Finite Group Databasebeta

← #201323

Subgroups — #201323

Order 512.

1385 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35227C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3C2×#1821C211yesmaximal, characteristic, complemented
4#17587C212yesmaximal
5#9464C214yesmaximal, complemented
6#9460C214yesmaximal, complemented
7#7943C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2849—21no—
12#2849—21no—
13#2623—22no—
14#2504C412yescomplemented
15#2503C2212yes—
16#2502C412yescomplemented
17#2501C2212yes—
18#2487—22no—
19#2486—22no—
20#2317C412yescomplemented
21#2313—24no—
22#2313—22no—
23#2313C2214yes—
24#2311C412yescomplemented
25C2×#387—24no—
26#2280C2211yescharacteristic
27C2×#381C2212yes—
28C2×#380—21no—
29C2×C4×C22⋊C4—21no—
30#1821—22no—
31#1821C2214yes—
32C2×#343—22no—
33#1516—24no—
34#1511C2214yes—
35#1510C2214yes—
36#1504—24no—
37#1284C2218yes—
38C22×#41C2×C411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43#525—22no—
44#525—22no—
45#525—22no—
46#525—22no—
47C2×#123—21no—
48C2×#123—22no—
49C2×#123—22no—
50C2×C22⋊Q8C2×C411yescharacteristic
51C2×C22⋊Q8C2311yescharacteristic
52C2×#121C2×C411yescharacteristic
53C2×#121C2311yescharacteristic
54C2×#120—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C412yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C42—22no—
72#407—22no—
73#407—22no—
74#407—22no—
75#407C2314yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78#400—24no—
79#399—24no—
80#398—24no—
81#397—24no—
82#391C2×C414yescomplemented
83#390C2×C414yescomplemented
84#389—24no—
85#387—216no—
86#387—216no—
87#386—22no—
88#386—22no—
89#386—22no—
90#386—22no—
91#385—22no—
92#385—22no—
93#385—22no—
94#385—22no—
95#384C2×C414yescomplemented
96#382C2×C414yescomplemented
97#381—22no—
98#381—22no—
99#381—22no—
100#381—22no—
101#381—21no—
102#381D414yes—
103#381C2314yes—
104#380—24no—
105#380—24no—
106#380—21no—
107#380—21no—
108#380—24no—
109#379—21no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4C2×C412yes—
113C4×C22⋊C4C2312yes—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D414yes—
118#343—28no—
119#328—28no—
120#327—28no—
121C25C2×D411yescharacteristic
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—44no—
125C22×Q8C22×C411yescharacteristic
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4C22×C411yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—44no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C2×M16—24no—
160C2×M16—42no—
161C2×M16—48no—
162C22×C8—42no—
163#123—42no—
164#123—42no—
165#123—84no—
166#123—84no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171#121—22no—
172#121—22no—
173#121—22no—
174#121—22no—
175#120—41no—
176#120—41no—
177#120—41no—
178#120—41no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183#117—22no—
184#117—22no—
185#117—44no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—84no—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C22⋊C414yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—48no—
224C2×C42—21no—
225C2×C42—28no—
226C2×C42—84no—
227C2×C42C22×C414yes—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C22⋊C8—42no—
231C22⋊C8—42no—
232#95—28no—
233#95—28no—
234#95—22no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95C2×D418yes—
244C24—24no—
245C24—21no—
246C24—42no—
247C24—42no—
248C24C2×C22⋊C411yescharacteristic
249C24C2×C22⋊C411yescharacteristic
250C24#12011yescharacteristic
251C24#12011yescharacteristic
252C24#12311yescharacteristic
253C24#12311yescharacteristic
254C24C22×D411yescharacteristic
255#41—84no—
256#41—84no—
257C2×Q8—22no—
258C2×Q8—42no—
259C2×D4—22no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—44no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—28no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—42no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C414yes—
319C22×C4C2×C22⋊C414yes—
320C22×C4C4×D412yes—
321C22×C4C4×D412yes—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4#12012yes—
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4#12312yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C2×#4112yes—
334M16—48no—
335M16—84no—
336C2×C8—44no—
337C2×C8—84no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—88no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C22⋊C4—44no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—44no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—84no—
364C22⋊C4—84no—
365C42—44no—
366C42—42no—
367C42—88no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—416no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23#38611yescharacteristic
394C23#38611yescharacteristic
395C23#38611yescharacteristic
396C23#38611yescharacteristic
397C23#41014yes—
398C23C22×C22⋊C411yescharacteristic
399C23C2×#12011yescharacteristic
400C23C2×#12311yescharacteristic
401Q8—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—82no—
406D4—82no—
407D4—82no—
408D4—82no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—48no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—816no—
455C2×C4#38612yes—
456C2×C4#38612yes—
457C2×C4C2×C4×D412yes—
458C2×C4C2×#12112yes—
459C8—84no—
460C22—28no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—84no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22#189912yes—
488C22#189912yes—
489C22#191212yes—
490C22C2×#38611yescharacteristic
491C22C2×#41012yes—
492C22#301912yes—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—44no—
498C4—44no—
499C4—84no—
500C4—88no—
501C4—81no—
502C4—81no—
503C4—81no—
504C2—22no—
505C2—22no—
506C2—22no—
507C2—22no—
508C2—22no—
509C2—41no—
510C2—41no—
511C2—161no—
512C2#2095311yescharacteristic
513C2C2×#189911yescharacteristic
514C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2313C2semidirect
3. C2×#1821C2semidirect
5. #9464C2semidirect
6. #9460C2semidirect
7. #7943C2semidirect
14. #2504C4semidirect
16. #2502C4semidirect
20. #2317C4semidirect
24. #2311C4semidirect
82. #391C2×C4semidirect
83. #390C2×C4semidirect
95. #384C2×C4semidirect
96. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#352272
C2×#23132
C2×#18212
#175872
#94642
#94602
#79432