Finite Group Databasebeta

← #201536

Subgroups — #201536

Order 512.

1231 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#34952C211yesmaximal, characteristic
3#20637C211yesmaximal, characteristic
4C2×#1487C212yesmaximal, complemented
5#9488C214yesmaximal, complemented
6#9487C214yesmaximal, complemented
7#7957C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2989—24no—
11#2983C2212yes—
12#2975C2212yes—
13#2973—22no—
14#2965—24no—
15#2920—22no—
16C2×#431C2212yes—
17C2×#423C2212yes—
18C2×#413—21no—
19#2582—21no—
20#2581C2212yes—
21#2580C2212yes—
22#2579—22no—
23#2538—21no—
24#2487C412yes—
25#2409C412yes—
26#2396C414yes—
27C4×C22⋊Q8—22no—
28#2297—21no—
29#2296—21no—
30#2285C2211yescharacteristic
31#1801—21no—
32#1801C2212yes—
33#1800—21no—
34#1800C2212yes—
35#1742—21no—
36#1487—24no—
37#1487C2218yes—
38C2×Q8⋊C8—24no—
39#1286C2218yes—
40C22×#41D411yescharacteristic
41C23×Q8C2×C411yescharacteristic
42C22×M16—22no—
43C2×C22⋊Q8—22no—
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2312yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4C2×C412yes—
59C2×C4×D4D412yes—
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×#117—21no—
63C2×#117C2×C412yes—
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C42—21no—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70#446—28no—
71#436—28no—
72#435—216no—
73#432—24no—
74#432—24no—
75#431—24no—
76#431—22no—
77#431—22no—
78#431D414yes—
79#431C2314yes—
80#423—24no—
81#423—22no—
82#423—22no—
83#423—22no—
84#423C2314yes—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88#413—44no—
89C2×#101—22no—
90C2×#100—22no—
91#407—24no—
92C2×C22⋊C8—22no—
93C4×M16—24no—
94C2×C4×C8D414yes—
95#400C2×C414yes—
96#399C2×C414yes—
97#398—24no—
98#396—24no—
99#395C2×C414yes—
100#393C2×C414yes—
101#391—22no—
102#391—22no—
103#390—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#387—24no—
108#386—22no—
109#386—22no—
110#385—24no—
111#385—22no—
112#385—22no—
113#384—21no—
114#382C2312yes—
115#380—22no—
116#380—21no—
117#380C2×C412yes—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C22⋊C4—24no—
122#343—22no—
123#343—22no—
124Q8⋊C8—216no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—24no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—44no—
134C22×Q8C22×C412yes—
135C22×Q8C2×D411yescharacteristic
136C22×D4C22⋊C412yes—
137C22×D4C2×D411yescharacteristic
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—28no—
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—44no—
150C2×M16—44no—
151C22×C8—24no—
152C22×C8—42no—
153C22⋊Q8—48no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176#117—22no—
177#117—22no—
178#117—24no—
179#117—24no—
180#117—42no—
181#117—42no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22×C412yes—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42C22⋊C412yes—
215C2×C42C22⋊C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42C2411yescharacteristic
225C4⋊C8—28no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232#101—44no—
233#100—44no—
234C22⋊C8—28no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237#97—28no—
238C4×C8—24no—
239C4×C8C2×D418yes—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95C22×C414yes—
246C24C2×C22⋊C412yes—
247C24C2×C22⋊C411yescharacteristic
248#41—44no—
249#41—44no—
250#41—44no—
251#41—44no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—21no—
255C2×Q8—22no—
256C2×Q8—21no—
257C2×Q8—28no—
258C2×Q8—24no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—48no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—22no—
271C2×D4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4#12012yes—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12312yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4#13712yes—
305C22×C4#13714yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—44no—
310M16—44no—
311M16—88no—
312M16—84no—
313C2×C8—28no—
314C2×C8—44no—
315C2×C8—84no—
316C2×C8—84no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—44no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—48no—
337C22⋊C4—44no—
338C22⋊C4—84no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—44no—
346C42—42no—
347C42—42no—
348C42—44no—
349C42C4×D412yes—
350C42C4×D412yes—
351C42#12011yescharacteristic
352C42#12011yescharacteristic
353C42#12112yes—
354C42#12112yes—
355C42#12311yescharacteristic
356C42#12311yescharacteristic
357C42C22×D412yes—
358C42C2×#4112yes—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23#38612yes—
367C23#41712yes—
368C23#41912yes—
369C23C22×C22⋊C411yescharacteristic
370Q8—28no—
371Q8—44no—
372Q8—44no—
373Q8—44no—
374Q8—42no—
375Q8—42no—
376D4—48no—
377D4—42no—
378D4—42no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—84no—
411C2×C4#38611yescharacteristic
412C2×C4#38612yes—
413C2×C4#38611yescharacteristic
414C2×C4#38611yescharacteristic
415C2×C4#38611yescharacteristic
416C2×C4#41712yes—
417C2×C4#41912yes—
418C2×C4#46814yes—
419C2×C4#46814yes—
420C2×C4#47014yes—
421C2×C4#47014yes—
422C2×C4C2×C4×D412yes—
423C2×C4C2×#12011yescharacteristic
424C2×C4C2×#12112yes—
425C2×C4C2×#12311yescharacteristic
426C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
427C2×C4C2×#13712yes—
428C2×C4C2×#13711yescharacteristic
429C8—48no—
430C8—84no—
431C22—21no—
432C22—22no—
433C22—21no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—44no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22#189414yes—
443C22C2×#38611yescharacteristic
444C22C2×#41911yescharacteristic
445C22C2×#41711yescharacteristic
446C4—21no—
447C4—21no—
448C4—22no—
449C4—21no—
450C4—22no—
451C4—42no—
452C4—44no—
453C4—42no—
454C4—42no—
455C4—84no—
456C4—82no—
457C4#189112yes—
458C4#189412yes—
459C4#189412yes—
460C4C2×#46812yes—
461C4C2×#47012yes—
462C2—21no—
463C2—21no—
464C2—24no—
465C2—82no—
466C2C2×#189111yescharacteristic
467C2C2×#189411yescharacteristic
468C2#2101211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1487C2semidirect
5. #9488C2semidirect
6. #9487C2semidirect
7. #7957C2semidirect

Maximal subgroups

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

subgroupindex
#602772
#349522
#206372
C2×#14872
#94882
#94872
#79572