Finite Group Databasebeta

← #116294

Subgroups — #116294

Order 512.

643 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21306C211yesmaximal, characteristic
2#21145C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4C2×#1481C211yesmaximal, characteristic
5#7981C211yesmaximal, characteristic
6#7893C211yesmaximal, characteristic
7#7882C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#424—21no—
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#2285C2211yescharacteristic
16#1986—22no—
17#1986—22no—
18#1940—22no—
19#1939—22no—
20#1733—21no—
21#1733—21no—
22#1729—21no—
23#1520—22no—
24#1518C414yescomplemented
25#1503—22no—
26#1481—22no—
27#1481—22no—
28C2×Q8⋊C8—22no—
29#1289—22no—
30#1279—22no—
31#1278—22no—
32C22×M16D411yescharacteristic
33C22×M16D411yescharacteristic
34C23×C8—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533D412yes—
43C2×C4⋊Q8Q811yescharacteristic
44C2×C4⋊Q8Q811yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431C2×C412yes—
62#424—42no—
63#424—42no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8D412yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8D412yes—
83#384—21no—
84#384—21no—
85#382—21no—
86#382C2×C412yes—
87#380—21no—
88#380—21no—
89#380—21no—
90#380D412yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4D412yes—
94C4×C4⋊C4D412yes—
95#339—24no—
96#339—24no—
97#336—24no—
98#328—28no—
99Q8⋊C8—28no—
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—21no—
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16SD1612yes—
116C2×M16Q1612yes—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—41no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8C4⋊C412yes—
135C4⋊Q8C4⋊C412yes—
136#125—22no—
137#125—22no—
138#125—41no—
139#125—41no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156#117—22no—
157#117—22no—
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—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—41no—
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—48no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211#97—41no—
212#97—41no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—41no—
217C4×C8—41no—
218C4×C8—41no—
219#95—22no—
220#95—22no—
221#95—41no—
222#95C22⋊C414yes—
223C24#12111yescharacteristic
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—82no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4C2×SD1611yescharacteristic
255C22×C4C2×Q1611yescharacteristic
256C22×C4#13511yescharacteristic
257C22×C4#13511yescharacteristic
258C22×C4AGL(1,ℤ/8)11yescharacteristic
259C22×C4#13711yescharacteristic
260C22×C4#13711yescharacteristic
261C22×C4#13711yescharacteristic
262M16—22no—
263M16—22no—
264M16—22no—
265M16—22no—
266M16—41no—
267M16—41no—
268M16—41no—
269M16—41no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—82no—
282C2×C8—81no—
283C2×C8—81no—
284C2×C8—81no—
285C2×C8—81no—
286C2×C8—82no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42C2×C4⋊C411yescharacteristic
320C42#11711yescharacteristic
321C42C4×D411yescharacteristic
322C42C4×D411yescharacteristic
323C42#12011yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42C22⋊Q811yescharacteristic
329C42#12311yescharacteristic
330C42#12311yescharacteristic
331C42#12411yescharacteristic
332C42#12411yescharacteristic
333C42#12611yescharacteristic
334C42#12711yescharacteristic
335C23—41no—
336C23—41no—
337C23#38611yescharacteristic
338C23#39411yescharacteristic
339C23#39611yescharacteristic
340C23#46111yescharacteristic
341C23#46211yescharacteristic
342C23#46311yescharacteristic
343C23#46411yescharacteristic
344Q8—44no—
345Q8—82no—
346Q8—82no—
347Q8—82no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4#38011yescharacteristic
371C2×C4#38511yescharacteristic
372C2×C4#38511yescharacteristic
373C2×C4#38811yescharacteristic
374C2×C4#38811yescharacteristic
375C2×C4#39011yescharacteristic
376C2×C4#39211yescharacteristic
377C2×C4#39311yescharacteristic
378C2×C4#39411yescharacteristic
379C2×C4#39611yescharacteristic
380C2×C4#39711yescharacteristic
381C2×C4#39911yescharacteristic
382C2×C4C4×SD1611yescharacteristic
383C2×C4C4×Q1611yescharacteristic
384C2×C4#44011yescharacteristic
385C2×C4#44111yescharacteristic
386C2×C4#44811yescharacteristic
387C2×C4#44911yescharacteristic
388C2×C4#45011yescharacteristic
389C2×C4#45111yescharacteristic
390C2×C4#45111yescharacteristic
391C2×C4#45211yescharacteristic
392C2×C4#45211yescharacteristic
393C2×C4#45211yescharacteristic
394C2×C4#46511yescharacteristic
395C2×C4#46711yescharacteristic
396C2×C4#46811yescharacteristic
397C2×C4#47011yescharacteristic
398C2×C4#47511yescharacteristic
399C2×C4#47711yescharacteristic
400C2×C4#47911yescharacteristic
401C2×C4#47911yescharacteristic
402C2×C4#48711yescharacteristic
403C2×C4#48811yescharacteristic
404C2×C4#48911yescharacteristic
405C2×C4#49111yescharacteristic
406C8—42no—
407C8—42no—
408C8—82no—
409C8—82no—
410C8—82no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22#143911yescharacteristic
422C22#187711yescharacteristic
423C22#187811yescharacteristic
424C22#204211yescharacteristic
425C22#204311yescharacteristic
426C22#206611yescharacteristic
427C22#206811yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—84no—
436C4—82no—
437C4—82no—
438C4—162no—
439C4#186311yescharacteristic
440C4#186511yescharacteristic
441C4#193311yescharacteristic
442C4#193511yescharacteristic
443C4#200011yescharacteristic
444C4#200111yescharacteristic
445C4#202411yescharacteristic
446C4#202611yescharacteristic
447C4#209812yes—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2#1106711yescharacteristic
455C2#1108411yescharacteristic
456C2#1109411yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#121
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
24. #1518C4semidirect

Maximal subgroups

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

subgroupindex
#213062
#211452
#175752
C2×#14812
#79812
#78932
#78822