Finite Group Databasebeta

← #239431

Subgroups — #239431

Order 512.

1248 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
1C2×#2627C211yesmaximal, characteristic
2#21938C211yesmaximal, characteristic
3#21759C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5C22×#324C211yesmaximal, characteristic
6#14339C211yesmaximal, characteristic
7#10401C218yesmaximal, complemented
8#8023C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2627—22no—
14#2627—22no—
15#2627—22no—
16#2627C2214yes—
17C2×#400—21no—
18C2×#398—22no—
19C2×#395—21no—
20C2×#385—21no—
21C2×#381—21no—
22C2×#381—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#1892—22no—
27#1872C2214yes—
28#1853—22no—
29#1853C2214yes—
30#1824—21no—
31#1807—22no—
32#1787—21no—
33#1520—24no—
34C2×#328C2214yes—
35C2×#324—24no—
36C2×#324C2214yes—
37#1441—22no—
38#1441C2214yes—
39C22⋊#95—21no—
40#1294C2214yes—
41C23×Q8D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8—21no—
46C23×C8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8D412yes—
54C2×C22⋊Q8C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C2×#106—24no—
76C2×#103—24no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8D414yes—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85#400—24no—
86#400—24no—
87#398—28no—
88#398—28no—
89#395—24no—
90#395—24no—
91#385—24no—
92#385—24no—
93#381—24no—
94#381—24no—
95#381—44no—
96#380—21no—
97#380—21no—
98#380—22no—
99#380—22no—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380D412yes—
105#380C2312yes—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95C2312yes—
114#340—24no—
115#340—24no—
116#328—28no—
117#324—28no—
118C25C2×D411yescharacteristic
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—82no—
123C22×Q8C2×D412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4SD1614yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—24no—
157C22×C8—22no—
158C22×C8—42no—
159C22×C8—82no—
160C22×C8—82no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—84no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D414yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222#106—84no—
223#103—84no—
224C22⋊C8—24no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—42no—
229C22⋊C8C2×D414yes—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—41no—
238#95—41no—
239#95—48no—
240#95—84no—
241C24—42no—
242C24—42no—
243C24#12011yescharacteristic
244C24#12011yescharacteristic
245C24#12112yes—
246C24#12312yes—
247C24C2×SD1612yes—
248C24#13512yes—
249C24#13712yes—
250C24#13712yes—
251C24C22×D411yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—44no—
255C2×Q8—42no—
256C2×Q8—82no—
257C2×Q8—84no—
258C2×Q8—84no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12312yes—
323C22×C4#12312yes—
324C22×C4#12312yes—
325C22×C4#12412yes—
326C22×C4#12412yes—
327C22×C4C2×SD1614yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C2×C8—24no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—42no—
338C2×C8—84no—
339C2×C8—84no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—82no—
356C4⋊C4—88no—
357C4⋊C4—84no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—84no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23#39412yes—
382C23#39412yes—
383C23#44812yes—
384C23#44812yes—
385C23#45212yes—
386C23#45212yes—
387C23#46514yes—
388C23#47014yes—
389C23#48314yes—
390C23#48514yes—
391C23C2×#12011yescharacteristic
392C23C2×#12111yescharacteristic
393C23C2×#12311yescharacteristic
394C23C22×SD1611yescharacteristic
395C23C2×#13511yescharacteristic
396C23C2×#13711yescharacteristic
397C23C2×#13711yescharacteristic
398Q8—44no—
399Q8—82no—
400Q8—84no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
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—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—88no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#39412yes—
455C2×C4#39412yes—
456C2×C4C2×#12111yescharacteristic
457C2×C4C2×#12111yescharacteristic
458C2×C4C2×#12311yescharacteristic
459C2×C4C2×#12411yescharacteristic
460C8—44no—
461C8—82no—
462C22—22no—
463C22—24no—
464C22—22no—
465C22—24no—
466C22—22no—
467C22—22no—
468C22—24no—
469C22—24no—
470C22—21no—
471C22—22no—
472C22—21no—
473C22—22no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—22no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22#203514yes—
483C22#203514yes—
484C22C2×#39411yescharacteristic
485C22C2×#44811yescharacteristic
486C22C2×#45211yescharacteristic
487C22#302012yes—
488C22#302612yes—
489C22C2×#46511yescharacteristic
490C22C2×#47011yescharacteristic
491C22C2×#48311yescharacteristic
492C22C2×#48511yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—44no—
500C4—84no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—162no—
505C4—162no—
506C2—21no—
507C2—21no—
508C2—21no—
509C2—21no—
510C2—24no—
511C2—24no—
512C2#1828912yes—
513C2#1829412yes—
514C2C2×#203511yescharacteristic
515C2#2349411yescharacteristic
516C2#2351511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10401C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
#219382
#217592
C2×#15202
C22×#3242
#143392
#104012
#80232