Finite Group Databasebeta

← #131939

Subgroups — #131939

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3383C211yesmaximal, characteristic
2#13282C211yesmaximal, characteristic
3#12812C211yesmaximal, characteristic
4C2×#1364C211yesmaximal, characteristic
5C2×Q8⋊C16C211yesmaximal, characteristic
6C2×#1332C211yesmaximal, characteristic
7#8645C211yesmaximal, characteristic
8C2×#557—21no—
9#3383—22no—
10#3383—22no—
11#3383—22no—
12#3383C414yescomplemented
13C2×#500C2211yescharacteristic
14C2×#499—21no—
15C2×#479—22no—
16C2×#475—21no—
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16—21no—
19C2×C4×Q16C2211yescharacteristic
20C2×C4⋊C16C2211yescharacteristic
21C2×#358—21no—
22C2×C4×C16C2211yescharacteristic
23#1845C2211yescharacteristic
24#1803—21no—
25C2×#335C2211yescharacteristic
26#1380—21no—
27#1380—22no—
28#1364—22no—
29Q8⋊C16—22no—
30#1332—22no—
31#1296—21no—
32C22×Q16D411yescharacteristic
33#557—24no—
34#557—24no—
35#557—42no—
36#557—42no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—41no—
39C2×#125—22no—
40C2×#125—42no—
41C2×C4×Q8—22no—
42C2×C4×Q8—21no—
43C2×C4×Q8—41no—
44C2×C4×Q8D411yescharacteristic
45C22×C16—21no—
46C22×C16—21no—
47#500—21no—
48#500—21no—
49#500C2×C412yes—
50#500D412yes—
51#499—24no—
52#499—24no—
53#479—28no—
54#479—28no—
55#475—24no—
56#475—24no—
57C8×Q8—21no—
58C8×Q8—21no—
59C8×Q8C2×C412yes—
60C8×Q8D412yes—
61C4×Q16—24no—
62C4×Q16—24no—
63C4×Q16—21no—
64C4×Q16—21no—
65C4×Q16C2×C412yes—
66C4×Q16D412yes—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107D412yes—
71C2×#107D411yescharacteristic
72C2×#106—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D411yescharacteristic
76C2×#103—22no—
77C2×#103—21no—
78C2×#103—41no—
79C2×C4×C8—21no—
80C2×C4×C8C2311yescharacteristic
81#384—21no—
82C4⋊C16—21no—
83C4⋊C16—21no—
84#358—42no—
85C4×C16—21no—
86C4×C16—21no—
87#340—41no—
88#335—21no—
89#335—21no—
90C22×Q8—21no—
91C22×Q8—41no—
92C22×Q8D811yescharacteristic
93C2×Q16—21no—
94C2×Q16—21no—
95C2×Q16—21no—
96C2×Q16C22⋊C412yes—
97C2×Q16SD1612yes—
98C2×Q16Q1612yes—
99C22×C8—21no—
100C22×C8—41no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—44no—
106C4⋊Q8—44no—
107C4⋊Q8—41no—
108C4⋊Q8—41no—
109#125—28no—
110#125—44no—
111#125—48no—
112#125—48no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×Q8—28no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—44no—
119C4×Q8—44no—
120C4×Q8—44no—
121C4×Q8—41no—
122C4×Q8—41no—
123C4×Q8C22⋊C412yes—
124C4×Q8SD1612yes—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4D811yescharacteristic
137C2×C42—21no—
138C2×C42—42no—
139C2×C42—41no—
140C2×C42C2×D411yescharacteristic
141C2×C16—22no—
142C2×C16—41no—
143C2×C16—41no—
144C2×C16—41no—
145C2×C16—41no—
146C2×C16—41no—
147#107—22no—
148#107—22no—
149#107—21no—
150#107—22no—
151#107—42no—
152#107—41no—
153#107—41no—
154#107C22⋊C412yes—
155#107C22⋊C414yes—
156#106—28no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—21no—
160C4⋊C8—41no—
161C4⋊C8—41no—
162C4⋊C8C22⋊C412yes—
163#103—28no—
164#103—22no—
165#103—22no—
166#103—82no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8C22×C411yescharacteristic
170C4×C8C2×D411yescharacteristic
171C4×C8C2×D411yescharacteristic
172C4×C8C2×D411yescharacteristic
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—81no—
187C2×Q8—81no—
188C2×Q8—81no—
189C2×Q8—81no—
190C2×Q8#10212yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—81no—
197C22×C4#12011yescharacteristic
198C22×C4C2×D811yescharacteristic
199C22×C4AGL(1,ℤ/8)11yescharacteristic
200Q16—22no—
201Q16—22no—
202Q16—22no—
203Q16#10314yescomplemented
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—22no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8—81no—
212C2×C8C2×C22⋊C411yescharacteristic
213C2×C8C2×C22⋊C411yescharacteristic
214C2×C8#12011yescharacteristic
215C2×C8#12011yescharacteristic
216C2×C8C2×SD1611yescharacteristic
217C2×C8C2×Q1611yescharacteristic
218C2×C8#13711yescharacteristic
219C2×C8#13711yescharacteristic
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—48no—
226C4⋊C4—48no—
227C4⋊C4—48no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—161no—
239C4⋊C4#10212yes—
240C42—22no—
241C42—22no—
242C42—48no—
243C42—42no—
244C42—42no—
245C42C2×C22⋊C411yescharacteristic
246C42#12011yescharacteristic
247C42C2×SD1611yescharacteristic
248C42AGL(1,ℤ/8)11yescharacteristic
249C16—42no—
250C16—81no—
251C23#44711yescharacteristic
252Q8—22no—
253Q8—22no—
254Q8—24no—
255Q8—42no—
256Q8—42no—
257Q8—42no—
258Q8—41no—
259Q8—41no—
260Q8—82no—
261Q8—82no—
262Q8—82no—
263Q8—82no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—161no—
282C2×C4—161no—
283C2×C4—161no—
284C2×C4#37911yescharacteristic
285C2×C4C2×#10211yescharacteristic
286C2×C4#41811yescharacteristic
287C2×C4#44711yescharacteristic
288C2×C4#44811yescharacteristic
289C2×C4#45011yescharacteristic
290C2×C4#45011yescharacteristic
291C2×C4#45111yescharacteristic
292C2×C4#45411yescharacteristic
293C2×C4#45611yescharacteristic
294C2×C4#50811yescharacteristic
295C2×C4#50811yescharacteristic
296C2×C4#50911yescharacteristic
297C2×C4#51011yescharacteristic
298C8—21no—
299C8—21no—
300C8—42no—
301C8—161no—
302C8C2×#10311yescharacteristic
303C8#41911yescharacteristic
304C8#44811yescharacteristic
305C8#45111yescharacteristic
306C22#162711yescharacteristic
307C22#164911yescharacteristic
308C22#178611yescharacteristic
309C22#218611yescharacteristic
310C22#218911yescharacteristic
311C22#219311yescharacteristic
312C22#219511yescharacteristic
313C4—21no—
314C4—21no—
315C4—42no—
316C4—42no—
317C4—44no—
318C4—84no—
319C4—82no—
320C4—321no—
321C4#163711yescharacteristic
322C4#165111yescharacteristic
323C4#178711yescharacteristic
324C4#179311yescharacteristic
325C4#213811yescharacteristic
326C4#213911yescharacteristic
327C4#218611yescharacteristic
328C4#218911yescharacteristic
329C2#844711yescharacteristic
330C2#1235311yescharacteristic
331C2#1235711yescharacteristic
332C2#1240511yescharacteristic
333C2#1240911yescharacteristic
334C2#1273311yescharacteristic
335C2#1274911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #3383C4semidirect
203. Q16#103semidirect

Maximal subgroups

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

subgroupindex
C2×#33832
#132822
#128122
C2×#13642
C2×Q8⋊C162
C2×#13322
#86452