Finite Group Databasebeta

← #114207

Subgroups — #114207

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3139C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1546C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#10857C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3139—22no—
11#3139—22no—
12#3139—22no—
13#3139C414yescomplemented
14C2×#491—22no—
15C2×#487—21no—
16C2×#487—21no—
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#419—22no—
20C2×C4×M16C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#1990—22no—
24#1990—22no—
25#1990—21no—
26#1956—21no—
27#1872—22no—
28#1803—22no—
29C2×#343—22no—
30C2×#340—22no—
31#1546—22no—
32#1546—22no—
33C2×#332—22no—
34#1493—22no—
35#1493—22no—
36C2×Q8⋊C8—22no—
37#1278—22no—
38C23×Q8—21no—
39C22×M16—21no—
40C22×M16D412yes—
41C23×C8—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D412yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D412yes—
75C22×C42C2311yescharacteristic
76#491—28no—
77#491—28no—
78#487—24no—
79#487—24no—
80#487—24no—
81#487—24no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#423—21no—
87#423—21no—
88#423—21no—
89#423D412yes—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8D412yes—
93#419—28no—
94#419—28no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×C22⋊C8—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16C2×C412yes—
102C4×M16D412yes—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8D412yes—
112#384—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116#343—44no—
117#340—24no—
118#340—28no—
119#332—28no—
120Q8⋊C8—28no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C23×C4—22no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4#4111yescharacteristic
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—41no—
145C2×M16—42no—
146C2×M16—42no—
147C2×M16C22⋊C414yes—
148C22×C8—24no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—42no—
153C22×C8—41no—
154C22×C8—41no—
155C22×C8—41no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—22no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177C4⋊Q8C22⋊C414yes—
178C4⋊Q8C22⋊C412yes—
179C4⋊Q8C22⋊C412yes—
180#125—22no—
181#125—22no—
182#125—41no—
183#125—41no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200#117—22no—
201#117—22no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—82no—
232C2×C22⋊C4—41no—
233C2×C42—41no—
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252#103—216no—
253#103—216no—
254#103—84no—
255C22⋊C8—41no—
256C22⋊C8—41no—
257#97—22no—
258#97—22no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—41no—
264C4×C8—41no—
265C4×C8—41no—
266C4×C8—41no—
267C24#12111yescharacteristic
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—82no—
287C2×Q8—82no—
288C2×Q8—82no—
289C2×Q8—82no—
290C2×Q8—81no—
291C2×Q8—81no—
292C2×Q8—81no—
293C2×Q8—81no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4#12311yescharacteristic
323C22×C4#12411yescharacteristic
324C22×C4#12411yescharacteristic
325C22×C4#12711yescharacteristic
326C22×C4#12711yescharacteristic
327C22×C4#13512yes—
328C22×C4#13512yes—
329C22×C4AGL(1,ℤ/8)12yes—
330C22×C4#13712yes—
331M16—24no—
332M16—24no—
333M16—42no—
334M16—42no—
335M16—44no—
336M16—82no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—42no—
340C2×C8—44no—
341C2×C8—44no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—81no—
350C2×C8—81no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C42—42no—
381C42—42no—
382C42C2×C22⋊C411yescharacteristic
383C42C2×C22⋊C411yescharacteristic
384C42C4×D411yescharacteristic
385C42C4×D411yescharacteristic
386C42#12011yescharacteristic
387C42#12011yescharacteristic
388C42#12011yescharacteristic
389C42#12011yescharacteristic
390C42#12111yescharacteristic
391C42#12111yescharacteristic
392C42#12111yescharacteristic
393C42#12111yescharacteristic
394C42#12111yescharacteristic
395C42#12111yescharacteristic
396C42#12311yescharacteristic
397C42#12311yescharacteristic
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23#39011yescharacteristic
403C23#39211yescharacteristic
404C23#39411yescharacteristic
405C23#46312yes—
406C23#46412yes—
407Q8—44no—
408Q8—44no—
409Q8—48no—
410Q8—82no—
411Q8—82no—
412Q8—82no—
413Q8—82no—
414Q8—82no—
415Q8—82no—
416Q8—84no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—84no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—164no—
444C2×C4#37911yescharacteristic
445C2×C4#38611yescharacteristic
446C2×C4#38611yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#39011yescharacteristic
450C2×C4#39211yescharacteristic
451C2×C4#39211yescharacteristic
452C2×C4#39211yescharacteristic
453C2×C4#39411yescharacteristic
454C2×C4#39411yescharacteristic
455C2×C4#39411yescharacteristic
456C2×C4#41612yes—
457C2×C4#41712yes—
458C2×C4#44912yes—
459C2×C4#44912yes—
460C2×C4#45212yes—
461C2×C4#45212yes—
462C2×C4#45512yes—
463C2×C4#45612yes—
464C2×C4#46312yes—
465C2×C4#46412yes—
466C2×C4#47112yes—
467C2×C4#47312yes—
468C2×C4#48812yes—
469C2×C4#48912yes—
470C2×C4#49511yescharacteristic
471C2×C4#49511yescharacteristic
472C2×C4#49611yescharacteristic
473C2×C4#49711yescharacteristic
474C8—44no—
475C8—44no—
476C8—82no—
477C8—82no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22#143811yescharacteristic
487C22#198312yes—
488C22#200411yescharacteristic
489C22#200511yescharacteristic
490C22#201211yescharacteristic
491C22#201511yescharacteristic
492C22#201911yescharacteristic
493C22#202011yescharacteristic
494C22#204212yes—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—84no—
500C4—82no—
501C4—82no—
502C4—162no—
503C4#178812yes—
504C4#188212yes—
505C4#190812yes—
506C4#195811yescharacteristic
507C4#195911yescharacteristic
508C4#200712yes—
509C4#201211yescharacteristic
510C4#201511yescharacteristic
511C4#204212yes—
512C4#205212yes—
513C2—22no—
514C2—22no—
515C2#1016911yescharacteristic
516C2#1017111yescharacteristic
517C2#1017311yescharacteristic
518C2#1017511yescharacteristic
519C2#1100911yescharacteristic
520C2#1101711yescharacteristic
521C2#1102111yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3139C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
#218602
C2×#19902
C2×#15462
C2×#14932
#108572
C2×#12782