Finite Group Databasebeta

← #116084

Subgroups — #116084

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1996C211yesmaximal, characteristic
2C2×#1520C211yesmaximal, characteristic
3C22×#328C211yesmaximal, characteristic
4#13965C211yesmaximal, characteristic
5C2×#1294C211yesmaximal, characteristic
6C2×#1289C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C2×#385—22no—
12C2×#380C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15C22×#95C2211yescharacteristic
16#1996—22no—
17#1996C414yescomplemented
18C2×#342—22no—
19#1520—22no—
20#1520—22no—
21C2×#328—22no—
22C2×#328—22no—
23C2×#328—22no—
24C2×#328—22no—
25#1444—22no—
26C4⋊#95—22no—
27#1294—22no—
28#1289—22no—
29#1282—22no—
30C23×Q8Q811yescharacteristic
31C24×C4—21no—
32C24×C4C2311yescharacteristic
33C23×C8—21no—
34C23×C8—21no—
35C2×C22⋊Q8—21no—
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×C22⋊Q8D412yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4Q811yescharacteristic
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4D412yes—
56C22×C42—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8D412yes—
68C2×C22⋊C8D412yes—
69#385—44no—
70#385—44no—
71#380—21no—
72#380—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380C2×C412yes—
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81C4×C22⋊C4C2×C412yes—
82C4×C22⋊C4D412yes—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—24no—
90C2×#95—21no—
91C2×#95D412yes—
92C2×#95D412yes—
93#342—28no—
94#328—28no—
95#328—28no—
96C25C2×D411yescharacteristic
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—82no—
103C22×Q8C4⋊C412yescomplemented
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—44no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—41no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—82no—
116C23×C4—82no—
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×Q811yescharacteristic
125C23×C4#4111yescharacteristic
126C23×C4#4111yescharacteristic
127C23×C4#4111yescharacteristic
128C23×C4#4111yescharacteristic
129C23×C4#4111yescharacteristic
130C23×C4#4111yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—82no—
138C22×C8—82no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—84no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C4⋊C412yescomplemented
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4C22⋊C414yes—
194C2×C22⋊C4SD1614yes—
195C2×C22⋊C4Q1614yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—82no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—48no—
215#95—41no—
216#95—41no—
217#95—41no—
218C24—42no—
219C24—42no—
220C24C2×C22⋊C411yescharacteristic
221C24#12011yescharacteristic
222C24#12111yescharacteristic
223C24C22⋊Q811yescharacteristic
224C24#12311yescharacteristic
225C24#12411yescharacteristic
226C24#12711yescharacteristic
227C24C2×SD1612yes—
228C24C2×Q1612yes—
229C24AGL(1,ℤ/8)12yes—
230C24#13712yes—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—82no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—82no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—88no—
280C22×C4—88no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C4⋊C411yescharacteristic
292C22×C4#11711yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4C22⋊Q811yescharacteristic
307C22×C4C22⋊Q811yescharacteristic
308C22×C4C22⋊Q811yescharacteristic
309C22×C4C22⋊Q811yescharacteristic
310C22×C4C22⋊Q811yescharacteristic
311C22×C4#12311yescharacteristic
312C22×C4#12311yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12411yescharacteristic
316C22×C4#12411yescharacteristic
317C22×C4#12611yescharacteristic
318C22×C4#12611yescharacteristic
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—82no—
322C2×C8—82no—
323C2×C8—82no—
324C2×C8—82no—
325C2×C8—82no—
326C2×C8—82no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—816no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—82no—
349C22⋊C4—81no—
350C22⋊C4—81no—
351C22⋊C4—81no—
352C22⋊C4—81no—
353C22⋊C4#10318yes—
354C42—82no—
355C42—82no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—44no—
369C23—44no—
370C23—41no—
371C23—41no—
372C23—44no—
373C23—44no—
374C23#38011yescharacteristic
375C23#38611yescharacteristic
376C23#39011yescharacteristic
377C23#39211yescharacteristic
378C23#39311yescharacteristic
379C23#39711yescharacteristic
380C23#39911yescharacteristic
381C23C2×#10312yes—
382C23#41712yes—
383C23#45012yes—
384C23#45112yes—
385C23#46012yes—
386C23#46212yes—
387C23#47512yes—
388C23#47712yes—
389C23#48312yes—
390C23#48412yes—
391C23#48712yes—
392C23#48912yes—
393C23#49211yescharacteristic
394C23#49411yescharacteristic
395C23#49611yescharacteristic
396C23#49711yescharacteristic
397Q8—82no—
398Q8—82no—
399Q8—82no—
400Q8—82no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—28no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—88no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—84no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—84no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—164no—
459C2×C4#38511yescharacteristic
460C2×C4#38511yescharacteristic
461C2×C4#38811yescharacteristic
462C2×C4#38811yescharacteristic
463C2×C4#39411yescharacteristic
464C2×C4#39411yescharacteristic
465C2×C4#39611yescharacteristic
466C2×C4#39611yescharacteristic
467C8—82no—
468C8—82no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—24no—
482C22—24no—
483C22—24no—
484C22—24no—
485C22—24no—
486C22—24no—
487C22—42no—
488C22—41no—
489C22—41no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—41no—
496C22—41no—
497C22#143911yescharacteristic
498C22#180312yes—
499C22#187212yes—
500C22#195611yescharacteristic
501C22#195911yescharacteristic
502C22#201411yescharacteristic
503C22#201611yescharacteristic
504C22#202412yes—
505C22#203112yes—
506C22#203212yes—
507C22#207212yes—
508C22#209511yescharacteristic
509C22#209611yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—48no—
519C4—84no—
520C4—82no—
521C4—82no—
522C4—162no—
523C4—162no—
524C2—22no—
525C2—21no—
526C2—21no—
527C2—22no—
528C2—22no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2#1037514yes—
534C2#1105011yescharacteristic
535C2#1108911yescharacteristic
536C2#1109111yescharacteristic
subgroupG/H
center, socleC23#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1996C4semidirect
103. C22×Q8C4⋊C4semidirect
175. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19962
C2×#15202
C22×#3282
#139652
C2×#12942
C2×#12892
C2×#12822