Finite Group Databasebeta

← #119814

Subgroups — #119814

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21400C211yesmaximal, characteristic
2#21306C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4#10848C211yesmaximal, characteristic
5#7981C211yesmaximal, characteristic
6#7897C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15#1990—21no—
16#1986—22no—
17#1986—22no—
18#1956—21no—
19#1892—22no—
20C4×#103—22no—
21C2×#344—21no—
22C2×#344—21no—
23#1733—21no—
24#1733—21no—
25#1550C414yescomplemented
26#1549—21no—
27#1548—21no—
28#1542—22no—
29#1289—22no—
30#1279—22no—
31#1278—22no—
32C22×M16—21no—
33C22×M16D411yescharacteristic
34C22×M16D411yescharacteristic
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533D412yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423C2×C412yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2×C412yes—
66C2×#103—24no—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8D412yes—
81#384—21no—
82#384—21no—
83#382—21no—
84#382—21no—
85#380—21no—
86#380—21no—
87#380—21no—
88#380D412yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D412yes—
93#344—42no—
94#344—42no—
95#340—24no—
96#339—24no—
97#339—24no—
98#333—24no—
99#332—24no—
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×M16—22no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16SD1612yes—
122C2×M16Q1612yes—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—44no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8C22⋊C412yes—
138C4⋊Q8C22⋊C412yes—
139#125—22no—
140#125—22no—
141#125—41no—
142#125—41no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159#117—22no—
160#117—22no—
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#117—41no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C42—22no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207#103—48no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210#97—41no—
211#97—41no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—41no—
217C4×C8—41no—
218C4×C8—41no—
219C4×C8—41no—
220#95—22no—
221#95—22no—
222#95—41no—
223#95—41no—
224C24#12111yescharacteristic
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—82no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4#11711yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12411yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4C2×SD1611yescharacteristic
256C22×C4C2×Q1611yescharacteristic
257C22×C4#13511yescharacteristic
258C22×C4#13511yescharacteristic
259C22×C4AGL(1,ℤ/8)11yescharacteristic
260C22×C4#13711yescharacteristic
261C22×C4#13711yescharacteristic
262C22×C4#13711yescharacteristic
263M16—22no—
264M16—22no—
265M16—22no—
266M16—22no—
267M16—41no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—42no—
272M16—42no—
273M16—81no—
274M16—81no—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—42no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—81no—
289C2×C8—81no—
290C2×C8—82no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—81no—
316C22⋊C4—81no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—81no—
322C42—81no—
323C42C2×C22⋊C411yescharacteristic
324C42#11711yescharacteristic
325C42C4×D411yescharacteristic
326C42C4×D411yescharacteristic
327C42#12111yescharacteristic
328C42#12111yescharacteristic
329C42#12111yescharacteristic
330C42#12311yescharacteristic
331C42#12311yescharacteristic
332C42#12311yescharacteristic
333C42#12311yescharacteristic
334C42#12311yescharacteristic
335C42#12411yescharacteristic
336C42#12611yescharacteristic
337C42#12611yescharacteristic
338C42#12611yescharacteristic
339C23—41no—
340C23—41no—
341C23#38811yescharacteristic
342C23#39411yescharacteristic
343C23#39411yescharacteristic
344C23#46111yescharacteristic
345C23#46211yescharacteristic
346C23#46311yescharacteristic
347C23#46411yescharacteristic
348Q8—44no—
349Q8—82no—
350Q8—82no—
351Q8—82no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4#38111yescharacteristic
375C2×C4#38611yescharacteristic
376C2×C4#38611yescharacteristic
377C2×C4#38811yescharacteristic
378C2×C4#38811yescharacteristic
379C2×C4#38811yescharacteristic
380C2×C4#39411yescharacteristic
381C2×C4#39411yescharacteristic
382C2×C4#39711yescharacteristic
383C2×C4#39711yescharacteristic
384C2×C4#39711yescharacteristic
385C2×C4#39711yescharacteristic
386C2×C4C4×SD1611yescharacteristic
387C2×C4C4×Q1611yescharacteristic
388C2×C4#44011yescharacteristic
389C2×C4#44111yescharacteristic
390C2×C4#44811yescharacteristic
391C2×C4#44911yescharacteristic
392C2×C4#45111yescharacteristic
393C2×C4#45211yescharacteristic
394C2×C4#46011yescharacteristic
395C2×C4#46211yescharacteristic
396C2×C4#46411yescharacteristic
397C2×C4#46411yescharacteristic
398C2×C4#46511yescharacteristic
399C2×C4#46711yescharacteristic
400C2×C4#46811yescharacteristic
401C2×C4#47011yescharacteristic
402C2×C4#48311yescharacteristic
403C2×C4#48411yescharacteristic
404C2×C4#48511yescharacteristic
405C2×C4#48511yescharacteristic
406C2×C4#48711yescharacteristic
407C2×C4#48811yescharacteristic
408C2×C4#48911yescharacteristic
409C2×C4#49111yescharacteristic
410C8—44no—
411C8—42no—
412C8—42no—
413C8—82no—
414C8—82no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22#144011yescharacteristic
426C22#192711yescharacteristic
427C22#192811yescharacteristic
428C22#204211yescharacteristic
429C22#204311yescharacteristic
430C22#205511yescharacteristic
431C22#205611yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—82no—
439C4—82no—
440C4—162no—
441C4—162no—
442C4#187811yescharacteristic
443C4#188011yescharacteristic
444C4#190612yes—
445C4#193311yescharacteristic
446C4#193511yescharacteristic
447C4#203511yescharacteristic
448C4#203611yescharacteristic
449C4#207211yescharacteristic
450C4#207411yescharacteristic
451C4#208012yes—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2#1115011yescharacteristic
459C2#1119011yescharacteristic
460C2#1119311yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1440
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
25. #1550C4semidirect

Maximal subgroups

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

subgroupindex
#214002
#213062
#177082
#108482
#79812
#78972
C2×#12782