Finite Group Databasebeta

← #117640

Subgroups — #117640

Order 512.

838 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21942C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C2×#1821C211yesmaximal, characteristic
5C2×#1298C211yesmaximal, characteristic
6#7915C211yesmaximal, characteristic
7#7915C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1853—21no—
16#1853—21no—
17#1852—21no—
18#1852—21no—
19#1824—22no—
20#1824—22no—
21#1823—22no—
22#1823—22no—
23#1821—22no—
24#1821C414yescomplemented
25#1809—22no—
26C2×#343—22no—
27C2×#340—21no—
28C2×#340—21no—
29C2×#340—21no—
30C2×#340—21no—
31#1298—22no—
32#1280—22no—
33#1280—22no—
34C24×C4—21no—
35C24×C4C2311yescharacteristic
36C22×M16—21no—
37C22×M16D412yes—
38C23×C8—21no—
39C23×C8—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4Q812yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C2×#107—22no—
61C2×#106—22no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407—21no—
66#407—21no—
67#407C2×C412yes—
68#407C2×C412yes—
69#407D412yes—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8D412yes—
85C2×C22⋊C8D412yes—
86#380—21no—
87#380—21no—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—21no—
92#380—21no—
93#380—21no—
94C2×#95—21no—
95C2×#95—21no—
96#343—28no—
97#340—22no—
98#340—22no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103C25#4111yescharacteristic
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—41no—
114C23×C4—41no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4—82no—
118C23×C4—82no—
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×Q811yescharacteristic
128C23×C4#4111yescharacteristic
129C23×C4#4111yescharacteristic
130C23×C4#4111yescharacteristic
131C23×C4#4111yescharacteristic
132C23×C4#4111yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—41no—
136C2×M16—42no—
137C2×M16—42no—
138C2×M16—42no—
139C2×M16—42no—
140C2×M16C22⋊C414yes—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—41no—
149C22×C8—41no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—41no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C4⋊C414yes—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199#107—82no—
200#106—82no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—41no—
226C24—41no—
227C24#11711yescharacteristic
228C24#12111yescharacteristic
229C24#12111yescharacteristic
230C24#12111yescharacteristic
231C24C22⋊Q811yescharacteristic
232C24#12611yescharacteristic
233C24#12611yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—42no—
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—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C4⋊C411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303C22×C4C22⋊Q811yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12411yescharacteristic
312C22×C4#12411yescharacteristic
313C22×C4#12411yescharacteristic
314C22×C4#12711yescharacteristic
315M16—44no—
316M16—44no—
317M16—82no—
318M16—82no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—82no—
330C2×C8—82no—
331C2×C8—82no—
332C2×C8—82no—
333C2×C8—81no—
334C2×C8—81no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23#38011yescharacteristic
386C23#38811yescharacteristic
387C23#38811yescharacteristic
388C23#39211yescharacteristic
389C23#39611yescharacteristic
390C23#39611yescharacteristic
391C23#39711yescharacteristic
392C23#47112yes—
393C23#47112yes—
394C23#47212yes—
395C23#47312yes—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—84no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—162no—
448C2×C4—162no—
449C2×C4—162no—
450C2×C4—162no—
451C2×C4#38511yescharacteristic
452C2×C4#38511yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4#39011yescharacteristic
455C2×C4#39311yescharacteristic
456C2×C4#39411yescharacteristic
457C2×C4#39411yescharacteristic
458C2×C4#39911yescharacteristic
459C2×C4#45812yes—
460C2×C4#45812yes—
461C8—44no—
462C8—82no—
463C8—82no—
464C8—82no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22#143911yescharacteristic
496C22#193012yes—
497C22#193112yes—
498C22#201811yescharacteristic
499C22#201911yescharacteristic
500C22#201911yescharacteristic
501C22#202011yescharacteristic
502C22#202211yescharacteristic
503C22#202211yescharacteristic
504C22#206312yes—
505C22#206412yes—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—84no—
513C4—161no—
514C4—161no—
515C4—161no—
516C4—161no—
517C4#189012yes—
518C4#203012yes—
519C2—22no—
520C2—22no—
521C2—22no—
522C2—22no—
523C2—41no—
524C2—41no—
525C2#999211yescharacteristic
526C2#999311yescharacteristic
527C2#999411yescharacteristic
528C2#999511yescharacteristic
529C2#1107011yescharacteristic
530C2#1114411yescharacteristic
531C2#1114511yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1821C4semidirect

Maximal subgroups

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

subgroupindex
#219422
C2×#18242
C2×#18232
C2×#18212
C2×#12982
#79152