Finite Group Databasebeta

← #190039

Subgroups — #190039

Order 512.

856 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34247C211yesmaximal, characteristic
2C2×#1473C211yesmaximal, characteristic
3C2×#1473C211yesmaximal, characteristic
4#8737C212yesmaximal, complemented
5#8736C212yesmaximal
6#8735C212yesmaximal, complemented
7#8734C212yesmaximal
8#7930C211yesmaximal, characteristic, complemented
9#7928C211yesmaximal, characteristic, complemented
10#7926C211yesmaximal, characteristic, complemented
11#7924C211yesmaximal, characteristic, complemented
12C22×#126C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2373—22no—
18#2373—22no—
19#2345C412yes—
20#2344C412yes—
21#2342C412yes—
22#2342C412yes—
23#2334—22no—
24#2313C2211yescharacteristic
25#2312C2211yescharacteristic
26C4×#126—22no—
27#2283C2211yescharacteristic
28#2274C2211yescharacteristic
29#2273C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1473—22no—
32#1473—22no—
33#1473—22no—
34#1473—22no—
35C2×#324C2212yes—
36C2×#324C2212yes—
37C2×#323C2212yes—
38C2×#323C2212yes—
39#1282C2212yes—
40#1282C2212yes—
41#1282C2212yes—
42#1282C2212yes—
43#1282C2212yes—
44#1282C2212yes—
45#1282C2212yes—
46#1282C2212yes—
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—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2312yes—
92C2×C22⋊C8C2312yes—
93C2×#97—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97#389—22no—
98#389—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#385—22no—
112#385—22no—
113#383—21no—
114#382—21no—
115#381C2×C412yes—
116#381C2×C412yes—
117#380C2×C412yes—
118#380C2×C412yes—
119#380C2×C412yes—
120#379C2×C412yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4C2×C412yes—
133C4×C22⋊C4C2×C412yes—
134C4×C22⋊C4C2×C412yes—
135C4×C22⋊C4C2311yescharacteristic
136C4×C22⋊C4C2311yescharacteristic
137C4×C22⋊C4C2311yescharacteristic
138C4×C22⋊C4C2311yescharacteristic
139#376—21no—
140#376—21no—
141C43—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
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—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—24no—
175#117—24no—
176#117—42no—
177#117—42no—
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—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
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—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4C22⋊C412yes—
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⋊C4C2×D412yes—
227C2×C22⋊C4C2×D412yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8—41no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265#97—41no—
266#97—41no—
267#97—41no—
268#97—41no—
269C4×C8—41no—
270C4×C8—41no—
271C4×C8—41no—
272C4×C8—41no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—81no—
292C24—81no—
293C24C2×C22⋊C412yes—
294C24C23×C411yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—44no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4#11711yescharacteristic
324C22×C4#11711yescharacteristic
325C22×C4#11711yescharacteristic
326C22×C4#11711yescharacteristic
327C22×C4#11711yescharacteristic
328C22×C4#11711yescharacteristic
329C22×C4#11711yescharacteristic
330C22×C4#11711yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—82no—
342C2×C8—82no—
343C2×C8—82no—
344C2×C8—82no—
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—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C22⋊C4—22no—
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—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4#13114yes—
385C42—44no—
386C42—44no—
387C42—82no—
388C42—81no—
389C42—81no—
390C42—81no—
391C42—81no—
392C42—82no—
393C42—82no—
394C42#12311yescharacteristic
395C42#12311yescharacteristic
396C42#12311yescharacteristic
397C42#12311yescharacteristic
398C42#12311yescharacteristic
399C42#12311yescharacteristic
400C42#12311yescharacteristic
401C42#12311yescharacteristic
402C42#12311yescharacteristic
403C42#12311yescharacteristic
404C42#12311yescharacteristic
405C42#12311yescharacteristic
406C42#12311yescharacteristic
407C42#12311yescharacteristic
408C42#12311yescharacteristic
409C42#12311yescharacteristic
410C23—22no—
411C23—22no—
412C23—22no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—162no—
425C23—162no—
426C23C22×C22⋊C411yescharacteristic
427C23C2×#11711yescharacteristic
428C23C2×#11711yescharacteristic
429C23C2×#13112yes—
430C23#56812yes—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—164no—
478C2×C4#38111yescharacteristic
479C2×C4#38111yescharacteristic
480C2×C4#38111yescharacteristic
481C2×C4#38111yescharacteristic
482C2×C4#38111yescharacteristic
483C2×C4#38111yescharacteristic
484C2×C4#38111yescharacteristic
485C2×C4#38111yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4C2×#12311yescharacteristic
490C8—82no—
491C8—82no—
492C8—82no—
493C8—82no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—161no—
508C22—161no—
509C22—162no—
510C22C2×#38111yescharacteristic
511C22#288012yes—
512C22#292211yescharacteristic
513C22#292311yescharacteristic
514C22#292512yes—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—44no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—82no—
532C4—82no—
533C4—82no—
534C2—22no—
535C2—21no—
536C2—21no—
537C2—22no—
538C2—82no—
539C2#2068411yescharacteristic
540C2#2071011yescharacteristic
541C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ 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
4. #8737C2semidirect
6. #8735C2semidirect
8. #7930C2semidirect
9. #7928C2semidirect
10. #7926C2semidirect
11. #7924C2semidirect

Maximal subgroups

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

subgroupindex
#342472
C2×#14732
#87372
#87362
#87352
#87342
#79302
#79282
#79262
#79242