Finite Group Databasebeta

← #201575

Subgroups — #201575

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#35334C211yesmaximal, characteristic
3#20637C211yesmaximal, characteristic
4#17486C212yesmaximal, complemented
5#9488C214yesmaximal, complemented
6#9487C214yesmaximal, complemented
7C2×#1286C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2989—24no—
11#2983C2212yes—
12#2975C2212yes—
13#2973—22no—
14#2965—24no—
15#2920—22no—
16C2×#431C2212yes—
17C2×#423C2212yes—
18C2×#413—21no—
19#2851—22no—
20#2628C412yes—
21#2628C412yes—
22#2581—21no—
23#2581C2212yes—
24#2580—21no—
25#2580C2212yes—
26#2564—22no—
27#2564C414yes—
28#2316—21no—
29#2311—21no—
30C2×#382C2211yescharacteristic
31#1801—21no—
32#1801C2212yes—
33#1800—21no—
34#1800C2212yes—
35#1742—21no—
36#1493—24no—
37#1487C2218yes—
38#1481—24no—
39#1286C2218yes—
40C22×#41D411yescharacteristic
41C22×M16—22no—
42#533—22no—
43#533—22no—
44#533—22no—
45#533—22no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2×C412yes—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2312yes—
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4D411yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117C2×C412yes—
65C2×#117D411yescharacteristic
66C22×C4⋊C4—21no—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69#446—28no—
70#436—28no—
71#435—216no—
72#432—24no—
73#432—24no—
74#431—24no—
75#431—22no—
76#431—22no—
77#431D414yes—
78#431C2314yes—
79#423—24no—
80#423—22no—
81#423—22no—
82#423—22no—
83#423C2314yes—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87#413—44no—
88C2×#101—22no—
89C2×#100—22no—
90#407—24no—
91C2×C22⋊C8—22no—
92C4×M16—24no—
93C2×C4×C8D414yes—
94#400—22no—
95#400C2×C414yes—
96#399—22no—
97#399C2×C414yes—
98#395—22no—
99#395C2×C414yes—
100#393—22no—
101#393C2×C414yes—
102#391—22no—
103#390—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#387—24no—
109#387—24no—
110#386—24no—
111#385—24no—
112#385—24no—
113#382—21no—
114#382—21no—
115#382C2×C412yes—
116#382C2312yes—
117#380—22no—
118#380—22no—
119#380—24no—
120C2×#95—21no—
121C2×#95C2×C412yes—
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×Q8C22×C412yes—
132C22×Q8C2×D411yescharacteristic
133C22×D4C22⋊C412yes—
134C22×D4C2×D411yescharacteristic
135C23×C4—22no—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×M16—28no—
143C2×M16—24no—
144C2×M16—42no—
145C2×M16—44no—
146C2×M16—44no—
147C22×C8—24no—
148C22×C8—42no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153#125—41no—
154#125—41no—
155#125—41no—
156#125—41no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—42no—
160C22⋊Q8—42no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177#117—22no—
178#117—22no—
179#117—24no—
180#117—24no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
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×C42C22⋊C412yes—
211C2×C42C22⋊C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42C2411yescharacteristic
221C4⋊C8—28no—
222C4⋊C8—24no—
223C4⋊C8—24no—
224C4⋊C8—24no—
225C4⋊C8—24no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228#101—44no—
229#100—44no—
230C22⋊C8—28no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233#97—28no—
234C4×C8—24no—
235C4×C8C2×D418yes—
236#95—22no—
237#95—22no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95C22×C414yes—
244C24C2×C22⋊C412yes—
245C24C2×C22⋊C411yescharacteristic
246#41—44no—
247#41—44no—
248#41—44no—
249#41—44no—
250C2×Q8—22no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—21no—
254C2×Q8—24no—
255C2×Q8—42no—
256C2×Q8—82no—
257C2×Q8—82no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—22no—
264C2×D4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—44no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C4×D412yes—
290C22×C4C4×D412yes—
291C22×C4#12012yes—
292C22×C4#12112yes—
293C22×C4#12112yes—
294C22×C4#12312yes—
295C22×C4AGL(1,ℤ/8)12yes—
296C22×C4#13712yes—
297C22×C4#13714yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301M16—44no—
302M16—44no—
303M16—88no—
304M16—84no—
305C2×C8—28no—
306C2×C8—44no—
307C2×C8—84no—
308C2×C8—84no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—84no—
327C4⋊C4—84no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—44no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—44no—
341C42C4×D412yes—
342C42C4×D412yes—
343C42#12011yescharacteristic
344C42#12011yescharacteristic
345C42#12112yes—
346C42#12112yes—
347C42#12311yescharacteristic
348C42#12311yescharacteristic
349C42C22×D412yes—
350C42C2×#4112yes—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23#38612yes—
359C23#41712yes—
360C23#41912yes—
361C23C22×C22⋊C411yescharacteristic
362Q8—48no—
363Q8—42no—
364Q8—42no—
365Q8—84no—
366D4—48no—
367D4—42no—
368D4—42no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—164no—
400C2×C4#38611yescharacteristic
401C2×C4#38611yescharacteristic
402C2×C4#38612yes—
403C2×C4#38611yescharacteristic
404C2×C4#38611yescharacteristic
405C2×C4#41712yes—
406C2×C4#41912yes—
407C2×C4#46814yes—
408C2×C4#46814yes—
409C2×C4#47014yes—
410C2×C4#47014yes—
411C2×C4C2×C4×D412yes—
412C2×C4C2×#12011yescharacteristic
413C2×C4C2×#12112yes—
414C2×C4C2×#12311yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4C2×#13712yes—
417C2×C4C2×#13711yescharacteristic
418C8—48no—
419C8—84no—
420C22—21no—
421C22—22no—
422C22—21no—
423C22—22no—
424C22—42no—
425C22—42no—
426C22—44no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—82no—
431C22#189414yes—
432C22C2×#38611yescharacteristic
433C22C2×#41911yescharacteristic
434C22C2×#41711yescharacteristic
435C4—21no—
436C4—21no—
437C4—22no—
438C4—21no—
439C4—21no—
440C4—22no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—84no—
445C4—82no—
446C4—82no—
447C4#189412yes—
448C4#189412yes—
449C4C2×#46812yes—
450C4C2×#47012yes—
451C2—21no—
452C2—21no—
453C2—24no—
454C2—82no—
455C2#2096411yescharacteristic
456C2C2×#189411yescharacteristic
457C2#2101211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17486C2semidirect
5. #9488C2semidirect
6. #9487C2semidirect
7. C2×#1286C2semidirect

Maximal subgroups

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

subgroupindex
#602772
#353342
#206372
#174862
#94882
#94872
C2×#12862