Finite Group Databasebeta

← #196431

Subgroups — #196431

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic
2#35248C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4C2×#1515C211yesmaximal, characteristic
5#14346C211yesmaximal, characteristic
6#9116C214yesmaximal
7#9112C214yesmaximal
8#8025C211yesmaximal, characteristic
9#8017C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2984—24no—
13#2984—24no—
14C2×#446—22no—
15C2×#446—21no—
16C2×C8×Q8—21no—
17C2×#432C2212yes—
18C2×#432C2211yescharacteristic
19C2×#431C2212yes—
20C2×#431C2211yescharacteristic
21#2855C2214yes—
22#2634—21no—
23#2634—21no—
24#2581C412yescomplemented
25#2581C412yescomplemented
26#2568C412yescomplemented
27#2546C412yescomplemented
28#2503—21no—
29#2484—22no—
30#2434—21no—
31#2327—22no—
32#2283C2211yescharacteristic
33C2×#383C2211yescharacteristic
34C2×#382C2211yescharacteristic
35#1518—24no—
36#1515—24no—
37C2×#328C2214yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40#1446—21no—
41#1445—21no—
42#1441—22no—
43#1434C2214yes—
44#1294C2214yes—
45#1294C2212yes—
46#1294C2212yes—
47C23×Q8D411yescharacteristic
48#533—22no—
49#533—22no—
50#533—22no—
51#533—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2312yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D412yes—
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4D411yescharacteristic
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76#446—216no—
77#446—28no—
78C8×Q8—28no—
79#432—22no—
80#432—24no—
81#432—24no—
82#432D414yes—
83#431—22no—
84#431—24no—
85#431—24no—
86#431D414yes—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2311yescharacteristic
94C2×#97—22no—
95C2×#97—21no—
96C2×C4×C8—22no—
97C2×C4×C8—21no—
98#400—22no—
99#400—22no—
100#400—22no—
101#400—22no—
102#398—22no—
103#398—22no—
104#395—22no—
105#395—22no—
106#391—22no—
107#391—22no—
108#391C2×C414yes—
109#391C2×C414yes—
110#389C2×C414yes—
111#389C2×C414yes—
112#387—24no—
113#387—24no—
114#387—24no—
115#387—24no—
116#385—24no—
117#383—21no—
118#383—22no—
119#383—22no—
120#382—22no—
121#382—22no—
122#382—22no—
123#382C2×C412yes—
124#381—24no—
125#381—21no—
126#380C2312yes—
127C4×C4⋊C4C2×C412yes—
128C4×C22⋊C4—21no—
129#376—21no—
130C2×#95—24no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—24no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137#328—28no—
138#328—24no—
139#328—24no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—44no—
144C22×Q8C22⋊C411yescharacteristic
145C22×Q8C22⋊C412yes—
146C22×Q8C22⋊C411yescharacteristic
147C22×Q8C22×C412yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—48no—
163#125—42no—
164#125—42no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—22no—
178C4×Q8—24no—
179C4×Q8—22no—
180C4×Q8—24no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—42no—
186C4×Q8—42no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C414yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42C22×C412yes—
220C2×C42C22×C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42#4112yes—
230C2×C42#4112yes—
231C4⋊C8—28no—
232C4⋊C8—28no—
233C4⋊C8—24no—
234C4⋊C8—24no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—22no—
238C22⋊C8C2×D412yes—
239C22⋊C8C2×D414yes—
240C22⋊C8C2×D412yes—
241#97—28no—
242#97—24no—
243C4×C8—28no—
244C4×C8—24no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—48no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—84no—
264C24C22×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—48no—
277C2×Q8—48no—
278C2×Q8—42no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C412yes—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—42no—
325C2×C8—44no—
326C2×C8—42no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—48no—
345C4⋊C4—84no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—44no—
359C42—44no—
360C42C4×D412yes—
361C42C4×D412yes—
362C42C4×D412yes—
363C42C4×D412yes—
364C42#12012yes—
365C42#12012yes—
366C42#12312yes—
367C42#12312yes—
368C23—22no—
369C23—22no—
370C23C22×C22⋊C411yescharacteristic
371C23C2×#12111yescharacteristic
372C23C2×#12111yescharacteristic
373C23#57712yes—
374C23#57812yes—
375Q8—28no—
376Q8—48no—
377Q8—44no—
378Q8—82no—
379Q8—82no—
380Q8—84no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4—88no—
421C2×C4—84no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#38612yes—
427C2×C4#38612yes—
428C2×C4#38612yes—
429C2×C4#38612yes—
430C2×C4#42114yes—
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×#12011yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C8—42no—
436C8—44no—
437C8—42no—
438C22—21no—
439C22—21no—
440C22—24no—
441C22—24no—
442C22—21no—
443C22—21no—
444C22C2×#38611yescharacteristic
445C22#288412yes—
446C22#289812yes—
447C22C2×#42112yes—
448C22#305711yescharacteristic
449C22#305911yescharacteristic
450C22#306312yes—
451C4—22no—
452C4—22no—
453C4—24no—
454C4—22no—
455C4—22no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—82no—
464C4—84no—
465C4—82no—
466C4—162no—
467C4—162no—
468C2—21no—
469C2—22no—
470C2—21no—
471C2#1760114yes—
472C2#2090211yescharacteristic
473C2#2097411yescharacteristic
474C2#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
24. #2581C4semidirect
25. #2581C4semidirect
26. #2568C4semidirect
27. #2546C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
#352482
C2×#15182
C2×#15152
#143462
#91162
#91122
#80252
#80172