Finite Group Databasebeta

← #120029

Subgroups — #120029

Order 512.

810 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21933C211yesmaximal, characteristic
2#20844C211yesmaximal, characteristic
3#10730C211yesmaximal, characteristic
4#10727C211yesmaximal, characteristic
5C2×#1305C211yesmaximal, characteristic
6#8016C211yesmaximal, characteristic
7C2×#1293C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#428—21no—
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2280C2211yescharacteristic
15C2×#380C2211yescharacteristic
16#1853—21no—
17#1853C412yescomplemented
18#1852—21no—
19#1852C412yescomplemented
20#1851—22no—
21#1850—22no—
22#1798—21no—
23#1798—21no—
24C2×#343—22no—
25#1736—21no—
26C2×#340—22no—
27#1305—22no—
28#1294—21no—
29#1294—21no—
30#1293—22no—
31C24×C4—21no—
32C24×C4C2311yescharacteristic
33C22×M16—21no—
34C22×M16—21no—
35C22×M16D412yes—
36C23×C8D412yes—
37C2×#117D411yescharacteristic
38C2×#117D411yescharacteristic
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—44no—
46C22×C4⋊C4C2×C412yes—
47C22×C22⋊C4—21no—
48#428—44no—
49C2×#107—22no—
50C2×#106—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53#407—21no—
54#407—21no—
55#407—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407D412yes—
63#407Q812yes—
64#407Q812yes—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
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⋊C8C2×C412yes—
77#381—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380C2×C411yescharacteristic
82#380C2×C411yescharacteristic
83#380D411yescharacteristic
84#380D411yescharacteristic
85#380Q812yes—
86#379—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C2×#95—21no—
90#343—44no—
91#340—22no—
92#340—22no—
93#340—44no—
94#336—22no—
95#336—22no—
96C25#4111yescharacteristic
97C23×C4—22no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—44no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—81no—
105C23×C4—81no—
106C23×C4—82no—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×Q811yescharacteristic
114C23×C4C2×Q811yescharacteristic
115C23×C4C2×Q811yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—41no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—42no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—41no—
141#117—42no—
142#117—42no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—81no—
173C2×C22⋊C4—81no—
174C2×C42—21no—
175C2×C42—21no—
176#107—44no—
177#106—44no—
178C4⋊C8—82no—
179C4⋊C8—82no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8C4⋊C414yes—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—22no—
201#95—41no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—41no—
207C24—41no—
208C24—41no—
209C24—41no—
210C24C4×D411yescharacteristic
211C24#12111yescharacteristic
212C24C22⋊Q811yescharacteristic
213C24#12311yescharacteristic
214C24#12511yescharacteristic
215C24#12511yescharacteristic
216C24#12611yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—88no—
251C22×C4—88no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C4⋊C411yescharacteristic
264C22×C4#11711yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4C4×Q811yescharacteristic
267C22×C4C4×Q811yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4C22⋊Q811yescharacteristic
279C22×C4C22⋊Q811yescharacteristic
280C22×C4C22⋊Q811yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4#12311yescharacteristic
283C22×C4#12411yescharacteristic
284C22×C4#12511yescharacteristic
285C22×C4#12511yescharacteristic
286C22×C4#12611yescharacteristic
287C22×C4C4⋊Q811yescharacteristic
288C22×C4C4⋊Q811yescharacteristic
289C22×C4C4⋊Q811yescharacteristic
290M16—44no—
291M16—44no—
292M16—44no—
293M16—44no—
294M16—82no—
295M16—82no—
296C2×C8—24no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—44no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—81no—
308C2×C8—81no—
309C2×C8—82no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—82no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C22⋊C4—81no—
329C22⋊C4—81no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C42—42no—
333C42—42no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23#38611yescharacteristic
375C23#38711yescharacteristic
376C23#38911yescharacteristic
377C23#39611yescharacteristic
378C23#39811yescharacteristic
379C23#39811yescharacteristic
380C23#40011yescharacteristic
381C23#47111yescharacteristic
382C23#47111yescharacteristic
383C23#47211yescharacteristic
384C23#47311yescharacteristic
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
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—44no—
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—41no—
427C2×C4—41no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—88no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—84no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—161no—
439C2×C4—161no—
440C2×C4—161no—
441C2×C4—161no—
442C2×C4#38011yescharacteristic
443C2×C4#38711yescharacteristic
444C2×C4#38911yescharacteristic
445C2×C4#39111yescharacteristic
446C2×C4#39411yescharacteristic
447C2×C4#39511yescharacteristic
448C2×C4#39611yescharacteristic
449C2×C4#39811yescharacteristic
450C2×C4B(4,𝔽2)12yes—
451C2×C4#45812yes—
452C8—44no—
453C8—82no—
454C8—82no—
455C8—82no—
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—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22#144111yescharacteristic
497C22#188911yescharacteristic
498C22#189011yescharacteristic
499C22#190211yescharacteristic
500C22#190311yescharacteristic
501C22#190711yescharacteristic
502C22#190811yescharacteristic
503C22#206311yescharacteristic
504C22#206411yescharacteristic
505C22#208911yescharacteristic
506C22#209011yescharacteristic
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—81no—
514C4—81no—
515C4—84no—
516C4—81no—
517C4—81no—
518C4—162no—
519C4#204412yes—
520C4#206212yes—
521C2—21no—
522C2—21no—
523C2—21no—
524C2—21no—
525C2—41no—
526C2—41no—
527C2—41no—
528C2—41no—
529C2#951911yescharacteristic
530C2#952011yescharacteristic
531C2#952111yescharacteristic
532C2#952211yescharacteristic
533C2#1121411yescharacteristic
534C2#1121611yescharacteristic
535C2#1122211yescharacteristic
subgroupG/H
center, socleC23#386
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
17. #1853C4semidirect
19. #1852C4semidirect

Maximal subgroups

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

subgroupindex
#219332
#208442
#107302
#107272
C2×#13052
#80162
C2×#12932