Finite Group Databasebeta

← #111206

Subgroups — #111206

Order 512.

942 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1873C211yesmaximal, characteristic
2#20379C211yesmaximal, characteristic
3C2×#1508C211yesmaximal, characteristic
4C2×#1472C211yesmaximal, characteristic
5#10733C211yesmaximal, characteristic
6C2×#1293C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C22×#103C412yescomplemented
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#405—21no—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1873—21no—
18#1873—22no—
19#1873—21no—
20#1750—22no—
21#1749—21no—
22C4×C22⋊C8—21no—
23C2×#340C412yescomplemented
24C2×#339—22no—
25#1729—21no—
26#1508—22no—
27#1508C414yescomplemented
28C2×#328—22no—
29#1472—22no—
30#1472C414yescomplemented
31C2×#324—21no—
32C2×#324—21no—
33#1293—22no—
34#1282—22no—
35C23×Q8C2×C411yescharacteristic
36C24×C4C2311yescharacteristic
37C22×M16—21no—
38C22×M16—21no—
39C23×C8—21no—
40C23×C8C2×C412yes—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8C2×C412yes—
53C2×#117—21no—
54C2×#117—21no—
55C2×#117—44no—
56C2×#117D411yescharacteristic
57C2×#117D412yes—
58C2×#117D411yescharacteristic
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42—21no—
66C22×C42—21no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—44no—
70#407—21no—
71#407—21no—
72#407—21no—
73#407—21no—
74#407—21no—
75#407C2×C412yes—
76#407C2×C412yes—
77#407D412yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8C2×C412yes—
91C2×C22⋊C8D412yes—
92C2×C22⋊C8D411yescharacteristic
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D412yes—
95#405—44no—
96C2×#97—22no—
97C2×C4×C8—22no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4C2×C412yes—
102C2×#95—21no—
103#340—22no—
104#340—22no—
105#339—44no—
106#336—22no—
107#336—22no—
108#328—28no—
109#324—24no—
110#324—24no—
111C25#4111yescharacteristic
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8C4212yes—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—41no—
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C23×C4#4111yescharacteristic
137C23×C4#4111yescharacteristic
138C23×C4#4111yescharacteristic
139C23×C4#4111yescharacteristic
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—42no—
144C2×M16—42no—
145C2×M16—41no—
146C2×M16—42no—
147C2×M16—42no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—41no—
159C22×C8—42no—
160C22×C8—42no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—41no—
170C22⋊Q8—41no—
171C22⋊Q8—41no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177#117—24no—
178#117—24no—
179#117—44no—
180#117—44no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4C4212yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—41no—
222#103—44no—
223#103—44no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234C22⋊C8—41no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8C22⋊C412yes—
240C22⋊C8C22⋊C412yes—
241C22⋊C8C22⋊C414yes—
242#97—44no—
243C4×C8—44no—
244#95—22no—
245#95—41no—
246C24—22no—
247C24—22no—
248C24—41no—
249C24—41no—
250C24—42no—
251C24#11711yescharacteristic
252C24C4×D411yescharacteristic
253C24C4×D411yescharacteristic
254C24#12111yescharacteristic
255C24#12311yescharacteristic
256C24#12411yescharacteristic
257C24#12611yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—82no—
267C2×Q8—82no—
268C2×Q8—81no—
269C2×Q8—81no—
270C22×C4—22no—
271C22×C4—22no—
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—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—82no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4—82no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4C2×C4211yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4#11711yescharacteristic
316C22×C4#11711yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4#12311yescharacteristic
334C22×C4#12311yescharacteristic
335C22×C4#12411yescharacteristic
336C22×C4#12411yescharacteristic
337C22×C4#12611yescharacteristic
338C22×C4#12711yescharacteristic
339M16—44no—
340M16—44no—
341M16—44no—
342M16—82no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—24no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—44no—
352C2×C8—44no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—81no—
359C2×C8—81no—
360C2×C8—82no—
361C2×C8—82no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4#9914yes—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C42—24no—
386C42—44no—
387C42—42no—
388C42—44no—
389C42—82no—
390C23—21no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23C4×C22⋊C411yescharacteristic
420C23#38611yescharacteristic
421C23#38811yescharacteristic
422C23#38811yescharacteristic
423C23#38811yescharacteristic
424C23#38811yescharacteristic
425C23#39011yescharacteristic
426C23#44312yes—
427C23#44412yes—
428Q8—44no—
429Q8—82no—
430Q8—82no—
431Q8—82no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4C4×C22⋊C411yescharacteristic
486C2×C4C4×C22⋊C411yescharacteristic
487C2×C4#38611yescharacteristic
488C2×C4#38611yescharacteristic
489C2×C4#38811yescharacteristic
490C2×C4#38811yescharacteristic
491C2×C4#39011yescharacteristic
492C2×C4#39011yescharacteristic
493C2×C4C2×#9912yes—
494C2×C4#41012yes—
495C8—42no—
496C8—42no—
497C8—44no—
498C8—82no—
499C8—82no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22#143711yescharacteristic
531C22#176411yescharacteristic
532C22#176511yescharacteristic
533C22#187612yes—
534C22#189811yescharacteristic
535C22#189911yescharacteristic
536C22#190212yes—
537C22#192912yes—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—44no—
545C4—84no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4C4×#9912yes—
550C4#196412yes—
551C2—22no—
552C2—21no—
553C2—21no—
554C2—41no—
555C2—41no—
556C2—41no—
557C2—41no—
558C2#923912yes—
559C2#924012yes—
560C2#1093411yescharacteristic
561C2#1095211yescharacteristic
562C2#1096011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C22×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
10. C22×#103C4semidirect
23. C2×#340C4semidirect
27. #1508C4semidirect
30. #1472C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18732
#203792
C2×#15082
C2×#14722
#107332
C2×#12932
C2×#12822