Finite Group Databasebeta

← #193197

Subgroups — #193197

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6#8926C214yesmaximal, complemented
7#8925C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16#2365—22no—
17#2365—22no—
18#2365—22no—
19#2365C2214yes—
20C2×#391—21no—
21C2×#389—21no—
22C2×#387—22no—
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990C2212yes—
28#1990C2212yes—
29#1988—22no—
30#1949—22no—
31#1948C2212yes—
32#1948C2212yes—
33#1948C2212yes—
34#1947C2212yes—
35#1853—21no—
36#1853—21no—
37#1823—22no—
38C4×#106—22no—
39#1736—21no—
40#1546C418yescomplemented
41C2×#332C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×Q8—21no—
45C22×M16—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C414yes—
51C2×C4⋊Q8D412yes—
52C2×C4⋊Q8Q814yes—
53C2×C4⋊Q8C2312yes—
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8—44no—
57C2×C4×Q8—44no—
58C2×#117—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C42—22no—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#431—22no—
71#431—22no—
72#431—22no—
73#431C2×C414yes—
74#428—44no—
75C2×#107—22no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—22no—
79#423—22no—
80#423C2×C414yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#391—24no—
90#391—24no—
91#389—24no—
92#389—24no—
93#387—28no—
94#387—28no—
95#384C2311yescharacteristic
96#384C2311yescharacteristic
97#382—21no—
98#380—21no—
99#380—21no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4C2312yes—
111C4×C4⋊C4C2312yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#332C2×C4116yescomplemented
123C22×Q8—24no—
124C22×Q8—22no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—42no—
141C2×M16—42no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8C4⋊C418yes—
153C4⋊Q8C22×C414yes—
154C4⋊Q8C2×Q814yes—
155C4×Q8—48no—
156C4×Q8—416no—
157C4×Q8—48no—
158#117—42no—
159#117—42no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4C4⋊C418yes—
184C2×C4⋊C4SD1614yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—41no—
191C2×C42—28no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—48no—
197C2×C42—44no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—41no—
201C2×C42C22×C412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42C2411yescharacteristic
209#107—44no—
210#106—44no—
211#106—44no—
212#106—44no—
213C4⋊C8—24no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—42no—
217C4⋊C8C22×C414yes—
218C4⋊C8C22×C414yes—
219C22⋊C8—24no—
220C22⋊C8—42no—
221C4×C8—24no—
222C4×C8—22no—
223C4×C8—22no—
224#95—28no—
225#95—28no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—41no—
231#95—41no—
232C24#12011yescharacteristic
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—28no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—48no—
241C2×Q8—48no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—48no—
245C2×Q8—48no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4C2×C4⋊C412yes—
273C22×C4C4×D412yes—
274C22×C4C4×D412yes—
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4C22⋊Q812yes—
279C22×C4C22⋊Q812yes—
280C22×C4#12312yes—
281C22×C4C2×SD1612yes—
282C22×C4#13512yes—
283C22×C4AGL(1,ℤ/8)12yes—
284C22×C4AGL(1,ℤ/8)12yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288M16—48no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—416no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4C2×C4⋊C418yes—
323C4⋊C4C2×SD1614yes—
324C4⋊C4#13714yes—
325C22⋊C4—82no—
326C42—44no—
327C42—416no—
328C42—42no—
329C42—42no—
330C42—44no—
331C42—44no—
332C42—82no—
333C42C2×C4⋊C412yes—
334C42C4×D412yes—
335C42C4×D412yes—
336C42C22⋊Q812yes—
337C42C22⋊Q812yes—
338C42#12312yes—
339C42C23×C411yescharacteristic
340C42C22×Q811yescharacteristic
341C42C2×#4111yescharacteristic
342C42C2×#4111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23#38512yes—
348C23#44912yes—
349C23#45012yes—
350C23C2×#12011yescharacteristic
351Q8—44no—
352Q8—416no—
353Q8—44no—
354Q8—44no—
355Q8—44no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—28no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—416no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4#38512yes—
393C2×C4#38512yes—
394C2×C4#38512yes—
395C2×C4C4×SD1614yes—
396C2×C4#44214yes—
397C2×C4#44912yes—
398C2×C4#45012yes—
399C2×C4#47614yes—
400C2×C4#47814yes—
401C2×C4C22×C4⋊C411yescharacteristic
402C2×C4C2×C4×D411yescharacteristic
403C2×C4C2×C4×D411yescharacteristic
404C2×C4C2×C22⋊Q811yescharacteristic
405C2×C4C2×C22⋊Q811yescharacteristic
406C2×C4C2×#12311yescharacteristic
407C2×C4C22×SD1611yescharacteristic
408C2×C4C2×#13511yescharacteristic
409C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
410C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
411C2×C4C2×#13712yes—
412C2×C4#57512yes—
413C8—44no—
414C8—44no—
415C8—82no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22#186414yes—
423C22C2×#38511yescharacteristic
424C22C2×#45011yescharacteristic
425C22C2×#44911yescharacteristic
426C22#300411yescharacteristic
427C22#300511yescharacteristic
428C22#302011yescharacteristic
429C22#302011yescharacteristic
430C4—21no—
431C4—21no—
432C4—22no—
433C4—22no—
434C4—44no—
435C4—44no—
436C4—48no—
437C4—82no—
438C4—82no—
439C4—161no—
440C4—161no—
441C4#186414yes—
442C4C2×C4×SD1611yescharacteristic
443C4C2×#44211yescharacteristic
444C4C4×#13712yes—
445C4C2×#47611yescharacteristic
446C4C2×#47811yescharacteristic
447C4#307812yes—
448C2—24no—
449C2#1875711yescharacteristic
450C2#1876011yescharacteristic
451C2#1877711yescharacteristic
452C2#1878011yescharacteristic
453C2#2085711yescharacteristic
454C2C2×#186411yescharacteristic
455C2#2087411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8926C2semidirect
7. #8925C2semidirect
40. #1546C4semidirect
122. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
#218332
#213592
#211702
C2×#15462
#89262
#89252
#79012
#78922