Finite Group Databasebeta

← #192049

Subgroups — #192049

Order 512.

1023 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#21976C212yesmaximal, complemented
3#21851C211yesmaximal, characteristic
4C4×#410C211yesmaximal, characteristic
5#8852C214yesmaximal, complemented
6#8851C214yesmaximal, complemented
7#7930C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2990—22no—
10#2990—22no—
11#2983—21no—
12#2973—21no—
13#2973C2212yes—
14#2972C2212yes—
15#2970—24no—
16#2920—21no—
17C2×#432—21no—
18C2×#432C2211yescharacteristic
19C2×#432C2212yes—
20C22×C4⋊C8C2211yescharacteristic
21C2×#410C412yes—
22C4×#117C2212yes—
23#1985—21no—
24#1955—21no—
25#1940—24no—
26#1869C2212yes—
27#1869C2212yes—
28#1850C2218yes—
29#1842—22no—
30#1831—22no—
31#1797—21no—
32#1797—21no—
33C4×#99C412yes—
34C4×#99C411yescharacteristic
35C4×#99C411yescharacteristic
36C4×#99C2212yes—
37C4×#99C2212yes—
38#1740C412yes—
39#1282C2218yes—
40C22×#41C2×C411yescharacteristic
41C23×C8C2312yes—
42C4×#41—22no—
43C4×#41—22no—
44C4×#41—22no—
45C4×#41—22no—
46C2×C4×Q8D411yescharacteristic
47C2×C4×Q8Q811yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4D411yescharacteristic
51C2×C4×D4Q811yescharacteristic
52C2×C4×D4C2311yescharacteristic
53C2×C4×D4C2311yescharacteristic
54C2×#117C2×C414yes—
55C2×#117C2×C411yescharacteristic
56C2×#117Q811yescharacteristic
57C2×#117Q811yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61#446—24no—
62#446—24no—
63#435—24no—
64#435—24no—
65#435—24no—
66#435—24no—
67C8×D4—24no—
68C8×D4—24no—
69#432—22no—
70#432—22no—
71#432—22no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432—22no—
76#432—22no—
77#432—22no—
78#432—22no—
79#431—22no—
80#431—22no—
81#423—21no—
82#423—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—28no—
91C2×C4⋊C8D412yes—
92C2×C4⋊C8D412yes—
93#410—28no—
94C2×#99C2×C412yes—
95C2×#99C2×C414yes—
96C2×#99C2×C412yes—
97#407—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8C2312yes—
102C2×C22⋊C8C2314yes—
103C4×M16—22no—
104C2×#97—22no—
105C2×#97—21no—
106C2×#97D414yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4C2×C412yes—
111C4×C22⋊C4C2314yes—
112#376—22no—
113C43—22no—
114#342C2×C412yes—
115#342C2×C414yes—
116#342C2×C412yes—
117#336—28no—
118#336—22no—
119#336—22no—
120C2×#41—24no—
121C2×#41—22no—
122C2×#41—22no—
123C2×#41—42no—
124C22×Q8C4⋊C411yescharacteristic
125C22×D4C4⋊C411yescharacteristic
126C22×D4C22×C411yescharacteristic
127C22×D4C22×C411yescharacteristic
128C23×C4C4⋊C411yescharacteristic
129C23×C4C4⋊C411yescharacteristic
130C23×C4C4⋊C411yescharacteristic
131C23×C4C4⋊C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C2×M16—22no—
138C2×M16—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—24no—
143C22×C8—22no—
144C22×C8—82no—
145C22×C8C2×D414yes—
146C22×C8C2×D414yes—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—28no—
170#117—44no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4C4⋊C411yescharacteristic
173C2×C4⋊C4C4⋊C411yescharacteristic
174C2×C4⋊C4C4⋊C411yescharacteristic
175C2×C4⋊C4C4⋊C411yescharacteristic
176C2×C4⋊C4C2×Q811yescharacteristic
177C2×C4⋊C4C2×Q811yescharacteristic
178C2×C22⋊C4C4⋊C411yescharacteristic
179C2×C22⋊C4C4⋊C411yescharacteristic
180C2×C22⋊C4C22×C414yes—
181C2×C22⋊C4C22×C411yescharacteristic
182C2×C22⋊C4C22×C411yescharacteristic
183C2×C22⋊C4C22×C414yes—
184C2×C22⋊C4C2×Q811yescharacteristic
185C2×C22⋊C4C2×Q811yescharacteristic
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42C4⋊C411yescharacteristic
199C2×C42C4⋊C411yescharacteristic
200C2×C42C4⋊C411yescharacteristic
201C2×C42C4⋊C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×Q811yescharacteristic
209C2×C42C2×Q811yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—41no—
221C4⋊C8—42no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—88no—
226#99—24no—
227#99—28no—
228#99M1618yescomplemented
229C22⋊C8—24no—
230C22⋊C8—24no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233C22⋊C8—24no—
234C22⋊C8—82no—
235C22⋊C8—82no—
236#97—24no—
237#97—24no—
238#97—24no—
239#97—42no—
240C4×C8—24no—
241#95—22no—
242#95—22no—
243#95—24no—
244C24C2×C4⋊C411yescharacteristic
245C24C2×C4⋊C411yescharacteristic
246C24C23×C411yescharacteristic
247#41—48no—
248#41—48no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—41no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4C2×M1614yes—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—24no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4C2×C4⋊C411yescharacteristic
291C22×C4C2×C4⋊C411yescharacteristic
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C2×C4⋊C411yescharacteristic
297C22×C4C2×C4⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4C4×D412yes—
303C22×C4C4×D412yes—
304C22×C4C22×D412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C2×#4112yes—
308M16—82no—
309M16—82no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—48no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—82no—
322C2×C8—82no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—48no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—24no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4C2×M1618yes—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—48no—
353C42—44no—
354C42—44no—
355C42—44no—
356C42—44no—
357C42#12011yescharacteristic
358C42#12012yes—
359C42#12011yescharacteristic
360C42C22⋊Q812yes—
361C42C22⋊Q812yes—
362C42C22⋊Q812yes—
363C42C22⋊Q812yes—
364C42#12312yes—
365C42#12311yescharacteristic
366C42#12311yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23#42312yes—
379C23C22×C4⋊C411yescharacteristic
380C23C2×C4×D412yes—
381C23C22×M1612yes—
382C23#56812yes—
383Q8—44no—
384D4—28no—
385D4—44no—
386D4—44no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4#38511yescharacteristic
437C2×C4#38511yescharacteristic
438C2×C4#38511yescharacteristic
439C2×C4#38512yes—
440C2×C4#38512yes—
441C2×C4#38511yescharacteristic
442C2×C4#42312yes—
443C2×C4C2×#12011yescharacteristic
444C2×C4C2×C22⋊Q812yes—
445C2×C4C2×#12311yescharacteristic
446C8—44no—
447C8—44no—
448C8—82no—
449C8—82no—
450C22—22no—
451C22—21no—
452C22—22no—
453C22—21no—
454C22—22no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—81no—
465C22—81no—
466C22C2×#38511yescharacteristic
467C22C2×#42311yescharacteristic
468C22#290511yescharacteristic
469C22D4×M1614yes—
470C4—21no—
471C4—21no—
472C4—22no—
473C4—22no—
474C4—21no—
475C4—22no—
476C4—21no—
477C4—22no—
478C4—48no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—84no—
485C4#186812yes—
486C4#302711yescharacteristic
487C4#302811yescharacteristic
488C2—24no—
489C2—21no—
490C2—21no—
491C2—42no—
492C2—81no—
493C2#2083711yescharacteristic
494C2#2089011yescharacteristic
495C2C2×#186811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21976C2semidirect
5. #8852C2semidirect
6. #8851C2semidirect
7. #7930C2semidirect
228. #99M16semidirect

Maximal subgroups

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

subgroupindex
#602822
#219762
#218512
C4×#4102
#88522
#88512
#79302