Finite Group Databasebeta

← #112358

Subgroups — #112358

Order 512.

990 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3C2×#1550C211yesmaximal, characteristic
4C2×#1549C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic
6#10852C211yesmaximal, characteristic
7#7891C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2868—21no—
14#2862—21no—
15#2660—22no—
16#2567—22no—
17#2546C414yescomplemented
18#2502—22no—
19#2434—22no—
20#2317—22no—
21C4×#117C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1987—22no—
24#1987—22no—
25C2×#340—22no—
26C2×#339—22no—
27#1550—22no—
28#1550—22no—
29#1549—22no—
30#1549—22no—
31C2×#333—21no—
32C2×#332—22no—
33C2×#332—21no—
34#1503—22no—
35#1503—22no—
36C2×#328—22no—
37#1279—22no—
38C23×Q8—21no—
39C22×M16—21no—
40C23×C8—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49C2×C4⋊Q8—24no—
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⋊Q8—21no—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D412yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8D412yes—
67C2×C4×Q8—21no—
68C2×C4×Q8C2×C412yes—
69C2×#117—21no—
70C2×#117D412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—21no—
77#431—21no—
78#431—21no—
79#431D412yes—
80#423—21no—
81#423—21no—
82#423—21no—
83#423D412yes—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8D412yes—
93C2×C22⋊C8—21no—
94C2×C4×C8—21no—
95#398—28no—
96#395—28no—
97#391—24no—
98#391—24no—
99#389—28no—
100#385—28no—
101#384—24no—
102#384—24no—
103#382—24no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112#376—21no—
113C2×#95—21no—
114C43—21no—
115#340—28no—
116#339—44no—
117#333—24no—
118#332—28no—
119#332—24no—
120#328—28no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—84no—
129C22×Q8C22⋊C412yes—
130C23×C4—22no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C2×M16—22no—
136C2×M16—41no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—44no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—41no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—22no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—88no—
170#125—41no—
171#125—41no—
172#125—41no—
173#125—41no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—41no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190#117—22no—
191#117—22no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117SD1614yes—
197#117Q1614yes—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
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—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—216no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C414yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—41no—
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C22⋊C8—22no—
255C22⋊C8—41no—
256C4×C8—22no—
257C4×C8—41no—
258#95—216no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—41no—
263C24#12011yescharacteristic
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—48no—
267C2×Q8—44no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—84no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12411yescharacteristic
312C22×C4#12711yescharacteristic
313C22×C4C2×SD1612yes—
314C22×C4C2×Q1612yes—
315C22×C4#13712yes—
316C22×C4#13712yes—
317M16—44no—
318M16—81no—
319M16—81no—
320C2×C8—44no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—82no—
325C2×C8—82no—
326C2×C8—82no—
327C2×C8—82no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—816no—
353C4⋊C4#10318yes—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C42—24no—
359C42—24no—
360C42—44no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—44no—
366C42—81no—
367C42—81no—
368C42C4×D411yescharacteristic
369C42C4×D411yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42#12011yescharacteristic
373C42#12011yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42#12111yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12311yescharacteristic
381C42#12311yescharacteristic
382C42#12411yescharacteristic
383C42#12711yescharacteristic
384C23—22no—
385C23—22no—
386C23#37911yescharacteristic
387C23#39211yescharacteristic
388C23#39411yescharacteristic
389C23#44812yes—
390C23#45112yes—
391Q8—82no—
392Q8—82no—
393Q8—84no—
394Q8—84no—
395Q8—84no—
396Q8—84no—
397Q8—81no—
398Q8—81no—
399Q8—82no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—88no—
424C2×C4—82no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—82no—
428C2×C4#38611yescharacteristic
429C2×C4#38611yescharacteristic
430C2×C4#38611yescharacteristic
431C2×C4#38611yescharacteristic
432C2×C4#39011yescharacteristic
433C2×C4#39011yescharacteristic
434C2×C4#39211yescharacteristic
435C2×C4#39211yescharacteristic
436C2×C4#39211yescharacteristic
437C2×C4#39411yescharacteristic
438C2×C4#39411yescharacteristic
439C2×C4#39411yescharacteristic
440C2×C4C2×#10312yes—
441C2×C4#41912yes—
442C2×C4#44812yes—
443C2×C4#45112yes—
444C2×C4#45412yes—
445C2×C4#45612yes—
446C2×C4#46112yes—
447C2×C4#46112yes—
448C2×C4#46212yes—
449C2×C4#46212yes—
450C2×C4#48712yes—
451C2×C4#49112yes—
452C2×C4#49211yescharacteristic
453C2×C4#49411yescharacteristic
454C2×C4#49711yescharacteristic
455C2×C4#49711yescharacteristic
456C8—82no—
457C8—82no—
458C8—82no—
459C22—24no—
460C22—24no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22#143811yescharacteristic
466C22#178712yes—
467C22#179312yes—
468C22#179512yes—
469C22#201311yescharacteristic
470C22#201611yescharacteristic
471C22#204312yes—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—48no—
479C4—84no—
480C4—84no—
481C4—82no—
482C4—82no—
483C4#195611yescharacteristic
484C4#196011yescharacteristic
485C4#198112yes—
486C4#200311yescharacteristic
487C4#200511yescharacteristic
488C4#201311yescharacteristic
489C4#201611yescharacteristic
490C4#203812yes—
491C4#204312yes—
492C2—21no—
493C2—21no—
494C2—22no—
495C2#842414yes—
496C2#1098411yescharacteristic
497C2#1099811yescharacteristic
498C2#1101011yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
17. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#352812
C2×#19872
C2×#15502
C2×#15492
C2×#15032
#108522
#78912