Finite Group Databasebeta

← #117517

Subgroups — #117517

Order 512.

732 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21804C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4C2×#1549C211yesmaximal, characteristic
5C2×#1301C211yesmaximal, characteristic
6#7908C211yesmaximal, characteristic
7#7893C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1987—22no—
17#1987—22no—
18#1981—22no—
19#1956—21no—
20#1956—21no—
21#1949—22no—
22#1948—21no—
23#1947—21no—
24#1787—22no—
25#1736—21no—
26C2×#340—21no—
27C2×#340—21no—
28#1549—22no—
29#1549C414yescomplemented
30C2×#332—22no—
31#1301—22no—
32#1280—22no—
33#1279—22no—
34C23×Q8—21no—
35C22×M16—21no—
36C23×C8—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⋊Q8D412yes—
48C2×C4⋊Q8Q812yes—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#428—42no—
65#428—42no—
66C2×#107—22no—
67C2×#106—22no—
68#423—21no—
69#423—21no—
70#423C2×C412yes—
71#423D412yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D412yes—
81C2×C4⋊C8D412yes—
82C2×#103—24no—
83C2×#103—24no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8D412yes—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88#384D411yescharacteristic
89#384D411yescharacteristic
90#382—21no—
91#380—21no—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C2×#95—21no—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#332—28no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—41no—
109C22×Q8—41no—
110C22×Q8—41no—
111C22×Q8—82no—
112C22×Q8—82no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—41no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—41no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149C4⋊Q8—41no—
150C4⋊Q8C4⋊C414yes—
151#117—41no—
152#117—41no—
153#117—41no—
154#117—41no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
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—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
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⋊C4SD1612yes—
184C2×C4⋊C4Q1612yes—
185C2×C22⋊C4—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—82no—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×Q811yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202#107—44no—
203#106—44no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8C22⋊C414yes—
216#103—84no—
217#103—84no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—41no—
222C4×C8—22no—
223C4×C8—41no—
224C4×C8—41no—
225C4×C8—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230C24#12111yescharacteristic
231C2×Q8—24no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—84no—
240C2×Q8—84no—
241C2×Q8—84no—
242C2×Q8—84no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4#11711yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4#12311yescharacteristic
278C22×C4#12411yescharacteristic
279C22×C4#12611yescharacteristic
280C22×C4#12611yescharacteristic
281C22×C4#12711yescharacteristic
282C22×C4C2×SD1611yescharacteristic
283C22×C4C2×Q1611yescharacteristic
284C22×C4#13511yescharacteristic
285C22×C4#13511yescharacteristic
286C22×C4AGL(1,ℤ/8)11yescharacteristic
287C22×C4#13711yescharacteristic
288C22×C4#13711yescharacteristic
289C22×C4#13711yescharacteristic
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—81no—
305C2×C8—81no—
306C2×C8—81no—
307C2×C8—81no—
308C2×C8—81no—
309C2×C8—81no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C42—82no—
337C42—81no—
338C42—81no—
339C42—82no—
340C42C2×C22⋊C411yescharacteristic
341C42C2×C4⋊C411yescharacteristic
342C42C4×D411yescharacteristic
343C42C4×D411yescharacteristic
344C42#12111yescharacteristic
345C42#12111yescharacteristic
346C42#12111yescharacteristic
347C42C22⋊Q811yescharacteristic
348C42C22⋊Q811yescharacteristic
349C42C22⋊Q811yescharacteristic
350C42#12311yescharacteristic
351C42#12311yescharacteristic
352C42#12311yescharacteristic
353C42#12311yescharacteristic
354C42#12411yescharacteristic
355C42#12411yescharacteristic
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23#38811yescharacteristic
361C23#39211yescharacteristic
362C23#39611yescharacteristic
363C23#46511yescharacteristic
364C23#46711yescharacteristic
365C23#46811yescharacteristic
366C23#47011yescharacteristic
367Q8—48no—
368Q8—84no—
369Q8—84no—
370Q8—84no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4#38011yescharacteristic
407C2×C4#38511yescharacteristic
408C2×C4#38511yescharacteristic
409C2×C4#38611yescharacteristic
410C2×C4#38811yescharacteristic
411C2×C4#39011yescharacteristic
412C2×C4#39311yescharacteristic
413C2×C4#39411yescharacteristic
414C2×C4#39411yescharacteristic
415C2×C4#39611yescharacteristic
416C2×C4#39711yescharacteristic
417C2×C4#39911yescharacteristic
418C2×C4C4×SD1611yescharacteristic
419C2×C4C4×Q1611yescharacteristic
420C2×C4#44011yescharacteristic
421C2×C4#44111yescharacteristic
422C2×C4#44811yescharacteristic
423C2×C4#44911yescharacteristic
424C2×C4#45011yescharacteristic
425C2×C4#45111yescharacteristic
426C2×C4#45111yescharacteristic
427C2×C4#45211yescharacteristic
428C2×C4#45211yescharacteristic
429C2×C4#45211yescharacteristic
430C2×C4#46111yescharacteristic
431C2×C4#46211yescharacteristic
432C2×C4#46311yescharacteristic
433C2×C4#46411yescharacteristic
434C2×C4#47112yes—
435C2×C4#47312yes—
436C2×C4#47511yescharacteristic
437C2×C4#47711yescharacteristic
438C2×C4#47911yescharacteristic
439C2×C4#47911yescharacteristic
440C2×C4#48711yescharacteristic
441C2×C4#48811yescharacteristic
442C2×C4#48911yescharacteristic
443C2×C4#49111yescharacteristic
444C8—44no—
445C8—82no—
446C8—82no—
447C8—82no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22#143911yescharacteristic
461C22#193311yescharacteristic
462C22#193511yescharacteristic
463C22#200011yescharacteristic
464C22#200111yescharacteristic
465C22#200811yescharacteristic
466C22#200811yescharacteristic
467C22#201911yescharacteristic
468C22#202011yescharacteristic
469C22#206611yescharacteristic
470C22#206811yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—84no—
478C4—82no—
479C4—82no—
480C4—161no—
481C4—161no—
482C4#186311yescharacteristic
483C4#186511yescharacteristic
484C4#187711yescharacteristic
485C4#187811yescharacteristic
486C4#193012yes—
487C4#202411yescharacteristic
488C4#202611yescharacteristic
489C4#204211yescharacteristic
490C4#204311yescharacteristic
491C4#206412yes—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2#1018011yescharacteristic
497C2#1018111yescharacteristic
498C2#1018211yescharacteristic
499C2#1018311yescharacteristic
500C2#1106711yescharacteristic
501C2#1112911yescharacteristic
502C2#1114511yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
29. #1549C4semidirect

Maximal subgroups

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

subgroupindex
#218042
C2×#19872
#211702
C2×#15492
C2×#13012
#79082
#78932