Finite Group Databasebeta

← #307366

Subgroups — #307366

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28531C211yesmaximal, characteristic
2#26280C212yesmaximal
3C4⋊C4×Q16C211yesmaximal, characteristic
4#10482C212yesmaximal
5#10415C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9656C211yesmaximal, characteristic
9#9615C211yesmaximal, characteristic
10#8975C212yesmaximal
11C8×#106C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×C4×Q16—21no—
15C2×C4×Q16C2211yescharacteristic
16#2567C2212yes—
17Q8×C4⋊C4C2212yes—
18#2099—22no—
19#2095—22no—
20#2084—22no—
21#1990C2212yes—
22#1987C2212yes—
23#1984C2212yes—
24#1969—21no—
25#1956C2212yes—
26#1948C2212yes—
27#1942—21no—
28#1920—22no—
29#1920—22no—
30#1863—22no—
31#1845—21no—
32#1845C2211yescharacteristic
33#1803C2212yes—
34C4×#106C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#334C2211yescharacteristic
39C2×#333C2212yes—
40C2×Q8⋊C8C2212yes—
41C2×C82C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2211yescharacteristic
45C22×Q16D411yescharacteristic
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8—22no—
48C2×C4×Q8C2312yes—
49#500—22no—
50#500—22no—
51#477—24no—
52#477—42no—
53C4×Q16—22no—
54C4×Q16—22no—
55C4×Q16—44no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—22no—
60C2×#107—21no—
61C2×#107D411yescharacteristic
62C2×#106—22no—
63C2×#106D411yescharacteristic
64C2×#106D411yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—42no—
71C2×#103D412yes—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#395—42no—
78#391—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—42no—
83#384—22no—
84#384—22no—
85#384C2312yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2311yescharacteristic
88C4×C4⋊C4C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340D411yescharacteristic
96#340D411yescharacteristic
97#336—22no—
98#336—21no—
99#336—22no—
100#336—21no—
101#335—22no—
102#334—22no—
103#333—42no—
104Q8⋊C8—42no—
105C82—22no—
106C22×Q8—22no—
107C22×Q8C2×D412yes—
108C2×Q16—22no—
109C2×Q16—22no—
110C2×Q16SD1614yes—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—44no—
128C4×Q8—82no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C4⋊C4C2×D411yescharacteristic
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42—42no—
148C2×C42C2411yescharacteristic
149#107—22no—
150#107—44no—
151#107—42no—
152#107—42no—
153#107—42no—
154#107—81no—
155#106—22no—
156#106—44no—
157#106Q1614yescomplemented
158C4⋊C8—21no—
159C4⋊C8—22no—
160C4⋊C8—21no—
161C4⋊C8—22no—
162C4⋊C8—21no—
163C4⋊C8#4112yes—
164#103—24no—
165#103—44no—
166#103—44no—
167#103—84no—
168C4×C8—21no—
169C4×C8—21no—
170C4×C8—21no—
171C4×C8—21no—
172C4×C8—42no—
173C4×C8—42no—
174C4×C8#4112yes—
175C4×C8#4112yes—
176#95—21no—
177#95—21no—
178#95—42no—
179#95—42no—
180#95—42no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8C2×SD1614yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4C22×D411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4H(5,𝔽2)11yescharacteristic
202Q16—28no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—22no—
208C2×C8—21no—
209C2×C8—22no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—41no—
222C2×C8#12312yes—
223C2×C8#12312yes—
224C2×C8C2×SD1612yes—
225C2×C8C2×Q1612yes—
226C2×C8#13712yes—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—84no—
245C4⋊C4—82no—
246C4⋊C4—162no—
247C4⋊C4—162no—
248C4⋊C4—162no—
249C4⋊C4C2×Q1612yes—
250C4⋊C4C2×Q1612yes—
251C4⋊C4#13512yes—
252C42—42no—
253C42—42no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—84no—
259C42C2×#4111yescharacteristic
260C42C2×#4111yescharacteristic
261C42C2×#4111yescharacteristic
262C42H(5,𝔽2)11yescharacteristic
263C23D4211yescharacteristic
264Q8—24no—
265Q8—24no—
266Q8—42no—
267Q8—42no—
268Q8—84no—
269Q8—162no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4C2×#12311yescharacteristic
297C2×C4#53211yescharacteristic
298C2×C4#54011yescharacteristic
299C2×C4#54611yescharacteristic
300C2×C4#55011yescharacteristic
301C2×C4#55911yescharacteristic
302C2×C4C22×SD1611yescharacteristic
303C2×C4C22×Q1611yescharacteristic
304C2×C4C2×#13511yescharacteristic
305C2×C4C2×#13711yescharacteristic
306C2×C4#57711yescharacteristic
307C2×C4#57811yescharacteristic
308C2×C4#57811yescharacteristic
309C2×C4#57811yescharacteristic
310C8—22no—
311C8—22no—
312C8—42no—
313C8—42no—
314C8—41no—
315C8—82no—
316C8—81no—
317C8#48314yes—
318C22#244511yescharacteristic
319C22#327611yescharacteristic
320C22D4×SD1611yescharacteristic
321C22#328511yescharacteristic
322C22D4×Q1611yescharacteristic
323C22#329011yescharacteristic
324C22#329211yescharacteristic
325C4—21no—
326C4—21no—
327C4—42no—
328C4—44no—
329C4—42no—
330C4—42no—
331C4—81no—
332C4—82no—
333C4—162no—
334C4—162no—
335C4C2×#48311yescharacteristic
336C4#309711yescharacteristic
337C4#317011yescharacteristic
338C4#317711yescharacteristic
339C4#329811yescharacteristic
340C4#333111yescharacteristic
341C4#339311yescharacteristic
342C4#339511yescharacteristic
343C2#1993011yescharacteristic
344C2#1993811yescharacteristic
345C2SD16×Q1611yescharacteristic
346C2#1998611yescharacteristic
347C2#2623411yescharacteristic
348C2#2625511yescharacteristic
349C2#2627211yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
157. #106Q16semidirect

Maximal subgroups

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

subgroupindex
#285312
#262802
C4⋊C4×Q162
#104822
#104152
#98232
#97302
#96562
#96152
#89752
C8×#1062