Finite Group Databasebeta

← #369328

Subgroups — #369328

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3C2×#1981C211yesmaximal, characteristic
4C2×#1960C211yesmaximal, characteristic
5#13925C211yesmaximal, characteristic
6#11077C212yesmaximal
7#11077C212yesmaximal
8#10928C212yesmaximal
9#10926C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#446—24no—
15C2×#446—24no—
16C2×#446—24no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2568—24no—
28#2568C418yescomplemented
29C2×#400—21no—
30C2×#400—21no—
31C2×#391—21no—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#382—21no—
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1981—24no—
39#1960—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44#1446—21no—
45#1445—21no—
46#1436—21no—
47#1436—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×Q8D411yescharacteristic
51C23×Q8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8C2×C414yes—
67C2×C4×Q8D414yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#446—232no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8D414yes—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#97—22no—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97—22no—
108C2×#97D414yes—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112#400—28no—
113#400—28no—
114#391—28no—
115#391—24no—
116#391—24no—
117#391C2×C418yes—
118#391C2×C418yes—
119#382—28no—
120#382—22no—
121#382—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2×C414yes—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138#340—24no—
139#340—24no—
140#340—42no—
141#340—42no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—42no—
159C22×Q8C22⋊C414yes—
160C22×Q8C22×C414yes—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8#4114yes—
192C4⋊Q8—28no—
193C4⋊Q8—28no—
194C4⋊Q8—44no—
195C4⋊Q8—44no—
196C4×Q8—28no—
197C4×Q8—28no—
198C4×Q8—28no—
199C4×Q8—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C22⋊C414yes—
229C2×C4⋊C4C22⋊C414yes—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4#4114yes—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—24no—
242C2×C42—44no—
243C2×C42—42no—
244C2×C42—82no—
245C2×C42C22×C412yes—
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C4⋊C8—28no—
256C4⋊C8—28no—
257C4⋊C8—28no—
258#103—48no—
259#103—48no—
260#103—48no—
261#103—48no—
262#97—28no—
263#97—28no—
264C4×C8—28no—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—44no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—44no—
275#95—82no—
276#95—82no—
277C24C22×D411yescharacteristic
278C2×Q8—28no—
279C2×Q8—28no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—44no—
287C2×Q8—48no—
288C2×Q8—48no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4#13714yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+412yes—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8#12414yes—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—48no—
425C42—84no—
426C42C4×D414yes—
427C42C4×D414yes—
428C42#12014yes—
429C42#12314yes—
430C23C22×C22⋊C411yescharacteristic
431C23C2×#12111yescharacteristic
432C23C2×#12111yescharacteristic
433C23#54812yes—
434C23#55012yes—
435C23C2×#13712yes—
436C23#57512yes—
437C23#57712yes—
438C23#57812yes—
439Q8—416no—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—44no—
444Q8—82no—
445Q8—82no—
446Q8—82no—
447Q8—82no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4#38614yes—
504C2×C4#38614yes—
505C2×C4#41914yes—
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C4×#4112yes—
509C2×C4#52012yes—
510C2×C4C2×#12011yescharacteristic
511C2×C4C2×#12311yescharacteristic
512C2×C4C2×#12411yescharacteristic
513C2×C4#52912yes—
514C2×C4#53211yescharacteristic
515C2×C4#53511yescharacteristic
516C2×C4#53611yescharacteristic
517C2×C4#54212yes—
518C2×C4#54612yes—
519C2×C4#54812yes—
520C2×C4#54812yes—
521C2×C4#55012yes—
522C2×C4#55612yes—
523C2×C4#55912yes—
524C8—24no—
525C8—42no—
526C8—42no—
527C8—42no—
528C22C2×#38611yescharacteristic
529C22#234112yes—
530C22#242612yes—
531C22#254511yescharacteristic
532C22#254811yescharacteristic
533C22C2×#41911yescharacteristic
534C22#289611yescharacteristic
535C22#289812yes—
536C22#305612yes—
537C22#305612yes—
538C22#305711yescharacteristic
539C22#305911yescharacteristic
540C22#306312yes—
541C22#337212yes—
542C22#337412yes—
543C22#337612yes—
544C22#337712yes—
545C22#339812yes—
546C22#340012yes—
547C22#340112yes—
548C22#340212yes—
549C4—24no—
550C4—24no—
551C4—24no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—84no—
556C4—84no—
557C4—84no—
558C4—162no—
559C4C4×#12411yescharacteristic
560C4#232611yescharacteristic
561C4#236712yes—
562C4#243111yescharacteristic
563C4#243611yescharacteristic
564C4#244812yes—
565C2#1402611yescharacteristic
566C2#2097211yescharacteristic
567C2#2097411yescharacteristic
568C2#2254712yes—
569C2#2254812yes—
570C2#2530612yes—
571C2#2530812yes—
572C2#2865212yes—
573C2#2874712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #2568C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4462
C2×#25682
C2×#19812
C2×#19602
#139252
#110772
#109282
#109262
#108502
#107032