Finite Group Databasebeta

← #117526

Subgroups — #117526

Order 512.

643 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21332C211yesmaximal, characteristic
2#21169C211yesmaximal, characteristic
3C2×#1542C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5#7981C211yesmaximal, characteristic
6#7893C211yesmaximal, characteristic
7#7866C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#427—21no—
12C22×C4⋊C8C2211yescharacteristic
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15C4×#117C2211yescharacteristic
16#1996—22no—
17#1996—22no—
18#1949—22no—
19#1948—21no—
20#1947—21no—
21#1740—21no—
22#1740—21no—
23#1735—21no—
24#1542—21no—
25#1542—21no—
26#1542—22no—
27C2×#333C412yescomplemented
28C2×#332C412yescomplemented
29#1520—22no—
30#1518—22no—
31#1503—22no—
32#1289—22no—
33#1279—22no—
34#1276—22no—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8D412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8Q812yes—
51C2×#117—21no—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C22×C42C2311yescharacteristic
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431D412yes—
65#427—42no—
66#427—42no—
67C2×#107—22no—
68C2×#106—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2×C412yes—
74C2×C4⋊C8C2×C412yes—
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8D412yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#384—21no—
83#384—21no—
84#382—21no—
85#382D412yes—
86#380—21no—
87#380—21no—
88#380—21no—
89#380D412yes—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98#376—21no—
99C43—21no—
100#342—24no—
101#342—24no—
102#340—22no—
103#340—22no—
104#333—22no—
105#333—22no—
106#332—22no—
107#332—22no—
108#328—28no—
109C22×Q8—21no—
110C22×Q8C4⋊C412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—41no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132#125—41no—
133#125—41no—
134#125—41no—
135#125—41no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152#117—21no—
153#117—21no—
154#117—21no—
155#117—21no—
156#117—21no—
157#117—21no—
158#117—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162#117—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170#117SD1612yes—
171#117Q1612yes—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C4⋊C412yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—41no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q811yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210#107—44no—
211#106—44no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—22no—
219C4⋊C8—41no—
220C4⋊C8C22⋊C412yes—
221C4⋊C8C22⋊C412yes—
222C22⋊C8—22no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C4×C8—22no—
227C4×C8—41no—
228C4×C8—41no—
229C4×C8—41no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—41no—
234#95—41no—
235#95—41no—
236C24#12011yescharacteristic
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—82no—
242C2×Q8—82no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12411yescharacteristic
268C22×C4#12711yescharacteristic
269C22×C4C2×SD1611yescharacteristic
270C22×C4C2×Q1611yescharacteristic
271C22×C4#13511yescharacteristic
272C22×C4#13511yescharacteristic
273C22×C4AGL(1,ℤ/8)11yescharacteristic
274C22×C4#13711yescharacteristic
275C22×C4#13711yescharacteristic
276C22×C4#13711yescharacteristic
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—82no—
290C2×C8—82no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—44no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—81no—
314C4⋊C4—81no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—81no—
324C22⋊C4—81no—
325C42—22no—
326C42—22no—
327C42—42no—
328C42—42no—
329C42—41no—
330C42—41no—
331C42—82no—
332C42—82no—
333C42—81no—
334C42—81no—
335C42—81no—
336C42—81no—
337C42C2×C22⋊C411yescharacteristic
338C42#11711yescharacteristic
339C42C4×D411yescharacteristic
340C42C4×D411yescharacteristic
341C42#12111yescharacteristic
342C42#12111yescharacteristic
343C42C22⋊Q811yescharacteristic
344C42C22⋊Q811yescharacteristic
345C42C22⋊Q811yescharacteristic
346C42C22⋊Q811yescharacteristic
347C42#12311yescharacteristic
348C42#12311yescharacteristic
349C42#12411yescharacteristic
350C42#12411yescharacteristic
351C42#12611yescharacteristic
352C42#12611yescharacteristic
353C23—41no—
354C23—41no—
355C23#38511yescharacteristic
356C23#39211yescharacteristic
357C23#39411yescharacteristic
358C23#44811yescharacteristic
359C23#44911yescharacteristic
360C23#45111yescharacteristic
361C23#45211yescharacteristic
362Q8—82no—
363Q8—82no—
364Q8—82no—
365Q8—82no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#38011yescharacteristic
395C2×C4#38511yescharacteristic
396C2×C4#38611yescharacteristic
397C2×C4#38811yescharacteristic
398C2×C4#38811yescharacteristic
399C2×C4#39011yescharacteristic
400C2×C4#39311yescharacteristic
401C2×C4#39411yescharacteristic
402C2×C4#39611yescharacteristic
403C2×C4#39611yescharacteristic
404C2×C4#39711yescharacteristic
405C2×C4#39911yescharacteristic
406C2×C4C4×SD1611yescharacteristic
407C2×C4C4×Q1611yescharacteristic
408C2×C4#44011yescharacteristic
409C2×C4#44111yescharacteristic
410C2×C4#45011yescharacteristic
411C2×C4#45111yescharacteristic
412C2×C4#45211yescharacteristic
413C2×C4#45211yescharacteristic
414C2×C4#46111yescharacteristic
415C2×C4#46211yescharacteristic
416C2×C4#46311yescharacteristic
417C2×C4#46411yescharacteristic
418C2×C4#46511yescharacteristic
419C2×C4#46711yescharacteristic
420C2×C4#46811yescharacteristic
421C2×C4#47011yescharacteristic
422C2×C4#47511yescharacteristic
423C2×C4#47711yescharacteristic
424C2×C4#47911yescharacteristic
425C2×C4#47911yescharacteristic
426C2×C4#48711yescharacteristic
427C2×C4#48811yescharacteristic
428C2×C4#48911yescharacteristic
429C2×C4#49111yescharacteristic
430C8—42no—
431C8—42no—
432C8—82no—
433C8—82no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22#143911yescharacteristic
445C22#186311yescharacteristic
446C22#186511yescharacteristic
447C22#200011yescharacteristic
448C22#200111yescharacteristic
449C22#204211yescharacteristic
450C22#204311yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—82no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—162no—
463C4—162no—
464C4#187711yescharacteristic
465C4#187811yescharacteristic
466C4#193311yescharacteristic
467C4#193511yescharacteristic
468C4#197612yes—
469C4#202411yescharacteristic
470C4#202611yescharacteristic
471C4#206611yescharacteristic
472C4#206811yescharacteristic
473C2—21no—
474C2—21no—
475C2—21no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2#1106711yescharacteristic
480C2#1112611yescharacteristic
481C2#1113811yescharacteristic
subgroupG/H
centerC2×C4#396
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. C2×#333C4semidirect
28. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#213322
#211692
C2×#15422
#175752
#79812
#78932
#78662