Finite Group Databasebeta

← #198826

Subgroups — #198826

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60966C211yesmaximal, characteristic
2#60249C211yesmaximal, characteristic, complemented
3#17541C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic, complemented
5#17494C211yesmaximal, characteristic
6#9282C214yesmaximal, complemented
7#9280C214yesmaximal, complemented
8#7968C211yesmaximal, characteristic, complemented
9#7855C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3252—22no—
13#3231C2212yes—
14#3230C411yescharacteristic, complemented
15#3227C411yescharacteristic, complemented
16#3210C2212yes—
17#3209C411yescharacteristic, complemented
18#3206C411yescharacteristic, complemented
19#3196—22no—
20#3185C412yescomplemented
21#3182C412yescomplemented
22#3117—22no—
23#3116—22no—
24#2987—22no—
25#2986—22no—
26#2985—22no—
27#2985C2214yes—
28Q8×M16—22no—
29Q8×M16—22no—
30#2921—22no—
31#2920C2211yescharacteristic
32#2920C2211yescharacteristic
33#2920C2211yescharacteristic
34C2×#432C2211yescharacteristic
35C2×#376C2211yescharacteristic
36#1506—22no—
37#1506C2212yes—
38#1506C2214yes—
39#1505C2212yes—
40#1504—22no—
41#1500—22no—
42C2×#328—22no—
43#1497C2212yes—
44#1496C2212yes—
45#1488—22no—
46#1483—22no—
47#1287C2214yes—
48#1274C2214yes—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533C2×C411yescharacteristic
54#533C2×C411yescharacteristic
55#533C2×C412yes—
56#533D412yes—
57#533D412yes—
58#533C2311yescharacteristic
59#533C2312yes—
60#532C2×C411yescharacteristic
61#532C2×C411yescharacteristic
62#532C2311yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65#529—21no—
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71C2×#125—21no—
72#525—21no—
73#525—21no—
74#525—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8D412yes—
77C4×#41—21no—
78C4×#41—21no—
79C4×#41—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8D412yes—
82C2×#117—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#500C2×C412yescomplemented
86#498C2×C412yescomplemented
87#494C2×C412yescomplemented
88#492C2×C412yescomplemented
89#487C2×C412yescomplemented
90#487C2×C412yescomplemented
91#487C2×C412yescomplemented
92#486C2×C412yescomplemented
93#485—24no—
94#484—22no—
95#481—22no—
96#479—24no—
97#477—22no—
98#476—22no—
99#470—24no—
100#468—24no—
101#464—24no—
102#462—22no—
103#461—22no—
104#452—24no—
105#451—22no—
106#448—22no—
107#446—24no—
108#446—28no—
109#446—28no—
110#446—24no—
111C8×Q8—24no—
112C8×Q8—24no—
113#441—24no—
114#440—24no—
115#433—28no—
116#433—28no—
117#432—21no—
118#432—21no—
119#432—21no—
120#432—21no—
121#432—21no—
122#432—21no—
123#432—21no—
124#432—21no—
125#432C2312yes—
126#432C2312yes—
127#431—21no—
128#431C2×C411yescharacteristic
129#431C2×C411yescharacteristic
130#431D412yes—
131#431D412yes—
132#431C2312yes—
133#431C2312yes—
134#423—24no—
135#423—21no—
136#423—21no—
137#423—21no—
138C2×C4⋊C8—21no—
139#407—21no—
140#407—21no—
141#407D412yes—
142C2×C22⋊C8—21no—
143C4×M16—24no—
144C4×M16—21no—
145C4×M16C2×C412yes—
146C4×M16D412yes—
147C2×#97—21no—
148#376—21no—
149#376—21no—
150#376—21no—
151#376C2312yes—
152C2×#95—21no—
153C2×#95—21no—
154#328—28no—
155#328—24no—
156#327—24no—
157Q8⋊C8—24no—
158D4⋊C8—24no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C22×Q8—21no—
163C22×Q8C22⋊C412yes—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C2×Q16—24no—
169C2×SD16—24no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C22×C8—41no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C22⋊C412yes—
179C4⋊Q8C22⋊C412yes—
180C4⋊Q8C22×C411yescharacteristic
181C4⋊Q8C22×C411yescharacteristic
182C4⋊Q8C22×C411yescharacteristic
183C4⋊Q8C22×C412yes—
184C4⋊Q8C22×C412yes—
185#127C22×C411yescharacteristic
186#126—41no—
187#126—41no—
188#125—22no—
189#125—21no—
190#125—21no—
191#125—22no—
192#125—22no—
193#125—21no—
194#125—22no—
195#124—21no—
196#124—21no—
197#124—21no—
198#123—41no—
199#123—41no—
200#123—41no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8C22⋊C414yes—
214C22⋊Q8C2×D414yes—
215#121—21no—
216#121—21no—
217#121—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221#120—41no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8C22⋊C414yes—
235C4×Q8C2×D414yes—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—41no—
243C4×D4—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117—22no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—41no—
256#117—41no—
257#117—41no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4C22⋊C412yes—
264C2×C22⋊C4—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—44no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42C2411yescharacteristic
282#107—24no—
283#106—24no—
284C4⋊C8—28no—
285C4⋊C8—28no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—41no—
295#103—24no—
296#103—24no—
297#103—24no—
298#102—24no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—41no—
307C22⋊C8C2×D414yes—
308#97—28no—
309#97—22no—
310#97—22no—
311#97—22no—
312#97—22no—
313#97—41no—
314C4×C8—28no—
315C4×C8—22no—
316C4×C8C22×C412yes—
317C4×C8C22×C412yes—
318C4×C8C2×D414yes—
319#95—22no—
320#95—22no—
321#95—41no—
322#95—41no—
323C24#12111yescharacteristic
324#41—82no—
325#41—82no—
326#41—82no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—82no—
334C2×Q8—82no—
335C2×Q8C2×C22⋊C414yes—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—81no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4#12011yescharacteristic
361C22×C4#12011yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12311yescharacteristic
368C22×C4#12311yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372Q16—84no—
373SD16—84no—
374M16—44no—
375M16—44no—
376M16—82no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—42no—
382C2×C8—82no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4C2×C22⋊C414yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—42no—
422C42—44no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42C2×C22⋊C411yescharacteristic
427C42C2×C22⋊C411yescharacteristic
428C42C4×D411yescharacteristic
429C42C4×D411yescharacteristic
430C42C4×D411yescharacteristic
431C42C4×D411yescharacteristic
432C42#12011yescharacteristic
433C42#12011yescharacteristic
434C42#12111yescharacteristic
435C42#12111yescharacteristic
436C42#12311yescharacteristic
437C42#12311yescharacteristic
438C42C23×C411yescharacteristic
439C42C22×D411yescharacteristic
440C42C22×D411yescharacteristic
441C42C2×#4111yescharacteristic
442C23—22no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23#38611yescharacteristic
447C23#38611yescharacteristic
448C23C2×#12111yescharacteristic
449Q8—44no—
450Q8—44no—
451Q8—42no—
452Q8—44no—
453Q8—42no—
454Q8—82no—
455Q8—81no—
456D4—42no—
457D4—42no—
458D4—81no—
459D4—82no—
460D4—82no—
461D4—82no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—88no—
494C2×C4#38611yescharacteristic
495C2×C4#38611yescharacteristic
496C2×C4#38611yescharacteristic
497C2×C4#38611yescharacteristic
498C2×C4#38611yescharacteristic
499C2×C4#38611yescharacteristic
500C2×C4C22×C22⋊C411yescharacteristic
501C2×C4C2×C4×D411yescharacteristic
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×#12011yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12311yescharacteristic
506C8—44no—
507C8—44no—
508C8—42no—
509C8—42no—
510C8—82no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—44no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22#190512yes—
524C22C2×#38611yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—44no—
537C4—44no—
538C4—84no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4#295612yes—
543C2—22no—
544C2—41no—
545C2—41no—
546C2—161no—
547C2#2091811yescharacteristic
548C2C2×#190511yescharacteristic
549C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60249C2semidirect
4. C2×#1506C2semidirect
6. #9282C2semidirect
7. #9280C2semidirect
8. #7968C2semidirect
9. #7855C2semidirect
14. #3230C4semidirect
15. #3227C4semidirect
17. #3209C4semidirect
18. #3206C4semidirect
20. #3185C4semidirect
21. #3182C4semidirect
85. #500C2×C4semidirect
86. #498C2×C4semidirect
87. #494C2×C4semidirect
88. #492C2×C4semidirect
89. #487C2×C4semidirect
90. #487C2×C4semidirect
91. #487C2×C4semidirect
92. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#609662
#602492
#175412
C2×#15062
#174942
#92822
#92802
#79682
#78552