Finite Group Databasebeta

← #200530

Subgroups — #200530

Order 512.

1159 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#34947C211yesmaximal, characteristic
3#20728C211yesmaximal, characteristic
4#17747C212yesmaximal, complemented
5#9404C212yesmaximal, complemented
6#9400C212yesmaximal, complemented
7#9395C212yesmaximal, complemented
8#9391C212yesmaximal, complemented
9#8098C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2990—24no—
13#2983C2212yes—
14#2973—22no—
15#2972C2212yes—
16#2970—24no—
17#2920C2212yes—
18#2920C2212yes—
19C2×#432—22no—
20#2568C2211yescharacteristic
21#2567C2211yescharacteristic
22#2566—21no—
23#2565—21no—
24#2564C414yes—
25#2563C2211yescharacteristic
26#2542C2211yescharacteristic
27#2417—22no—
28#2396C414yes—
29C4×C22⋊Q8—22no—
30#2297—21no—
31#2285—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1822—21no—
34#1801C2212yes—
35#1800C2212yes—
36#1785—21no—
37C4×#101—21no—
38C4×#100—21no—
39#1559—24no—
40#1554—24no—
41#1549C2214yes—
42#1548C2214yes—
43#1303C2214yes—
44#1303C2214yes—
45C22×#41D411yescharacteristic
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8C2×C412yes—
56C4×#41—22no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4C2×C412yes—
65C2×C4×D4D412yes—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117C2×C412yes—
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C42—21no—
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75#446—28no—
76#435—216no—
77C8×D4—28no—
78#432—24no—
79#432—24no—
80#432—22no—
81#432—24no—
82#432—22no—
83#432—22no—
84#432D414yes—
85#431—22no—
86#431—22no—
87#431—22no—
88#431C2314yes—
89#423D414yes—
90#423C2314yes—
91C2×C4⋊C8—24no—
92C2×#101—22no—
93C2×#100—22no—
94#407—22no—
95#407—22no—
96C2×C22⋊C8—24no—
97C4×M16—22no—
98C4×M16—22no—
99C2×#97—24no—
100#400C2×C414yes—
101#399C2×C414yes—
102#398—24no—
103#396—24no—
104#395C2×C414yes—
105#393C2×C414yes—
106#391—21no—
107#391—21no—
108#391—22no—
109#390—21no—
110#390—21no—
111#389—22no—
112#387—24no—
113#386—24no—
114#385—24no—
115#384—21no—
116#384—21no—
117#382—21no—
118#382—21no—
119#380—22no—
120#380—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—24no—
128C2×#95C2×C412yes—
129#343—22no—
130#343—22no—
131#333—28no—
132#332—28no—
133C2×#41—24no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—42no—
137C22×Q8—21no—
138C22×Q8C22×C412yes—
139C22×Q8C2×D411yescharacteristic
140C22×D4C22⋊C412yes—
141C22×D4C2×D411yescharacteristic
142C23×C4—22no—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151C22×C8—28no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156#125—42no—
157#125—42no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—48no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181#117—22no—
182#117—22no—
183#117—24no—
184#117—24no—
185#117—42no—
186#117—42no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22×C412yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—44no—
218C2×C42C22⋊C412yes—
219C2×C42C22⋊C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229C4⋊C8—28no—
230C4⋊C8—24no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8C2×D418yes—
234#101—44no—
235#100—44no—
236C22⋊C8—28no—
237C22⋊C8—24no—
238C22⋊C8—24no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241#97—28no—
242#97—24no—
243#97—42no—
244C4×C8—24no—
245C4×C8—42no—
246#95—22no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95C22×C412yes—
252#95C22×C412yes—
253C24C2×C22⋊C411yescharacteristic
254C24C2×C22⋊C412yes—
255#41—48no—
256#41—48no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—44no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—82no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—42no—
270C2×D4#13714yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—81no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4#12012yes—
301C22×C4#12112yes—
302C22×C4#12112yes—
303C22×C4#12312yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307M16—44no—
308M16—44no—
309M16—84no—
310C2×C8—28no—
311C2×C8—44no—
312C2×C8—88no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—48no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—84no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—48no—
334C22⋊C4—44no—
335C22⋊C4—84no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—44no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42C4×D412yes—
348C42C4×D412yes—
349C42#12011yescharacteristic
350C42#12011yescharacteristic
351C42#12112yes—
352C42#12112yes—
353C42#12311yescharacteristic
354C42#12311yescharacteristic
355C42C22×D412yes—
356C42C2×#4112yes—
357C23—21no—
358C23—21no—
359C23—22no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23#38612yes—
366C23#41914yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—44no—
369Q8—84no—
370Q8—82no—
371Q8—82no—
372D4—28no—
373D4—48no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—88no—
410C2×C4#38611yescharacteristic
411C2×C4#38611yescharacteristic
412C2×C4#38611yescharacteristic
413C2×C4#38611yescharacteristic
414C2×C4#38612yes—
415C2×C4C2×C4×D412yes—
416C2×C4C2×#12011yescharacteristic
417C2×C4C2×#12112yes—
418C2×C4C2×#12311yescharacteristic
419C2×C4C2×#13712yes—
420C2×C4#57512yes—
421C8—48no—
422C8—84no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—24no—
429C22—44no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—82no—
435C22C2×#38611yescharacteristic
436C22C2×#41911yescharacteristic
437C22#289611yescharacteristic
438C4—21no—
439C4—21no—
440C4—22no—
441C4—21no—
442C4—22no—
443C4—21no—
444C4—21no—
445C4—44no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—82no—
450C4—82no—
451C4—82no—
452C4#301812yes—
453C4#305614yes—
454C2—21no—
455C2—21no—
456C2—42no—
457C2—44no—
458C2#2095211yescharacteristic
459C2#2097211yescharacteristic
460C2#2101711yescharacteristic
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. #17747C2semidirect
5. #9404C2semidirect
6. #9400C2semidirect
7. #9395C2semidirect
8. #9391C2semidirect
9. #8098C2semidirect

Maximal subgroups

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

subgroupindex
#602822
#349472
#207282
#177472
#94042
#94002
#93952
#93912
#80982