Finite Group Databasebeta

← #196449

Subgroups — #196449

Order 512.

1204 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic
2#35179C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13942C211yesmaximal, characteristic
6#9116C218yesmaximal
7#8026C211yesmaximal, characteristic
8#8014C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#2983—24no—
12#2983—24no—
13C2×#446—22no—
14C2×#446—22no—
15C2×#432C2214yes—
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2855C2214yes—
19#2627—22no—
20#2582C414yescomplemented
21#2581—22no—
22#2568C412yescomplemented
23#2567C412yescomplemented
24#2565—22no—
25#2317—21no—
26#2310—21no—
27C2×#391—21no—
28#2285C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1518—24no—
32#1503—24no—
33C2×#328C2214yes—
34C2×#328C2214yes—
35#1441—22no—
36C4⋊#95—21no—
37#1434C2214yes—
38#1294C2214yes—
39#1294C2214yes—
40C23×Q8D411yescharacteristic
41#533—22no—
42#533—22no—
43#533—22no—
44#533—22no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2312yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8D412yes—
59C2×C4×Q8D412yes—
60C2×#117—21no—
61C2×#117—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D412yes—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#446—216no—
70#446—216no—
71#432—28no—
72#432—28no—
73#431—22no—
74#431—22no—
75#431D414yes—
76#431D414yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—24no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×#97—24no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#400—24no—
86#398—28no—
87#395—24no—
88#391—22no—
89#391—22no—
90#391—24no—
91#391—44no—
92#391C2×C418yes—
93#389—24no—
94#389C2×C418yes—
95#387—28no—
96#385—24no—
97#384—22no—
98#384C2×C412yes—
99#382—22no—
100#382—21no—
101#382—22no—
102#382—22no—
103#381—24no—
104#380—21no—
105#380C2312yes—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C412yes—
110C2×#95—24no—
111C2×#95—24no—
112C2×#95—21no—
113C2×#95C2312yes—
114C2×#95C2312yes—
115#328—28no—
116#328—28no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8C22⋊C412yes—
125C22×Q8C22⋊C412yes—
126C22×Q8C22×C412yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140#125—42no—
141#125—42no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—42no—
159C4×Q8—42no—
160#117—42no—
161#117—42no—
162#117—42no—
163#117—42no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C414yes—
184C2×C4⋊C4C22⋊C414yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C414yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42C22×C414yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D414yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4112yes—
207C2×C42#4112yes—
208C4⋊C8—216no—
209C4⋊C8—24no—
210C4⋊C8—24no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8C2×D414yes—
214C22⋊C8C2×D414yes—
215#97—216no—
216C4×C8—24no—
217C4×C8—24no—
218#95—22no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—48no—
228#95—48no—
229#95—42no—
230C24C22×D411yescharacteristic
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—44no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—82no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C2×C22⋊C414yes—
273C22×C4C2×C22⋊C414yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4#12112yes—
279C22×C4#12112yes—
280C22×C4#12112yes—
281C22×C4#12112yes—
282C22×C4AGL(1,ℤ/8)12yes—
283C22×C4#13712yes—
284C22×C4#13712yes—
285C22×C4#13712yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—44no—
295C2×C8—44no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—82no—
312C4⋊C4—88no—
313C4⋊C4—84no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—44no—
323C42—44no—
324C42—84no—
325C42C4×D412yes—
326C42C4×D412yes—
327C42C4×D412yes—
328C42C4×D412yes—
329C42#12012yes—
330C42#12012yes—
331C42#12312yes—
332C42#12312yes—
333C23—22no—
334C23—22no—
335C23C22×C22⋊C411yescharacteristic
336C23C2×#12111yescharacteristic
337C23C2×#12111yescharacteristic
338C23C2×AGL(1,ℤ/8)11yescharacteristic
339C23C2×#13711yescharacteristic
340C23C2×#13711yescharacteristic
341C23C2×#13711yescharacteristic
342Q8—48no—
343Q8—48no—
344Q8—82no—
345Q8—82no—
346Q8—84no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4#38612yes—
388C2×C4#38612yes—
389C2×C4#38612yes—
390C2×C4#38612yes—
391C2×C4#41712yes—
392C2×C4#41712yes—
393C2×C4#41912yes—
394C2×C4#41912yes—
395C2×C4#46814yes—
396C2×C4#46814yes—
397C2×C4#47014yes—
398C2×C4#47014yes—
399C2×C4C2×C4×D411yescharacteristic
400C2×C4C2×C4×D411yescharacteristic
401C2×C4C2×#12011yescharacteristic
402C2×C4C2×#12311yescharacteristic
403C8—44no—
404C8—44no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22C2×#38611yescharacteristic
414C22#288412yes—
415C22C2×#41911yescharacteristic
416C22C2×#41711yescharacteristic
417C22#290112yes—
418C22C2×#46811yescharacteristic
419C22C2×#46811yescharacteristic
420C22C2×#47011yescharacteristic
421C22C2×#47011yescharacteristic
422C4—22no—
423C4—22no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—44no—
429C4—44no—
430C4—84no—
431C4—82no—
432C4—84no—
433C4—82no—
434C4—162no—
435C4#189414yes—
436C4#189414yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2#1762812yes—
442C2#1763212yes—
443C2#2090311yescharacteristic
444C2C2×#189411yescharacteristic
445C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #2582C4semidirect
22. #2568C4semidirect
23. #2567C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#351792
C2×#15182
C2×#15032
#139422
#91162
#80262
#80142