Finite Group Databasebeta

← #117507

Subgroups — #117507

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1980C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic, complemented
3C2×#1540C211yesmaximal, characteristic
4C2×#1492C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic, complemented
6C2×#1279C211yesmaximal, characteristic, complemented
7#7848C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C22×#108—21no—
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#104—21no—
14C22×#104—21no—
15C2×C4×M16C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1980—22no—
19#1980—22no—
20#1950—22no—
21#1950—22no—
22C2×#341—21no—
23#1540—22no—
24#1540—22no—
25C2×#332C411yescharacteristic, complemented
26C2×#332C411yescharacteristic, complemented
27C2×#331C411yescharacteristic, complemented
28C2×#331C411yescharacteristic, complemented
29#1492—22no—
30#1492—22no—
31C2×D4⋊C8—22no—
32#1303—22no—
33#1279—22no—
34#1273—22no—
35C22×#41—21no—
36C22×M16—21no—
37C22×M16D411yescharacteristic
38C22×M16D411yescharacteristic
39C23×C8—21no—
40C23×C8—21no—
41#532—21no—
42#532—21no—
43#532—21no—
44#532—21no—
45#532—21no—
46#532—21no—
47#532D412yes—
48#532Q812yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#124—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C22×C4⋊C4—21no—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63C2×#108—24no—
64C2×#108—42no—
65C2×#108—42no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2×C411yescharacteristic
76C2×C4⋊C8C2×C411yescharacteristic
77C2×C4⋊C8C2×C411yescharacteristic
78C2×C4⋊C8C2×C411yescharacteristic
79C2×C4⋊C8D412yes—
80C2×C4⋊C8D412yes—
81C2×#104—24no—
82C2×#104—24no—
83C2×#104—42no—
84C2×#104—42no—
85C2×#104—42no—
86C2×#104—42no—
87C2×C22⋊C8—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16—21no—
91C4×M16—21no—
92C2×#97—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8D412yes—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D412yes—
105C2×#95—21no—
106#341—24no—
107#332—22no—
108#332—22no—
109#331—22no—
110#331—22no—
111D4⋊C8—28no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—41no—
115C2×#41—41no—
116C2×#41—41no—
117C2×#41—41no—
118C2×#41—41no—
119C2×#41—41no—
120C2×#41—41no—
121C2×#41—41no—
122C2×#41—41no—
123C2×#41—41no—
124C22×Q8—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—41no—
129C23×C4—41no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C23×C4#4111yescharacteristic
134C2×M16—21no—
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—41no—
146C2×M16—41no—
147C2×M16—41no—
148C2×M16—41no—
149C2×M16—41no—
150C2×M16D812yes—
151C2×M16SD1612yes—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—44no—
162C22×C8—41no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—41no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C4⋊C412yes—
173#127—21no—
174#127—21no—
175#127—21no—
176#127C4⋊C412yes—
177#124—22no—
178#124—22no—
179#124—41no—
180#124—41no—
181#121—22no—
182#121—22no—
183#121—22no—
184#121—22no—
185#121—41no—
186#121—41no—
187#121—41no—
188#121—41no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C42—22no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42—41no—
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q811yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42#4111yescharacteristic
223#108—48no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—41no—
235C4⋊C8—41no—
236C4⋊C8C22⋊C412yes—
237C4⋊C8C22⋊C412yes—
238#104—48no—
239#104—48no—
240C22⋊C8—41no—
241C22⋊C8—41no—
242#97—41no—
243#97—41no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8—41no—
247C4×C8—41no—
248C4×C8—41no—
249C4×C8—41no—
250C4×C8—41no—
251C4×C8—41no—
252#95—22no—
253#95—41no—
254C24—41no—
255C24#12411yescharacteristic
256#41—42no—
257#41—42no—
258#41—42no—
259#41—42no—
260#41—42no—
261#41—42no—
262#41—42no—
263#41—42no—
264#41—81no—
265#41—81no—
266#41—81no—
267#41—81no—
268#41—81no—
269#41—81no—
270#41—81no—
271#41—81no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—82no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4—81no—
296C2×D4—81no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—82no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4#11711yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4C22⋊Q811yescharacteristic
322C22×C4C22⋊Q811yescharacteristic
323C22×C4C22⋊Q811yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4#12311yescharacteristic
327C22×C4#12411yescharacteristic
328C22×C4#12411yescharacteristic
329C22×C4#12611yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4C2×SD1611yescharacteristic
332C22×C4#13511yescharacteristic
333C22×C4#13511yescharacteristic
334C22×C4AGL(1,ℤ/8)11yescharacteristic
335C22×C4AGL(1,ℤ/8)11yescharacteristic
336C22×C4AGL(1,ℤ/8)11yescharacteristic
337C22×C4#13711yescharacteristic
338M16—22no—
339M16—22no—
340M16—22no—
341M16—22no—
342M16—41no—
343M16—41no—
344M16—41no—
345M16—41no—
346M16—42no—
347M16—42no—
348M16—81no—
349M16—81no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—44no—
357C2×C8—44no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—81no—
377C2×C8—81no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C42—42no—
390C42—42no—
391C42—81no—
392C42—81no—
393C42C2×C22⋊C411yescharacteristic
394C42C2×C4⋊C411yescharacteristic
395C42C4×D411yescharacteristic
396C42C4×D411yescharacteristic
397C42#12011yescharacteristic
398C42#12111yescharacteristic
399C42#12111yescharacteristic
400C42#12111yescharacteristic
401C42#12111yescharacteristic
402C42C22⋊Q811yescharacteristic
403C42C22⋊Q811yescharacteristic
404C42C22⋊Q811yescharacteristic
405C42#12311yescharacteristic
406C42#12311yescharacteristic
407C42#12611yescharacteristic
408C42#12711yescharacteristic
409C23—22no—
410C23—22no—
411C23—42no—
412C23—82no—
413C23—82no—
414C23—81no—
415C23—81no—
416C23#38811yescharacteristic
417C23#39311yescharacteristic
418C23#39411yescharacteristic
419C23#48611yescharacteristic
420C23#48811yescharacteristic
421C23#48911yescharacteristic
422C23#49011yescharacteristic
423Q8—42no—
424Q8—42no—
425Q8—81no—
426Q8—81no—
427D4—42no—
428D4—42no—
429D4—44no—
430D4—82no—
431D4—82no—
432D4—82no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—81no—
437D4—81no—
438D4—81no—
439D4—81no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4#38011yescharacteristic
466C2×C4#38511yescharacteristic
467C2×C4#38511yescharacteristic
468C2×C4#38611yescharacteristic
469C2×C4#38811yescharacteristic
470C2×C4#39011yescharacteristic
471C2×C4#39211yescharacteristic
472C2×C4#39411yescharacteristic
473C2×C4#39611yescharacteristic
474C2×C4#39611yescharacteristic
475C2×C4#39711yescharacteristic
476C2×C4#39911yescharacteristic
477C2×C4C4×D811yescharacteristic
478C2×C4C4×SD1611yescharacteristic
479C2×C4#44011yescharacteristic
480C2×C4#44211yescharacteristic
481C2×C4#44312yes—
482C2×C4#44412yes—
483C2×C4#44711yescharacteristic
484C2×C4#44711yescharacteristic
485C2×C4#44811yescharacteristic
486C2×C4#44911yescharacteristic
487C2×C4#44911yescharacteristic
488C2×C4#44911yescharacteristic
489C2×C4#45011yescharacteristic
490C2×C4#45211yescharacteristic
491C2×C4#45911yescharacteristic
492C2×C4#46011yescharacteristic
493C2×C4#46311yescharacteristic
494C2×C4#46411yescharacteristic
495C2×C4#46511yescharacteristic
496C2×C4#46611yescharacteristic
497C2×C4#46811yescharacteristic
498C2×C4#46911yescharacteristic
499C2×C4#47411yescharacteristic
500C2×C4#47611yescharacteristic
501C2×C4#47811yescharacteristic
502C2×C4#47811yescharacteristic
503C8—44no—
504C8—42no—
505C8—42no—
506C8—42no—
507C8—42no—
508C8—81no—
509C8—81no—
510C8—82no—
511C22—21no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—21no—
518C22—21no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—162no—
523C22—162no—
524C22#143911yescharacteristic
525C22#192912yes—
526C22#193212yes—
527C22#193411yescharacteristic
528C22#193611yescharacteristic
529C22#202311yescharacteristic
530C22#202511yescharacteristic
531C22#204111yescharacteristic
532C22#204211yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—84no—
540C4—82no—
541C4#186211yescharacteristic
542C4#186411yescharacteristic
543C4#186612yes—
544C4#187911yescharacteristic
545C4#188011yescharacteristic
546C4#197411yescharacteristic
547C4#197511yescharacteristic
548C4#199911yescharacteristic
549C4#200011yescharacteristic
550C4#206511yescharacteristic
551C4#206711yescharacteristic
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—162no—
557C2#959012yes—
558C2#959212yes—
559C2#1106611yescharacteristic
560C2#1112511yescharacteristic
561C2#1114611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#388
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1950C2semidirect
5. C2×#1303C2semidirect
6. C2×#1279C2semidirect
7. #7848C2semidirect
25. C2×#332C4semidirect
26. C2×#332C4semidirect
27. C2×#331C4semidirect
28. C2×#331C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19802
C2×#19502
C2×#15402
C2×#14922
C2×#13032
C2×#12792
#78482