Finite Group Databasebeta

← #196731

Subgroups — #196731

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2988C211yesmaximal, characteristic, complemented
2#35292C211yesmaximal, characteristic
3#20721C211yesmaximal, characteristic, complemented
4C2×#1553C211yesmaximal, characteristic, complemented
5C2×#1540C211yesmaximal, characteristic, complemented
6#9134C212yesmaximal, complemented
7#9133C212yesmaximal, complemented
8#9132C212yesmaximal, complemented
9#9131C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#8038C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#2988—24no—
15#2988—24no—
16C2×C8×Q8—21no—
17C2×C8×D4—22no—
18C2×C8×D4—21no—
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2212yes—
22C2×#431C2211yescharacteristic
23C22×C4⋊C8C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25#2857C2212yes—
26#2856C2212yes—
27#2624—21no—
28#2615—21no—
29#2592—21no—
30#2580C412yes—
31#2540—21no—
32#2535C412yes—
33#2414C414yes—
34#2407—22no—
35#2324—22no—
36#2285C2211yescharacteristic
37#1818—21no—
38#1797C2212yes—
39#1797C2212yes—
40#1780—21no—
41#1750—22no—
42#1553—24no—
43#1540—24no—
44C2×#332C2212yes—
45C2×#332C2211yescharacteristic
46C2×#332C2211yescharacteristic
47C2×#331C2212yes—
48C2×#331C2211yescharacteristic
49C2×#331C2211yescharacteristic
50#1296C2212yes—
51#1296C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54C22×#41—21no—
55C22×#41D411yescharacteristic
56C23×C8—22no—
57C23×C8—21no—
58#532—22no—
59#532—22no—
60#532—22no—
61#532—22no—
62C2×C4⋊Q8C2311yescharacteristic
63C2×#127C2311yescharacteristic
64C2×#124—21no—
65C2×C22⋊Q8—22no—
66C2×#121—22no—
67C2×#121—21no—
68C2×#121—22no—
69C2×#121C2×C412yes—
70C2×#121C2×C414yes—
71C4×#41—22no—
72C4×#41—22no—
73C4×#41—22no—
74C4×#41—22no—
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4C2×C412yes—
79C2×C4×D4C2×C412yes—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4D412yes—
82C2×C4×D4D411yescharacteristic
83C2×#117C2×C412yes—
84C2×#117D412yes—
85C2×#117D411yescharacteristic
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88C8×Q8—28no—
89C8×D4—216no—
90C8×D4—28no—
91#432—22no—
92#432—22no—
93#431—22no—
94#431—24no—
95#431—24no—
96#431—22no—
97#431—22no—
98#431D414yescomplemented
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—24no—
103C2×C4⋊C8—24no—
104C2×C4⋊C8D414yescomplemented
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2311yescharacteristic
111C2×C22⋊C8—22no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×#97—21no—
118C2×C4×C8—22no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121#399—22no—
122#397—22no—
123#394—24no—
124#393—22no—
125#392—24no—
126#392—22no—
127#391—22no—
128#390—24no—
129#390—22no—
130#390—22no—
131#390C2×C414yes—
132#389—22no—
133#389C2×C414yes—
134#388—24no—
135#386—24no—
136#385—24no—
137#384C2312yes—
138#382C2×C412yes—
139#380—22no—
140#380—21no—
141#380—21no—
142#380C2×C414yes—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145#336—22no—
146#336—22no—
147#332—24no—
148#332—22no—
149#332—22no—
150#331—24no—
151#331—22no—
152#331—22no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—24no—
156C2×#41—24no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—82no—
160C2×#41—82no—
161C22×Q8—21no—
162C22×Q8C22⋊C411yescharacteristic
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C22⋊C411yescharacteristic
168C22×D4C22⋊C412yes—
169C22×D4C22×C412yes—
170C23×C4—22no—
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—28no—
181C22×C8—24no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—82no—
186C22×C8—82no—
187C22×C8—82no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192#127—21no—
193#127—21no—
194#127—21no—
195#127—21no—
196#124—42no—
197#124—42no—
198C22⋊Q8—82no—
199C22⋊Q8—82no—
200#121—24no—
201#121—24no—
202#121—42no—
203#121—42no—
204#121—44no—
205#121—44no—
206#121—82no—
207#121—82no—
208#121—84no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—42no—
224C4×D4—42no—
225#117—22no—
226#117—22no—
227#117—24no—
228#117—24no—
229#117—82no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4C22⋊C412yes—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C414yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4C22⋊C412yes—
248C2×C22⋊C4C22⋊C412yes—
249C2×C22⋊C4C22⋊C412yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42C22×C412yes—
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42#4112yes—
264C2×C42#4112yes—
265C2×C42C2411yescharacteristic
266C4⋊C8—28no—
267C4⋊C8—24no—
268C4⋊C8—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—22no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—22no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8C2×D412yes—
278C4⋊C8C2×D414yes—
279C4⋊C8C2×D412yes—
280C22⋊C8—28no—
281C22⋊C8—24no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C22⋊C8—44no—
285C22⋊C8—44no—
286#97—42no—
287C4×C8—28no—
288C4×C8—24no—
289C4×C8—42no—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—24no—
294C24—22no—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C412yes—
297#41—216no—
298#41—28no—
299#41—28no—
300#41—88no—
301#41—84no—
302#41—84no—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—24no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—84no—
329C2×D4—84no—
330C2×D4—84no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—84no—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C412yes—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#12012yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12312yes—
367C22×C4#13514yes—
368C22×C4#13514yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—84no—
387C2×C8—88no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—44no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—84no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C22⋊C4—82no—
411C22⋊C4—84no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—82no—
419C42C4×D412yes—
420C42C4×D412yes—
421C42#12012yes—
422C42#12112yes—
423C42#12112yes—
424C42#12312yes—
425C42C22×D411yescharacteristic
426C42C22×D411yescharacteristic
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C23—21no—
430C23—22no—
431C23—21no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23#38612yes—
445C23#41614yes—
446C23C22×C22⋊C411yescharacteristic
447Q8—28no—
448Q8—82no—
449Q8—81no—
450Q8—81no—
451D4—28no—
452D4—216no—
453D4—82no—
454D4—82no—
455D4—88no—
456D4—82no—
457D4—81no—
458D4—81no—
459D4—82no—
460D4—81no—
461D4—81no—
462D4—168no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—88no—
501C2×C4#38612yes—
502C2×C4#38612yes—
503C2×C4#38612yes—
504C2×C4#41614yes—
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×#12011yescharacteristic
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×#12111yescharacteristic
510C2×C4C2×#12311yescharacteristic
511C2×C4C2×#13512yes—
512C2×C4C2×#13512yes—
513C2×C4#57612yes—
514C2×C4#57712yes—
515C8—44no—
516C8—42no—
517C8—42no—
518C8—82no—
519C8—81no—
520C8—81no—
521C22—22no—
522C22—24no—
523C22—24no—
524C22—22no—
525C22—22no—
526C22—24no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—82no—
534C22—84no—
535C22—84no—
536C22—82no—
537C22—82no—
538C22C2×#38611yescharacteristic
539C22#288811yescharacteristic
540C22#288911yescharacteristic
541C22C2×#41612yes—
542C22#289712yes—
543C4—21no—
544C4—21no—
545C4—22no—
546C4—22no—
547C4—22no—
548C4—24no—
549C4—42no—
550C4—42no—
551C4—84no—
552C4—82no—
553C4#300811yescharacteristic
554C4#301011yescharacteristic
555C4#305012yes—
556C4#305011yescharacteristic
557C4#305011yescharacteristic
558C4#309612yes—
559C2—42no—
560C2—44no—
561C2—162no—
562C2#1772412yes—
563C2#1772712yes—
564C2#2091411yescharacteristic
565C2#2096711yescharacteristic
566C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35292C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2988C2semidirect
3. #20721C2semidirect
4. C2×#1553C2semidirect
5. C2×#1540C2semidirect
6. #9134C2semidirect
7. #9133C2semidirect
8. #9132C2semidirect
9. #9131C2semidirect
10. #8044C2semidirect
11. #8038C2semidirect
98. #431D4semidirect
104. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29882
#352922
#207212
C2×#15532
C2×#15402
#91342
#91332
#91322
#91312
#80442
#80382