Finite Group Databasebeta

← #190197

Subgroups — #190197

Order 512.

802 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34381C211yesmaximal, characteristic
2#17398C211yesmaximal, characteristic
3#17398C211yesmaximal, characteristic
4#8759C212yesmaximal
5#8758C212yesmaximal, complemented
6#8757C212yesmaximal, complemented
7#8756C212yesmaximal
8#8110C211yesmaximal, characteristic, complemented
9#8109C211yesmaximal, characteristic, complemented
10#8106C211yesmaximal, characteristic, complemented
11#8105C211yesmaximal, characteristic, complemented
12C22×#126C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15#2920C2211yescharacteristic
16#2920C2211yescharacteristic
17#2383—22no—
18#2373—22no—
19#2345C412yes—
20#2344C412yes—
21#2342C412yes—
22#2342C412yes—
23#2334—22no—
24#2334—22no—
25#2316C2211yescharacteristic
26#2315C2211yescharacteristic
27#2313C2211yescharacteristic
28#2312C2211yescharacteristic
29#2285C2211yescharacteristic
30#2283C2211yescharacteristic
31#1473—22no—
32#1473—22no—
33#1473—22no—
34#1473—22no—
35#1463C2212yes—
36#1463C2212yes—
37#1462C2212yes—
38#1462C2212yes—
39#1305C2212yes—
40#1305C2212yes—
41#1305C2212yes—
42#1305C2212yes—
43#1305C2212yes—
44#1305C2212yes—
45#1305C2212yes—
46#1305C2212yes—
47C2×#126—21no—
48C2×#126—21no—
49C2×#126—21no—
50C2×#126—21no—
51C2×#126—21no—
52C2×#126—21no—
53C2×#126—21no—
54C2×#126—21no—
55C2×#126D412yes—
56C2×#126D412yes—
57C2×#126D412yes—
58C2×#126D412yes—
59C2×#117C2×C412yes—
60C2×#117C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#432—21no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423—21no—
89#407C2312yes—
90#407C2312yes—
91#407C2312yes—
92#407C2312yes—
93C4×M16—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#384—21no—
118#383—21no—
119#382—21no—
120#382—21no—
121#381C2×C412yes—
122#381C2×C412yes—
123#380C2×C412yes—
124#380C2×C412yes—
125#380C2×C412yes—
126#380C2311yescharacteristic
127#380C2311yescharacteristic
128#380C2311yescharacteristic
129#380C2311yescharacteristic
130#379C2×C412yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4C2×C412yes—
139C4×C22⋊C4C2×C412yes—
140C4×C22⋊C4C2×C412yes—
141#376—21no—
142#324—24no—
143#324—24no—
144#323—24no—
145#323—24no—
146C25C22×C411yescharacteristic
147C23×C4—22no—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C2411yescharacteristic
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158#126—22no—
159#126—22no—
160#126—22no—
161#126—22no—
162#126—22no—
163#126—22no—
164#126—22no—
165#126—22no—
166#126—22no—
167#126—22no—
168#126—22no—
169#126—22no—
170#126—22no—
171#126—22no—
172#126—22no—
173#126—22no—
174#117—82no—
175#117—82no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22×C412yes—
224C2×C22⋊C4C22×C412yes—
225C2×C22⋊C4C22×C412yes—
226C2×C22⋊C4C22×C412yes—
227C2×C22⋊C4C2×D412yes—
228C2×C22⋊C4C2×D412yes—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262#97—41no—
263#97—41no—
264#97—41no—
265#97—41no—
266C4×C8—41no—
267C4×C8—41no—
268C4×C8—41no—
269C4×C8—41no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284C24—21no—
285C24—21no—
286C24—41no—
287C24—41no—
288C24—81no—
289C24—81no—
290C24C2×C22⋊C412yes—
291C24C23×C411yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4#11711yescharacteristic
321C22×C4#11711yescharacteristic
322C22×C4#11711yescharacteristic
323C22×C4#11711yescharacteristic
324C22×C4#11711yescharacteristic
325C22×C4#11711yescharacteristic
326C22×C4#11711yescharacteristic
327C22×C4#11711yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334M16—82no—
335M16—82no—
336M16—82no—
337M16—82no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C42—82no—
383C42—82no—
384C42—82no—
385C42—82no—
386C42—82no—
387C42#12311yescharacteristic
388C42#12311yescharacteristic
389C42#12311yescharacteristic
390C42#12311yescharacteristic
391C42#12311yescharacteristic
392C42#12311yescharacteristic
393C42#12311yescharacteristic
394C42#12311yescharacteristic
395C42#12311yescharacteristic
396C42#12311yescharacteristic
397C42#12311yescharacteristic
398C42#12311yescharacteristic
399C42#12311yescharacteristic
400C42#12311yescharacteristic
401C42#12311yescharacteristic
402C42#12311yescharacteristic
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—162no—
419C23—162no—
420C23C22×C22⋊C411yescharacteristic
421C23C2×#11711yescharacteristic
422C23C2×#11711yescharacteristic
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
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—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—164no—
469C2×C4#38111yescharacteristic
470C2×C4#38111yescharacteristic
471C2×C4#38111yescharacteristic
472C2×C4#38111yescharacteristic
473C2×C4#38111yescharacteristic
474C2×C4#38111yescharacteristic
475C2×C4#38111yescharacteristic
476C2×C4#38111yescharacteristic
477C2×C4C2×#12311yescharacteristic
478C2×C4C2×#12311yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4C2×#12311yescharacteristic
481C8—82no—
482C8—82no—
483C8—82no—
484C8—82no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—82no—
499C22—82no—
500C22—162no—
501C22—162no—
502C22C2×#38111yescharacteristic
503C22#288811yescharacteristic
504C22#288911yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—82no—
524C2—21no—
525C2—21no—
526C2—41no—
527C2—41no—
528C2—82no—
529C2#2069511yescharacteristic
530C2#2071011yescharacteristic
531C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8758C2semidirect
6. #8757C2semidirect
8. #8110C2semidirect
9. #8109C2semidirect
10. #8106C2semidirect
11. #8105C2semidirect

Maximal subgroups

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

subgroupindex
#343812
#173982
#87592
#87582
#87572
#87562
#81102
#81092
#81062
#81052