no—
29#2538—22no—
30#2536—22no—
31C2×#398C2214yes—
32#2311—21no—
33C2×#391—21no—
34C2×#389—22no—
35#2285C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1990—22no—
38#1988—22no—
39#1823C2214yes—
40#1502—24no—
41C2×#327C2214yes—
42#1461—24no—
43C2×#324C2214yes—
44#1294C2214yes—
45C22×#41—21no—
46C23×Q8—21no—
47C23×Q8D411yescharacteristic
48#536—44no—
49#536—44no—
50#532—22no—
51#532—22no—
52#532—22no—
53#532—22no—
54C2×C4⋊Q8—28no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—44no—
64C2×C4⋊Q8—44no—
65C2×C4⋊Q8D411yescharacteristic
66C2×C4⋊Q8D414yes—
67C2×#127D411yescharacteristic
68C2×#124—22no—
69C2×#124—21no—
70C2×#123—24no—
71C2×C22⋊Q8—24no—
72C2×#121—21no—
73C2×#121C2312yes—
74C2×C4×Q8—22no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8—21no—
81C2×C4×Q8—22no—
82C2×C4×D4—21no—
83C2×C4×D4D412yes—
84C2×#117—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C42—21no—
92C22×C42D412yes—
93C22×C42C2311yescharacteristic
94#500—28no—
95#486—28no—
96#480—28no—
97#480—28no—
98#474—216no—
99#431—22no—
100#431—22no—
101#431—22no—
102#431D414yes—
103C2×#107—24no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×#102—24no—
107C2×C22⋊C8C2312yes—
108C2×C22⋊C8C2312yes—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111#398—24no—
112#398—28no—
113#398—28no—
114#396—28no—
115#394—24no—
116#391—22no—
117#391—28no—
118#390—22no—
119#390—24no—
120#389—24no—
121#389—216no—
122#388—28no—
123#386—24no—
124#384D412yes—
125#382—21no—
126#380—24no—
127#380—21no—
128#380C2312yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4D414yes—
133C2×#95—24no—
134C2×#95C2312yes—
135#340—28no—
136#327—28no—
137#324—28no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8D814yes—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C2×D412yes—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—22no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8—22no—
181C4⋊Q8—432no—
182C4⋊Q8—48no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187#127—22no—
188#127—22no—
189#124—48no—
190#124—42no—
191#124—42no—
192#123—48no—
193#123—48no—
194C22⋊Q8—416no—
195#121—22no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—42no—
200#121—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—42no—
212C4×D4—42no—
213#117—42no—
214#117—42no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—28no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267#107—216no—
268C4⋊C8—24no—
269C4⋊C8—42no—
270#102—216no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8C2×D414yes—
275C4×C8—24no—
276C4×C8—42no—
277#95—216no—
278#95—28no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95C2×D414yes—
283C24—22no—
284C24C22×D411yescharacteristic
285#41—84no—
286#41—84no—
287#41—84no—
288#41—84no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—28no—
292C2×Q8—28no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—48no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—44no—
303C2×Q8—48no—
304C2×Q8—42no—
305C2×Q8—44no—
306C2×Q8—82no—
307C2×Q8—82no—
308C2×Q8—82no—
309C2×Q8—82no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C22×C4—24no—
327C22×C4—28no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4#12012yes—
356C22×C4#12011yescharacteristic
357C22×C4#12012yes—
358C22×C4#12011yescharacteristic
359C22×C4#12011yescharacteristic
360C22×C4#12011yescharacteristic
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4C2×D814yes—
364C22×C4C2×D812yes—
365C22×C4#13512yes—
366C22×C4AGL(1,ℤ/8)12yes—
367C22×C4AGL(1,ℤ/8)12yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C2×C8—24no—
375C2×C8—44no—
376C2×C8—44no—
377C2×C8—42no—
378C4⋊C4—216no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—44no—
384C4⋊C4—432no—
385C4⋊C4—416no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C22⋊C4—416no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—88no—
406C42—44no—
407C42—44no—
408C42—48no—
409C42—44no—
410C42#12012yes—
411C42#12012yes—
412C42#12112yes—
413C42#12112yes—
414C42#12112yes—
415C42#12112yes—
416C42#12712yes—
417C42#12712yes—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—44no—
425C23C2×#12011yescharacteristic
426C23C2×#12011yescharacteristic
427C23C2×#12111yescharacteristic
428C23C22×D811yescharacteristic
429C23C2×#13511yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×AGL(1,ℤ/8)11yescharacteristic
432Q8—48no—
433Q8—48no—
434Q8—88no—
435Q8—88no—
436Q8—84no—
437D4—44no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4—82no—
443D4—84no—
444D4—84no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—88no—
483C2×C4—82no—
484C2×C4#39212yes—
485C2×C4#39212yes—
486C2×C4#39212yes—
487C2×C4#39212yes—
488C2×C4#44712yes—
489C2×C4#44714yes—
490C2×C4#44712yes—
491C2×C4#44912yes—
492C2×C4#44912yes—
493C2×C4#44914yes—
494C2×C4#45614yes—
495C2×C4#46614yes—
496C2×C4#46914yes—
497C2×C4C2×#12011yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12711yescharacteristic
501C8—44no—
502C8—82no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—84no—
515C22—84no—
516C22C2×#39211yescharacteristic
517C22C2×#44711yescharacteristic
518C22C2×#44711yescharacteristic
519C22C2×#44911yescharacteristic
520C22C2×#44911yescharacteristic
521C22C2×#45612yes—
522C22#302812yes—
523C22C2×#46611yescharacteristic
524C22C2×#46911yescharacteristic
525C4—22no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—48no—
531C4—44no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4—84no—
536C4#199914yes—
537C4#199914yes—
538C4#200514yes—
539C2—21no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—162no—
544C2#1839812yes—
545C2#1840812yes—
546C2C2×#199911yescharacteristic
547C2C2×#200511yescharacteristic
548C2#2344611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C23×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
3. C2×#2546C2semidirect
4. #21833C2semidirect
6. C2×#1461C2semidirect
7. #9919C2semidirect
8. #8014C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#351782
C2×#25462
#218332
C2×#15022
C2×#14612
#99192
#80142