Finite Group Databasebeta

← #106917

Subgroups — #106917

Order 512.

902 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic
2#20740C211yesmaximal, characteristic
3C2×#1763C211yesmaximal, characteristic
4C2×#1302C211yesmaximal, characteristic
5#8034C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—22no—
10#2984—22no—
11#2984—22no—
12#2984C414yescomplemented
13C2×#446—22no—
14C2×#446—21no—
15C2×C8×Q8—21no—
16C2×#432—21no—
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431—21no—
20C2×#431C2211yescharacteristic
21C22×#103—21no—
22C22×#97C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1823—21no—
26#1803C412yescomplemented
27#1803C412yescomplemented
28#1787—21no—
29#1763—21no—
30#1763—21no—
31#1763—22no—
32#1763C414yescomplemented
33C2×#340—21no—
34#1302—22no—
35#1295—22no—
36#1294—21no—
37#1294—21no—
38#1285—22no—
39C23×Q8D411yescharacteristic
40C23×C8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8Q812yes—
55C2×#117—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C42—21no—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#446—28no—
63#446—24no—
64#446—24no—
65#446—28no—
66C8×Q8—24no—
67C8×Q8—24no—
68#432—24no—
69#432—21no—
70#432—21no—
71#432—24no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432C2×C412yes—
77#432D412yes—
78#431—24no—
79#431—21no—
80#431—21no—
81#431—24no—
82#431C2×C412yes—
83#431D412yes—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2×C412yes—
89C2×#103—28no—
90C2×#103—24no—
91C2×#103—22no—
92C2×#103—22no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8D411yescharacteristic
96C2×C22⋊C8D411yescharacteristic
97C2×#97—22no—
98C2×#97—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97C2×C412yes—
105C2×#97D412yes—
106C2×C4×C8—22no—
107C2×C4×C8—21no—
108#384C2×C412yes—
109#382—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C2×#95—21no—
119#340—22no—
120#340—22no—
121#340—24no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—24no—
126C22×Q8—42no—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22⋊C412yes—
129C22×Q8C22⋊C411yescharacteristic
130C22×Q8C4⋊C412yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—41no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4#4111yescharacteristic
137C22×C8—24no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—42no—
144C22×C8—41no—
145C22×C8—41no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—22no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—22no—
156C4×Q8—21no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—22no—
162C4×Q8C4212yes—
163C4×Q8C4212yes—
164C4×Q8C22⋊C412yes—
165C4×Q8C22⋊C412yes—
166C4×Q8C4⋊C412yes—
167C4×Q8C4⋊C412yes—
168C4×Q8C4⋊C412yes—
169C4×Q8C4⋊C412yes—
170#117—81no—
171#117—81no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C4⋊C412yes—
194C2×C22⋊C4—41no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—81no—
207C2×C42—81no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×Q811yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42#4111yescharacteristic
220C4⋊C8—28no—
221C4⋊C8—28no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228#103—216no—
229#103—44no—
230#103—44no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8C22⋊C412yes—
238C22⋊C8C22⋊C412yes—
239#97—28no—
240#97—22no—
241#97—22no—
242#97—22no—
243#97—41no—
244#97—41no—
245#97—41no—
246#97—41no—
247#97C22⋊C414yes—
248C4×C8—28no—
249C4×C8—22no—
250C4×C8—22no—
251#95—22no—
252#95—41no—
253#95—41no—
254#95—41no—
255C24#12111yescharacteristic
256C2×Q8—22no—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—21no—
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—44no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8#9512yes—
271C2×Q8#9512yes—
272C2×Q8#10414yes—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—82no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C4⋊C411yescharacteristic
303C22×C4#11711yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4C22⋊Q811yescharacteristic
314C22×C4C22⋊Q811yescharacteristic
315C22×C4#12311yescharacteristic
316C22×C4#12311yescharacteristic
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—24no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4#9512yes—
351C4⋊C4#9512yes—
352C22⋊C4—81no—
353C22⋊C4—81no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—82no—
363C42—82no—
364C42—82no—
365C42C2×C4211yescharacteristic
366C42C2×C22⋊C411yescharacteristic
367C42C2×C22⋊C411yescharacteristic
368C42C2×C4⋊C411yescharacteristic
369C42C2×C4⋊C411yescharacteristic
370C42#11711yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42#12011yescharacteristic
376C42#12011yescharacteristic
377C42C22⋊Q811yescharacteristic
378C42C22⋊Q811yescharacteristic
379C42#12311yescharacteristic
380C42#12311yescharacteristic
381C23—22no—
382C23—22no—
383C23#38011yescharacteristic
384C23#38611yescharacteristic
385C23#38611yescharacteristic
386C23#47111yescharacteristic
387C23#47111yescharacteristic
388C23#47211yescharacteristic
389C23#47311yescharacteristic
390Q8—24no—
391Q8—22no—
392Q8—22no—
393Q8—22no—
394Q8—44no—
395Q8—44no—
396Q8—42no—
397Q8#33914yes—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4C2×#9511yescharacteristic
448C2×C4C4×C22⋊C411yescharacteristic
449C2×C4C4×C22⋊C411yescharacteristic
450C2×C4#37911yescharacteristic
451C2×C4#38011yescharacteristic
452C2×C4#38111yescharacteristic
453C2×C4#38511yescharacteristic
454C2×C4#38511yescharacteristic
455C2×C4#38511yescharacteristic
456C2×C4#38511yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4C2×#10412yes—
460C2×C4#42112yes—
461C8—42no—
462C8—42no—
463C8—44no—
464C8—44no—
465C8—82no—
466C22—21no—
467C22—21no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—21no—
473C22—21no—
474C22C22⋊#9511yescharacteristic
475C22#181511yescharacteristic
476C22#181611yescharacteristic
477C22#187112yes—
478C22#190512yes—
479C22#190711yescharacteristic
480C22#190711yescharacteristic
481C22#190811yescharacteristic
482C22#190811yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—24no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—84no—
498C4—162no—
499C4C2×#33911yescharacteristic
500C4#173311yescharacteristic
501C4#187112yes—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—21no—
506C2#916412yes—
507C2#916612yes—
508C2#1073211yescharacteristic
509C2#1077411yescharacteristic
510C2#1078611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2984C4semidirect
26. #1803C4semidirect
27. #1803C4semidirect
32. #1763C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
#207402
C2×#17632
C2×#13022
#80342
#80142
C2×#12852