Finite Group Databasebeta

← #110972

Subgroups — #110972

Order 512.

867 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21759C211yesmaximal, characteristic
2C2×#1996C211yesmaximal, characteristic
3#20526C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5#10854C211yesmaximal, characteristic
6#8023C211yesmaximal, characteristic
7C2×#1289C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C2×#380C2211yescharacteristic
12C2×#380C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1996—22no—
16#1996C414yescomplemented
17#1892—22no—
18#1872—22no—
19#1824C412yescomplemented
20#1787C412yescomplemented
21#1752—22no—
22#1752—22no—
23#1749—22no—
24C2×#342—22no—
25C2×#342—22no—
26#1520—22no—
27#1520—22no—
28C2×#328C414yescomplemented
29#1294—22no—
30#1289—22no—
31C23×Q8C2×C411yescharacteristic
32C24×C4C2311yescharacteristic
33C23×C8—21no—
34C23×C8D412yes—
35C2×C22⋊Q8—21no—
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×C22⋊Q8C2×C412yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4D412yes—
57C22×C42—21no—
58C2×#103—24no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8C2×C412yes—
67C2×C22⋊C8C2×C412yes—
68C2×C22⋊C8D412yes—
69C2×C22⋊C8D412yes—
70C2×C22⋊C8D412yes—
71C2×#97—24no—
72#380—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380C2×C412yes—
83#380C2×C412yes—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4C2×C412yes—
87C4×C22⋊C4D412yes—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95C2×C412yes—
92#342—28no—
93#342—44no—
94#340—24no—
95#336—24no—
96#328—28no—
97C25C2×D411yescharacteristic
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—82no—
104C22×Q8C4212yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—82no—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C23×C4#4111yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—82no—
132C22×C8—82no—
133C22×C8C22⋊C414yes—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C4212yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4C22⋊C414yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—82no—
198#103—84no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8C22⋊C414yes—
207#97—84no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—41no—
215#95—41no—
216C24—81no—
217C24—81no—
218C24C2×C22⋊C411yescharacteristic
219C24C4×D411yescharacteristic
220C24C4×D411yescharacteristic
221C24#12111yescharacteristic
222C24#12111yescharacteristic
223C24#12411yescharacteristic
224C24#12711yescharacteristic
225C24AGL(1,ℤ/8)12yes—
226C24#13712yes—
227C24#13712yes—
228C24#13712yes—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—44no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×Q8—82no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4C2×C4211yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4#11711yescharacteristic
283C22×C4#11711yescharacteristic
284C22×C4#11711yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12311yescharacteristic
301C22×C4#12311yescharacteristic
302C22×C4#12311yescharacteristic
303C22×C4#12411yescharacteristic
304C22×C4#12411yescharacteristic
305C22×C4#12611yescharacteristic
306C22×C4#12611yescharacteristic
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—82no—
313C2×C8—82no—
314C2×C8—82no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—81no—
342C22⋊C4—81no—
343C22⋊C4—81no—
344C22⋊C4—81no—
345C42—44no—
346C42—82no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23C4×C22⋊C411yescharacteristic
366C23#38611yescharacteristic
367C23#38811yescharacteristic
368C23#38811yescharacteristic
369C23#39011yescharacteristic
370C23#39011yescharacteristic
371C23#39011yescharacteristic
372C23#41712yes—
373C23#41912yes—
374C23#44012yes—
375C23#44012yes—
376C23#44112yes—
377C23#44112yes—
378C23#46812yes—
379C23#46812yes—
380C23#47012yes—
381C23#47012yes—
382C23#48912yes—
383C23#49112yes—
384C23#49611yescharacteristic
385C23#49711yescharacteristic
386C23#49711yescharacteristic
387C23#49711yescharacteristic
388Q8—44no—
389Q8—82no—
390Q8—82no—
391Q8—82no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—164no—
443C2×C4—162no—
444C2×C4—162no—
445C2×C4—164no—
446C2×C4C4×C22⋊C411yescharacteristic
447C2×C4C4×C22⋊C411yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#38811yescharacteristic
451C2×C4#38811yescharacteristic
452C2×C4#38811yescharacteristic
453C2×C4#38811yescharacteristic
454C8—44no—
455C8—44no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—41no—
485C22—41no—
486C22#143711yescharacteristic
487C22#176312yes—
488C22#189412yes—
489C22#193512yes—
490C22#193512yes—
491C22#195911yescharacteristic
492C22#196011yescharacteristic
493C22#197111yescharacteristic
494C22#197111yescharacteristic
495C22#197311yescharacteristic
496C22#197311yescharacteristic
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4—162no—
508C4—162no—
509C4—162no—
510C4#191312yes—
511C2—22no—
512C2—21no—
513C2—21no—
514C2—21no—
515C2—22no—
516C2—22no—
517C2—21no—
518C2—21no—
519C2—21no—
520C2—22no—
521C2#1092811yescharacteristic
522C2#1093811yescharacteristic
523C2#1095111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1996C4semidirect
19. #1824C4semidirect
20. #1787C4semidirect
28. C2×#328C4semidirect

Maximal subgroups

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

subgroupindex
#217592
C2×#19962
#205262
C2×#15202
#108542
#80232
C2×#12892