Finite Group Databasebeta

← #214653

Subgroups — #214653

Order 512.

757 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37392C211yesmaximal, characteristic
2#21832C211yesmaximal, characteristic
3#21828C211yesmaximal, characteristic
4#9461C212yesmaximal
5#9460C212yesmaximal
6#9444C212yesmaximal
7#9444C212yesmaximal
8#8741C211yesmaximal, characteristic
9#8740C211yesmaximal, characteristic
10#8521C211yesmaximal, characteristic
11#8521C211yesmaximal, characteristic
12C2×#544C2211yescharacteristic
13C2×#541C2211yescharacteristic
14#2921C2212yes—
15#2862C411yescharacteristic, complemented
16#2859C411yescharacteristic, complemented
17#2719—22no—
18#2713—22no—
19#2698—22no—
20#2689—22no—
21#2660C411yescharacteristic, complemented
22#2642C411yescharacteristic, complemented
23#2568C411yescharacteristic, complemented
24#2567C411yescharacteristic, complemented
25#2565C411yescharacteristic, complemented
26#2565C411yescharacteristic, complemented
27#2503C2211yescharacteristic
28#2502C2211yescharacteristic
29#2499C2211yescharacteristic
30#2499C2211yescharacteristic
31#2313C2212yes—
32#2298C2211yescharacteristic
33#1998—22no—
34#1997—22no—
35#1996—22no—
36#1996—22no—
37#1822C2211yescharacteristic
38#1822C2211yescharacteristic
39#1821C2212yes—
40#1820C2211yescharacteristic
41#1820C2211yescharacteristic
42#1820C2212yes—
43#1511C2212yes—
44#1511C2212yes—
45#1511C2212yes—
46#1511C2212yes—
47#1284C2212yes—
48#1284C2212yes—
49#1284C2212yes—
50#1284C2212yes—
51#544—42no—
52#544—42no—
53#541—42no—
54#541—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8D412yes—
59C2×#126—21no—
60C2×#126—21no—
61C2×#125—21no—
62C2×#125—21no—
63C2×#125C2×C411yescharacteristic
64C2×#125C2×C411yescharacteristic
65C2×#125C2×C411yescharacteristic
66C2×#125D412yes—
67C2×C22⋊Q8D412yes—
68C2×C22⋊Q8D412yes—
69C2×C22⋊Q8C2311yescharacteristic
70C2×C22⋊Q8C2311yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76#433—42no—
77#433—42no—
78#423—22no—
79#407C2312yes—
80#407C2312yes—
81C4×M16—22no—
82#400—22no—
83#400—22no—
84#400—22no—
85#400—22no—
86#398—22no—
87#398—22no—
88#398—22no—
89#398—22no—
90#397—22no—
91#397—22no—
92#395—22no—
93#395—22no—
94#391—22no—
95#391—22no—
96#391C2×C411yescharacteristic
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#389—22no—
105#389—22no—
106#389—22no—
107#388—21no—
108#388—21no—
109#387—22no—
110#387—22no—
111#387—21no—
112#387—21no—
113#387—22no—
114#387—22no—
115#387—42no—
116#387—42no—
117#385—22no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#384C2×C411yescharacteristic
122#382—22no—
123#382C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#381—22no—
128#381C2312yes—
129#381C2312yes—
130#380—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—22no—
134C4×C22⋊C4C2311yescharacteristic
135C4×C22⋊C4C2311yescharacteristic
136#376C2×C411yescharacteristic
137C43C2×C411yescharacteristic
138#343—24no—
139#343—21no—
140#343—21no—
141#343—21no—
142#343—21no—
143#342—21no—
144#342—21no—
145#342—21no—
146#342—21no—
147#342—24no—
148#328—42no—
149#328—42no—
150#328—42no—
151#328—42no—
152C22×Q8C22⋊C412yes—
153C22×Q8C22⋊C412yes—
154C22×Q8C22×C411yescharacteristic
155C22×Q8C22×C411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2411yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168#126—44no—
169#126—44no—
170#125—41no—
171#125—41no—
172#125—41no—
173#125—41no—
174#125—41no—
175#125—41no—
176#125—42no—
177#125—42no—
178#125—44no—
179#125—44no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192#117—22no—
193#117—22no—
194#117—22no—
195#117—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4C2×D411yescharacteristic
235C2×C22⋊C4C2×D411yescharacteristic
236C2×C22⋊C4C2×D412yes—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—42no—
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C2×D412yes—
249C4⋊C8—44no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254#97—42no—
255C4×C8—42no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—42no—
279#95—42no—
280C24C22×D411yescharacteristic
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—82no—
292C2×Q8—82no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—82no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4H(5,𝔽2)12yes—
319C22×C4H(5,𝔽2)12yes—
320M16—42no—
321M16—42no—
322M16—42no—
323M16—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4—162no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—81no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—22no—
380C42—22no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—81no—
389C42—162no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23C22×C22⋊C411yescharacteristic
395C23#53412yes—
396Q8—81no—
397Q8—81no—
398Q8—81no—
399Q8—81no—
400Q8—162no—
401Q8—162no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#51811yescharacteristic
455C2×C4#51811yescharacteristic
456C2×C4#53412yes—
457C8—82no—
458C8—82no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—81no—
465C22—81no—
466C22#230811yescharacteristic
467C22#288411yescharacteristic
468C22#288411yescharacteristic
469C4—22no—
470C4—21no—
471C4—21no—
472C4—42no—
473C4—41no—
474C4—41no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—41no—
479C4—41no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—162no—
489C4—162no—
490C2—21no—
491C2—21no—
492C2—42no—
493C2#2175111yescharacteristic
494C2#2178311yescharacteristic
495C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2862C4semidirect
16. #2859C4semidirect
21. #2660C4semidirect
22. #2642C4semidirect
23. #2568C4semidirect
24. #2567C4semidirect
25. #2565C4semidirect
26. #2565C4semidirect

Maximal subgroups

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

subgroupindex
#373922
#218322
#218282
#94612
#94602
#94442
#87412
#87402
#85212