Finite Group Databasebeta

← #117548

Subgroups — #117548

Order 512.

786 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21834C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic
4#14161C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1990—21no—
17#1990—21no—
18#1987—22no—
19#1949—22no—
20#1948—21no—
21#1947—21no—
22#1824—22no—
23#1736—21no—
24#1549—22no—
25#1549C414yescomplemented
26C2×#332—22no—
27#1446—22no—
28C4⋊#95—21no—
29C4⋊#95—21no—
30C4⋊#95—22no—
31#1303—22no—
32#1294—22no—
33#1279—22no—
34C23×Q8—21no—
35C22×M16—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#428—42no—
66#428—42no—
67C2×#107—22no—
68C2×#106—22no—
69#423—21no—
70#423—21no—
71#423C2×C412yes—
72#423D412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76C2×C4⋊C8D412yes—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#384D411yescharacteristic
82#384D411yescharacteristic
83#382—21no—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C2×#95—21no—
97C2×#95—24no—
98C2×#95—24no—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—22no—
103#340—22no—
104#332—28no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—41no—
108C22×Q8—41no—
109C22×Q8—41no—
110C22×Q8—41no—
111C22×Q8—82no—
112C22×Q8—82no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—41no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145#117—41no—
146#117—41no—
147#117—41no—
148#117—41no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4C4⋊C414yes—
181C2×C4⋊C4SD1612yes—
182C2×C4⋊C4Q1612yes—
183C2×C22⋊C4—41no—
184C2×C42—44no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×Q811yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206#107—44no—
207#106—44no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—41no—
215C4⋊C8C22⋊C414yes—
216C22⋊C8—22no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220C4×C8—22no—
221C4×C8—41no—
222C4×C8—41no—
223C4×C8—41no—
224#95—22no—
225#95—22no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—84no—
231#95—84no—
232C24#12011yescharacteristic
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8—84no—
244C2×Q8—84no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C2×C4⋊C411yescharacteristic
274C22×C4C4×D411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4C22⋊Q811yescharacteristic
285C22×C4C22⋊Q811yescharacteristic
286C22×C4#12311yescharacteristic
287C22×C4#12311yescharacteristic
288C22×C4#12311yescharacteristic
289C22×C4#12411yescharacteristic
290C22×C4#12711yescharacteristic
291C22×C4C2×SD1611yescharacteristic
292C22×C4C2×Q1611yescharacteristic
293C22×C4#13511yescharacteristic
294C22×C4#13511yescharacteristic
295C22×C4AGL(1,ℤ/8)11yescharacteristic
296C22×C4#13711yescharacteristic
297C22×C4#13711yescharacteristic
298C22×C4#13711yescharacteristic
299M16—42no—
300M16—42no—
301M16—42no—
302M16—42no—
303C2×C8—24no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—81no—
311C2×C8—81no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C42—82no—
341C42—82no—
342C42—81no—
343C42—81no—
344C42—81no—
345C42—81no—
346C42—81no—
347C42—81no—
348C42C2×C22⋊C411yescharacteristic
349C42#11711yescharacteristic
350C42C4×D411yescharacteristic
351C42C4×D411yescharacteristic
352C42#12111yescharacteristic
353C42#12111yescharacteristic
354C42C22⋊Q811yescharacteristic
355C42C22⋊Q811yescharacteristic
356C42C22⋊Q811yescharacteristic
357C42C22⋊Q811yescharacteristic
358C42#12311yescharacteristic
359C42#12311yescharacteristic
360C42#12411yescharacteristic
361C42#12411yescharacteristic
362C42#12611yescharacteristic
363C42#12611yescharacteristic
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23#38511yescharacteristic
369C23#39211yescharacteristic
370C23#39411yescharacteristic
371C23#44811yescharacteristic
372C23#44911yescharacteristic
373C23#45111yescharacteristic
374C23#45211yescharacteristic
375Q8—84no—
376Q8—84no—
377Q8—84no—
378Q8—84no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—84no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#38011yescharacteristic
422C2×C4#38511yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38811yescharacteristic
425C2×C4#38811yescharacteristic
426C2×C4#39011yescharacteristic
427C2×C4#39311yescharacteristic
428C2×C4#39411yescharacteristic
429C2×C4#39611yescharacteristic
430C2×C4#39611yescharacteristic
431C2×C4#39711yescharacteristic
432C2×C4#39911yescharacteristic
433C2×C4C4×SD1611yescharacteristic
434C2×C4C4×Q1611yescharacteristic
435C2×C4#44011yescharacteristic
436C2×C4#44111yescharacteristic
437C2×C4#45011yescharacteristic
438C2×C4#45111yescharacteristic
439C2×C4#45211yescharacteristic
440C2×C4#45211yescharacteristic
441C2×C4#45412yes—
442C2×C4#45612yes—
443C2×C4#46111yescharacteristic
444C2×C4#46211yescharacteristic
445C2×C4#46311yescharacteristic
446C2×C4#46411yescharacteristic
447C2×C4#46511yescharacteristic
448C2×C4#46711yescharacteristic
449C2×C4#46811yescharacteristic
450C2×C4#47011yescharacteristic
451C2×C4#47511yescharacteristic
452C2×C4#47711yescharacteristic
453C2×C4#47911yescharacteristic
454C2×C4#47911yescharacteristic
455C2×C4#48711yescharacteristic
456C2×C4#48811yescharacteristic
457C2×C4#48911yescharacteristic
458C2×C4#49111yescharacteristic
459C8—44no—
460C8—82no—
461C8—82no—
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#143911yescharacteristic
475C22#186311yescharacteristic
476C22#186511yescharacteristic
477C22#200011yescharacteristic
478C22#200111yescharacteristic
479C22#200311yescharacteristic
480C22#200511yescharacteristic
481C22#200811yescharacteristic
482C22#200811yescharacteristic
483C22#204211yescharacteristic
484C22#204311yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—84no—
492C4—82no—
493C4—82no—
494C4—84no—
495C4—161no—
496C4—161no—
497C4#185712yes—
498C4#187711yescharacteristic
499C4#187811yescharacteristic
500C4#193311yescharacteristic
501C4#193511yescharacteristic
502C4#202411yescharacteristic
503C4#202611yescharacteristic
504C4#203812yes—
505C4#206611yescharacteristic
506C4#206811yescharacteristic
507C2—21no—
508C2—21no—
509C2—21no—
510C2—21no—
511C2#1024111yescharacteristic
512C2#1024511yescharacteristic
513C2#1024911yescharacteristic
514C2#1025311yescharacteristic
515C2#1106711yescharacteristic
516C2#1112911yescharacteristic
517C2#1113611yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×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
25. #1549C4semidirect

Maximal subgroups

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

subgroupindex
#218342
#211702
C2×#15492
#141612
C2×#13032
#80142
#78922