Finite Group Databasebeta

← #201169

Subgroups — #201169

Order 512.

1065 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#35179C211yesmaximal, characteristic
3#17664C212yesmaximal
4#17365C211yesmaximal, characteristic
5#9450C218yesmaximal, complemented
6#7953C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—24no—
10#2983—21no—
11#2983C2214yes—
12#2982—21no—
13Q8×M16—24no—
14#2920—22no—
15#2920C2212yes—
16#2920C2212yes—
17#2855—22no—
18#2627—22no—
19#2582—21no—
20#2582—21no—
21#2581—21no—
22#2581—21no—
23#2568C412yes—
24#2567C412yes—
25#2565C2214yes—
26#2317C412yes—
27#2310C412yes—
28#2285C2211yescharacteristic
29#1537—24no—
30#1534C2218yes—
31#1526—24no—
32#1467C2214yes—
33#1463—22no—
34#1461—21no—
35#1461—21no—
36#1285C2218yes—
37C23×Q8C2×C411yescharacteristic
38#533—22no—
39#533—22no—
40#533—22no—
41#533—22no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125C2312yes—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8C2312yes—
59C2×#117C2×C412yes—
60C2×#117C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4D412yes—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65#446—28no—
66#446—216no—
67C8×Q8—28no—
68#432—24no—
69#432—24no—
70#432—22no—
71#432D414yes—
72#432C2314yes—
73#432C2314yes—
74#431—24no—
75#431—24no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#423—24no—
81#423—22no—
82#423—22no—
83#407—22no—
84#407—22no—
85C4×M16—24no—
86C4×M16—22no—
87C4×M16D414yes—
88#400—22no—
89#400—22no—
90#398—24no—
91#398—24no—
92#395—22no—
93#395—22no—
94#391—22no—
95#391—22no—
96#391—22no—
97#391—22no—
98#391C2×C414yes—
99#391C2×C414yes—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#387—24no—
105#387—24no—
106#385—24no—
107#384—21no—
108#384C2×C414yes—
109#382—21no—
110#382C2×C414yes—
111#381—24no—
112#380—21no—
113#380—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2312yes—
116#330—216no—
117#324—24no—
118#324—24no—
119C22×Q8—24no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—42no—
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22×C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C2×M16—24no—
134C2×M16—42no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139#125—24no—
140#125—24no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—42no—
144C22⋊Q8—42no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163#117—24no—
164#117—24no—
165#117—44no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22×C414yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—28no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—24no—
213C4⋊C8—42no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218#97—28no—
219#97—24no—
220#97C2×D418yes—
221C4×C8—28no—
222C4×C8—24no—
223C4×C8—42no—
224#95—22no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—22no—
232C24C2×C22⋊C411yescharacteristic
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8—48no—
242C2×Q8—48no—
243C2×Q8—44no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—82no—
248C2×Q8#10114yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—81no—
262C22×C4—82no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12112yes—
275C22×C4#12112yes—
276C22×C4#12311yescharacteristic
277C22×C4#12311yescharacteristic
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281M16—48no—
282M16—84no—
283C2×C8—28no—
284C2×C8—44no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—44no—
302C4⋊C4—82no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—44no—
317C42—44no—
318C42C4×D412yes—
319C42C4×D412yes—
320C42#12012yes—
321C42#12112yes—
322C42#12112yes—
323C42#12312yes—
324C42C22×D412yes—
325C42C2×#4112yes—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23#38611yescharacteristic
331C23#38611yescharacteristic
332C23C22×C22⋊C411yescharacteristic
333Q8—28no—
334Q8—48no—
335Q8—44no—
336Q8—82no—
337Q8—84no—
338Q8—82no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4#38611yescharacteristic
375C2×C4#38612yes—
376C2×C4#38612yes—
377C2×C4#38611yescharacteristic
378C2×C4C2×#10114yes—
379C2×C4C2×#10112yes—
380C2×C4#41312yes—
381C2×C4C2×C4×D412yes—
382C2×C4C2×#12011yescharacteristic
383C2×C4C2×#12112yes—
384C2×C4C2×#12311yescharacteristic
385C8—48no—
386C8—84no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—42no—
396C22—42no—
397C22C2×#38611yescharacteristic
398C22C22×#10111yescharacteristic
399C22C2×#41311yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—24no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—84no—
411C4—82no—
412C4—82no—
413C4#190114yes—
414C4#191212yes—
415C2—21no—
416C2—21no—
417C2—41no—
418C2—41no—
419C2#2094711yescharacteristic
420C2C2×#190111yescharacteristic
421C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
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
5. #9450C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#351792
#176642
#173652
#94502
#79532